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• Random Assignment in Experiments | Introduction & Examples

Random Assignment in Experiments | Introduction & Examples

Published on March 8, 2021 by Pritha Bhandari . Revised on June 22, 2023.

In experimental research, random assignment is a way of placing participants from your sample into different treatment groups using randomization.

With simple random assignment, every member of the sample has a known or equal chance of being placed in a control group or an experimental group. Studies that use simple random assignment are also called completely randomized designs .

Random assignment is a key part of experimental design . It helps you ensure that all groups are comparable at the start of a study: any differences between them are due to random factors, not research biases like sampling bias or selection bias .

Table of contents

Why does random assignment matter, random sampling vs random assignment, how do you use random assignment, when is random assignment not used, other interesting articles, frequently asked questions about random assignment.

Random assignment is an important part of control in experimental research, because it helps strengthen the internal validity of an experiment and avoid biases.

In experiments, researchers manipulate an independent variable to assess its effect on a dependent variable, while controlling for other variables. To do so, they often use different levels of an independent variable for different groups of participants.

This is called a between-groups or independent measures design.

You use three groups of participants that are each given a different level of the independent variable:

• a control group that’s given a placebo (no dosage, to control for a placebo effect ),
• an experimental group that’s given a low dosage,
• a second experimental group that’s given a high dosage.

Random assignment to helps you make sure that the treatment groups don’t differ in systematic ways at the start of the experiment, as this can seriously affect (and even invalidate) your work.

If you don’t use random assignment, you may not be able to rule out alternative explanations for your results.

• participants recruited from cafes are placed in the control group ,
• participants recruited from local community centers are placed in the low dosage experimental group,
• participants recruited from gyms are placed in the high dosage group.

With this type of assignment, it’s hard to tell whether the participant characteristics are the same across all groups at the start of the study. Gym-users may tend to engage in more healthy behaviors than people who frequent cafes or community centers, and this would introduce a healthy user bias in your study.

Although random assignment helps even out baseline differences between groups, it doesn’t always make them completely equivalent. There may still be extraneous variables that differ between groups, and there will always be some group differences that arise from chance.

Most of the time, the random variation between groups is low, and, therefore, it’s acceptable for further analysis. This is especially true when you have a large sample. In general, you should always use random assignment in experiments when it is ethically possible and makes sense for your study topic.

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Random sampling and random assignment are both important concepts in research, but it’s important to understand the difference between them.

Random sampling (also called probability sampling or random selection) is a way of selecting members of a population to be included in your study. In contrast, random assignment is a way of sorting the sample participants into control and experimental groups.

While random sampling is used in many types of studies, random assignment is only used in between-subjects experimental designs.

Some studies use both random sampling and random assignment, while others use only one or the other.

Random sampling enhances the external validity or generalizability of your results, because it helps ensure that your sample is unbiased and representative of the whole population. This allows you to make stronger statistical inferences .

You use a simple random sample to collect data. Because you have access to the whole population (all employees), you can assign all 8000 employees a number and use a random number generator to select 300 employees. These 300 employees are your full sample.

Random assignment enhances the internal validity of the study, because it ensures that there are no systematic differences between the participants in each group. This helps you conclude that the outcomes can be attributed to the independent variable .

• a control group that receives no intervention.
• an experimental group that has a remote team-building intervention every week for a month.

You use random assignment to place participants into the control or experimental group. To do so, you take your list of participants and assign each participant a number. Again, you use a random number generator to place each participant in one of the two groups.

To use simple random assignment, you start by giving every member of the sample a unique number. Then, you can use computer programs or manual methods to randomly assign each participant to a group.

• Random number generator: Use a computer program to generate random numbers from the list for each group.
• Lottery method: Place all numbers individually in a hat or a bucket, and draw numbers at random for each group.
• Flip a coin: When you only have two groups, for each number on the list, flip a coin to decide if they’ll be in the control or the experimental group.
• Use a dice: When you have three groups, for each number on the list, roll a dice to decide which of the groups they will be in. For example, assume that rolling 1 or 2 lands them in a control group; 3 or 4 in an experimental group; and 5 or 6 in a second control or experimental group.

This type of random assignment is the most powerful method of placing participants in conditions, because each individual has an equal chance of being placed in any one of your treatment groups.

Random assignment in block designs

In more complicated experimental designs, random assignment is only used after participants are grouped into blocks based on some characteristic (e.g., test score or demographic variable). These groupings mean that you need a larger sample to achieve high statistical power .

For example, a randomized block design involves placing participants into blocks based on a shared characteristic (e.g., college students versus graduates), and then using random assignment within each block to assign participants to every treatment condition. This helps you assess whether the characteristic affects the outcomes of your treatment.

In an experimental matched design , you use blocking and then match up individual participants from each block based on specific characteristics. Within each matched pair or group, you randomly assign each participant to one of the conditions in the experiment and compare their outcomes.

Sometimes, it’s not relevant or ethical to use simple random assignment, so groups are assigned in a different way.

When comparing different groups

Sometimes, differences between participants are the main focus of a study, for example, when comparing men and women or people with and without health conditions. Participants are not randomly assigned to different groups, but instead assigned based on their characteristics.

In this type of study, the characteristic of interest (e.g., gender) is an independent variable, and the groups differ based on the different levels (e.g., men, women, etc.). All participants are tested the same way, and then their group-level outcomes are compared.

When it’s not ethically permissible

When studying unhealthy or dangerous behaviors, it’s not possible to use random assignment. For example, if you’re studying heavy drinkers and social drinkers, it’s unethical to randomly assign participants to one of the two groups and ask them to drink large amounts of alcohol for your experiment.

When you can’t assign participants to groups, you can also conduct a quasi-experimental study . In a quasi-experiment, you study the outcomes of pre-existing groups who receive treatments that you may not have any control over (e.g., heavy drinkers and social drinkers). These groups aren’t randomly assigned, but may be considered comparable when some other variables (e.g., age or socioeconomic status) are controlled for.

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If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval
• Quartiles & Quantiles
• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Prospective cohort study

Research bias

• Implicit bias
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hindsight bias
• Affect heuristic
• Social desirability bias

In experimental research, random assignment is a way of placing participants from your sample into different groups using randomization. With this method, every member of the sample has a known or equal chance of being placed in a control group or an experimental group.

Random selection, or random sampling , is a way of selecting members of a population for your study’s sample.

In contrast, random assignment is a way of sorting the sample into control and experimental groups.

Random sampling enhances the external validity or generalizability of your results, while random assignment improves the internal validity of your study.

Random assignment is used in experiments with a between-groups or independent measures design. In this research design, there’s usually a control group and one or more experimental groups. Random assignment helps ensure that the groups are comparable.

In general, you should always use random assignment in this type of experimental design when it is ethically possible and makes sense for your study topic.

To implement random assignment , assign a unique number to every member of your study’s sample .

Then, you can use a random number generator or a lottery method to randomly assign each number to a control or experimental group. You can also do so manually, by flipping a coin or rolling a dice to randomly assign participants to groups.

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Random Assignment in Psychology: Definition & Examples

Julia Simkus

Editor at Simply Psychology

BA (Hons) Psychology, Princeton University

Julia Simkus is a graduate of Princeton University with a Bachelor of Arts in Psychology. She is currently studying for a Master's Degree in Counseling for Mental Health and Wellness in September 2023. Julia's research has been published in peer reviewed journals.

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In psychology, random assignment refers to the practice of allocating participants to different experimental groups in a study in a completely unbiased way, ensuring each participant has an equal chance of being assigned to any group.

In experimental research, random assignment, or random placement, organizes participants from your sample into different groups using randomization. 

Random assignment uses chance procedures to ensure that each participant has an equal opportunity of being assigned to either a control or experimental group.

The control group does not receive the treatment in question, whereas the experimental group does receive the treatment.

When using random assignment, neither the researcher nor the participant can choose the group to which the participant is assigned. This ensures that any differences between and within the groups are not systematic at the onset of the study. 

In a study to test the success of a weight-loss program, investigators randomly assigned a pool of participants to one of two groups.

Group A participants participated in the weight-loss program for 10 weeks and took a class where they learned about the benefits of healthy eating and exercise.

Group B participants read a 200-page book that explains the benefits of weight loss. The investigator randomly assigned participants to one of the two groups.

The researchers found that those who participated in the program and took the class were more likely to lose weight than those in the other group that received only the book.

Importance 

Random assignment ensures that each group in the experiment is identical before applying the independent variable.

In experiments , researchers will manipulate an independent variable to assess its effect on a dependent variable, while controlling for other variables. Random assignment increases the likelihood that the treatment groups are the same at the onset of a study.

Thus, any changes that result from the independent variable can be assumed to be a result of the treatment of interest. This is particularly important for eliminating sources of bias and strengthening the internal validity of an experiment.

Random assignment is the best method for inferring a causal relationship between a treatment and an outcome.

Random Selection vs. Random Assignment 

Random selection (also called probability sampling or random sampling) is a way of randomly selecting members of a population to be included in your study.

On the other hand, random assignment is a way of sorting the sample participants into control and treatment groups. 

Random selection ensures that everyone in the population has an equal chance of being selected for the study. Once the pool of participants has been chosen, experimenters use random assignment to assign participants into groups. 

Random assignment is only used in between-subjects experimental designs, while random selection can be used in a variety of study designs.

Random Assignment vs Random Sampling

Random sampling refers to selecting participants from a population so that each individual has an equal chance of being chosen. This method enhances the representativeness of the sample.

Random assignment, on the other hand, is used in experimental designs once participants are selected. It involves allocating these participants to different experimental groups or conditions randomly.

This helps ensure that any differences in results across groups are due to manipulating the independent variable, not preexisting differences among participants.

When to Use Random Assignment

Random assignment is used in experiments with a between-groups or independent measures design.

In these research designs, researchers will manipulate an independent variable to assess its effect on a dependent variable, while controlling for other variables.

There is usually a control group and one or more experimental groups. Random assignment helps ensure that the groups are comparable at the onset of the study.

How to Use Random Assignment

There are a variety of ways to assign participants into study groups randomly. Here are a handful of popular methods: 

• Random Number Generator : Give each member of the sample a unique number; use a computer program to randomly generate a number from the list for each group.
• Lottery : Give each member of the sample a unique number. Place all numbers in a hat or bucket and draw numbers at random for each group.
• Flipping a Coin : Flip a coin for each participant to decide if they will be in the control group or experimental group (this method can only be used when you have just two groups) 
• Roll a Die : For each number on the list, roll a dice to decide which of the groups they will be in. For example, assume that rolling 1, 2, or 3 places them in a control group and rolling 3, 4, 5 lands them in an experimental group.

When is Random Assignment not used?

• When it is not ethically permissible: Randomization is only ethical if the researcher has no evidence that one treatment is superior to the other or that one treatment might have harmful side effects. 
• When answering non-causal questions : If the researcher is just interested in predicting the probability of an event, the causal relationship between the variables is not important and observational designs would be more suitable than random assignment. 
• When studying the effect of variables that cannot be manipulated: Some risk factors cannot be manipulated and so it would not make any sense to study them in a randomized trial. For example, we cannot randomly assign participants into categories based on age, gender, or genetic factors.

Drawbacks of Random Assignment

While randomization assures an unbiased assignment of participants to groups, it does not guarantee the equality of these groups. There could still be extraneous variables that differ between groups or group differences that arise from chance. Additionally, there is still an element of luck with random assignments.

Thus, researchers can not produce perfectly equal groups for each specific study. Differences between the treatment group and control group might still exist, and the results of a randomized trial may sometimes be wrong, but this is absolutely okay.

Scientific evidence is a long and continuous process, and the groups will tend to be equal in the long run when data is aggregated in a meta-analysis.

Additionally, external validity (i.e., the extent to which the researcher can use the results of the study to generalize to the larger population) is compromised with random assignment.

Random assignment is challenging to implement outside of controlled laboratory conditions and might not represent what would happen in the real world at the population level. 

Random assignment can also be more costly than simple observational studies, where an investigator is just observing events without intervening with the population.

Randomization also can be time-consuming and challenging, especially when participants refuse to receive the assigned treatment or do not adhere to recommendations. 

What is the difference between random sampling and random assignment?

Random sampling refers to randomly selecting a sample of participants from a population. Random assignment refers to randomly assigning participants to treatment groups from the selected sample.

Does random assignment increase internal validity?

Yes, random assignment ensures that there are no systematic differences between the participants in each group, enhancing the study’s internal validity .

Does random assignment reduce sampling error?

Yes, with random assignment, participants have an equal chance of being assigned to either a control group or an experimental group, resulting in a sample that is, in theory, representative of the population.

Random assignment does not completely eliminate sampling error because a sample only approximates the population from which it is drawn. However, random sampling is a way to minimize sampling errors. 

When is random assignment not possible?

Random assignment is not possible when the experimenters cannot control the treatment or independent variable.

For example, if you want to compare how men and women perform on a test, you cannot randomly assign subjects to these groups.

Participants are not randomly assigned to different groups in this study, but instead assigned based on their characteristics.

Does random assignment eliminate confounding variables?

Yes, random assignment eliminates the influence of any confounding variables on the treatment because it distributes them at random among the study groups. Randomization invalidates any relationship between a confounding variable and the treatment.

Why is random assignment of participants to treatment conditions in an experiment used?

Random assignment is used to ensure that all groups are comparable at the start of a study. This allows researchers to conclude that the outcomes of the study can be attributed to the intervention at hand and to rule out alternative explanations for study results.

Further Reading

• Bogomolnaia, A., & Moulin, H. (2001). A new solution to the random assignment problem .  Journal of Economic theory ,  100 (2), 295-328.
• Krause, M. S., & Howard, K. I. (2003). What random assignment does and does not do .  Journal of Clinical Psychology ,  59 (7), 751-766.

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The Definition of Random Assignment According to Psychology

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Emily is a board-certified science editor who has worked with top digital publishing brands like Voices for Biodiversity, Study.com, GoodTherapy, Vox, and Verywell.

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Random assignment refers to the use of chance procedures in psychology experiments to ensure that each participant has the same opportunity to be assigned to any given group in a study to eliminate any potential bias in the experiment at the outset. Participants are randomly assigned to different groups, such as the treatment group versus the control group. In clinical research, randomized clinical trials are known as the gold standard for meaningful results.

Simple random assignment techniques might involve tactics such as flipping a coin, drawing names out of a hat, rolling dice, or assigning random numbers to a list of participants. It is important to note that random assignment differs from random selection .

While random selection refers to how participants are randomly chosen from a target population as representatives of that population, random assignment refers to how those chosen participants are then assigned to experimental groups.

Random Assignment In Research

To determine if changes in one variable will cause changes in another variable, psychologists must perform an experiment. Random assignment is a critical part of the experimental design that helps ensure the reliability of the study outcomes.

Researchers often begin by forming a testable hypothesis predicting that one variable of interest will have some predictable impact on another variable.

The variable that the experimenters will manipulate in the experiment is known as the independent variable , while the variable that they will then measure for different outcomes is known as the dependent variable. While there are different ways to look at relationships between variables, an experiment is the best way to get a clear idea if there is a cause-and-effect relationship between two or more variables.

Once researchers have formulated a hypothesis, conducted background research, and chosen an experimental design, it is time to find participants for their experiment. How exactly do researchers decide who will be part of an experiment? As mentioned previously, this is often accomplished through something known as random selection.

Random Selection

In order to generalize the results of an experiment to a larger group, it is important to choose a sample that is representative of the qualities found in that population. For example, if the total population is 60% female and 40% male, then the sample should reflect those same percentages.

Choosing a representative sample is often accomplished by randomly picking people from the population to be participants in a study. Random selection means that everyone in the group stands an equal chance of being chosen to minimize any bias. Once a pool of participants has been selected, it is time to assign them to groups.

By randomly assigning the participants into groups, the experimenters can be fairly sure that each group will have the same characteristics before the independent variable is applied.

Participants might be randomly assigned to the control group , which does not receive the treatment in question. The control group may receive a placebo or receive the standard treatment. Participants may also be randomly assigned to the experimental group , which receives the treatment of interest. In larger studies, there can be multiple treatment groups for comparison.

There are simple methods of random assignment, like rolling the die. However, there are more complex techniques that involve random number generators to remove any human error.

There can also be random assignment to groups with pre-established rules or parameters. For example, if you want to have an equal number of men and women in each of your study groups, you might separate your sample into two groups (by sex) before randomly assigning each of those groups into the treatment group and control group.

Random assignment is essential because it increases the likelihood that the groups are the same at the outset. With all characteristics being equal between groups, other than the application of the independent variable, any differences found between group outcomes can be more confidently attributed to the effect of the intervention.

Example of Random Assignment

Imagine that a researcher is interested in learning whether or not drinking caffeinated beverages prior to an exam will improve test performance. After randomly selecting a pool of participants, each person is randomly assigned to either the control group or the experimental group.

The participants in the control group consume a placebo drink prior to the exam that does not contain any caffeine. Those in the experimental group, on the other hand, consume a caffeinated beverage before taking the test.

Participants in both groups then take the test, and the researcher compares the results to determine if the caffeinated beverage had any impact on test performance.

A Word From Verywell

Random assignment plays an important role in the psychology research process. Not only does this process help eliminate possible sources of bias, but it also makes it easier to generalize the results of a tested sample of participants to a larger population.

Random assignment helps ensure that members of each group in the experiment are the same, which means that the groups are also likely more representative of what is present in the larger population of interest. Through the use of this technique, psychology researchers are able to study complex phenomena and contribute to our understanding of the human mind and behavior.

Lin Y, Zhu M, Su Z. The pursuit of balance: An overview of covariate-adaptive randomization techniques in clinical trials . Contemp Clin Trials. 2015;45(Pt A):21-25. doi:10.1016/j.cct.2015.07.011

Sullivan L. Random assignment versus random selection . In: The SAGE Glossary of the Social and Behavioral Sciences. SAGE Publications, Inc.; 2009. doi:10.4135/9781412972024.n2108

Alferes VR. Methods of Randomization in Experimental Design . SAGE Publications, Inc.; 2012. doi:10.4135/9781452270012

Nestor PG, Schutt RK. Research Methods in Psychology: Investigating Human Behavior. (2nd Ed.). SAGE Publications, Inc.; 2015.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Random Assignment in Psychology (Intro for Students)

Random assignment is a research procedure used to randomly assign participants to different experimental conditions (or ‘groups’). This introduces the element of chance, ensuring that each participant has an equal likelihood of being placed in any condition group for the study.

It is absolutely essential that the treatment condition and the control condition are the same in all ways except for the variable being manipulated.

Using random assignment to place participants in different conditions helps to achieve this.

It ensures that those conditions are the same in regards to all potential confounding variables and extraneous factors .

Why Researchers Use Random Assignment

Researchers use random assignment to control for confounds in research.

Confounds refer to unwanted and often unaccounted-for variables that might affect the outcome of a study. These confounding variables can skew the results, rendering the experiment unreliable.

For example, below is a study with two groups. Note how there are more ‘red’ individuals in the first group than the second:

There is likely a confounding variable in this experiment explaining why more red people ended up in the treatment condition and less in the control condition. The red people might have self-selected, for example, leading to a skew of them in one group over the other.

Ideally, we’d want a more even distribution, like below:

To achieve better balance in our two conditions, we use randomized sampling.

Fact File: Experiments 101

Random assignment is used in the type of research called the experiment.

An experiment involves manipulating the level of one variable and examining how it affects another variable. These are the independent and dependent variables :

• Independent Variable: The variable manipulated is called the independent variable (IV)
• Dependent Variable: The variable that it is expected to affect is called the dependent variable (DV).

The most basic form of the experiment involves two conditions: the treatment and the control .

• The Treatment Condition: The treatment condition involves the participants being exposed to the IV.
• The Control Condition: The control condition involves the absence of the IV. Therefore, the IV has two levels: zero and some quantity.

Researchers utilize random assignment to determine which participants go into which conditions.

Methods of Random Assignment

There are several procedures that researchers can use to randomly assign participants to different conditions.

1. Random number generator

There are several websites that offer computer-generated random numbers. Simply indicate how many conditions are in the experiment and then click. If there are 4 conditions, the program will randomly generate a number between 1 and 4 each time it is clicked.

2. Flipping a coin

If there are two conditions in an experiment, then the simplest way to implement random assignment is to flip a coin for each participant. Heads means being assigned to the treatment and tails means being assigned to the control (or vice versa).

3. Rolling a die

Rolling a single die is another way to randomly assign participants. If the experiment has three conditions, then numbers 1 and 2 mean being assigned to the control; numbers 3 and 4 mean treatment condition one; and numbers 5 and 6 mean treatment condition two.

4. Condition names in a hat

In some studies, the researcher will write the name of the treatment condition(s) or control on slips of paper and place them in a hat. If there are 4 conditions and 1 control, then there are 5 slips of paper.

The researcher closes their eyes and selects one slip for each participant. That person is then assigned to one of the conditions in the study and that slip of paper is placed back in the hat. Repeat as necessary.

There are other ways of trying to ensure that the groups of participants are equal in all ways with the exception of the IV. However, random assignment is the most often used because it is so effective at reducing confounds.

Read About More Methods and Examples of Random Assignment Here

Potential Confounding Effects

Random assignment is all about minimizing confounding effects.

Here are six types of confounds that can be controlled for using random assignment:

• Individual Differences: Participants in a study will naturally vary in terms of personality, intelligence, mood, prior knowledge, and many other characteristics. If one group happens to have more people with a particular characteristic, this could affect the results. Random assignment ensures that these individual differences are spread out equally among the experimental groups, making it less likely that they will unduly influence the outcome.
• Temporal or Time-Related Confounds: Events or situations that occur at a particular time can influence the outcome of an experiment. For example, a participant might be tested after a stressful event, while another might be tested after a relaxing weekend. Random assignment ensures that such effects are equally distributed among groups, thus controlling for their potential influence.
• Order Effects: If participants are exposed to multiple treatments or tests, the order in which they experience them can influence their responses. Randomly assigning the order of treatments for different participants helps control for this.
• Location or Environmental Confounds: The environment in which the study is conducted can influence the results. One group might be tested in a noisy room, while another might be in a quiet room. Randomly assigning participants to different locations can control for these effects.
• Instrumentation Confounds: These occur when there are variations in the calibration or functioning of measurement instruments across conditions. If one group’s responses are being measured using a slightly different tool or scale, it can introduce a confound. Random assignment can ensure that any such potential inconsistencies in instrumentation are equally distributed among groups.
• Experimenter Effects: Sometimes, the behavior or expectations of the person administering the experiment can unintentionally influence the participants’ behavior or responses. For instance, if an experimenter believes one treatment is superior, they might unconsciously communicate this belief to participants. Randomly assigning experimenters or using a double-blind procedure (where neither the participant nor the experimenter knows the treatment being given) can help control for this.

Random assignment helps balance out these and other potential confounds across groups, ensuring that any observed differences are more likely due to the manipulated independent variable rather than some extraneous factor.

Limitations of the Random Assignment Procedure

Although random assignment is extremely effective at eliminating the presence of participant-related confounds, there are several scenarios in which it cannot be used.

• Ethics: The most obvious scenario is when it would be unethical. For example, if wanting to investigate the effects of emotional abuse on children, it would be unethical to randomly assign children to either received abuse or not.  Even if a researcher were to propose such a study, it would not receive approval from the Institutional Review Board (IRB) which oversees research by university faculty.
• Practicality: Other scenarios involve matters of practicality. For example, randomly assigning people to specific types of diet over a 10-year period would be interesting, but it would be highly unlikely that participants would be diligent enough to make the study valid. This is why examining these types of subjects has to be carried out through observational studies . The data is correlational, which is informative, but falls short of the scientist’s ultimate goal of identifying causality.
• Small Sample Size: The smaller the sample size being assigned to conditions, the more likely it is that the two groups will be unequal. For example, if you flip a coin many times in a row then you will notice that sometimes there will be a string of heads or tails that come up consecutively. This means that one condition may have a build-up of participants that share the same characteristics. However, if you continue flipping the coin, over the long-term, there will be a balance of heads and tails. Unfortunately, how large a sample size is necessary has been the subject of considerable debate (Bloom, 2006; Shadish et al., 2002).

“It is well known that larger sample sizes reduce the probability that random assignment will result in conditions that are unequal” (Goldberg, 2019, p. 2).

Applications of Random Assignment

The importance of random assignment has been recognized in a wide range of scientific and applied disciplines (Bloom, 2006).

Random assignment began as a tool in agricultural research by Fisher (1925, 1935). After WWII, it became extensively used in medical research to test the effectiveness of new treatments and pharmaceuticals (Marks, 1997).

Today it is widely used in industrial engineering (Box, Hunter, and Hunter, 2005), educational research (Lindquist, 1953; Ong-Dean et al., 2011)), psychology (Myers, 1972), and social policy studies (Boruch, 1998; Orr, 1999).

One of the biggest obstacles to the validity of an experiment is the confound. If the group of participants in the treatment condition are substantially different from the group in the control condition, then it is impossible to determine if the IV has an affect or if the confound has an effect.

Thankfully, random assignment is highly effective at eliminating confounds that are known and unknown. Because each participant has an equal chance of being placed in each condition, they are equally distributed.

There are several ways of implementing random assignment, including flipping a coin or using a random number generator.

Random assignment has become an essential procedure in research in a wide range of subjects such as psychology, education, and social policy.

Alferes, V. R. (2012). Methods of randomization in experimental design . Sage Publications.

Bloom, H. S. (2008). The core analytics of randomized experiments for social research. The SAGE Handbook of Social Research Methods , 115-133.

Boruch, R. F. (1998). Randomized controlled experiments for evaluation and planning. Handbook of applied social research methods , 161-191.

Box, G. E., Hunter, W. G., & Hunter, J. S. (2005). Design of experiments: Statistics for Experimenters: Design, Innovation and Discovery.

Dehue, T. (1997). Deception, efficiency, and random groups: Psychology and the gradual origination of the random group design. Isis , 88 (4), 653-673.

Fisher, R.A. (1925). Statistical methods for research workers (11th ed. rev.). Oliver and Boyd: Edinburgh.

Fisher, R. A. (1935). The Design of Experiments. Edinburgh: Oliver and Boyd.

Goldberg, M. H. (2019). How often does random assignment fail? Estimates and recommendations. Journal of Environmental Psychology , 66 , 101351.

Jamison, J. C. (2019). The entry of randomized assignment into the social sciences. Journal of Causal Inference , 7 (1), 20170025.

Lindquist, E. F. (1953). Design and analysis of experiments in psychology and education . Boston: Houghton Mifflin Company.

Marks, H. M. (1997). The progress of experiment: Science and therapeutic reform in the United States, 1900-1990 . Cambridge University Press.

Myers, J. L. (1972). Fundamentals of experimental design (2nd ed.). Allyn & Bacon.

Ong-Dean, C., Huie Hofstetter, C., & Strick, B. R. (2011). Challenges and dilemmas in implementing random assignment in educational research. American Journal of Evaluation , 32 (1), 29-49.

Orr, L. L. (1999). Social experiments: Evaluating public programs with experimental methods . Sage.

Shadish, W. R., Cook, T. D., & Campbell, D. T. (2002). Quasi-experiments: interrupted time-series designs. Experimental and quasi-experimental designs for generalized causal inference , 171-205.

Stigler, S. M. (1992). A historical view of statistical concepts in psychology and educational research. American Journal of Education , 101 (1), 60-70.

Dave Cornell (PhD)

Dr. Cornell has worked in education for more than 20 years. His work has involved designing teacher certification for Trinity College in London and in-service training for state governments in the United States. He has trained kindergarten teachers in 8 countries and helped businessmen and women open baby centers and kindergartens in 3 countries.

• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ 25 Positive Punishment Examples
• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ 25 Dissociation Examples (Psychology)
• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ 15 Zone of Proximal Development Examples
• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ Perception Checking: 15 Examples and Definition

Chris Drew (PhD)

This article was peer-reviewed and edited by Chris Drew (PhD). The review process on Helpful Professor involves having a PhD level expert fact check, edit, and contribute to articles. Reviewers ensure all content reflects expert academic consensus and is backed up with reference to academic studies. Dr. Drew has published over 20 academic articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education and holds a PhD in Education from ACU.

• Chris Drew (PhD) #molongui-disabled-link 25 Positive Punishment Examples
• Chris Drew (PhD) #molongui-disabled-link 25 Dissociation Examples (Psychology)
• Chris Drew (PhD) #molongui-disabled-link 15 Zone of Proximal Development Examples
• Chris Drew (PhD) #molongui-disabled-link Perception Checking: 15 Examples and Definition

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Random Assignment in Psychology (Definition + 40 Examples)

Have you ever wondered how researchers discover new ways to help people learn, make decisions, or overcome challenges? A hidden hero in this adventure of discovery is a method called random assignment, a cornerstone in psychological research that helps scientists uncover the truths about the human mind and behavior.

Random Assignment is a process used in research where each participant has an equal chance of being placed in any group within the study. This technique is essential in experiments as it helps to eliminate biases, ensuring that the different groups being compared are similar in all important aspects.

By doing so, researchers can be confident that any differences observed are likely due to the variable being tested, rather than other factors.

In this article, we’ll explore the intriguing world of random assignment, diving into its history, principles, real-world examples, and the impact it has had on the field of psychology.

History of Random Assignment

Stepping back in time, we delve into the origins of random assignment, which finds its roots in the early 20th century.

The pioneering mind behind this innovative technique was Sir Ronald A. Fisher , a British statistician and biologist. Fisher introduced the concept of random assignment in the 1920s, aiming to improve the quality and reliability of experimental research .

His contributions laid the groundwork for the method's evolution and its widespread adoption in various fields, particularly in psychology.

Fisher’s groundbreaking work on random assignment was motivated by his desire to control for confounding variables – those pesky factors that could muddy the waters of research findings.

By assigning participants to different groups purely by chance, he realized that the influence of these confounding variables could be minimized, paving the way for more accurate and trustworthy results.

Early Studies Utilizing Random Assignment

Following Fisher's initial development, random assignment started to gain traction in the research community. Early studies adopting this methodology focused on a variety of topics, from agriculture (which was Fisher’s primary field of interest) to medicine and psychology.

The approach allowed researchers to draw stronger conclusions from their experiments, bolstering the development of new theories and practices.

One notable early study utilizing random assignment was conducted in the field of educational psychology. Researchers were keen to understand the impact of different teaching methods on student outcomes.

By randomly assigning students to various instructional approaches, they were able to isolate the effects of the teaching methods, leading to valuable insights and recommendations for educators.

Evolution of the Methodology

As the decades rolled on, random assignment continued to evolve and adapt to the changing landscape of research.

Advances in technology introduced new tools and techniques for implementing randomization, such as computerized random number generators, which offered greater precision and ease of use.

The application of random assignment expanded beyond the confines of the laboratory, finding its way into field studies and large-scale surveys.

Researchers across diverse disciplines embraced the methodology, recognizing its potential to enhance the validity of their findings and contribute to the advancement of knowledge.

From its humble beginnings in the early 20th century to its widespread use today, random assignment has proven to be a cornerstone of scientific inquiry.

Its development and evolution have played a pivotal role in shaping the landscape of psychological research, driving discoveries that have improved lives and deepened our understanding of the human experience.

Principles of Random Assignment

Delving into the heart of random assignment, we uncover the theories and principles that form its foundation.

The method is steeped in the basics of probability theory and statistical inference, ensuring that each participant has an equal chance of being placed in any group, thus fostering fair and unbiased results.

Basic Principles of Random Assignment

Understanding the core principles of random assignment is key to grasping its significance in research. There are three principles: equal probability of selection, reduction of bias, and ensuring representativeness.

The first principle, equal probability of selection , ensures that every participant has an identical chance of being assigned to any group in the study. This randomness is crucial as it mitigates the risk of bias and establishes a level playing field.

The second principle focuses on the reduction of bias . Random assignment acts as a safeguard, ensuring that the groups being compared are alike in all essential aspects before the experiment begins.

This similarity between groups allows researchers to attribute any differences observed in the outcomes directly to the independent variable being studied.

Lastly, ensuring representativeness is a vital principle. When participants are assigned randomly, the resulting groups are more likely to be representative of the larger population.

This characteristic is crucial for the generalizability of the study’s findings, allowing researchers to apply their insights broadly.

Theoretical Foundation

The theoretical foundation of random assignment lies in probability theory and statistical inference .

Probability theory deals with the likelihood of different outcomes, providing a mathematical framework for analyzing random phenomena. In the context of random assignment, it helps in ensuring that each participant has an equal chance of being placed in any group.

Statistical inference, on the other hand, allows researchers to draw conclusions about a population based on a sample of data drawn from that population. It is the mechanism through which the results of a study can be generalized to a broader context.

Random assignment enhances the reliability of statistical inferences by reducing biases and ensuring that the sample is representative.

Differentiating Random Assignment from Random Selection

It’s essential to distinguish between random assignment and random selection, as the two terms, while related, have distinct meanings in the realm of research.

Random assignment refers to how participants are placed into different groups in an experiment, aiming to control for confounding variables and help determine causes.

In contrast, random selection pertains to how individuals are chosen to participate in a study. This method is used to ensure that the sample of participants is representative of the larger population, which is vital for the external validity of the research.

While both methods are rooted in randomness and probability, they serve different purposes in the research process.

Understanding the theories, principles, and distinctions of random assignment illuminates its pivotal role in psychological research.

This method, anchored in probability theory and statistical inference, serves as a beacon of reliability, guiding researchers in their quest for knowledge and ensuring that their findings stand the test of validity and applicability.

Methodology of Random Assignment

Implementing random assignment in a study is a meticulous process that involves several crucial steps.

The initial step is participant selection, where individuals are chosen to partake in the study. This stage is critical to ensure that the pool of participants is diverse and representative of the population the study aims to generalize to.

Once the pool of participants has been established, the actual assignment process begins. In this step, each participant is allocated randomly to one of the groups in the study.

Researchers use various tools, such as random number generators or computerized methods, to ensure that this assignment is genuinely random and free from biases.

Monitoring and adjusting form the final step in the implementation of random assignment. Researchers need to continuously observe the groups to ensure that they remain comparable in all essential aspects throughout the study.

If any significant discrepancies arise, adjustments might be necessary to maintain the study’s integrity and validity.

Tools and Techniques Used

The evolution of technology has introduced a variety of tools and techniques to facilitate random assignment.

Random number generators, both manual and computerized, are commonly used to assign participants to different groups. These generators ensure that each individual has an equal chance of being placed in any group, upholding the principle of equal probability of selection.

In addition to random number generators, researchers often use specialized computer software designed for statistical analysis and experimental design.

These software programs offer advanced features that allow for precise and efficient random assignment, minimizing the risk of human error and enhancing the study’s reliability.

Ethical Considerations

The implementation of random assignment is not devoid of ethical considerations. Informed consent is a fundamental ethical principle that researchers must uphold.

Informed consent means that every participant should be fully informed about the nature of the study, the procedures involved, and any potential risks or benefits, ensuring that they voluntarily agree to participate.

Beyond informed consent, researchers must conduct a thorough risk and benefit analysis. The potential benefits of the study should outweigh any risks or harms to the participants.

Safeguarding the well-being of participants is paramount, and any study employing random assignment must adhere to established ethical guidelines and standards.

Conclusion of Methodology

The methodology of random assignment, while seemingly straightforward, is a multifaceted process that demands precision, fairness, and ethical integrity. From participant selection to assignment and monitoring, each step is crucial to ensure the validity of the study’s findings.

The tools and techniques employed, coupled with a steadfast commitment to ethical principles, underscore the significance of random assignment as a cornerstone of robust psychological research.

Benefits of Random Assignment in Psychological Research

The impact and importance of random assignment in psychological research cannot be overstated. It is fundamental for ensuring the study is accurate, allowing the researchers to determine if their study actually caused the results they saw, and making sure the findings can be applied to the real world.

Facilitating Causal Inferences

When participants are randomly assigned to different groups, researchers can be more confident that the observed effects are due to the independent variable being changed, and not other factors.

This ability to determine the cause is called causal inference .

This confidence allows for the drawing of causal relationships, which are foundational for theory development and application in psychology.

Ensuring Internal Validity

One of the foremost impacts of random assignment is its ability to enhance the internal validity of an experiment.

Internal validity refers to the extent to which a researcher can assert that changes in the dependent variable are solely due to manipulations of the independent variable , and not due to confounding variables.

By ensuring that each participant has an equal chance of being in any condition of the experiment, random assignment helps control for participant characteristics that could otherwise complicate the results.

Enhancing Generalizability

Beyond internal validity, random assignment also plays a crucial role in enhancing the generalizability of research findings.

When done correctly, it ensures that the sample groups are representative of the larger population, so can allow researchers to apply their findings more broadly.

This representative nature is essential for the practical application of research, impacting policy, interventions, and psychological therapies.

Limitations of Random Assignment

Potential for implementation issues.

While the principles of random assignment are robust, the method can face implementation issues.

One of the most common problems is logistical constraints. Some studies, due to their nature or the specific population being studied, find it challenging to implement random assignment effectively.

For instance, in educational settings, logistical issues such as class schedules and school policies might stop the random allocation of students to different teaching methods .

Ethical Dilemmas

Random assignment, while methodologically sound, can also present ethical dilemmas.

In some cases, withholding a potentially beneficial treatment from one of the groups of participants can raise serious ethical questions, especially in medical or clinical research where participants' well-being might be directly affected.

Researchers must navigate these ethical waters carefully, balancing the pursuit of knowledge with the well-being of participants.

Generalizability Concerns

Even when implemented correctly, random assignment does not always guarantee generalizable results.

The types of people in the participant pool, the specific context of the study, and the nature of the variables being studied can all influence the extent to which the findings can be applied to the broader population.

Researchers must be cautious in making broad generalizations from studies, even those employing strict random assignment.

Practical and Real-World Limitations

In the real world, many variables cannot be manipulated for ethical or practical reasons, limiting the applicability of random assignment.

For instance, researchers cannot randomly assign individuals to different levels of intelligence, socioeconomic status, or cultural backgrounds.

This limitation necessitates the use of other research designs, such as correlational or observational studies , when exploring relationships involving such variables.

Response to Critiques

In response to these critiques, people in favor of random assignment argue that the method, despite its limitations, remains one of the most reliable ways to establish cause and effect in experimental research.

They acknowledge the challenges and ethical considerations but emphasize the rigorous frameworks in place to address them.

The ongoing discussion around the limitations and critiques of random assignment contributes to the evolution of the method, making sure it is continuously relevant and applicable in psychological research.

While random assignment is a powerful tool in experimental research, it is not without its critiques and limitations. Implementation issues, ethical dilemmas, generalizability concerns, and real-world limitations can pose significant challenges.

However, the continued discourse and refinement around these issues underline the method's enduring significance in the pursuit of knowledge in psychology.

By being careful with how we do things and doing what's right, random assignment stays a really important part of studying how people act and think.

Real-World Applications and Examples

Random assignment has been employed in many studies across various fields of psychology, leading to significant discoveries and advancements.

Here are some real-world applications and examples illustrating the diversity and impact of this method:

• Medicine and Health Psychology: Randomized Controlled Trials (RCTs) are the gold standard in medical research. In these studies, participants are randomly assigned to either the treatment or control group to test the efficacy of new medications or interventions.
• Educational Psychology: Studies in this field have used random assignment to explore the effects of different teaching methods, classroom environments, and educational technologies on student learning and outcomes.
• Cognitive Psychology: Researchers have employed random assignment to investigate various aspects of human cognition, including memory, attention, and problem-solving, leading to a deeper understanding of how the mind works.
• Social Psychology: Random assignment has been instrumental in studying social phenomena, such as conformity, aggression, and prosocial behavior, shedding light on the intricate dynamics of human interaction.

Let's get into some specific examples. You'll need to know one term though, and that is "control group." A control group is a set of participants in a study who do not receive the treatment or intervention being tested , serving as a baseline to compare with the group that does, in order to assess the effectiveness of the treatment.

• Smoking Cessation Study: Researchers used random assignment to put participants into two groups. One group received a new anti-smoking program, while the other did not. This helped determine if the program was effective in helping people quit smoking.
• Math Tutoring Program: A study on students used random assignment to place them into two groups. One group received additional math tutoring, while the other continued with regular classes, to see if the extra help improved their grades.
• Exercise and Mental Health: Adults were randomly assigned to either an exercise group or a control group to study the impact of physical activity on mental health and mood.
• Diet and Weight Loss: A study randomly assigned participants to different diet plans to compare their effectiveness in promoting weight loss and improving health markers.
• Sleep and Learning: Researchers randomly assigned students to either a sleep extension group or a regular sleep group to study the impact of sleep on learning and memory.
• Classroom Seating Arrangement: Teachers used random assignment to place students in different seating arrangements to examine the effect on focus and academic performance.
• Music and Productivity: Employees were randomly assigned to listen to music or work in silence to investigate the effect of music on workplace productivity.
• Medication for ADHD: Children with ADHD were randomly assigned to receive either medication, behavioral therapy, or a placebo to compare treatment effectiveness.
• Mindfulness Meditation for Stress: Adults were randomly assigned to a mindfulness meditation group or a waitlist control group to study the impact on stress levels.
• Video Games and Aggression: A study randomly assigned participants to play either violent or non-violent video games and then measured their aggression levels.
• Online Learning Platforms: Students were randomly assigned to use different online learning platforms to evaluate their effectiveness in enhancing learning outcomes.
• Hand Sanitizers in Schools: Schools were randomly assigned to use hand sanitizers or not to study the impact on student illness and absenteeism.
• Caffeine and Alertness: Participants were randomly assigned to consume caffeinated or decaffeinated beverages to measure the effects on alertness and cognitive performance.
• Green Spaces and Well-being: Neighborhoods were randomly assigned to receive green space interventions to study the impact on residents’ well-being and community connections.
• Pet Therapy for Hospital Patients: Patients were randomly assigned to receive pet therapy or standard care to assess the impact on recovery and mood.
• Yoga for Chronic Pain: Individuals with chronic pain were randomly assigned to a yoga intervention group or a control group to study the effect on pain levels and quality of life.
• Flu Vaccines Effectiveness: Different groups of people were randomly assigned to receive either the flu vaccine or a placebo to determine the vaccine’s effectiveness.
• Reading Strategies for Dyslexia: Children with dyslexia were randomly assigned to different reading intervention strategies to compare their effectiveness.
• Physical Environment and Creativity: Participants were randomly assigned to different room setups to study the impact of physical environment on creative thinking.
• Laughter Therapy for Depression: Individuals with depression were randomly assigned to laughter therapy sessions or control groups to assess the impact on mood.
• Financial Incentives for Exercise: Participants were randomly assigned to receive financial incentives for exercising to study the impact on physical activity levels.
• Art Therapy for Anxiety: Individuals with anxiety were randomly assigned to art therapy sessions or a waitlist control group to measure the effect on anxiety levels.
• Natural Light in Offices: Employees were randomly assigned to workspaces with natural or artificial light to study the impact on productivity and job satisfaction.
• School Start Times and Academic Performance: Schools were randomly assigned different start times to study the effect on student academic performance and well-being.
• Horticulture Therapy for Seniors: Older adults were randomly assigned to participate in horticulture therapy or traditional activities to study the impact on cognitive function and life satisfaction.
• Hydration and Cognitive Function: Participants were randomly assigned to different hydration levels to measure the impact on cognitive function and alertness.
• Intergenerational Programs: Seniors and young people were randomly assigned to intergenerational programs to study the effects on well-being and cross-generational understanding.
• Therapeutic Horseback Riding for Autism: Children with autism were randomly assigned to therapeutic horseback riding or traditional therapy to study the impact on social communication skills.
• Active Commuting and Health: Employees were randomly assigned to active commuting (cycling, walking) or passive commuting to study the effect on physical health.
• Mindful Eating for Weight Management: Individuals were randomly assigned to mindful eating workshops or control groups to study the impact on weight management and eating habits.
• Noise Levels and Learning: Students were randomly assigned to classrooms with different noise levels to study the effect on learning and concentration.
• Bilingual Education Methods: Schools were randomly assigned different bilingual education methods to compare their effectiveness in language acquisition.
• Outdoor Play and Child Development: Children were randomly assigned to different amounts of outdoor playtime to study the impact on physical and cognitive development.
• Social Media Detox: Participants were randomly assigned to a social media detox or regular usage to study the impact on mental health and well-being.
• Therapeutic Writing for Trauma Survivors: Individuals who experienced trauma were randomly assigned to therapeutic writing sessions or control groups to study the impact on psychological well-being.
• Mentoring Programs for At-risk Youth: At-risk youth were randomly assigned to mentoring programs or control groups to assess the impact on academic achievement and behavior.
• Dance Therapy for Parkinson’s Disease: Individuals with Parkinson’s disease were randomly assigned to dance therapy or traditional exercise to study the effect on motor function and quality of life.
• Aquaponics in Schools: Schools were randomly assigned to implement aquaponics programs to study the impact on student engagement and environmental awareness.
• Virtual Reality for Phobia Treatment: Individuals with phobias were randomly assigned to virtual reality exposure therapy or traditional therapy to compare effectiveness.
• Gardening and Mental Health: Participants were randomly assigned to engage in gardening or other leisure activities to study the impact on mental health and stress reduction.

Each of these studies exemplifies how random assignment is utilized in various fields and settings, shedding light on the multitude of ways it can be applied to glean valuable insights and knowledge.

Real-world Impact of Random Assignment

Random assignment is like a key tool in the world of learning about people's minds and behaviors. It’s super important and helps in many different areas of our everyday lives. It helps make better rules, creates new ways to help people, and is used in lots of different fields.

Health and Medicine

In health and medicine, random assignment has helped doctors and scientists make lots of discoveries. It’s a big part of tests that help create new medicines and treatments.

By putting people into different groups by chance, scientists can really see if a medicine works.

This has led to new ways to help people with all sorts of health problems, like diabetes, heart disease, and mental health issues like depression and anxiety.

Schools and education have also learned a lot from random assignment. Researchers have used it to look at different ways of teaching, what kind of classrooms are best, and how technology can help learning.

This knowledge has helped make better school rules, develop what we learn in school, and find the best ways to teach students of all ages and backgrounds.

Workplace and Organizational Behavior

Random assignment helps us understand how people act at work and what makes a workplace good or bad.

Studies have looked at different kinds of workplaces, how bosses should act, and how teams should be put together. This has helped companies make better rules and create places to work that are helpful and make people happy.

Environmental and Social Changes

Random assignment is also used to see how changes in the community and environment affect people. Studies have looked at community projects, changes to the environment, and social programs to see how they help or hurt people’s well-being.

This has led to better community projects, efforts to protect the environment, and programs to help people in society.

Technology and Human Interaction

In our world where technology is always changing, studies with random assignment help us see how tech like social media, virtual reality, and online stuff affect how we act and feel.

This has helped make better and safer technology and rules about using it so that everyone can benefit.

The effects of random assignment go far and wide, way beyond just a science lab. It helps us understand lots of different things, leads to new and improved ways to do things, and really makes a difference in the world around us.

From making healthcare and schools better to creating positive changes in communities and the environment, the real-world impact of random assignment shows just how important it is in helping us learn and make the world a better place.

So, what have we learned? Random assignment is like a super tool in learning about how people think and act. It's like a detective helping us find clues and solve mysteries in many parts of our lives.

From creating new medicines to helping kids learn better in school, and from making workplaces happier to protecting the environment, it’s got a big job!

This method isn’t just something scientists use in labs; it reaches out and touches our everyday lives. It helps make positive changes and teaches us valuable lessons.

Whether we are talking about technology, health, education, or the environment, random assignment is there, working behind the scenes, making things better and safer for all of us.

In the end, the simple act of putting people into groups by chance helps us make big discoveries and improvements. It’s like throwing a small stone into a pond and watching the ripples spread out far and wide.

Thanks to random assignment, we are always learning, growing, and finding new ways to make our world a happier and healthier place for everyone!

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Chapter 6: Data Collection Strategies

6.1.1 Random Assignation

As previously mentioned, one of the characteristics of a true experiment is that researchers use a random process to decide which participants are tested under which conditions. Random assignation is a powerful research technique that addresses the assumption of pre-test equivalence – that the experimental and control group are equal in all respects before the administration of the independent variable (Palys & Atchison, 2014).

Random assignation is the primary way that researchers attempt to control extraneous variables across conditions. Random assignation is associated with experimental research methods. In its strictest sense, random assignment should meet two criteria.  One is that each participant has an equal chance of being assigned to each condition (e.g., a 50% chance of being assigned to each of two conditions). The second is that each participant is assigned to a condition independently of other participants. Thus, one way to assign participants to two conditions would be to flip a coin for each one. If the coin lands on the heads side, the participant is assigned to Condition A, and if it lands on the tails side, the participant is assigned to Condition B. For three conditions, one could use a computer to generate a random integer from 1 to 3 for each participant. If the integer is 1, the participant is assigned to Condition A; if it is 2, the participant is assigned to Condition B; and, if it is 3, the participant is assigned to Condition C. In practice, a full sequence of conditions—one for each participant expected to be in the experiment—is usually created ahead of time, and each new participant is assigned to the next condition in the sequence as he or she is tested.

However, one problem with coin flipping and other strict procedures for random assignment is that they are likely to result in unequal sample sizes in the different conditions. Unequal sample sizes are generally not a serious problem, and you should never throw away data you have already collected to achieve equal sample sizes. However, for a fixed number of participants, it is statistically most efficient to divide them into equal-sized groups. It is standard practice, therefore, to use a kind of modified random assignment that keeps the number of participants in each group as similar as possible.

One approach is block randomization. In block randomization, all the conditions occur once in the sequence before any of them is repeated. Then they all occur again before any of them is repeated again. Within each of these “blocks,” the conditions occur in a random order. Again, the sequence of conditions is usually generated before any participants are tested, and each new participant is assigned to the next condition in the sequence. When the procedure is computerized, the computer program often handles the random assignment, which is obviously much easier. You can also find programs online to help you randomize your random assignation. For example, the Research Randomizer website will generate block randomization sequences for any number of participants and conditions ( Research Randomizer ).

Random assignation is not guaranteed to control all extraneous variables across conditions. It is always possible that, just by chance, the participants in one condition might turn out to be substantially older, less tired, more motivated, or less depressed on average than the participants in another condition. However, there are some reasons that this may not be a major concern. One is that random assignment works better than one might expect, especially for large samples. Another is that the inferential statistics that researchers use to decide whether a difference between groups reflects a difference in the population take the “fallibility” of random assignment into account. Yet another reason is that even if random assignment does result in a confounding variable and therefore produces misleading results, this confound is likely to be detected when the experiment is replicated. The upshot is that random assignment to conditions—although not infallible in terms of controlling extraneous variables—is always considered a strength of a research design. Note: Do not confuse random assignation with random sampling. Random sampling is a method for selecting a sample from a population; we will talk about this in Chapter 7.

Research Methods for the Social Sciences: An Introduction Copyright © 2020 by Valerie Sheppard is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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What Is Random Assignment in Psychology?

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Random assignment means that every participant has the same chance of being chosen for the experimental or control group. It involves using procedures that rely on chance to assign participants to groups. Doing this means that every participant in a study has an equal opportunity to be assigned to any group.

For example, in a psychology experiment, participants might be assigned to either a control or experimental group. Some experiments might only have one experimental group, while others may have several treatment variations.

Using random assignment means that each participant has the same chance of being assigned to any of these groups.

Table of Contents

How to Use Random Assignment

So what type of procedures might psychologists utilize for random assignment? Strategies can include:

• Flipping a coin
• Assigning random numbers
• Rolling dice
• Drawing names out of a hat

How Does Random Assignment Work?

A psychology experiment aims to determine if changes in one variable lead to changes in another variable. Researchers will first begin by coming up with a hypothesis. Once researchers have an idea of what they think they might find in a population, they will come up with an experimental design and then recruit participants for their study.

Once they have a pool of participants representative of the population they are interested in looking at, they will randomly assign the participants to their groups.

• Control group : Some participants will end up in the control group, which serves as a baseline and does not receive the independent variables.
• Experimental group : Other participants will end up in the experimental groups that receive some form of the independent variables.

By using random assignment, the researchers make it more likely that the groups are equal at the start of the experiment. Since the groups are the same on other variables, it can be assumed that any changes that occur are the result of varying the independent variables.

After a treatment has been administered, the researchers will then collect data in order to determine if the independent variable had any impact on the dependent variable.

Random Assignment vs. Random Selection

It is important to remember that random assignment is not the same thing as random selection , also known as random sampling.

Random selection instead involves how people are chosen to be in a study. Using random selection, every member of a population stands an equal chance of being chosen for a study or experiment.

So random sampling affects how participants are chosen for a study, while random assignment affects how participants are then assigned to groups.

Examples of Random Assignment

Imagine that a psychology researcher is conducting an experiment to determine if getting adequate sleep the night before an exam results in better test scores.

Forming a Hypothesis

They hypothesize that participants who get 8 hours of sleep will do better on a math exam than participants who only get 4 hours of sleep.

Obtaining Participants

The researcher starts by obtaining a pool of participants. They find 100 participants from a local university. Half of the participants are female, and half are male.

Randomly Assign Participants to Groups

The researcher then assigns random numbers to each participant and uses a random number generator to randomly assign each number to either the 4-hour or 8-hour sleep groups.

Conduct the Experiment

Those in the 8-hour sleep group agree to sleep for 8 hours that night, while those in the 4-hour group agree to wake up after only 4 hours. The following day, all of the participants meet in a classroom.

Collect and Analyze Data

Everyone takes the same math test. The test scores are then compared to see if the amount of sleep the night before had any impact on test scores.

Why Is Random Assignment Important in Psychology Research?

Random assignment is important in psychology research because it helps improve a study’s internal validity. This means that the researchers are sure that the study demonstrates a cause-and-effect relationship between an independent and dependent variable.

Random assignment improves the internal validity by minimizing the risk that there are systematic differences in the participants who are in each group.

Key Points to Remember About Random Assignment

• Random assignment in psychology involves each participant having an equal chance of being chosen for any of the groups, including the control and experimental groups.
• It helps control for potential confounding variables, reducing the likelihood of pre-existing differences between groups.
• This method enhances the internal validity of experiments, allowing researchers to draw more reliable conclusions about cause-and-effect relationships.
• Random assignment is crucial for creating comparable groups and increasing the scientific rigor of psychological studies.

Purpose and Limitations of Random Assignment

In an experimental study, random assignment is a process by which participants are assigned, with the same chance, to either a treatment or a control group. The goal is to assure an unbiased assignment of participants to treatment options.

Random assignment is considered the gold standard for achieving comparability across study groups, and therefore is the best method for inferring a causal relationship between a treatment (or intervention or risk factor) and an outcome.

Random assignment of participants produces comparable groups regarding the participants’ initial characteristics, thereby any difference detected in the end between the treatment and the control group will be due to the effect of the treatment alone.

How does random assignment produce comparable groups?

1. random assignment prevents selection bias.

Randomization works by removing the researcher’s and the participant’s influence on the treatment allocation. So the allocation can no longer be biased since it is done at random, i.e. in a non-predictable way.

This is in contrast with the real world, where for example, the sickest people are more likely to receive the treatment.

2. Random assignment prevents confounding

A confounding variable is one that is associated with both the intervention and the outcome, and thus can affect the outcome in 2 ways:

Either directly:

Or indirectly through the treatment:

This indirect relationship between the confounding variable and the outcome can cause the treatment to appear to have an influence on the outcome while in reality the treatment is just a mediator of that effect (as it happens to be on the causal pathway between the confounder and the outcome).

Random assignment eliminates the influence of the confounding variables on the treatment since it distributes them at random between the study groups, therefore, ruling out this alternative path or explanation of the outcome.

3. Random assignment also eliminates other threats to internal validity

By distributing all threats (known and unknown) at random between study groups, participants in both the treatment and the control group become equally subject to the effect of any threat to validity. Therefore, comparing the outcome between the 2 groups will bypass the effect of these threats and will only reflect the effect of the treatment on the outcome.

These threats include:

• History: This is any event that co-occurs with the treatment and can affect the outcome.
• Maturation: This is the effect of time on the study participants (e.g. participants becoming wiser, hungrier, or more stressed with time) which might influence the outcome.
• Regression to the mean: This happens when the participants’ outcome score is exceptionally good on a pre-treatment measurement, so the post-treatment measurement scores will naturally regress toward the mean — in simple terms, regression happens since an exceptional performance is hard to maintain. This effect can bias the study since it represents an alternative explanation of the outcome.

Note that randomization does not prevent these effects from happening, it just allows us to control them by reducing their risk of being associated with the treatment.

What if random assignment produced unequal groups?

Question: What should you do if after randomly assigning participants, it turned out that the 2 groups still differ in participants’ characteristics? More precisely, what if randomization accidentally did not balance risk factors that can be alternative explanations between the 2 groups? (For example, if one group includes more male participants, or sicker, or older people than the other group).

Short answer: This is perfectly normal, since randomization only assures an unbiased assignment of participants to groups, i.e. it produces comparable groups, but it does not guarantee the equality of these groups.

A more complete answer: Randomization will not and cannot create 2 equal groups regarding each and every characteristic. This is because when dealing with randomization there is still an element of luck. If you want 2 perfectly equal groups, you better match them manually as is done in a matched pairs design (for more information see my article on matched pairs design ).

This is similar to throwing a die: If you throw it 10 times, the chance of getting a specific outcome will not be 1/6. But it will approach 1/6 if you repeat the experiment a very large number of times and calculate the average number of times the specific outcome turned up.

So randomization will not produce perfectly equal groups for each specific study, especially if the study has a small sample size. But do not forget that scientific evidence is a long and continuous process, and the groups will tend to be equal in the long run when a meta-analysis aggregates the results of a large number of randomized studies.

So for each individual study, differences between the treatment and control group will exist and will influence the study results. This means that the results of a randomized trial will sometimes be wrong, and this is absolutely okay.

BOTTOM LINE:

Although the results of a particular randomized study are unbiased, they will still be affected by a sampling error due to chance. But the real benefit of random assignment will be when data is aggregated in a meta-analysis.

Limitations of random assignment

Randomized designs can suffer from:

1. Ethical issues:

Randomization is ethical only if the researcher has no evidence that one treatment is superior to the other.

Also, it would be unethical to randomly assign participants to harmful exposures such as smoking or dangerous chemicals.

2. Low external validity:

With random assignment, external validity (i.e. the generalizability of the study results) is compromised because the results of a study that uses random assignment represent what would happen under “ideal” experimental conditions, which is in general very different from what happens at the population level.

In the real world, people who take the treatment might be very different from those who don’t – so the assignment of participants is not a random event, but rather under the influence of all sort of external factors.

External validity can be also jeopardized in cases where not all participants are eligible or willing to accept the terms of the study.

3. Higher cost of implementation:

An experimental design with random assignment is typically more expensive than observational studies where the investigator’s role is just to observe events without intervening.

Experimental designs also typically take a lot of time to implement, and therefore are less practical when a quick answer is needed.

4. Impracticality when answering non-causal questions:

A randomized trial is our best bet when the question is to find the causal effect of a treatment or a risk factor.

Sometimes however, the researcher is just interested in predicting the probability of an event or a disease given some risk factors. In this case, the causal relationship between these variables is not important, making observational designs more suitable for such problems.

5. Impracticality when studying the effect of variables that cannot be manipulated:

The usual objective of studying the effects of risk factors is to propose recommendations that involve changing the level of exposure to these factors.

However, some risk factors cannot be manipulated, and so it does not make any sense to study them in a randomized trial. For example it would be impossible to randomly assign participants to age categories, gender, or genetic factors.

6. Difficulty to control participants:

These difficulties include:

• Participants refusing to receive the assigned treatment.
• Participants not adhering to recommendations.
• Differential loss to follow-up between those who receive the treatment and those who don’t.

All of these issues might occur in a randomized trial, but might not affect an observational study.

• Shadish WR, Cook TD, Campbell DT. Experimental and Quasi-Experimental Designs for Generalized Causal Inference . 2nd edition. Cengage Learning; 2001.
• Friedman LM, Furberg CD, DeMets DL, Reboussin DM, Granger CB. Fundamentals of Clinical Trials . 5th ed. 2015 edition. Springer; 2015.

Further reading

• Posttest-Only Control Group Design
• Pretest-Posttest Control Group Design
• Randomized Block Design

Chapter 6: Experimental Research

6.2 experimental design, learning objectives.

• Explain the difference between between-subjects and within-subjects experiments, list some of the pros and cons of each approach, and decide which approach to use to answer a particular research question.
• Define random assignment, distinguish it from random sampling, explain its purpose in experimental research, and use some simple strategies to implement it.
• Define what a control condition is, explain its purpose in research on treatment effectiveness, and describe some alternative types of control conditions.
• Define several types of carryover effect, give examples of each, and explain how counterbalancing helps to deal with them.

In this section, we look at some different ways to design an experiment. The primary distinction we will make is between approaches in which each participant experiences one level of the independent variable and approaches in which each participant experiences all levels of the independent variable. The former are called between-subjects experiments and the latter are called within-subjects experiments.

Between-Subjects Experiments

In a between-subjects experiment , each participant is tested in only one condition. For example, a researcher with a sample of 100 college students might assign half of them to write about a traumatic event and the other half write about a neutral event. Or a researcher with a sample of 60 people with severe agoraphobia (fear of open spaces) might assign 20 of them to receive each of three different treatments for that disorder. It is essential in a between-subjects experiment that the researcher assign participants to conditions so that the different groups are, on average, highly similar to each other. Those in a trauma condition and a neutral condition, for example, should include a similar proportion of men and women, and they should have similar average intelligence quotients (IQs), similar average levels of motivation, similar average numbers of health problems, and so on. This is a matter of controlling these extraneous participant variables across conditions so that they do not become confounding variables.

Random Assignment

The primary way that researchers accomplish this kind of control of extraneous variables across conditions is called random assignment , which means using a random process to decide which participants are tested in which conditions. Do not confuse random assignment with random sampling. Random sampling is a method for selecting a sample from a population, and it is rarely used in psychological research. Random assignment is a method for assigning participants in a sample to the different conditions, and it is an important element of all experimental research in psychology and other fields too.

In its strictest sense, random assignment should meet two criteria. One is that each participant has an equal chance of being assigned to each condition (e.g., a 50% chance of being assigned to each of two conditions). The second is that each participant is assigned to a condition independently of other participants. Thus one way to assign participants to two conditions would be to flip a coin for each one. If the coin lands heads, the participant is assigned to Condition A, and if it lands tails, the participant is assigned to Condition B. For three conditions, one could use a computer to generate a random integer from 1 to 3 for each participant. If the integer is 1, the participant is assigned to Condition A; if it is 2, the participant is assigned to Condition B; and if it is 3, the participant is assigned to Condition C. In practice, a full sequence of conditions—one for each participant expected to be in the experiment—is usually created ahead of time, and each new participant is assigned to the next condition in the sequence as he or she is tested. When the procedure is computerized, the computer program often handles the random assignment.

One problem with coin flipping and other strict procedures for random assignment is that they are likely to result in unequal sample sizes in the different conditions. Unequal sample sizes are generally not a serious problem, and you should never throw away data you have already collected to achieve equal sample sizes. However, for a fixed number of participants, it is statistically most efficient to divide them into equal-sized groups. It is standard practice, therefore, to use a kind of modified random assignment that keeps the number of participants in each group as similar as possible. One approach is block randomization . In block randomization, all the conditions occur once in the sequence before any of them is repeated. Then they all occur again before any of them is repeated again. Within each of these “blocks,” the conditions occur in a random order. Again, the sequence of conditions is usually generated before any participants are tested, and each new participant is assigned to the next condition in the sequence. Table 6.2 “Block Randomization Sequence for Assigning Nine Participants to Three Conditions” shows such a sequence for assigning nine participants to three conditions. The Research Randomizer website ( http://www.randomizer.org ) will generate block randomization sequences for any number of participants and conditions. Again, when the procedure is computerized, the computer program often handles the block randomization.

Table 6.2 Block Randomization Sequence for Assigning Nine Participants to Three Conditions

Random assignment is not guaranteed to control all extraneous variables across conditions. It is always possible that just by chance, the participants in one condition might turn out to be substantially older, less tired, more motivated, or less depressed on average than the participants in another condition. However, there are some reasons that this is not a major concern. One is that random assignment works better than one might expect, especially for large samples. Another is that the inferential statistics that researchers use to decide whether a difference between groups reflects a difference in the population takes the “fallibility” of random assignment into account. Yet another reason is that even if random assignment does result in a confounding variable and therefore produces misleading results, this is likely to be detected when the experiment is replicated. The upshot is that random assignment to conditions—although not infallible in terms of controlling extraneous variables—is always considered a strength of a research design.

Treatment and Control Conditions

Between-subjects experiments are often used to determine whether a treatment works. In psychological research, a treatment is any intervention meant to change people’s behavior for the better. This includes psychotherapies and medical treatments for psychological disorders but also interventions designed to improve learning, promote conservation, reduce prejudice, and so on. To determine whether a treatment works, participants are randomly assigned to either a treatment condition , in which they receive the treatment, or a control condition , in which they do not receive the treatment. If participants in the treatment condition end up better off than participants in the control condition—for example, they are less depressed, learn faster, conserve more, express less prejudice—then the researcher can conclude that the treatment works. In research on the effectiveness of psychotherapies and medical treatments, this type of experiment is often called a randomized clinical trial .

There are different types of control conditions. In a no-treatment control condition , participants receive no treatment whatsoever. One problem with this approach, however, is the existence of placebo effects. A placebo is a simulated treatment that lacks any active ingredient or element that should make it effective, and a placebo effect is a positive effect of such a treatment. Many folk remedies that seem to work—such as eating chicken soup for a cold or placing soap under the bedsheets to stop nighttime leg cramps—are probably nothing more than placebos. Although placebo effects are not well understood, they are probably driven primarily by people’s expectations that they will improve. Having the expectation to improve can result in reduced stress, anxiety, and depression, which can alter perceptions and even improve immune system functioning (Price, Finniss, & Benedetti, 2008).

Placebo effects are interesting in their own right (see Note 6.28 “The Powerful Placebo” ), but they also pose a serious problem for researchers who want to determine whether a treatment works. Figure 6.2 “Hypothetical Results From a Study Including Treatment, No-Treatment, and Placebo Conditions” shows some hypothetical results in which participants in a treatment condition improved more on average than participants in a no-treatment control condition. If these conditions (the two leftmost bars in Figure 6.2 “Hypothetical Results From a Study Including Treatment, No-Treatment, and Placebo Conditions” ) were the only conditions in this experiment, however, one could not conclude that the treatment worked. It could be instead that participants in the treatment group improved more because they expected to improve, while those in the no-treatment control condition did not.

Figure 6.2 Hypothetical Results From a Study Including Treatment, No-Treatment, and Placebo Conditions

Fortunately, there are several solutions to this problem. One is to include a placebo control condition , in which participants receive a placebo that looks much like the treatment but lacks the active ingredient or element thought to be responsible for the treatment’s effectiveness. When participants in a treatment condition take a pill, for example, then those in a placebo control condition would take an identical-looking pill that lacks the active ingredient in the treatment (a “sugar pill”). In research on psychotherapy effectiveness, the placebo might involve going to a psychotherapist and talking in an unstructured way about one’s problems. The idea is that if participants in both the treatment and the placebo control groups expect to improve, then any improvement in the treatment group over and above that in the placebo control group must have been caused by the treatment and not by participants’ expectations. This is what is shown by a comparison of the two outer bars in Figure 6.2 “Hypothetical Results From a Study Including Treatment, No-Treatment, and Placebo Conditions” .

Of course, the principle of informed consent requires that participants be told that they will be assigned to either a treatment or a placebo control condition—even though they cannot be told which until the experiment ends. In many cases the participants who had been in the control condition are then offered an opportunity to have the real treatment. An alternative approach is to use a waitlist control condition , in which participants are told that they will receive the treatment but must wait until the participants in the treatment condition have already received it. This allows researchers to compare participants who have received the treatment with participants who are not currently receiving it but who still expect to improve (eventually). A final solution to the problem of placebo effects is to leave out the control condition completely and compare any new treatment with the best available alternative treatment. For example, a new treatment for simple phobia could be compared with standard exposure therapy. Because participants in both conditions receive a treatment, their expectations about improvement should be similar. This approach also makes sense because once there is an effective treatment, the interesting question about a new treatment is not simply “Does it work?” but “Does it work better than what is already available?”

The Powerful Placebo

Many people are not surprised that placebos can have a positive effect on disorders that seem fundamentally psychological, including depression, anxiety, and insomnia. However, placebos can also have a positive effect on disorders that most people think of as fundamentally physiological. These include asthma, ulcers, and warts (Shapiro & Shapiro, 1999). There is even evidence that placebo surgery—also called “sham surgery”—can be as effective as actual surgery.

Medical researcher J. Bruce Moseley and his colleagues conducted a study on the effectiveness of two arthroscopic surgery procedures for osteoarthritis of the knee (Moseley et al., 2002). The control participants in this study were prepped for surgery, received a tranquilizer, and even received three small incisions in their knees. But they did not receive the actual arthroscopic surgical procedure. The surprising result was that all participants improved in terms of both knee pain and function, and the sham surgery group improved just as much as the treatment groups. According to the researchers, “This study provides strong evidence that arthroscopic lavage with or without débridement [the surgical procedures used] is not better than and appears to be equivalent to a placebo procedure in improving knee pain and self-reported function” (p. 85).

Research has shown that patients with osteoarthritis of the knee who receive a “sham surgery” experience reductions in pain and improvement in knee function similar to those of patients who receive a real surgery.

Army Medicine – Surgery – CC BY 2.0.

Within-Subjects Experiments

In a within-subjects experiment , each participant is tested under all conditions. Consider an experiment on the effect of a defendant’s physical attractiveness on judgments of his guilt. Again, in a between-subjects experiment, one group of participants would be shown an attractive defendant and asked to judge his guilt, and another group of participants would be shown an unattractive defendant and asked to judge his guilt. In a within-subjects experiment, however, the same group of participants would judge the guilt of both an attractive and an unattractive defendant.

The primary advantage of this approach is that it provides maximum control of extraneous participant variables. Participants in all conditions have the same mean IQ, same socioeconomic status, same number of siblings, and so on—because they are the very same people. Within-subjects experiments also make it possible to use statistical procedures that remove the effect of these extraneous participant variables on the dependent variable and therefore make the data less “noisy” and the effect of the independent variable easier to detect. We will look more closely at this idea later in the book.

Carryover Effects and Counterbalancing

The primary disadvantage of within-subjects designs is that they can result in carryover effects. A carryover effect is an effect of being tested in one condition on participants’ behavior in later conditions. One type of carryover effect is a practice effect , where participants perform a task better in later conditions because they have had a chance to practice it. Another type is a fatigue effect , where participants perform a task worse in later conditions because they become tired or bored. Being tested in one condition can also change how participants perceive stimuli or interpret their task in later conditions. This is called a context effect . For example, an average-looking defendant might be judged more harshly when participants have just judged an attractive defendant than when they have just judged an unattractive defendant. Within-subjects experiments also make it easier for participants to guess the hypothesis. For example, a participant who is asked to judge the guilt of an attractive defendant and then is asked to judge the guilt of an unattractive defendant is likely to guess that the hypothesis is that defendant attractiveness affects judgments of guilt. This could lead the participant to judge the unattractive defendant more harshly because he thinks this is what he is expected to do. Or it could make participants judge the two defendants similarly in an effort to be “fair.”

Carryover effects can be interesting in their own right. (Does the attractiveness of one person depend on the attractiveness of other people that we have seen recently?) But when they are not the focus of the research, carryover effects can be problematic. Imagine, for example, that participants judge the guilt of an attractive defendant and then judge the guilt of an unattractive defendant. If they judge the unattractive defendant more harshly, this might be because of his unattractiveness. But it could be instead that they judge him more harshly because they are becoming bored or tired. In other words, the order of the conditions is a confounding variable. The attractive condition is always the first condition and the unattractive condition the second. Thus any difference between the conditions in terms of the dependent variable could be caused by the order of the conditions and not the independent variable itself.

There is a solution to the problem of order effects, however, that can be used in many situations. It is counterbalancing , which means testing different participants in different orders. For example, some participants would be tested in the attractive defendant condition followed by the unattractive defendant condition, and others would be tested in the unattractive condition followed by the attractive condition. With three conditions, there would be six different orders (ABC, ACB, BAC, BCA, CAB, and CBA), so some participants would be tested in each of the six orders. With counterbalancing, participants are assigned to orders randomly, using the techniques we have already discussed. Thus random assignment plays an important role in within-subjects designs just as in between-subjects designs. Here, instead of randomly assigning to conditions, they are randomly assigned to different orders of conditions. In fact, it can safely be said that if a study does not involve random assignment in one form or another, it is not an experiment.

There are two ways to think about what counterbalancing accomplishes. One is that it controls the order of conditions so that it is no longer a confounding variable. Instead of the attractive condition always being first and the unattractive condition always being second, the attractive condition comes first for some participants and second for others. Likewise, the unattractive condition comes first for some participants and second for others. Thus any overall difference in the dependent variable between the two conditions cannot have been caused by the order of conditions. A second way to think about what counterbalancing accomplishes is that if there are carryover effects, it makes it possible to detect them. One can analyze the data separately for each order to see whether it had an effect.

When 9 Is “Larger” Than 221

Researcher Michael Birnbaum has argued that the lack of context provided by between-subjects designs is often a bigger problem than the context effects created by within-subjects designs. To demonstrate this, he asked one group of participants to rate how large the number 9 was on a 1-to-10 rating scale and another group to rate how large the number 221 was on the same 1-to-10 rating scale (Birnbaum, 1999). Participants in this between-subjects design gave the number 9 a mean rating of 5.13 and the number 221 a mean rating of 3.10. In other words, they rated 9 as larger than 221! According to Birnbaum, this is because participants spontaneously compared 9 with other one-digit numbers (in which case it is relatively large) and compared 221 with other three-digit numbers (in which case it is relatively small).

Simultaneous Within-Subjects Designs

So far, we have discussed an approach to within-subjects designs in which participants are tested in one condition at a time. There is another approach, however, that is often used when participants make multiple responses in each condition. Imagine, for example, that participants judge the guilt of 10 attractive defendants and 10 unattractive defendants. Instead of having people make judgments about all 10 defendants of one type followed by all 10 defendants of the other type, the researcher could present all 20 defendants in a sequence that mixed the two types. The researcher could then compute each participant’s mean rating for each type of defendant. Or imagine an experiment designed to see whether people with social anxiety disorder remember negative adjectives (e.g., “stupid,” “incompetent”) better than positive ones (e.g., “happy,” “productive”). The researcher could have participants study a single list that includes both kinds of words and then have them try to recall as many words as possible. The researcher could then count the number of each type of word that was recalled. There are many ways to determine the order in which the stimuli are presented, but one common way is to generate a different random order for each participant.

Between-Subjects or Within-Subjects?

Almost every experiment can be conducted using either a between-subjects design or a within-subjects design. This means that researchers must choose between the two approaches based on their relative merits for the particular situation.

Between-subjects experiments have the advantage of being conceptually simpler and requiring less testing time per participant. They also avoid carryover effects without the need for counterbalancing. Within-subjects experiments have the advantage of controlling extraneous participant variables, which generally reduces noise in the data and makes it easier to detect a relationship between the independent and dependent variables.

A good rule of thumb, then, is that if it is possible to conduct a within-subjects experiment (with proper counterbalancing) in the time that is available per participant—and you have no serious concerns about carryover effects—this is probably the best option. If a within-subjects design would be difficult or impossible to carry out, then you should consider a between-subjects design instead. For example, if you were testing participants in a doctor’s waiting room or shoppers in line at a grocery store, you might not have enough time to test each participant in all conditions and therefore would opt for a between-subjects design. Or imagine you were trying to reduce people’s level of prejudice by having them interact with someone of another race. A within-subjects design with counterbalancing would require testing some participants in the treatment condition first and then in a control condition. But if the treatment works and reduces people’s level of prejudice, then they would no longer be suitable for testing in the control condition. This is true for many designs that involve a treatment meant to produce long-term change in participants’ behavior (e.g., studies testing the effectiveness of psychotherapy). Clearly, a between-subjects design would be necessary here.

Remember also that using one type of design does not preclude using the other type in a different study. There is no reason that a researcher could not use both a between-subjects design and a within-subjects design to answer the same research question. In fact, professional researchers often do exactly this.

Key Takeaways

• Experiments can be conducted using either between-subjects or within-subjects designs. Deciding which to use in a particular situation requires careful consideration of the pros and cons of each approach.
• Random assignment to conditions in between-subjects experiments or to orders of conditions in within-subjects experiments is a fundamental element of experimental research. Its purpose is to control extraneous variables so that they do not become confounding variables.
• Experimental research on the effectiveness of a treatment requires both a treatment condition and a control condition, which can be a no-treatment control condition, a placebo control condition, or a waitlist control condition. Experimental treatments can also be compared with the best available alternative.

Discussion: For each of the following topics, list the pros and cons of a between-subjects and within-subjects design and decide which would be better.

• You want to test the relative effectiveness of two training programs for running a marathon.
• Using photographs of people as stimuli, you want to see if smiling people are perceived as more intelligent than people who are not smiling.
• In a field experiment, you want to see if the way a panhandler is dressed (neatly vs. sloppily) affects whether or not passersby give him any money.
• You want to see if concrete nouns (e.g., dog ) are recalled better than abstract nouns (e.g., truth ).
• Discussion: Imagine that an experiment shows that participants who receive psychodynamic therapy for a dog phobia improve more than participants in a no-treatment control group. Explain a fundamental problem with this research design and at least two ways that it might be corrected.

Birnbaum, M. H. (1999). How to show that 9 > 221: Collect judgments in a between-subjects design. Psychological Methods, 4 , 243–249.

Moseley, J. B., O’Malley, K., Petersen, N. J., Menke, T. J., Brody, B. A., Kuykendall, D. H., … Wray, N. P. (2002). A controlled trial of arthroscopic surgery for osteoarthritis of the knee. The New England Journal of Medicine, 347 , 81–88.

Price, D. D., Finniss, D. G., & Benedetti, F. (2008). A comprehensive review of the placebo effect: Recent advances and current thought. Annual Review of Psychology, 59 , 565–590.

Shapiro, A. K., & Shapiro, E. (1999). The powerful placebo: From ancient priest to modern physician . Baltimore, MD: Johns Hopkins University Press.

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As previously mentioned, one of the characteristics of a true experiment is that researchers use a random process to decide which participants are tested under which conditions. Random assignation is a powerful research technique that addresses the assumption of pre-test equivalence – that the experimental and control group are equal in all respects before the administration of the independent variable (Palys & Atchison, 2014).

Random assignation is the primary way that researchers attempt to control extraneous variables across conditions. Random assignation is associated with experimental research methods. In its strictest sense, random assignment should meet two criteria.  One is that each participant has an equal chance of being assigned to each condition (e.g., a 50% chance of being assigned to each of two conditions). The second is that each participant is assigned to a condition independently of other participants. Thus, one way to assign participants to two conditions would be to flip a coin for each one. If the coin lands on the heads side, the participant is assigned to Condition A, and if it lands on the tails side, the participant is assigned to Condition B. For three conditions, one could use a computer to generate a random integer from 1 to 3 for each participant. If the integer is 1, the participant is assigned to Condition A; if it is 2, the participant is assigned to Condition B; and, if it is 3, the participant is assigned to Condition C. In practice, a full sequence of conditions—one for each participant expected to be in the experiment—is usually created ahead of time, and each new participant is assigned to the next condition in the sequence as he or she is tested.

However, one problem with coin flipping and other strict procedures for random assignment is that they are likely to result in unequal sample sizes in the different conditions. Unequal sample sizes are generally not a serious problem, and you should never throw away data you have already collected to achieve equal sample sizes. However, for a fixed number of participants, it is statistically most efficient to divide them into equal-sized groups. It is standard practice, therefore, to use a kind of modified random assignment that keeps the number of participants in each group as similar as possible.

One approach is block randomization. In block randomization, all the conditions occur once in the sequence before any of them is repeated. Then they all occur again before any of them is repeated again. Within each of these “blocks,” the conditions occur in a random order. Again, the sequence of conditions is usually generated before any participants are tested, and each new participant is assigned to the next condition in the sequence. When the procedure is computerized, the computer program often handles the random assignment, which is obviously much easier. You can also find programs online to help you randomize your random assignation. For example, the Research Randomizer website will generate block randomization sequences for any number of participants and conditions ( Research Randomizer ).

Random assignation is not guaranteed to control all extraneous variables across conditions. It is always possible that, just by chance, the participants in one condition might turn out to be substantially older, less tired, more motivated, or less depressed on average than the participants in another condition. However, there are some reasons that this may not be a major concern. One is that random assignment works better than one might expect, especially for large samples. Another is that the inferential statistics that researchers use to decide whether a difference between groups reflects a difference in the population take the “fallibility” of random assignment into account. Yet another reason is that even if random assignment does result in a confounding variable and therefore produces misleading results, this confound is likely to be detected when the experiment is replicated. The upshot is that random assignment to conditions—although not infallible in terms of controlling extraneous variables—is always considered a strength of a research design. Note: Do not confuse random assignation with random sampling. Random sampling is a method for selecting a sample from a population; we will talk about this in Chapter 7.

Research Methods, Data Collection and Ethics Copyright © 2020 by Valerie Sheppard is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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• v.43(2); Mar-Apr 2008

Issues in Outcomes Research: An Overview of Randomization Techniques for Clinical Trials

Minsoo kang.

1 Middle Tennessee State University, Murfreesboro, TN

Brian G Ragan

2 University of Northern Iowa, Cedar Falls, IA

Jae-Hyeon Park

3 Korea National Sport University, Seoul, Korea

To review and describe randomization techniques used in clinical trials, including simple, block, stratified, and covariate adaptive techniques.

Background:

Clinical trials are required to establish treatment efficacy of many athletic training procedures. In the past, we have relied on evidence of questionable scientific merit to aid the determination of treatment choices. Interest in evidence-based practice is growing rapidly within the athletic training profession, placing greater emphasis on the importance of well-conducted clinical trials. One critical component of clinical trials that strengthens results is random assignment of participants to control and treatment groups. Although randomization appears to be a simple concept, issues of balancing sample sizes and controlling the influence of covariates a priori are important. Various techniques have been developed to account for these issues, including block, stratified randomization, and covariate adaptive techniques.

Advantages:

Athletic training researchers and scholarly clinicians can use the information presented in this article to better conduct and interpret the results of clinical trials. Implementing these techniques will increase the power and validity of findings of athletic medicine clinical trials, which will ultimately improve the quality of care provided.

Outcomes research is critical in the evidence-based health care environment because it addresses scientific questions concerning the efficacy of treatments. Clinical trials are considered the “gold standard” for outcomes in biomedical research. In athletic training, calls for more evidence-based medical research, specifically clinical trials, have been issued. 1 , 2

The strength of clinical trials is their superior ability to measure change over time from a treatment. Treatment differences identified from cross-sectional observational designs rather than experimental clinical trials have methodologic weaknesses, including confounding, cohort effects, and selection bias. 3 For example, using a nonrandomized trial to examine the effectiveness of prophylactic knee bracing to prevent medial collateral ligament injuries may suffer from confounders and jeopardize the results. One possible confounder is a history of knee injuries. Participants with a history of knee injuries may be more likely to wear braces than those with no such history. Participants with a history of injury are more likely to suffer additional knee injuries, unbalancing the groups and influencing the results of the study.

The primary goal of comparative clinical trials is to provide comparisons of treatments with maximum precision and validity. 4 One critical component of clinical trials is random assignment of participants into groups. Randomizing participants helps remove the effect of extraneous variables (eg, age, injury history) and minimizes bias associated with treatment assignment. Randomization is considered by most researchers to be the optimal approach for participant assignment in clinical trials because it strengthens the results and data interpretation. 4 – , 9

One potential problem with small clinical trials (n < 100) 7 is that conventional simple randomization methods, such as flipping a coin, may result in imbalanced sample size and baseline characteristics (ie, covariates) among treatment and control groups. 9 , 10 This imbalance of baseline characteristics can influence the comparison between treatment and control groups and introduce potential confounding factors. Many procedures have been proposed for random group assignment of participants in clinical trials. 11 Simple, block, stratified, and covariate adaptive randomizations are some examples. Each technique has advantages and disadvantages, which must be carefully considered before a method is selected. Our purpose is to introduce the concept and significance of randomization and to review several conventional and relatively new randomization techniques to aid in the design and implementation of valid clinical trials.

What Is Randomization?

Randomization is the process of assigning participants to treatment and control groups, assuming that each participant has an equal chance of being assigned to any group. 12 Randomization has evolved into a fundamental aspect of scientific research methodology. Demands have increased for more randomized clinical trials in many areas of biomedical research, such as athletic training. 2 , 13 In fact, in the last 2 decades, internationally recognized major medical journals, such as the Journal of the American Medical Association and the BMJ , have been increasingly interested in publishing studies reporting results from randomized controlled trials. 5

Since Fisher 14 first introduced the idea of randomization in a 1926 agricultural study, the academic community has deemed randomization an essential tool for unbiased comparisons of treatment groups. Five years after Fisher's introductory paper, the first randomized clinical trial involving tuberculosis was conducted. 15 A total of 24 participants were paired (ie, 12 comparable pairs), and by a flip of a coin, each participant within the pair was assigned to either the control or treatment group. By employing randomization, researchers offer each participant an equal chance of being assigned to groups, which makes the groups comparable on the dependent variable by eliminating potential bias. Indeed, randomization of treatments in clinical trials is the only means of avoiding systematic characteristic bias of participants assigned to different treatments. Although randomization may be accomplished with a simple coin toss, more appropriate and better methods are often needed, especially in small clinical trials. These other methods will be discussed in this review.

Why Randomize?

Researchers demand randomization for several reasons. First, participants in various groups should not differ in any systematic way. In a clinical trial, if treatment groups are systematically different, trial results will be biased. Suppose that participants are assigned to control and treatment groups in a study examining the efficacy of a walking intervention. If a greater proportion of older adults is assigned to the treatment group, then the outcome of the walking intervention may be influenced by this imbalance. The effects of the treatment would be indistinguishable from the influence of the imbalance of covariates, thereby requiring the researcher to control for the covariates in the analysis to obtain an unbiased result. 16

Second, proper randomization ensures no a priori knowledge of group assignment (ie, allocation concealment). That is, researchers, participants, and others should not know to which group the participant will be assigned. Knowledge of group assignment creates a layer of potential selection bias that may taint the data. Schulz and Grimes 17 stated that trials with inadequate or unclear randomization tended to overestimate treatment effects up to 40% compared with those that used proper randomization. The outcome of the trial can be negatively influenced by this inadequate randomization.

Statistical techniques such as analysis of covariance (ANCOVA), multivariate ANCOVA, or both, are often used to adjust for covariate imbalance in the analysis stage of the clinical trial. However, the interpretation of this postadjustment approach is often difficult because imbalance of covariates frequently leads to unanticipated interaction effects, such as unequal slopes among subgroups of covariates. 18 , 19 One of the critical assumptions in ANCOVA is that the slopes of regression lines are the same for each group of covariates (ie, homogeneity of regression slopes). The adjustment needed for each covariate group may vary, which is problematic because ANCOVA uses the average slope across the groups to adjust the outcome variable. Thus, the ideal way of balancing covariates among groups is to apply sound randomization in the design stage of a clinical trial (before the adjustment procedure) instead of after data collection. In such instances, random assignment is necessary and guarantees validity for statistical tests of significance that are used to compare treatments.

How To Randomize?

Many procedures have been proposed for the random assignment of participants to treatment groups in clinical trials. In this article, common randomization techniques, including simple randomization, block randomization, stratified randomization, and covariate adaptive randomization, are reviewed. Each method is described along with its advantages and disadvantages. It is very important to select a method that will produce interpretable, valid results for your study.

Simple Randomization

Randomization based on a single sequence of random assignments is known as simple randomization. 10 This technique maintains complete randomness of the assignment of a person to a particular group. The most common and basic method of simple randomization is flipping a coin. For example, with 2 treatment groups (control versus treatment), the side of the coin (ie, heads  =  control, tails  =  treatment) determines the assignment of each participant. Other methods include using a shuffled deck of cards (eg, even  =  control, odd  =  treatment) or throwing a die (eg, below and equal to 3  =  control, over 3  =  treatment). A random number table found in a statistics book or computer-generated random numbers can also be used for simple randomization of participants.

This randomization approach is simple and easy to implement in a clinical trial. In large trials (n > 200), simple randomization can be trusted to generate similar numbers of participants among groups. However, randomization results could be problematic in relatively small sample size clinical trials (n < 100), resulting in an unequal number of participants among groups. For example, using a coin toss with a small sample size (n  =  10) may result in an imbalance such that 7 participants are assigned to the control group and 3 to the treatment group ( Figure 1 ).

Block Randomization

The block randomization method is designed to randomize participants into groups that result in equal sample sizes. This method is used to ensure a balance in sample size across groups over time. Blocks are small and balanced with predetermined group assignments, which keeps the numbers of participants in each group similar at all times. According to Altman and Bland, 10 the block size is determined by the researcher and should be a multiple of the number of groups (ie, with 2 treatment groups, block size of either 4 or 6). Blocks are best used in smaller increments as researchers can more easily control balance. 7 After block size has been determined, all possible balanced combinations of assignment within the block (ie, equal number for all groups within the block) must be calculated. Blocks are then randomly chosen to determine the participants' assignment into the groups.

For a clinical trial with control and treatment groups involving 40 participants, a randomized block procedure would be as follows: (1) a block size of 4 is chosen, (2) possible balanced combinations with 2 C (control) and 2 T (treatment) subjects are calculated as 6 (TTCC, TCTC, TCCT, CTTC, CTCT, CCTT), and (3) blocks are randomly chosen to determine the assignment of all 40 participants (eg, one random sequence would be [TTCC / TCCT / CTTC / CTTC / TCCT / CCTT / TTCC / TCTC / CTCT / TCTC]). This procedure results in 20 participants in both the control and treatment groups ( Figure 2 ).

Although balance in sample size may be achieved with this method, groups may be generated that are rarely comparable in terms of certain covariates. 6 For example, one group may have more participants with secondary diseases (eg, diabetes, multiple sclerosis, cancer) that could confound the data and may negatively influence the results of the clinical trial. Pocock and Simon 11 stressed the importance of controlling for these covariates because of serious consequences to the interpretation of the results. Such an imbalance could introduce bias in the statistical analysis and reduce the power of the study. 4 , 6 , 8 Hence, sample size and covariates must be balanced in small clinical trials.

Stratified Randomization

The stratified randomization method addresses the need to control and balance the influence of covariates. This method can be used to achieve balance among groups in terms of participants' baseline characteristics (covariates). Specific covariates must be identified by the researcher who understands the potential influence each covariate has on the dependent variable. Stratified randomization is achieved by generating a separate block for each combination of covariates, and participants are assigned to the appropriate block of covariates. After all participants have been identified and assigned into blocks, simple randomization occurs within each block to assign participants to one of the groups.

The stratified randomization method controls for the possible influence of covariates that would jeopardize the conclusions of the clinical trial. For example, a clinical trial of different rehabilitation techniques after a surgical procedure will have a number of covariates. It is well known that the age of the patient affects the rate of healing. Thus, age could be a confounding variable and influence the outcome of the clinical trial. Stratified randomization can balance the control and treatment groups for age or other identified covariates.

For example, with 2 groups involving 40 participants, the stratified randomization method might be used to control the covariates of sex (2 levels: male, female) and body mass index (3 levels: underweight, normal, overweight) between study arms. With these 2 covariates, possible block combinations total 6 (eg, male, underweight). A simple randomization procedure, such as flipping a coin, is used to assign the participants within each block to one of the treatment groups ( Figure 3 ).

Although stratified randomization is a relatively simple and useful technique, especially for smaller clinical trials, it becomes complicated to implement if many covariates must be controlled. 20 For example, too many block combinations may lead to imbalances in overall treatment allocations because a large number of blocks can generate small participant numbers within the block. Therneau 21 purported that a balance in covariates begins to fail when the number of blocks approaches half the sample size. If another 4-level covariate was added to the example, the number of block combinations would increase from 6 to 24 (2 × 3 × 4), for an average of fewer than 2 (40 / 24  =  1.7) participants per block, reducing the usefulness of the procedure to balance the covariates and jeopardizing the validity of the clinical trial. In small studies, it may not be feasible to stratify more than 1 or 2 covariates because the number of blocks can quickly approach the number of participants. 10

Stratified randomization has another limitation: it works only when all participants have been identified before group assignment. This method is rarely applicable, however, because clinical trial participants are often enrolled one at a time on a continuous basis. When baseline characteristics of all participants are not available before assignment, using stratified randomization is difficult. 7

Covariate Adaptive Randomization

Covariate adaptive randomization has been recommended by many researchers as a valid alternative randomization method for clinical trials. 9 , 22 In covariate adaptive randomization, a new participant is sequentially assigned to a particular treatment group by taking into account the specific covariates and previous assignments of participants. 9 , 12 , 18 , 23 , 24 Covariate adaptive randomization uses the method of minimization by assessing the imbalance of sample size among several covariates. This covariate adaptive approach was first described by Taves. 23

The Taves covariate adaptive randomization method allows for the examination of previous participant group assignments to make a case-by-case decision on group assignment for each individual who enrolls in the study. Consider again the example of 2 groups involving 40 participants, with sex (2 levels: male, female) and body mass index (3 levels: underweight, normal, overweight) as covariates. Assume the first 9 participants have already been randomly assigned to groups by flipping a coin. The 9 participants' group assignments are broken down by covariate level in Figure 4 . Now the 10th participant, who is male and underweight, needs to be assigned to a group (ie, control versus treatment). Based on the characteristics of the 10th participant, the Taves method adds marginal totals of the corresponding covariate categories for each group and compares the totals. The participant is assigned to the group with the lower covariate total to minimize imbalance. In this example, the appropriate categories are male and underweight, which results in the total of 3 (2 for male category + 1 for underweight category) for the control group and a total of 5 (3 for male category + 2 for underweight category) for the treatment group. Because the sum of marginal totals is lower for the control group (3 < 5), the 10th participant is assigned to the control group ( Figure 5 ).

The Pocock and Simon method 11 of covariate adaptive randomization is similar to the method Taves 23 described. The difference in this approach is the temporary assignment of participants to both groups. This method uses the absolute difference between groups to determine group assignment. To minimize imbalance, the participant is assigned to the group determined by the lowest sum of the absolute differences among the covariates between the groups. For example, using the previous situation in assigning the 10th participant to a group, the Pocock and Simon method would (1) assign the 10th participant temporarily to the control group, resulting in marginal totals of 3 for male category and 2 for underweight category; (2) calculate the absolute difference between control and treatment group (males: 3 control – 3 treatment  =  0; underweight: 2 control – 2 treatment  =  0) and sum (0 + 0  =  0); (3) temporarily assign the 10th participant to the treatment group, resulting in marginal totals of 4 for male category and 3 for underweight category; (4) calculate the absolute difference between control and treatment group (males: 2 control – 4 treatment  =  2; underweight: 1 control – 3 treatment  =  2) and sum (2 + 2  =  4); and (5) assign the 10th participant to the control group because of the lowest sum of absolute differences (0 < 4).

Pocock and Simon 11 also suggested using a variance approach. Instead of calculating absolute difference among groups, this approach calculates the variance among treatment groups. Although the variance method performs similarly to the absolute difference method, both approaches suffer from the limitation of handling only categorical covariates. 25

Frane 18 introduced a covariate adaptive randomization for both continuous and categorical types. Frane used P values to identify imbalance among treatment groups: a smaller P value represents more imbalance among treatment groups.

The Frane method for assigning participants to either the control or treatment group would include (1) temporarily assigning the participant to both the control and treatment groups; (2) calculating P values for each of the covariates using a t test and analysis of variance (ANOVA) for continuous variables and goodness-of-fit χ 2 test for categorical variables; (3) determining the minimum P value for each control or treatment group, which indicates more imbalance among treatment groups; and (4) assigning the participant to the group with the larger minimum P value (ie, try to avoid more imbalance in groups).

Going back to the previous example of assigning the 10th participant (male and underweight) to a group, the Frane method would result in the assignment to the control group. The steps used to make this decision were calculating P values for each of the covariates using the χ 2 goodness-of-fit test represented in the Table . The t tests and ANOVAs were not used because the covariates in this example were categorical. Based on the Table , the lowest minimum P values were 1.0 for the control group and 0.317 for the treatment group. The 10th participant was assigned to the control group because of the higher minimum P value, which indicates better balance in the control group (1.0 > 0.317).

Probabilities From χ 2 Goodness-of-Fit Tests for the Example Shown in Figure 5 (Frane 18 Method)

Covariate adaptive randomization produces less imbalance than other conventional randomization methods and can be used successfully to balance important covariates among control and treatment groups. 6 Although the balance of covariates among groups using the stratified randomization method begins to fail when the number of blocks approaches half the sample size, covariate adaptive randomization can better handle the problem of increasing numbers of covariates (ie, increased block combinations). 9

One concern of these covariate adaptive randomization methods is that treatment assignments sometimes become highly predictable. Investigators using covariate adaptive randomization sometimes come to believe that group assignment for the next participant can be readily predicted, going against the basic concept of randomization. 12 , 26 , 27 This predictability stems from the ongoing assignment of participants to groups wherein the current allocation of participants may suggest future participant group assignment. In their review, Scott et al 9 argued that this predictability is also true of other methods, including stratified randomization, and it should not be overly penalized. Zielhuis et al 28 and Frane 18 suggested a practical approach to prevent predictability: a small number of participants should be randomly assigned into the groups before the covariate adaptive randomization technique being applied.

The complicated computation process of covariate adaptive randomization increases the administrative burden, thereby limiting its use in practice. A user-friendly computer program for covariate adaptive randomization is available (free of charge) upon request from the authors (M.K., B.G.R., or J.H.P.). 29

Conclusions

Our purpose was to introduce randomization, including its concept and significance, and to review several randomization techniques to guide athletic training researchers and practitioners to better design their randomized clinical trials. Many factors can affect the results of clinical research, but randomization is considered the gold standard in most clinical trials. It eliminates selection bias, ensures balance of sample size and baseline characteristics, and is an important step in guaranteeing the validity of statistical tests of significance used to compare treatment groups.

Before choosing a randomization method, several factors need to be considered, including the size of the clinical trial; the need for balance in sample size, covariates, or both; and participant enrollment. 16 Figure 6 depicts a flowchart designed to help select an appropriate randomization technique. For example, a power analysis for a clinical trial of different rehabilitation techniques after a surgical procedure indicated a sample size of 80. A well-known covariate for this study is age, which must be balanced among groups. Because of the nature of the study with postsurgical patients, participant recruitment and enrollment will be continuous. Using the flowchart, the appropriate randomization technique is covariate adaptive randomization technique.

Simple randomization works well for a large trial (eg, n > 200) but not for a small trial (n < 100). 7 To achieve balance in sample size, block randomization is desirable. To achieve balance in baseline characteristics, stratified randomization is widely used. Covariate adaptive randomization, however, can achieve better balance than other randomization methods and can be successfully used for clinical trials in an effective manner.

Acknowledgments

This study was partially supported by a Faculty Grant (FRCAC) from the College of Graduate Studies, at Middle Tennessee State University, Murfreesboro, TN.

Minsoo Kang, PhD; Brian G. Ragan, PhD, ATC; and Jae-Hyeon Park, PhD, contributed to conception and design; acquisition and analysis and interpretation of the data; and drafting, critical revision, and final approval of the article.

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• Random Assignment in Experiments | Introduction & Examples

Random Assignment in Experiments | Introduction & Examples

Published on 6 May 2022 by Pritha Bhandari . Revised on 13 February 2023.

In experimental research, random assignment is a way of placing participants from your sample into different treatment groups using randomisation.

With simple random assignment, every member of the sample has a known or equal chance of being placed in a control group or an experimental group. Studies that use simple random assignment are also called completely randomised designs .

Random assignment is a key part of experimental design . It helps you ensure that all groups are comparable at the start of a study: any differences between them are due to random factors.

Table of contents

Why does random assignment matter, random sampling vs random assignment, how do you use random assignment, when is random assignment not used, frequently asked questions about random assignment.

Random assignment is an important part of control in experimental research, because it helps strengthen the internal validity of an experiment.

In experiments, researchers manipulate an independent variable to assess its effect on a dependent variable, while controlling for other variables. To do so, they often use different levels of an independent variable for different groups of participants.

This is called a between-groups or independent measures design.

You use three groups of participants that are each given a different level of the independent variable:

• A control group that’s given a placebo (no dosage)
• An experimental group that’s given a low dosage
• A second experimental group that’s given a high dosage

Random assignment to helps you make sure that the treatment groups don’t differ in systematic or biased ways at the start of the experiment.

If you don’t use random assignment, you may not be able to rule out alternative explanations for your results.

• Participants recruited from pubs are placed in the control group
• Participants recruited from local community centres are placed in the low-dosage experimental group
• Participants recruited from gyms are placed in the high-dosage group

With this type of assignment, it’s hard to tell whether the participant characteristics are the same across all groups at the start of the study. Gym users may tend to engage in more healthy behaviours than people who frequent pubs or community centres, and this would introduce a healthy user bias in your study.

Although random assignment helps even out baseline differences between groups, it doesn’t always make them completely equivalent. There may still be extraneous variables that differ between groups, and there will always be some group differences that arise from chance.

Most of the time, the random variation between groups is low, and, therefore, it’s acceptable for further analysis. This is especially true when you have a large sample. In general, you should always use random assignment in experiments when it is ethically possible and makes sense for your study topic.

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Random sampling and random assignment are both important concepts in research, but it’s important to understand the difference between them.

Random sampling (also called probability sampling or random selection) is a way of selecting members of a population to be included in your study. In contrast, random assignment is a way of sorting the sample participants into control and experimental groups.

While random sampling is used in many types of studies, random assignment is only used in between-subjects experimental designs.

Some studies use both random sampling and random assignment, while others use only one or the other.

Random sampling enhances the external validity or generalisability of your results, because it helps to ensure that your sample is unbiased and representative of the whole population. This allows you to make stronger statistical inferences .

You use a simple random sample to collect data. Because you have access to the whole population (all employees), you can assign all 8,000 employees a number and use a random number generator to select 300 employees. These 300 employees are your full sample.

Random assignment enhances the internal validity of the study, because it ensures that there are no systematic differences between the participants in each group. This helps you conclude that the outcomes can be attributed to the independent variable .

• A control group that receives no intervention
• An experimental group that has a remote team-building intervention every week for a month

You use random assignment to place participants into the control or experimental group. To do so, you take your list of participants and assign each participant a number. Again, you use a random number generator to place each participant in one of the two groups.

To use simple random assignment, you start by giving every member of the sample a unique number. Then, you can use computer programs or manual methods to randomly assign each participant to a group.

• Random number generator: Use a computer program to generate random numbers from the list for each group.
• Lottery method: Place all numbers individually into a hat or a bucket, and draw numbers at random for each group.
• Flip a coin: When you only have two groups, for each number on the list, flip a coin to decide if they’ll be in the control or the experimental group.
• Use a dice: When you have three groups, for each number on the list, roll a die to decide which of the groups they will be in. For example, assume that rolling 1 or 2 lands them in a control group; 3 or 4 in an experimental group; and 5 or 6 in a second control or experimental group.

This type of random assignment is the most powerful method of placing participants in conditions, because each individual has an equal chance of being placed in any one of your treatment groups.

Random assignment in block designs

In more complicated experimental designs, random assignment is only used after participants are grouped into blocks based on some characteristic (e.g., test score or demographic variable). These groupings mean that you need a larger sample to achieve high statistical power .

For example, a randomised block design involves placing participants into blocks based on a shared characteristic (e.g., college students vs graduates), and then using random assignment within each block to assign participants to every treatment condition. This helps you assess whether the characteristic affects the outcomes of your treatment.

In an experimental matched design , you use blocking and then match up individual participants from each block based on specific characteristics. Within each matched pair or group, you randomly assign each participant to one of the conditions in the experiment and compare their outcomes.

Sometimes, it’s not relevant or ethical to use simple random assignment, so groups are assigned in a different way.

When comparing different groups

Sometimes, differences between participants are the main focus of a study, for example, when comparing children and adults or people with and without health conditions. Participants are not randomly assigned to different groups, but instead assigned based on their characteristics.

In this type of study, the characteristic of interest (e.g., gender) is an independent variable, and the groups differ based on the different levels (e.g., men, women). All participants are tested the same way, and then their group-level outcomes are compared.

When it’s not ethically permissible

When studying unhealthy or dangerous behaviours, it’s not possible to use random assignment. For example, if you’re studying heavy drinkers and social drinkers, it’s unethical to randomly assign participants to one of the two groups and ask them to drink large amounts of alcohol for your experiment.

When you can’t assign participants to groups, you can also conduct a quasi-experimental study . In a quasi-experiment, you study the outcomes of pre-existing groups who receive treatments that you may not have any control over (e.g., heavy drinkers and social drinkers).

These groups aren’t randomly assigned, but may be considered comparable when some other variables (e.g., age or socioeconomic status) are controlled for.

In experimental research, random assignment is a way of placing participants from your sample into different groups using randomisation. With this method, every member of the sample has a known or equal chance of being placed in a control group or an experimental group.

Random selection, or random sampling , is a way of selecting members of a population for your study’s sample.

In contrast, random assignment is a way of sorting the sample into control and experimental groups.

Random sampling enhances the external validity or generalisability of your results, while random assignment improves the internal validity of your study.

Random assignment is used in experiments with a between-groups or independent measures design. In this research design, there’s usually a control group and one or more experimental groups. Random assignment helps ensure that the groups are comparable.

In general, you should always use random assignment in this type of experimental design when it is ethically possible and makes sense for your study topic.

To implement random assignment , assign a unique number to every member of your study’s sample .

Then, you can use a random number generator or a lottery method to randomly assign each number to a control or experimental group. You can also do so manually, by flipping a coin or rolling a die to randomly assign participants to groups.

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Random Assignment – A Simple Introduction with Examples

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Completing a research or thesis paper is more work than most students imagine. For instance, you must conduct experiments before coming up with conclusions. Random assignment, a key methodology in academic research, ensures every participant has an equal chance of being placed in any group within an experiment. In experimental studies, the random assignment of participants is a vital element, which this article will discuss.

Inhaltsverzeichnis

• 1 Random Assignment – In a Nutshell
• 2 Definition: Random assignment
• 3 Importance of random assignment
• 4 Random assignment vs. random sampling
• 5 How to use random assignment
• 6 When random assignment is not used

Random Assignment – In a Nutshell

• Random assignment is where you randomly place research participants into specific groups.
• This method eliminates bias in the results by ensuring that all participants have an equal chance of getting into either group.
• Random assignment is usually used in independent measures or between-group experiment designs.

Definition: Random assignment

Pearson Correlation is a descriptive statistical procedure that describes the measure of linear dependence between two variables. It entails a sample, control group , experimental design , and randomized design. In this statistical procedure, random assignment is used. Random assignment is the random placement of participants into different groups in experimental research.

Importance of random assignment

Random assessment is essential for strengthening the internal validity of experimental research. Internal validity helps make a casual relationship’s conclusions reliable and trustworthy.

In experimental research, researchers isolate independent variables and manipulate them as they assess the impact while managing other variables. To achieve this, an independent variable for diverse member groups is vital. This experimental design is called an independent or between-group design.

Example: Different levels of independent variables

• In a medical study, you can research the impact of nutrient supplements on the immune (nutrient supplements = independent variable, immune = dependent variable)

Three independent participant levels are applicable here:

• Control group (given 0 dosages of iron supplements)
• The experimental group (low dosage)
• The second experimental group (high dosage)

This assignment technique in experiments ensures no bias in the treatment sets at the beginning of the trials. Therefore, if you do not use this technique, you won’t be able to exclude any alternate clarifications for your findings.

In the research experiment above, you can recruit participants randomly by handing out flyers at public spaces like gyms, cafés, and community centers. Then:

• Place the group from cafés in the control group
• Community center group in the low prescription trial group
• Gym group in the high-prescription group

Even with random participant assignment, other extraneous variables may still create bias in experiment results. However, these variations are usually low, hence should not hinder your research. Therefore, using random placement in experiments is highly necessary, especially where it is ethically required or makes sense for your research subject.

Random assignment vs. random sampling

Simple random sampling is a method of choosing the participants for a study. On the other hand, the random assignment involves sorting the participants selected through random sampling. Another difference between random sampling and random assignment is that the former is used in several types of studies, while the latter is only applied in between-subject experimental designs.

Your study researches the impact of technology on productivity in a specific company.

In such a case, you have contact with the entire staff. So, you can assign each employee a quantity and apply a random number generator to pick a specific sample.

For instance, from 500 employees, you can pick 200. So, the full sample is 200.

Random sampling enhances external validity, as it guarantees that the study sample is unbiased, and that an entire population is represented. This way, you can conclude that the results of your studies can be accredited to the autonomous variable.

After determining the full sample, you can break it down into two groups using random assignment. In this case, the groups are:

• The control group (does get access to technology)
• The experimental group (gets access to technology)

Using random assignment assures you that any differences in the productivity results for each group are not biased and will help the company make a decision.

How to use random assignment

Firstly, give each participant a unique number as an identifier. Then, use a specific tool to simplify assigning the participants to the sample groups. Some tools you can use are:

Random member assignment is a prevailing technique for placing participants in specific groups because each person has a fair opportunity of being put in either group.

Random assignment in block experimental designs

In complex experimental designs , you must group your participants into blocks before using the random assignment technique.

You can create participant blocks depending on demographic variables, working hours, or scores. However, the blocks imply that you will require a bigger sample to attain high statistical power.

After grouping the participants in blocks, you can use random assignments inside each block to allocate the members to a specific treatment condition. Doing this will help you examine if quality impacts the result of the treatment.

Depending on their unique characteristics, you can also use blocking in experimental matched designs before matching the participants in each block. Then, you can randomly allot each partaker to one of the treatments in the research and examine the results.

When random assignment is not used

As powerful a tool as it is, random assignment does not apply in all situations. Like the following:

Comparing different groups

When the purpose of your study is to assess the differences between the participants, random member assignment may not work.

If you want to compare teens and the elderly with and without specific health conditions, you must ensure that the participants have specific characteristics. Therefore, you cannot pick them randomly.

In such a study, the medical condition (quality of interest) is the independent variable, and the participants are grouped based on their ages (different levels). Also, all partakers are tried similarly to ensure they have the medical condition, and their outcomes are tested per group level.

No ethical justifiability

Another situation where you cannot use random assignment is if it is ethically not permitted.

If your study involves unhealthy or dangerous behaviors or subjects, such as drug use. Instead of assigning random partakers to sets, you can conduct quasi-experimental research.

When using a quasi-experimental design , you examine the conclusions of pre-existing groups you have no control over, such as existing drug users. While you cannot randomly assign them to groups, you can use variables like their age, years of drug use, or socioeconomic status to group the participants.

What is the definition of random assignment?

It is an experimental research technique that involves randomly placing participants from your samples into different groups. It ensures that every sample member has the same opportunity of being in whichever group (control or experimental group).

When is random assignment applicable?

You can use this placement technique in experiments featuring an independent measures design. It helps ensure that all your sample groups are comparable.

What is the importance of random assignment?

It can help you enhance your study’s validity . This technique also helps ensure that every sample has an equal opportunity of being assigned to a control or trial group.

When should you NOT use random assignment

You should not use this technique if your study focuses on group comparisons or if it is not legally ethical.

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Mathematics > Statistics Theory

Title: asymptotic mutual information in quadratic estimation problems over compact groups.

Abstract: Motivated by applications to group synchronization and quadratic assignment on random data, we study a general problem of Bayesian inference of an unknown ``signal'' belonging to a high-dimensional compact group, given noisy pairwise observations of a featurization of this signal. We establish a quantitative comparison between the signal-observation mutual information in any such problem with that in a simpler model with linear observations, using interpolation methods. For group synchronization, our result proves a replica formula for the asymptotic mutual information and Bayes-optimal mean-squared-error. Via analyses of this replica formula, we show that the conjectural phase transition threshold for computationally-efficient weak recovery of the signal is determined by a classification of the real-irreducible components of the observed group representation(s), and we fully characterize the information-theoretic limits of estimation in the example of angular/phase synchronization over $SO(2)$/$U(1)$. For quadratic assignment, we study observations given by a kernel matrix of pairwise similarities and a randomly permutated and noisy counterpart, and we show in a bounded signal-to-noise regime that the asymptotic mutual information coincides with that in a Bayesian spiked model with i.i.d. signal prior.

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Arakelov class groups of random number fields

• Open access
• Published: 16 April 2024

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• Alex Bartel   ORCID: orcid.org/0000-0002-3114-2541 1 ,
• Henri Johnston   ORCID: orcid.org/0000-0001-5764-0840 2 &
• Hendrik W. Lenstra Jr.   ORCID: orcid.org/0000-0001-7733-0444 3  

The main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen–Lenstra–Martinet heuristic on ideal class groups. To that end, we show that Chinburg’s \(\Omega (3)\) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups. As a consequence, we construct a new infinite series of counterexamples to the Cohen–Lenstra–Martinet heuristic, which have the novel feature that their Galois groups are non-abelian.

Avoid common mistakes on your manuscript.

1 Introduction

It has been an area of active research over the past few decades to understand the distribution of class groups \({{\,\textrm{Cl}\,}}_F\) of “random” algebraic number fields F . Specifically, we let K be a number field, and let G be a finite group. Let \(\Lambda \) be the quotient of the group ring \(\mathbb {Z}[\tfrac{1}{2\cdot \#G}][G]\) by the two-sided ideal generated by \(\sum _{g\in G}g.\) One studies the behaviour of the \(\Lambda \) -module \(\Lambda \otimes _{\mathbb {Z}[G]}{{\,\textrm{Cl}\,}}_F,\) as F runs over a “natural” family \(\mathcal {F}\) of G -extensions of K .

An equally classical invariant that one attaches to F is the unit group \(\mathcal {O}_F^\times \) of the ring of integers \(\mathcal {O}_F\) of F ,  viewed as a \(\mathbb {Z}[G]\) -module. Its isomorphism class, unlike that of the class group, has only finitely many possibilities, as F ranges over \(\mathcal {F}.\) The statistical properties of \(\mathcal {O}_F^\times \) have, however, been much less extensively studied.

In the present paper we make the case that, in this context, \({{\,\textrm{Cl}\,}}_F\) and \(\mathcal {O}_F^\times \) are most naturally studied in combination, since their distributions need, by all appearances, not be independent. Their dependence is best expressed by means of the Arakelov class group . It is a compact abelian group attached to F ,  and we will recall its definition in Sect.  2 . For number fields it plays the rôle that the Jacobian of a curve plays for function fields over finite fields. It can be broken up into two pieces, one being \({{\,\textrm{Cl}\,}}_F\) and the other coming from \(\mathcal {O}_F^\times ,\) but in several ways it is better than the sum of its parts. We find it convenient to replace the Arakelov class group by its Pontryagin dual \({{\,\textrm{Ar}\,}}_F.\) This is a finitely generated abelian group that fits into a short exact sequence

In other words, the torsion subgroup of \({{\,\textrm{Ar}\,}}_F\) is the Pontryagin dual of the class group, and its torsion-free part is the \(\mathbb {Z}\) -linear dual of the unit group. This exact sequence, being canonically associated with F ,  is an exact sequence of \(\mathbb {Z}[G]\) -modules.

Let \({{\,\textrm{G}\,}}_0(\Lambda )\) denote the Grothendieck group of the category of finitely generated \(\Lambda \) -modules; see Sect.  2.3 for the definition. Let F / K be a Galois extension with Galois group G ,  let \(S_{\infty }\) be the G -set of Archimedean places of F ,  and let \(\mathbb {Z}^{S_\infty }\) be the corresponding permutation module over \(\mathbb {Z}[G];\) it is a property of \(\mathcal {F}\) that the isomorphism class of the G -set \(S_{\infty }\) is independent of F when \(F \in \mathcal {F}.\) The difference of the classes \([\Lambda \otimes _{\mathbb {Z}[G]}{{\,\textrm{Ar}\,}}_F]\) and \([\Lambda \otimes _{\mathbb {Z}[G]} \mathbb {Z}^{S_\infty }]\) in \({{\,\textrm{G}\,}}_{0}(\Lambda )\) lies in the torsion subgroup \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}\) of \({{\,\textrm{G}\,}}_{0}(\Lambda ),\) as can easily be deduced from Lemma  5.10 . This torsion subgroup is a finite abelian group, which can be thought of as a “class group” of \(\Lambda .\) The following result will be proven at the end of Sect.  6 as a consequence of Proposition  6.5 .

Theorem 1.1

With the notation just introduced,  suppose that Chinburg’s \(\Omega (3)\) conjecture,  Conjecture  6.3 , holds for F / K . Suppose,  moreover,  that for every prime number p not dividing \(2\cdot \#G,\) each primitive p -th root of unity in F is in K . Then the equality

holds in \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}.\)

In fact, we will prove the conclusion of Theorem  1.1 under a weaker hypothesis; see Theorem  6.7 .

In Sect.  3 we will define the families \(\mathcal {F}\) that we consider. Together, Theorem  1.1 and Propositions  3.3 and  5.3 imply that as F ranges over \(\mathcal {F},\) the class \([\Lambda \otimes _{\mathbb {Z}[G]}\mathcal {O}_F^\times ] - [\Lambda \otimes _{\mathbb {Z}[G]}{{\,\textrm{Cl}\,}}_F]\) in \({{\,\textrm{G}\,}}_0(\Lambda )\) is conjectured to be constant. We have no reason to expect this to be true for either of the two terms individually.

There are cases in which the conclusion of Theorem  1.1 can be proven unconditionally. This includes Galois extensions of \(\mathbb {Q}\) of degree less than 112 (see Proposition  7.9 ), but also, more interestingly, a class of fields that can be used to construct a new series of counterexamples to the Cohen–Lenstra–Martinet heuristic [ 8 , 10 ] on class groups of number fields. Informally, this heuristic reads as follows.

Heuristic 1.2

(Vague version) Let F vary in a natural family of Galois number fields. Then the Galois module \({{\,\textrm{Ar}\,}}_F,\) after inverting the “bad” prime numbers,  behaves “randomly” with respect to a probability distribution that assigns to a suitable Galois module M a probability weight that is inversely proportional to the “size” of the automorphism group of M .

The precise version that we shall use will be formulated as Heuristic  3.2 in Sect.  3 . The same section explains what we mean by the term “natural family”. It also allows the ring \(\Lambda \) to be more general than the ring considered above.

Heuristic  3.2 represents, in several respects, a corrected version of the original Cohen–Lenstra–Martinet heuristic. Nevertheless, it is known to be invalid, counterexamples for certain abelian Galois groups G having been provided in [ 4 , Theorem 1.1, Proposition 4.4]. One of the main results of the present paper is a new series of counterexamples, this time with non-abelian  G . The other main achievement is a proposed correction to the heuristic.

The new counterexamples make use of groups G of order \(2^p\cdot p,\) where p is an odd prime number. Their abelianisations \(G/G',\) which are cyclic of order 2 p ,  coincide with the groups on which the abelian counterexamples in [ 4 ] depend. Our groups have centres Z of order 2,  and writing \(Z = \langle \gamma \rangle ,\) we shall make use of the ring \(\Lambda = \mathbb {Z}[\tfrac{1}{2p}][G]/(1+\gamma ).\) In Sect.  8 we prove the following theorem.

Theorem 1.3

For infinitely many odd prime numbers p there is a group G with the properties just listed such that the following is true. With \(\Lambda \) as just defined,  the group \({{\,\textrm{G}\,}}_0(\Lambda )_{{{\,\textrm{tors}\,}}}\) is non-trivial,  whereas there does exist a natural family of G -extensions of \(\mathbb {Q}\) such that for all members F of the family the class of \(\Lambda \otimes _{\mathbb {Z}[G]} {{\,\textrm{Cl}\,}}_F\) in \({{\,\textrm{G}\,}}_0(\Lambda )_{{{\,\textrm{tors}\,}}}\) is trivial.

The families in Theorem  1.3 necessarily violate Heuristic  3.2 , since the latter would imply equidistribution of \(\Lambda \otimes _{\mathbb {Z}[G]} {{\,\textrm{Cl}\,}}_F\) in \({{\,\textrm{G}\,}}_0(\Lambda )_{{{\,\textrm{tors}\,}}}\) as F ranges over the family. This is proven in Sect.  8 , to which we also refer for more information on the groups and families appearing in Theorem  1.3 .

The probability weight that is inversely proportional to the “size” of the automorphism group, as referred to in Heuristic  1.2 , reflects an attractive feature of the Arakelov class groups. The general principle behind many heuristics is that algebraic objects in natural families tend to be “as random as they can be”, with respect to a probability distribution that assigns to an algebraic object X a probability weight that is proportional to \(1/\# {{\,\textrm{Aut}\,}}X.\) The original Cohen–Lenstra–Martinet heuristic did, initially, look like an exception to the rule just mentioned, but this changed when it was reformulated in terms of \({{\,\textrm{Ar}\,}}_F.\) This is discussed in more detail in [ 4 ]; see also Sect.  3 below.

Theorem  1.1 restricts how random Arakelov class groups can be. We propose a correction to Heuristic  1.2 that takes this restriction into account.

Heuristic 1.4

(Vague version) Let F vary in a natural family of Galois number fields. Then the Galois module \({{\,\textrm{Ar}\,}}_F,\) after inverting the “bad” prime numbers,  behaves “randomly” with respect to a probability distribution that assigns to a suitable Galois module M a probability weight that is inversely proportional to the “size” of the automorphism group of M ,  restricted to those modules that satisfy Eq. ( 5.12 ) with M in place of \({{\,\textrm{Ar}\,}}_F.\)

A precise version of Heuristic  1.4 will be formulated as Heuristic  3.4 in Sect.  3 .

Note that the only difference between Heuristics  1.2 and  1.4 is the reference to ( 5.12 ). It expresses the restriction that Chinburg’s \(\Omega (3)\) conjecture imposes on the class of \({{\,\textrm{Ar}\,}}_F\) in the class group of \(\Lambda \) as a consequence of Theorem  1.1 . It is also important to point out in which way Heuristic  1.4 differs from Conjecture 1.5 formulated in [ 4 ]. The latter conjecture is only concerned with the local structure of \({{\,\textrm{Ar}\,}}_F\) at a finite set of prime numbers; in that case, the class group of \(\Lambda \) is trivial, so that Chinburg’s \(\Omega (3)\) conjecture imposes no restriction. On the other hand, Heuristic  1.4 considers almost all prime numbers, and in this global situation \(\Lambda \) may have a non-trivial class group. In particular, one should be able to extract explicit information on the distribution of \(\mathcal {O}_F^\times \) as a Galois module from Heuristic  1.4 , which is not possible with Conjecture 1.5 of [ 4 ].

The structure of the paper is as follows. After some preliminaries in Sect.  2 , we formulate in Sect.  3 the old and the new heuristics. Some basic material on Grothendieck groups of orders is treated in Sect.  4 . Section  5 is devoted to Arakelov class groups as Galois modules, and Sect.  6 to the implications of Chinburg’s \(\Omega (3)\) conjecture for these Galois modules. In Sect.  7 we collect some cases in which the conclusion of Theorem  1.1 is known, and use these in Sect.  8 to construct a new series of counterexamples to the Cohen–Lenstra–Martinet heuristic.

2 Preliminaries

In this section we recall material that we will use for the formulation of Heuristic  3.4 . In particular, we recall from [ 21 ] the definition of the Arakelov class group and of the oriented Arakelov class group of a number field.

2.1 Pontryagin duality

We briefly recall some facts on Pontryagin duality and refer the reader to [ 18 , Chapter 1, §1] for a more detailed overview. If A and B are abelian topological groups, then \({{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}(A,B)\) denotes the group of continuous group homomorphisms from A to B . Let \(\mathcal {C}\) be the category of Hausdorff locally compact abelian topological groups. If A is an object of \(\mathcal {C},\) then its Pontryagin dual is defined to be \({{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}(A,\mathbb {R}/\mathbb {Z}).\) This defines an anti-equivalence of \(\mathcal {C}\) with itself, of which the square is isomorphic to the identity functor. It induces an anti-equivalence between the full subcategories of compact abelian groups and of discrete abelian groups. If M is a finitely generated abelian group, then \({{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}(M \otimes _{\mathbb {Z}} (\mathbb {R}/\mathbb {Z}),\mathbb {R}/\mathbb {Z})\) is canonically isomorphic to \({{\,\textrm{Hom}\,}}(M,\mathbb {Z}).\) In particular, we have \({{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}(\mathbb {R}/\mathbb {Z},\mathbb {R}/\mathbb {Z}) \cong \mathbb {Z}.\)

2.2 The (oriented) Arakelov class group

Let F be a number field. Let \({{\,\textrm{Id}\,}}_F\) be the group of fractional ideals of the ring of integers \(\mathcal {O}_F\) of F ,  let \(S_{\infty }\) denote the set of Archimedean places of F ,  and let \(F_\mathbb {R}\) denote the étale \(\mathbb {R}\) -algebra \(F\otimes _{\mathbb {Q}}\mathbb {R}=\prod _{w\in S_{\infty }}F_w,\) where \(F_w\) denotes the completion of F at w . We have canonical maps \(\mathfrak {N}:{{\,\textrm{Id}\,}}_F\rightarrow \mathbb {R}_{>0}\) and \(|\!{{\,\textrm{Nm}\,}}\!|:F_{\mathbb {R}}^\times \rightarrow \mathbb {R}_{>0},\) the first given by the ideal norm, and the second given by the absolute value of the \(\mathbb {R}\) -algebra norm. Let \({{\,\textrm{Id}\,}}_F\times _{\mathbb {R}_{>0}} F_{\mathbb {R}}^\times \) denote the fibre product with respect to these maps. The oriented Arakelov class group \(\widetilde{{{\,\textrm{Pic}\,}}}^0_{F}\) of F is defined as the cokernel of the map \(F^\times \rightarrow {{\,\textrm{Id}\,}}_F\times _{\mathbb {R}_{>0}} F_{\mathbb {R}}^\times \) that sends \(\alpha \in F^\times \) to \((\alpha \mathcal {O}_F,\alpha ).\) It follows from Dirichlet’s unit theorem and the finiteness of the class group of \(\mathcal {O}_F,\) that this is a compact abelian group.

For every \(w\in S_{\infty }\) we have a direct product decomposition \(F_{w}^\times \cong \mathbb {R}_{>0} \times {{\,\textrm{c}\,}}(F_w^\times ),\) where \({{\,\textrm{c}\,}}(F_w^\times )\) is the maximal compact subgroup of \(F_w^\times ,\) which is equal to \(\{\pm 1\}\) if w is real, and to the circle group in \(F_w\) if w is complex. The maximal compact subgroup \({{\,\textrm{c}\,}}(F_{\mathbb {R}}^\times )=\prod _{w\in S_{\infty }}{{\,\textrm{c}\,}}(F_w^\times )\) of \(F_{\mathbb {R}}^\times \) is contained in the kernel of the map \(|\!{{\,\textrm{Nm}\,}}\!|.\) Define the Arakelov class group \({{\,\textrm{Pic}\,}}^0_F\) of F to be the quotient of \(\widetilde{{{\,\textrm{Pic}\,}}}^0_{F}\) by the image of \(\{1\}\times {{\,\textrm{c}\,}}(F_{\mathbb {R}}^\times )\subset {{\,\textrm{Id}\,}}_F\times _{\mathbb {R}_{>0}}F_{\mathbb {R}}^\times \) in \(\widetilde{{{\,\textrm{Pic}\,}}}^0_{F}.\)

Proposition 2.1

There is a short exact sequence

By [ 21 , Proposition 2.2] there is an exact sequence

The desired result follows by taking the Pontryagin dual of this sequence and recalling from Sect.  2.1 that we have an isomorphism \({{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}((\mathcal {O}_{F}^\times /\mu _{F})\otimes _{\mathbb {Z}}(\mathbb {R}/\mathbb {Z}),\mathbb {R}/\mathbb {Z}) \cong {{\,\textrm{Hom}\,}}(\mathcal {O}_F^\times /\mu _F,\mathbb {Z}).\) \(\square \)

Note that the canonical map \({{\,\textrm{Hom}\,}}(\mathcal {O}_F^\times / \mu _{F},\mathbb {Z})\rightarrow {{\,\textrm{Hom}\,}}(\mathcal {O}_F^\times ,\mathbb {Z})\) is an isomorphism.

2.3 Modules and Grothendieck groups

Henceforth all modules will be assumed to be left modules unless stated otherwise. If G is a group, S is a finite G -set, and R is a ring, in the remainder of the paper \(R^S\) will denote the free R -module on the set S with the induced R -linear R [ G ]-action. We will refer to such R [ G ]-modules as permutation modules .

Recall that for a ring T ,  the Grothendieck group \({{\,\textrm{G}\,}}_{0}(T)\) of the category of finitely generated T -modules is the additive group generated by expressions [ M ],  one for each isomorphism class of finitely generated T -modules M ,  with a relation \([L]+[N]=[M]\) whenever there exists a short exact sequence

of finitely generated T -modules.

If P is a set of prime numbers, then we define

If \(T\rightarrow T'\) is a ring homomorphism such that \(T'\) is a flat right T -module, then the functor \(T' \otimes _T \bullet \) from the category of finitely generated T -modules to that of finitely generated \(T'\) -modules induces a group homomorphism \({{\,\textrm{G}\,}}_0(T)\rightarrow {{\,\textrm{G}\,}}_0(T').\) The following two examples of this construction will be relevant to us. If G is a finite group and P is a set of prime numbers, then the flat localisation map \(\mathbb {Z}\rightarrow \mathbb {Z}_{(P)}\) induces a group homomorphism \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G])\rightarrow {{\,\textrm{G}\,}}_0(\mathbb {Z}_{(P)}[G]).\) Moreover, if we have a direct product decomposition \(T\cong U\times W\) of rings, then the right T -module W is projective, and in particular flat, so the quotient map \(T\rightarrow W\) induces a group homomorphism \({{\,\textrm{G}\,}}_0(T)\rightarrow {{\,\textrm{G}\,}}_0(W).\)

3 Cohen–Lenstra–Martinet heuristic

In this section we propose a correction of the Cohen–Lenstra–Martinet heuristic [ 8 , 10 ]. The notation and assumptions introduced in the next three paragraphs will remain in force throughout this section.

Let G be a finite group, let P be a set of prime numbers not dividing \(2\cdot \#G,\) let the \(\mathbb {Q}\) -algebra A be a quotient of \(\mathbb {Q}[G]\) by a two-sided ideal containing \(\sum _{g\in G}g,\) and let \(\Lambda \) be the image of \(\mathbb {Z}_{(P)}[G]\) in A ;  note that the ring \(\Lambda \) in the introduction is a special case of this. Next, let V be a finitely generated \(\mathbb {Q}[G]\) -module, and, for brevity, set \(V_A=A\otimes _{\mathbb {Q}[G]}V.\) Let \(\mathcal {M}_V\) be a set of finitely generated \(\Lambda \) -modules M that satisfy \(A\otimes _{\Lambda } M \cong _A V_A,\) and with the property that for every finitely generated \(\Lambda \) -module \(M'\) satisfying \(A\otimes _{\Lambda } M'\cong _A V_A\) there exists a unique \(M\in \mathcal {M}_V\) such that \(M'\cong M.\) Note that the set \(\mathcal {M}_V\) is countable. If M is a finitely generated \(\Lambda \) -module satisfying \(A\otimes _\Lambda M\cong _A V_A,\) and f is a function defined on \(\mathcal {M}_V,\) then we write f ( M ) for the value of f on the unique element of \(\mathcal {M}_V\) that is isomorphic to M . In [ 3 ] it was shown that there is a unique “automorphism index” function \({{\,\textrm{ia}\,}}:\mathcal {M}_V\times \mathcal {M}_V\rightarrow \mathbb {Q}_{>0}\) that behaves, in a precise sense explained in [ 3 , Theorem 1.1], like \((L,M)\mapsto \frac{\#{{\,\textrm{Aut}\,}}M}{\#{{\,\textrm{Aut}\,}}L},\) even when the automorphism groups of M and of L are infinite. Fix \(M\in \mathcal {M}_V.\) If \(\mathcal {N}\) is a subset of \(\mathcal {M}_V\) and X is a positive real number, let \(\mathcal {N}_X\) be the finite set of all \(L\in \mathcal {N}\) whose torsion subgroup \(L_{{{\,\textrm{tors}\,}}}\) has order less than X . For \(\mathcal {N}\subset \mathcal {M}_V\) and for a function \(f:\mathcal {N}\rightarrow \mathbb {C},\) define the expected value of f on \(\mathcal {N}\) by

when the limit exists. One of the defining properties of the function \({{\,\textrm{ia}\,}}\) is that for all L ,  M ,  and \(N\in \mathcal {M}_V\) we have \({{\,\textrm{ia}\,}}(L,M){{\,\textrm{ia}\,}}(M,N)={{\,\textrm{ia}\,}}(L,N),\) whence it follows that whether or not \(\mathbb {E}_{\mathcal {N}}(f)\) is defined is independent of the choice of M ,  and so is its value when it is defined.

Expected values behave well under passing to quotients in the following sense. Let \(G_1\rightarrow G_2\) be a surjective group homomorphism, let P be a set of prime numbers not dividing \(2\cdot \#G_1,\) let \(A_2\) be a quotient of \(\mathbb {Q}[G_2]\) as above, and let \(V_2\) be a finitely generated \(\mathbb {Q}[G_2]\) -module; let \(A_1\) be the same as \(A_2\) but viewed as a quotient of \(\mathbb {Q}[G_1],\) and let \(V_1\) be the same as \(V_2\) but viewed as a \(\mathbb {Q}[G_1]\) -module. For \(i\in \{1,2\},\) let \(\Lambda _i\) be the image of \(\mathbb {Z}_{(P)}[G_i]\) in \(A_i.\) Note that in particular the map \(G_1\rightarrow G_2\) induces a ring isomorphism \(\Lambda _1\rightarrow \Lambda _2.\) For \(i\in \{1,2\},\) define sets \(\mathcal {M}_{V_i}\) of \(\Lambda _i\) -modules as above. For brevity, write \(\mathcal {M}_i=\mathcal {M}_{V_i}\) for \(i\in \{1,2\}.\) Let \(f_2:\mathcal {M}_{2}\rightarrow \mathbb {C}\) be a function, and let \(f_1:\mathcal {M}_{1}\rightarrow \mathbb {C}\) be given by \(M\mapsto f_2(\Lambda _2\otimes _{\Lambda _1}M).\) Then one has \( \mathbb {E}_{\mathcal {M}_1}(f_1) = \mathbb {E}_{\mathcal {M}_2}(f_2). \)

Let K be a number field, and let \(\bar{K}\) be an algebraic closure of K . Given a pair ( F ,  i ),  where \(F\subset \bar{K}\) is a Galois extension of K and i is an isomorphism between the Galois group of F / K and G ,  we view \({{\,\textrm{Gal}\,}}(F/K)\) -modules as G -modules via i . Let \(\mathcal {F}\) be the set of all such pairs ( F ,  i ) for which F contains no primitive p -th root of unity for any prime number \(p\in P,\) and for which there is an isomorphism \(\mathbb {Q}\otimes _{\mathbb {Z}} \mathcal {O}_F^\times \cong V\) of \(\mathbb {Q}[G]\) -modules. Assume that \(\mathcal {F}\) is infinite. Such an \(\mathcal {F}\) is what we called a “natural family” in the introduction. Note that this family is a special case of the families considered in [ 4 , §2].

For \((F,i)\in \mathcal {F},\) let \(c_{F/K}\) be the ideal norm of the product of the prime ideals of \(\mathcal {O}_K\) that ramify in F / K . For a positive real number B ,  let \(\mathcal {F}_{c\le B}=\{(F,i)\in \mathcal {F}: c_{F/K}\le B\}.\) The following version of the Cohen–Lenstra–Martinet heuristic is a variant of [ 4 , Heuristic 2.1] phrased in terms of Arakelov class groups. It differs in several ways from the heuristic formulated in [ 8 , 10 ], but none of those differences shall concern us in the present paper.

Heuristic 3.2

Let f be a “reasonable” \(\mathbb {C}\) -valued function on \(\mathcal {M}_V.\) Then the limit

exists,  and is equal to \(\mathbb {E}_{\mathcal {M}_V}(f).\)

The notion of a reasonable function is left intentionally vague. The functions considered in [ 9 ] give rise to many examples of presumably reasonable functions on \(\mathcal {M}_V\) that factor through \(M\mapsto M_{{{\,\textrm{tors}\,}}}.\) An example of a function not of that form that we would consider reasonable, and which depends on the Galois module structure of both the class group and the unit group of the ring of integers, is \(M\mapsto \#{{\,\textrm{Hom}\,}}_{\Lambda }(M/M_{{{\,\textrm{tors}\,}}},M_{{{\,\textrm{tors}\,}}}).\)

If the set P is infinite, then Conjecture  5.11 can be an obstruction to the conclusions of Heuristic  3.2 . For example it was shown in [ 4 , §4], as a consequence of a proven special case of Conjecture  5.11 , that the conclusion of Heuristic  3.2 does not, in general, hold for functions of the form \(M\mapsto \chi ([M]),\) where \(\chi :{{\,\textrm{G}\,}}_{0}(\Lambda )\rightarrow \mathbb {C}^\times \) is a homomorphism of finite order. In [ 4 ] a corrected heuristic was proposed in which P was assumed to be finite. In the remainder of the section, we formulate a Cohen–Lenstra–Martinet heuristic without the hypothesis that P be finite.

Proposition 3.3

Let ( F ,  i ) and \((F',i')\in \mathcal {F},\) and let \(S_{\infty }\) and \(S_{\infty }'\) be the sets of Archimedean places of F and \(F',\) respectively. Then the equality

holds in \({{\,\textrm{G}\,}}_{0}(\Lambda ).\)

By definition of the family \(\mathcal {F},\) we have an isomorphism \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathcal {O}_F^\times \cong \mathbb {Q}\otimes _{\mathbb {Z}}\mathcal {O}_{F'}^\times \) of \(\mathbb {Q}[G]\) -modules. By a theorem of Herbrand there is an isomorphism \((\mathbb {Q}\otimes _{\mathbb {Z}}\mathcal {O}_F^\times ) \oplus \mathbb {Q}\cong \mathbb {Q}^{S_{\infty }}\) of \(\mathbb {Q}[G]\) -modules, see for example [ 24 , Chapter I, 4.3], and similarly for \(\mathcal {O}_{F'}^\times .\) Thus, we have an isomorphism \(\mathbb {Q}^{S_{\infty }}\cong \mathbb {Q}^{S_{\infty }'}.\)

Since all point stabilisers for \(S_{\infty }\) and \(S_{\infty }'\) are inertia groups at Archimedean places, they are all cyclic. It follows from Artin’s induction theorem (e.g. by combining [ 22 , §13.1, Corollary 1 and Theorem 30] and comparing dimensions) that if S and \(S'\) are finite G -sets with cyclic point stabilisers such that there is an isomorphism \(\mathbb {Q}^S\cong \mathbb {Q}^{S'}\) of \(\mathbb {Q}[G]\) -modules, then the G -sets S and \(S'\) are isomorphic. In particular, there is then an isomorphism \(\mathbb {Z}^S\cong \mathbb {Z}^{S'}\) of \(\mathbb {Z}[G]\) -modules. The result follows by applying this observation to the G -sets \(S_{\infty }\) and \(S_{\infty }'.\) \(\square \)

We define \(C(\mathcal {F})\) to be the common class of \(\Lambda \otimes _{\mathbb {Z}[G]} \mathbb {Z}^{S_{\infty }}\) in \({{\,\textrm{G}\,}}_{0}(\Lambda )\) for all \((F,i)\in \mathcal {F},\) where \(S_{\infty }\) is the G -set of Archimedean places of F .

Heuristic 3.4

Let \(\mathcal {N}=\{M\in \mathcal {M}_V: [M]=C(\mathcal {F})\text { in }{{\,\textrm{G}\,}}_{0}(\Lambda )\},\) and let f be a “reasonable” \(\mathbb {C}\) -valued function on \(\mathcal {M}_V.\) Then the limit

exists,  and is equal to \(\mathbb {E}_{\mathcal {N}}(f).\)

4 Grothendieck groups of orders

In this section we review some standard facts about Grothendieck groups of orders, and examine the effect of some duality operations upon these Grothendieck groups.

Let R be a Dedekind domain and let k be the field of fractions of R . An R -order is an R -algebra that is finitely generated and projective as an R -module. For example if G is a finite group, then the group ring \(\Lambda =R[G]\) is an R -order.

Let \(\Lambda \) be an R -order. A finitely generated \(\Lambda \) -module that is projective over R will be referred to as a \(\Lambda \) -lattice . Let \({{\,\textrm{G}\,}}_{0}^{R}(\Lambda )\) denote the Grothendieck group of the category of \(\Lambda \) -lattices. By definition, \({{\,\textrm{G}\,}}_{0}^{R}(\Lambda )\) is the additive group generated by expressions [ M ],  one for each isomorphism class of \(\Lambda \) -lattices M ,  with a relation \([L]+[N]=[M]\) whenever there exists a short exact sequence

of \(\Lambda \) -lattices. Recall from Sect.  2.3 that if T is a ring, then we similarly define the Grothendieck group \({{\,\textrm{G}\,}}_0(T)\) of the category of finitely generated T -modules by replacing, in the above definition, “ \(\Lambda \) -lattice” by “finitely generated T -module”. By [13, Theorem (38.42)], the inclusion of the category of \(\Lambda \) -lattices into the category of all finitely generated \(\Lambda \) -modules induces a canonical isomorphism \({{\,\textrm{G}\,}}_{0}^{R}(\Lambda )\cong {{\,\textrm{G}\,}}_{0}(\Lambda ).\)

Let \(\Lambda ^{{{\,\textrm{op}\,}}}\) denote the opposite ring of \(\Lambda .\) If M is a \(\Lambda \) -lattice, then \(M^*={{\,\textrm{Hom}\,}}_R(M,R)\) is a \(\Lambda ^{{{\,\textrm{op}\,}}}\) -lattice. This defines a contravariant functor from the category of \(\Lambda \) -lattices to the category of \(\Lambda ^{{{\,\textrm{op}\,}}}\) -lattices, given on objects by \(M\mapsto M^*\) for every \(\Lambda \) -lattice M ,  and on morphisms by \(f\mapsto (\nu \mapsto \nu \circ f)\in M^*\) for every morphism \(f:M\rightarrow N\) of \(\Lambda \) -lattices and every \(\nu \in N^*.\) This functor is easily seen to be exact, and to induce a group isomorphism \({{\,\textrm{G}\,}}_0^R(\Lambda )\rightarrow {{\,\textrm{G}\,}}_0^R(\Lambda ^{{{\,\textrm{op}\,}}}),\) and hence an isomorphism

For a \(\Lambda \) -module N ,  we define \(N^{\vee }={{\,\textrm{Hom}\,}}_R(N,k/R),\) which is also a \(\Lambda ^{{{\,\textrm{op}\,}}}\) -module. If N is a finitely generated \(\Lambda \) -module that is R -torsion, then \(N^{\vee }\) is finitely generated over \(\Lambda ^{{{\,\textrm{op}\,}}}\) and R -torsion.

In the special case that \(\Lambda =R[G],\) where G is a finite group, the ring \(\Lambda \) is equipped with an involution \(\iota \) induced by \(g\mapsto g^{-1}\) for all \(g\in G.\) If M is a finitely generated R [ G ]-module, then we view the \(R[G]^{{{\,\textrm{op}\,}}}\) -module \(M^{*}\) as a finitely generated R [ G ]-module via \(\iota ,\) and we view the map \(\textrm{j}\) as an automorphism of \({{\,\textrm{G}\,}}_0(R[G]).\) Specialising further to \(R=\mathbb {Z},\) if A is a Hausdorff locally compact abelian topological group on which G acts by continuous automorphisms, then we view its Pontryagin dual \({{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}(A,\mathbb {R}/\mathbb {Z})\) as a \(\mathbb {Z}[G]\) -module via \(\iota .\) If N is a \(\mathbb {Z}[G]\) -module of finite cardinality, then so is \({{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}(N,\mathbb {R}/\mathbb {Z})={{\,\textrm{Hom}\,}}_{\mathbb {Z}}(N,\mathbb {Q}/\mathbb {Z})=N^{\vee }.\)

Proposition 4.1

Let N be a finitely generated \(\Lambda \) -module that is R -torsion. Then the equality

holds in \({{\,\textrm{G}\,}}_{0}(\Lambda ^{{{\,\textrm{op}\,}}}).\)

Since R is a Dedekind domain, every \(\Lambda \) -submodule of a \(\Lambda \) -lattice is itself a \(\Lambda \) -lattice. Hence there exists a presentation \(0\rightarrow M_1\rightarrow M_2\rightarrow N\rightarrow 0\) of N by \(\Lambda \) -lattices, so that \([N] = [M_2]-[M_1]\) in \({{\,\textrm{G}\,}}_{0}(\Lambda ).\) We claim that \(N^\vee \) is canonically isomorphic as a \(\Lambda ^{{{\,\textrm{op}\,}}}\) -module to \(M_1^*/M_2^*.\)

Since \(M_1,\) \(M_2\) are projective over R ,  applying the functors \({{\,\textrm{Hom}\,}}_R(M_i,\bullet )\) for \(i=1,\) 2 to the short exact sequence

yields the commutative diagram with exact rows

where the vertical maps are induced by the injection \(M_1\rightarrow M_2.\) Of these, the middle map \({{\,\textrm{Hom}\,}}_R(M_2,k)\rightarrow {{\,\textrm{Hom}\,}}_R(M_1,k)\) is an isomorphism. Indeed, it is the k -linear dual of the map \(k\otimes _R M_1\rightarrow k\otimes _R M_2,\) which is clearly an isomorphism, since the cokernel N of \(M_1\rightarrow M_2\) is R -torsion. The snake lemma therefore gives an isomorphism of right \(\Lambda \) -modules from the kernel of \({{\,\textrm{Hom}\,}}_R(M_2,k/R)\rightarrow {{\,\textrm{Hom}\,}}_R(M_1,k/R)\) to the cokernel of \({{\,\textrm{Hom}\,}}_R(M_2,R)\rightarrow {{\,\textrm{Hom}\,}}_R(M_1,R).\) Since \({{\,\textrm{Hom}\,}}_R(\bullet ,k/R)\) is left exact, that kernel is exactly \(N^\vee ,\) while the cokernel is precisely \(M_1^*/M_2^*,\) as claimed. The proposition immediately follows. \(\square \)

5 Oriented Arakelov class groups as Galois modules

In this section we prove some properties of oriented Arakelov class groups as Galois modules, and formulate the main working hypothesis that motivates the statistical heuristic in Sect.  3 .

Let F / K be a finite Galois extension of number fields,  let G be the Galois group,  let \(S_{\infty }\) be the set of Archimedean places of F ,  and let d be the degree of K over \(\mathbb {Q}.\) Then the equality

holds in \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G]),\) where \({{\,\textrm{c}\,}}(F_{\mathbb {R}}^\times )\) is as defined in Sect.  2.2 .

If v is an Archimedean place of K ,  let \(I_v\subset G\) denote an inertia subgroup at v ,  and let \(\tau _v\) be a \(\mathbb {Z}[I_v]\) -module defined as follows: if v is real and \(I_v\) is the trivial group, then \(\tau _v=\mathbb {F}_2;\) if v is real and \(I_v\) has order 2,  then \(\tau _v\) is free over \(\mathbb {Z}\) of rank 1,  and with the generator of \(I_v\) acting by \(-1;\) and if v is complex, so that \(I_v\) is necessarily trivial, then \(\tau _v=\mathbb {Z}.\) Then it is easy to see that we have an isomorphism

of \(\mathbb {Z}[G]\) -modules, where the direct sum runs over the Archimedean places of K ,  and \({{\,\textrm{Ind}\,}}^G_{I_v}\) denotes induction from \(I_v\) to G .

If v is a real place of K such that \(I_v\) is trivial, then the exact sequence

shows that \([{{\,\textrm{Ind}\,}}^G_{I_v}\tau _v] =0\) in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]).\) We deduce that for all Archimedean places v of K we have

where \(\delta _v=1\) if v is real, and \(\delta _v=2\) is v is complex. The result follows by summing ( 5.2 ) over all Archimedean places v of K . \(\square \)

Proposition 5.3

hold in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]).\)

The rest of the proof is devoted to the derivation of the expression for \(.\) Let \(S_{{{\,\textrm{f}\,}}}=S{\setminus } S_{\infty }\) denote the set of non-Archimedean places in S . The subgroup of \({{\,\textrm{Id}\,}}_F\) generated by the prime ideals corresponding to the places in \(S_{{{\,\textrm{f}\,}}}\) is free abelian on the set \(S_{{{\,\textrm{f}\,}}}.\) Below, when we write \(\mathbb {Z}^{S_{{{\,\textrm{f}\,}}}},\) we will mean that subgroup. Let \({{\,\textrm{Id}\,}}_{F,S}\) be the quotient of \({{\,\textrm{Id}\,}}_F\) by the subgroup \(\mathbb {Z}^{S_{{{\,\textrm{f}\,}}}}.\) It is naturally isomorphic to the group of fractional ideals of \(\mathcal {O}_{F,S}.\) The preimage of

under the inclusion map \(F^\times \rightarrow {{\,\textrm{Id}\,}}_F\times _{\mathbb {R}_{>0}}F_{\mathbb {R}}^\times \) is \(\mathcal {O}_{F,S}^{\times }.\) There is thus a commutative diagram of \(\mathbb {Z}[G]\) -modules with exact rows and columns

where \(T_{F,S}\) is defined by the exactness of the last column, and the exactness of the first row follows from the snake lemma.

The group \(\mathbb {Z}^{S_{{{\,\textrm{f}\,}}}}\!\times _{\mathbb {R}_{>0}}F_{\mathbb {R}}^\times \) can be explicitly described as follows: we have

where \(\mathfrak {N}\) and \(|\!{{\,\textrm{Nm}\,}}\!|\) are as defined in Sect.  2.2 . That group naturally embeds into \(\mathbb {R}^{S_{{{\,\textrm{f}\,}}}}\!\times _{\mathbb {R}_{>0}}F_{\mathbb {R}}^\times ,\) where the fibre product is taken with respect to the map

and to the same map \(|\!{{\,\textrm{Nm}\,}}\!|:F_{\mathbb {R}}^\times \rightarrow \mathbb {R}_{>0}\) as before, so that we have an exact sequence

The preimage of \(\{0\}\times {{\,\textrm{c}\,}}(F_{\mathbb {R}}^\times )\subset \mathbb {Z}^{S_{{{\,\textrm{f}\,}}}}\!\times _{\mathbb {R}_{>0}}F_{\mathbb {R}}^\times \) in \(\mathcal {O}_{F,S}^\times \) is \(\mu _F.\) We therefore deduce from Dirichlet’s ( S -)unit theorem [ 16 , Chapter V, §1], that the middle term of ( 5.4 ) is an extension of the form

In particular, it is a compact abelian group whose Pontryagin dual

is finitely generated, therefore the same is true of the closed subgroup \(T_{F,S}.\)

Taking the Pontryagin dual of the right column of the commutative diagram above, we deduce that there is an equality

in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]).\)

Combining ( 5.6 ) with the Pontryagin duals of ( 5.4 ) and ( 5.5 ) and applying Lemma  5.1 , we see that there is an equality

in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]).\) We now deduce the result by applying Proposition  4.1 and noting that there are \(\mathbb {Z}[G]\) -module isomorphisms \((\mathbb {Z}^{S_{{{\,\textrm{f}\,}}}})^* \cong \mathbb {Z}^{S_{{{\,\textrm{f}\,}}}}\) and \(\mathbb {Z}^S \cong \mathbb {Z}^{S_{\infty }} \oplus \mathbb {Z}^{S_{{{\,\textrm{f}\,}}}}.\) \(\square \)

Corollary 5.7

Let F / K be a finite Galois extension of number fields,  let G be the Galois group,  let d be the degree of K over \(\mathbb {Q},\) and let S be a finite G -stable set of places of F containing all Archimedean places,  and large enough for \({{\,\textrm{Cl}\,}}_{F,S}\) to be trivial. Then the equality

holds in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]).\)

The result follows by combining Proposition  5.3 with the observation that \(\textrm{j}[\mathbb {Z}^S]=[\mathbb {Z}^S].\) \(\square \)

Corollary 5.8

Let F / K be a finite Galois extension of number fields,  let G be the Galois group,  and let S and \(S'\) be two finite G -stable sets of places of F ,  both containing all Archimedean places. Then the equality

The result follows by combining Proposition  5.3 with the observation that \(\textrm{j}[\mathbb {Z}^S] = [\mathbb {Z}^S]\) and \(\textrm{j}[\mathbb {Z}^{S'}] = [\mathbb {Z}^{S'}].\) \(\square \)

As we will see in the next section, the \(\Omega (3)\) conjecture, a standard conjecture in the theory of Galois module structures, implies the following simpler expressions for the classes of \({{\,\textrm{Ar}\,}}_F\) and \(.\)

Conjecture 5.9

Let F / K be a finite Galois extension of number fields,  let G be the Galois group,  let d be the degree of K over \(\mathbb {Q},\) let \(S_{\infty }\) be the set of Archimedean places of F ,  let \(\mu _F\) be the group of roots of unity in F ,  and let \(\textrm{j}\) be the automorphism of \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G])\) induced by the involution \(g\mapsto g^{-1}\) on \(\mathbb {Z}[G],\) as defined in Sect.  4 . Then the following equalities hold in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]){:}\)

\([{{\,\textrm{Ar}\,}}_F] = [\mathbb {Z}^{S_{\infty }}] - [\mathbb {Z}] - \textrm{j}[\mu _F].\)

The next result shows that, in the two equations in Conjecture  5.9 , the difference between the left hand side and the right hand side is the same. Hence each of (a) and (b) implies the other, and is therefore equivalent to the entire conjecture.

With the same notation as in Conjecture  5.9 , we have

in \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G]).\) Moreover,  the common difference between the left hand side and the right hand side of the two equations in Conjecture  5.9 lies in the torsion subgroup \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G])_{{{\,\textrm{tors}\,}}}\) of \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G]).\)

The first assertion follows from Proposition  5.3 with \(S=S_{\infty }.\)

We now prove the second assertion. By a theorem of Herbrand there is an isomorphism \((\mathbb {Q}\otimes _{\mathbb {Z}}\mathcal {O}_F^\times ) \oplus \mathbb {Q}\cong \mathbb {Q}^{S_{\infty }}\) of \(\mathbb {Q}[G]\) -modules, see for example [ 24 , Chapter I, 4.3]. Let \(\theta : {{\,\textrm{G}\,}}_0(\mathbb {Z}[G]) \rightarrow {{\,\textrm{G}\,}}_0(\mathbb {Q}[G])\) be the map induced by the flat ring homomorphism \(\mathbb {Z}[G] \rightarrow \mathbb {Q}[G].\) For every finitely generated \(\mathbb {Z}[G]\) -module M we have \(\theta (\textrm{j}[M])=\theta ([M]).\) Therefore applying \(\theta \) to the first equality of Proposition  5.3 , we deduce the equality

in \({{\,\textrm{G}\,}}_0(\mathbb {Q}[G]).\) This shows that the difference between the left hand side and the right hand side of Conjecture  5.9 (b) lies in \(\ker (\theta ),\) which is equal to \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G])_{{{\,\textrm{tors}\,}}}\) by [13, Theorem (39.14)]. The conclusion for Conjecture  5.9 (a) follows from this and the first assertion. \(\square \)

For every ring R that is flat over \(\mathbb {Z},\) the functor \(R\otimes _{\mathbb {Z}}\bullet \) induces a homomorphism \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G])\rightarrow {{\,\textrm{G}\,}}_0(R[G]).\) Hence, Conjecture  5.9 implies the following conjecture, which will suffice for the applications to the Cohen–Lenstra–Martinet heuristic in Sect.  3 .

Conjecture 5.11

Let F / K be a finite Galois extension of number fields,  let G be the Galois group,  let d be the degree of K over \(\mathbb {Q},\) and let P be a set of prime numbers not dividing \(2 \cdot \# G.\) Then the equalities

hold in \({{\,\textrm{G}\,}}_0(\mathbb {Z}_{(P)}[G]).\)

Remark 5.13

Conjecture  5.11 is equivalent to an affirmative answer to [ 4 , Question 5.5]. Thus in the special case that G is abelian and \(K=\mathbb {Q},\) Conjecture  5.11 was proven unconditionally in [ 4 , Theorem 5.4].

Remark 5.14

Let F / K be a finite Galois extension of number fields and let G be the Galois group. Lemma  5.10 shows that if \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G])_{{{\,\textrm{tors}\,}}}\) vanishes, then Conjecture  5.9 holds for F / K . There are many cases in which \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G])_{{{\,\textrm{tors}\,}}}\) has been calculated, and among these are numerous instances in which it is trivial. In the case that G is abelian, a formula for \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G])\) is given in [ 17 ]; this result has been extended to the case that G is nilpotent in [ 15 ]. Analogous observations also apply to Conjecture  5.11 .

6 The relation to Chinburg’s \(\Omega (3)\) conjecture

In this section we explain that Conjecture  5.9 , and a fortiori Conjecture  5.11 , follow from Conjecture  6.4 below, which is a weaker variant of Chinburg’s \(\Omega (3)\) conjecture, Conjecture  6.3 [ 6 , Conjecture 3.1].

We first briefly review some material from algebraic \({{\,\textrm{K}\,}}\) -theory. We refer the reader to [13, §38, §39, §49] for further details.

Let R be a Dedekind domain and let \(\Lambda \) be an R -order. Let \({{\,\textrm{K}\,}}_{0}(\Lambda )\) denote the Grothendieck group of the category of finitely generated projective \(\Lambda \) -modules. By definition, \({{\,\textrm{K}\,}}_{0}(\Lambda )\) is the additive group generated by expressions [ P ],  one for each isomorphism class of finitely generated projective \(\Lambda \) -modules P ,  with relations \([P_{1} \oplus P_{2}] = [P_{1}] + [P_{2}]\) for all such modules \(P_{1}, P_{2}.\) By the group \({{\,\textrm{K}\,}}_{0}(\Lambda )\) can be identified with the Grothendieck group of the category of finitely generated \(\Lambda \) -modules of finite projective dimension.

For each maximal ideal \(\mathfrak {p}\) of R ,  let \(R_{\mathfrak {p}}\) denote the localisation of R at \(\mathfrak {p},\) and define \(\Lambda _{\mathfrak {p}} = R_{\mathfrak {p}} \otimes _{R} \Lambda .\) A \(\Lambda \) -lattice M is said to be locally free if \(\Lambda _{\mathfrak {p}} \otimes _{\Lambda } M\) is free over \(\Lambda _{\mathfrak {p}}\) for every such \(\mathfrak {p}.\) Note that every locally free \(\Lambda \) -lattice is projective by [ 12 , Proposition (8.19)]. Let \({{\,\textrm{K}\,}}_{0}(\Lambda ) \rightarrow {{\,\textrm{K}\,}}_{0}(\Lambda _{\mathfrak {p}})\) be the map induced by the ring homomorphism \(\Lambda \rightarrow \Lambda _{\mathfrak {p}}.\) Then the locally free class group \({{\,\textrm{Cl}\,}}(\Lambda )\) is defined to be the kernel of the homomorphism \({{\,\textrm{K}\,}}_{0}(\Lambda ) \rightarrow \prod _{\mathfrak {p}} {{\,\textrm{K}\,}}_{0}(\Lambda _{\mathfrak {p}}),\) where the product runs over all maximal ideals \(\mathfrak {p}\) of R (see [13, Definition (39.12)]). By [13, (39.13)] we have

Note that there are several equivalent definitions of \({{\,\textrm{Cl}\,}}(\Lambda )\) (see [13, §49A], particularly [13, p. 223]).

We now recall the statement of Chinburg’s \(\Omega (3)\) conjecture. For the rest of the section, let F / K be a finite Galois extension of number fields and let G be the Galois group. For any finite G -stable set S of places of F ,  let \(X_{S}\) be the kernel of the augmentation map \(\mathbb {Z}^S \rightarrow \mathbb {Z}\) of \(\mathbb {Z}[G]\) -lattices. Henceforth let S be a finite G -stable set of places of F such that

S contains the set \(S_{\infty }\) of Archimedean places of F , 

S contains the ramified places of F / K ,  and

for every subfield N of F containing K ,  the ideal class group of N is generated by the classes \(\{[\mathfrak {p}\cap \mathcal {O}_N] :\mathfrak {p}\in S{\setminus } S_{\infty }\}.\)

Tate [ 23 , p. 711] defined a canonical class \(\alpha = \alpha _{S} \in {{\,\textrm{Ext}\,}}_{\mathbb {Z}[G]}^{2}(X_{S}, \mathcal {O}_{F,S}^{\times }),\) and showed the existence of so-called Tate sequences [ 24 , II, Théorème 5.1], that is, four term exact sequences of finitely generated \(\mathbb {Z}[G]\) -modules

representing \(\alpha ,\) where A and B are of finite projective dimension. In [ 6 ], Chinburg defined \(\Omega (F/K,3) = [A]-[B] \in {{\,\textrm{K}\,}}_{0}(\mathbb {Z}[G]).\) Moreover, he showed that \(\Omega (F/K,3)\) lies in the locally free class group \({{\,\textrm{Cl}\,}}(\mathbb {Z}[G]),\) and depends only on the extension F / K ;  in particular, it does not depend on the choice of S or on the choice of exact sequence ( 6.2 ).

The root number class \(W_{F/K} \in {{\,\textrm{Cl}\,}}(\mathbb {Z}[G])\) was defined by Ph. Cassou-Noguès in the case that F / K is at most tamely ramified, and was generalised to wildly ramified extensions F / K by Fröhlich [ 13 ]. It is an element of order at most 2,  and is defined in terms of the Artin root numbers of the irreducible symplectic characters of G . Moreover, if G has no irreducible symplectic characters (for example, if G is abelian or of odd order), then \(W_{F/K}\) is trivial by definition.

Conjecture 6.3

(Chinburg’s \(\Omega (3)\) conjecture) The equality

holds in \({{\,\textrm{Cl}\,}}(\mathbb {Z}[G]).\)

Fix, for the rest of the section, a maximal order \(\mathcal {M}\) in \(\mathbb {Q}[G]\) containing \(\mathbb {Z}[G],\) let \(\rho :{{\,\textrm{Cl}\,}}(\mathbb {Z}[G])\rightarrow {{\,\textrm{Cl}\,}}(\mathcal {M})\) be the map induced by the ring homomorphism \(\mathbb {Z}[G] \rightarrow \mathcal {M},\) and define the kernel subgroup \({{\,\textrm{D}\,}}(\mathbb {Z}[G])\) of \({{\,\textrm{Cl}\,}}(\mathbb {Z}[G])\) to be the kernel of \(\rho .\) If \(\mathcal {M}'\) is any other maximal order in \(\mathbb {Q}[G]\) containing \(\mathbb {Z}[G],\) and \(\rho ':{{\,\textrm{Cl}\,}}(\mathbb {Z}[G])\rightarrow {{\,\textrm{Cl}\,}}(\mathcal {M}')\) is the analogous map, then by [ 25 , Theorem 1.6] the kernel of \(\rho \) is equal to that of \(\rho '.\)

Conjecture 6.4

(Chinburg’s \(\Omega (3)\) conjecture modulo the kernel group) We have \(\Omega (F/K,3) \equiv W_{F/K} \bmod {{\,\textrm{D}\,}}(\mathbb {Z}[G]).\)

The next result will be the main ingredient in the proof of Theorem  1.1 from the introduction.

Proposition 6.5

Conjecture  6.4 for F / K implies Conjecture  5.9 for F / K .

In order to prove this result, we will make use of the following lemma. Let

Conjecture  5.9 for F / K is equivalent to the vanishing of \(\Psi (F/K),\) while Conjecture  5.11 is equivalent to the assertion that for all sets P of prime numbers not dividing \(2\cdot \#G\) the image of \(\Psi (F/K)\) under the map \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G])\rightarrow {{\,\textrm{G}\,}}_0(\mathbb {Z}_{(P)}[G])\) is 0. Moreover,  we always have \(\Psi (F/K) \in {{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G])_{{{\,\textrm{tors}\,}}}.\)

By Lemma  5.10 , Conjectures  5.9 (a) and (b) are equivalent to each other. By Corollary  5.7 , Conjecture  5.9 (a) is equivalent to the assertion that we have

in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]).\) Since \(\textrm{j}[\mathbb {Z}]=[\mathbb {Z}],\) this is equivalent to

By Corollary  5.8 with \(S'=S_{\infty },\) this in turn is equivalent to \(\Psi (F/K)=0.\)

The proof of the claim regarding Conjecture  5.11 is completely analogous.

The last claim follows from Lemma  5.10 . \(\square \)

In light of Lemma  6.6 , Proposition  6.5 amounts to the statement that Conjecture  6.4 implies the vanishing of \(\Psi (F/K).\) This statement is well known to experts in Galois module theory; see [ 5 , III], [ 11 , §4, Proposition 6] or [ 7 , §1]. We will recall the argument in the following proof and give some additional references.

Proof of Proposition 6.5

Let \(\mu : {{\,\textrm{Cl}\,}}(\mathbb {Z}[G]) \rightarrow {{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G])\) denote the restriction of the Cartan map \({{\,\textrm{K}\,}}_{0}(\mathbb {Z}[G]) \rightarrow {{\,\textrm{G}\,}}_{0}(\mathbb {Z}[G]),\) which is induced by letting \(\mu ([M])=[M]\) if M is a finitely generated projective \(\mathbb {Z}[G]\) -module.

Let \(\xi \in {{\,\textrm{Cl}\,}}(\mathbb {Z}[G]).\) Write \(\xi = [\mathbb {Z}[G]] - [L],\) where L is a locally free left ideal of \(\mathbb {Z}[G],\) which we may do by ( 6.1 ). By [ 12 , Exercise 31.10] we have

Hence we have the following commutative diagram of abelian groups:

where \(\alpha \) is induced by restriction, and \(\mu '\) is defined analogously to \(\mu .\)

Let A and B be as in the exact sequence ( 6.2 ). Then it follows from the definition of \(\Omega (F/K,3),\) from the exact sequence ( 6.2 ), and from Corollary  5.8 with \(S'=S_{\infty },\) that we have the equalities

Moreover, a special case of a result of Queyrut [ 19 , Proposition 2.3] shows that \(\mu (W_{F/K})=0.\)

Now suppose that Conjecture  6.4 holds for F / K . By definition of \({{\,\textrm{D}\,}}(\mathbb {Z}[G]),\) this is equivalent to \(\rho (\Omega (F/K,3))=\rho (W_{F/K}).\) Since \(\mu \) factors via \(\rho ,\) we have

The result now follows from Lemma  6.6 . \(\square \)

We close the section by proving Theorem  1.1 from the introduction. In fact, we prove the following stronger statement.

Theorem 6.7

Let F / K be a Galois extension of number fields,  let G be the Galois group,  let \(S_{\infty }\) be the G -set of Archimedean places of F ,  and let \(\Lambda \) be the quotient of the group ring \(\mathbb {Z}[\tfrac{1}{2\cdot \#G}][G]\) by the two-sided ideal generated by \(\sum _{g\in G}g.\) Suppose that Conjecture  6.4 holds for F / K ,  and that for every prime number p not dividing \(2\cdot \#G,\) each primitive p -th root of unity in F is in K . Then the equality

Note that the fact that \([\Lambda \otimes _{\mathbb {Z}[G]}{{\,\textrm{Ar}\,}}_F] - [\Lambda \otimes _{\mathbb {Z}[G]} \mathbb {Z}^{S_\infty }]\) indeed lies in \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}\) follows from Lemma  5.10 .

The ring homomorphism \(\mathbb {Z}[G]\rightarrow \Lambda \) is a composition of two homomorphisms of the types discussed at the end of Sect.  2.3 , so that it induces a group homomorphism \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G])\rightarrow {{\,\textrm{G}\,}}_0(\Lambda ).\) By Proposition  6.5 , the hypotheses of the theorem imply that we have the equality

in \({{\,\textrm{G}\,}}_0(\Lambda ),\) where \(\textrm{j}\) is the automorphism of \({{\,\textrm{G}\,}}_0(\Lambda )\) induced by the involution \(g\mapsto g^{-1}\) on G . Write temporarily \(\mathbb {Z}'=\mathbb {Z}[\tfrac{1}{2\cdot \#G}].\) Since the element \(\tfrac{1}{\#G}\sum _{g\in G}g\) of \(\mathbb {Z}'[G]\) acts trivially on the \(\mathbb {Z}'[G]\) -module \(\mathbb {Z}',\) but is 0 in the quotient \(\Lambda ,\) we have \([\Lambda \otimes _{\mathbb {Z}[G]}\mathbb {Z}]=0.\)

Next, the hypothesis on roots of unity implies that for all prime numbers p not dividing \(2\cdot \#G\) and all \(k\in \mathbb {Z}_{\ge 0},\) every \(p^k\) -th root of unity in F is in K . This implies that G acts trivially on \(\mathbb {Z}'\otimes _{\mathbb {Z}}\mu _F,\) hence, by the same argument as above, we have \([\Lambda \otimes _{\mathbb {Z}[G]}\mu _F]=0.\) This completes the proof. \(\square \)

7 Some known cases of the conjectures

In the present section we collect some situations in which Conjecture  5.11 is known. The main result of the section is Theorem  7.2 .

Definition 7.1

For a complex irreducible character \(\chi \) of a finite group G ,  let \(\mathbb {Q}(\chi )\) denote the field generated by the values of \(\chi ,\) and let \(C(\chi )\) be the narrow class group of \(\mathbb {Q}(\chi )\) if \(\chi \) is symplectic, and the usual ideal class group of \(\mathbb {Q}(\chi )\) otherwise.

Theorem 7.2

Let F / K be a finite Galois extension of number fields,  let G be the Galois group,  let \(G'\) be its commutator subgroup,  and let P be a set of prime numbers not dividing \(2\cdot \#G.\) Suppose that \(F^{G'}\!\!/\mathbb {Q}\) is abelian,  and that for every complex irreducible character \(\chi \) of G satisfying \(\chi (1)>1,\) the group \(C(\chi )\) is generated by the classes of the non-zero prime ideals of \(\mathcal {O}_{\mathbb {Q}(\chi )}\) not dividing any element of P . Then Conjecture  5.11 holds for F / K and P .

The rest of the section is devoted to the proof of Theorem  7.2 and some consequences.

Theorem 7.3

Let F / K be a finite Galois extension of number fields and let G be the Galois group. Suppose that \(F/\mathbb {Q}\) is abelian. Then we have \(\Omega (F/K,3)=W_{F/K}=0\) in \({{\,\textrm{Cl}\,}}(\mathbb {Z}[G]).\)

This is a special case of [ 2 , Corollary 1.4]. \(\square \)

Corollary 7.4

Let F / K be a finite Galois extension of number fields such that \(F/\mathbb {Q}\) is abelian. Then Conjecture  5.9 holds for F / K .

The assertion immediately follows from Theorem  7.3 and Proposition  6.5 . \(\square \)

If G is a finite group, we denote by \({{\,\textrm{Irr}\,}}_{{{\,\textrm{na}\,}}}(G)\) the set of complex irreducible characters of G of degree greater than 1,  and for \(\chi ,\) \(\chi '\in {{\,\textrm{Irr}\,}}_{{{\,\textrm{na}\,}}}(G)\) we write \(\chi \sim \chi '\) if \(\chi \) and \(\chi '\) are in the same Galois orbit, i.e., if there exists \(\tau \in {{\,\textrm{Gal}\,}}(\bar{\mathbb {Q}}/\mathbb {Q})\) such that \(\chi =\tau \circ \chi '.\)

Let G be a finite group and let \(G'\) be its commutator subgroup. Then there is a direct product decomposition of \(\mathbb {Q}\) -algebras

where the product is taken over a full set of representatives of Galois orbits of non-abelian characters of G ,  and each \(A_\chi \) is a simple \(\mathbb {Q}\) -algebra. Moreover,  for a set P of prime numbers not dividing \(\# G,\) there is a corresponding direct product decomposition

where each \(R_{\chi }\) is a maximal \(\mathbb {Z}_{(P)}\) -order in \(A_{\chi }.\)

The decomposition ( 7.6 ) is standard. By [ 12 , Proposition (27.1)], the \(\mathbb {Z}_{(P)}\) -order \(\mathbb {Z}_{(P)}[G]\) is maximal in \(\mathbb {Q}[G],\) so the decomposition ( 7.7 ) follows from [ 20 , Theorem (10.5)]. \(\square \)

Let G be a finite group,  let P be a set of prime numbers,  let \(\chi \) be a complex irreducible character of G ,  let \(A_{\chi }\) be the simple quotient of \(\mathbb {Q}[G]\) corresponding to the Galois orbit of \(\chi ,\) let \(R_{\chi }\) be the image of \(\mathbb {Z}_{(P)}[G]\) in \(A_{\chi },\) and let \(\mathbb {Q}(\chi )\) and \(C(\chi )\) be as in Definition  7.1 . Then \({{\,\textrm{G}\,}}_{0}(R_{\chi })_{{{\,\textrm{tors}\,}}}\) is isomorphic to the quotient of \(C(\chi )\) by the subgroup generated by the non-zero prime ideals of \(\mathcal {O}_{\mathbb {Q}(\chi )}\) not dividing any element of P .

This follows immediately from [13, Theorem (38.67)]. \(\square \)

Proof of Theorem 7.2

For an element m of \({{\,\textrm{G}\,}}_0(\mathbb {Z}[G]),\) let \(\mathbb {Z}_{(P)}\otimes m\) denote the image of m in \({{\,\textrm{G}\,}}_0(\mathbb {Z}_{(P)}[G]).\) Then Lemma  6.6 implies that we have \(\mathbb {Z}_{(P)}\otimes \Psi (F/K)\in {{\,\textrm{G}\,}}_0(\mathbb {Z}_{(P)}[G])_{{{\,\textrm{tors}\,}}},\) and that the claim is equivalent to the assertion that \(\mathbb {Z}_{(P)}\otimes \Psi (F/K)=0.\) Moreover, by Lemma  7.5 , there is a direct sum decomposition of abelian groups

Therefore the assertion that \(\mathbb {Z}_{(P)}\otimes \Psi (F/K)=0\) is in turn equivalent to the claim that the image of \(\Psi (F/K)\) in each of the above summands is 0.

Since P does not contain any prime divisors of \(\#G',\) it is easily seen that the image of \(\Psi (F/K)\) in \({{\,\textrm{G}\,}}_0(\mathbb {Z}_{(P)}[G/G'])_{{{\,\textrm{tors}\,}}}\) is equal to the image of \(\Psi (F^{G'}/K).\) Since \(F^{G'}/\mathbb {Q}\) is abelian, that image is 0 by Corollary  7.4 .

By Lemma  7.8 , the hypotheses of the theorem imply that \({{\,\textrm{G}\,}}_0(R_\chi )_{{{\,\textrm{tors}\,}}}\) is trivial for every \(\chi \in {{\,\textrm{Irr}\,}}_{{{\,\textrm{na}\,}}}(G),\) so the result follows. \(\square \)

We conclude the section with a summary of what we know about Conjecture  5.11 for small groups.

Proposition 7.9

Let \(F/\mathbb {Q}\) be a Galois extension of degree less than 112,  let G be the Galois group,  and let P be a set of prime numbers not dividing \(2\cdot \#G.\) Then Conjecture  5.11 holds for F / K and P ,  with \(K=\mathbb {Q}.\)

A direct computation, e.g. using the computational algebra system Magma [ 1 ], shows that the assumption that \(\#G<112\) implies that the hypotheses of Theorem  7.2 , with \(K=\mathbb {Q}\) and P equal to the set of prime numbers not dividing \(2\cdot \#G,\) are satisfied for F / K . A fortiori, the assumption is also satisfied with P being any smaller set. \(\square \)

Remark 7.10

There are exactly two groups of order 112 that have an irreducible character \(\chi \) of degree greater than 1 such that, in the notation of the proof of Theorem  7.2 , the group \({{\,\textrm{G}\,}}_0(R_{\chi })_{{{\,\textrm{tors}\,}}}\) is non-trivial. Each has exactly one Galois orbit of such characters, in both cases of degree 2. The two groups are both semidirect products of a normal cyclic subgroup C of order 56 and a group H of order 2. Let \(x\in C\) be an element of order 7,  and let \(y\in C\) be an element of order 8. In one semidirect product, the non-trivial element of H acts on C by \(x\mapsto x^{-1}\) and \(y\mapsto y^5;\) and in the other it acts on C by \(x\mapsto x^{-1}\) and \(y\mapsto y^3.\) These are the two smallest Galois groups G for which we do not currently know Conjecture  5.11 with \(K=\mathbb {Q}\) and with P being the set of all prime numbers not dividing \(2\cdot \#G.\)

8 Counterexamples to the Cohen–Lenstra–Martinet heuristic

In this section we prove Theorem  1.3 from the introduction and explain how it disproves Heuristic  3.2 . The following notation will remain in force throughout the section.

Notation 8.1

Let p be an odd prime number, let \(C_2\) and \(C_p\) be cyclic groups of orders 2 and p ,  respectively, and let \(G=C_2\wr C_p=C_2^p\rtimes C_p,\) where \(C_p\) acts on \(C_2^p\) via its regular permutation action. Let \(Z\) be the centre of G ,  which is cyclic of order 2,  and let \(\gamma \in Z\) be the unique non-trivial central element. Let P be the set of all prime numbers not dividing 2 p ,  and let \(\Lambda \) be the quotient of \(\mathbb {Z}_{(P)}[G]\) by the ideal generated by \(1+\gamma .\) Let \(\mathcal {F}\) be the family of all pairs ( F ,  i ),  where F is a Galois number field satisfying \(\mu _F=\{\pm 1\},\) and i is an isomorphism from G to the Galois group of F sending \(Z\) to the inertia group of every Archimedean place.

Proposition 8.2

All complex irreducible characters \(\chi \) of G with \(\chi (1)>1\) satisfy \(\mathbb {Q}(\chi )=\mathbb {Q}.\)

Let \(\chi \) be a complex irreducible character of G with \(\chi (1)>1.\) Then by [ 22 , §8.2] we have \(\chi ={{\,\textrm{Ind}\,}}_{C_2^p}^G\psi \) for some irreducible character \(\psi \) of \(C_2^p.\) Every such character \(\psi \) satisfies \(\mathbb {Q}(\psi )=\mathbb {Q},\) so we also have \(\mathbb {Q}(\chi )=\mathbb {Q}.\) \(\square \)

Corollary 8.3

Let \(K=\mathbb {Q},\) and let F be a Galois extension of \(\mathbb {Q}\) with Galois group isomorphic to G . Then Conjecture  5.11 holds for F / K and P .

This follows from Proposition  8.2 and Theorem  7.2 . \(\square \)

There exist,  up to isomorphism,  infinitely many Galois number fields F with Galois group isomorphic to G such that the inertia groups at infinity map to \(Z\) and such that \(\mu _{F}=\{\pm 1\}.\)

Let \(L/\mathbb {Q}\) be a cyclic extension of degree p and let H be the Galois group. Since \([L:\mathbb {Q}]\) is odd and \(L/\mathbb {Q}\) is Galois, L must be totally real. Let l be a prime number that splits completely in L ,  and let \(\mathfrak {l}\) be a place of L above l . By weak approximation, there exists \(a\in L^\times \) such that a is totally negative, has \(\mathfrak {l}\) -adic normalised valuation 1,  and for all \(\sigma \in H{\setminus }\{1\}\) has \(\sigma (\mathfrak {l})\) -adic valuation 0. In particular, a has the property that for every non-empty subset \(\Sigma \subseteq H\) the product \(\prod _{\sigma \in \Sigma } \sigma (a)\) is not a square in \(L^{\times }.\) The Galois closure \(F=L(\{\sqrt{\sigma (a)} : \sigma \in H \})\) of \(L(\sqrt{a})\) then has Galois group isomorphic to G such that the inertia groups at infinity map to \(Z.\) Moreover, the maximal abelian extension inside F is \(F^{{{\,\textrm{ab}\,}}}= L(\sqrt{{{\,\textrm{Norm}\,}}(a)})\) where \({{\,\textrm{Norm}\,}}(a)=\prod _{\sigma \in H} \sigma (a) \in \mathbb {Q}^{\times },\) which has l -adic valuation 1. Thus \(F^{{{\,\textrm{ab}\,}}}/\mathbb {Q}\) is ramified at l ,  so as l varies, we obtain infinitely many extensions F . Of these, only finitely many can contain a non-trivial cyclotomic field, which completes the proof. \(\square \)

Proposition 8.5

For all \((F,i)\in \mathcal {F}\) the class of \(\Lambda \otimes _{\mathbb {Z}[G]}{{\,\textrm{Cl}\,}}_F\) in \({{\,\textrm{G}\,}}_{0}(\Lambda )\) is trivial.

Let \((F,i)\in \mathcal {F}.\) Under the hypotheses the extension \(F/F^{Z}\) is a totally imaginary quadratic extension of a totally real field, and and we have \(\mu _{F} = \{ \pm 1 \}\) . These observations imply that \(\Lambda \otimes _{\mathbb {Z}[G]} \mathcal {O}_F^\times \) is trivial. Now Corollary  8.3 implies that Conjecture  5.11 holds in for F / K and P ,  and thus one has \([\mathbb {Z}_{(P)}\otimes _{\mathbb {Z}}{{\,\textrm{Ar}\,}}_F] = [(\mathbb {Z}_{(P)})^{S_\infty }] - [\mathbb {Z}_{(P)}]\) in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}_{(P)}[G]).\) Since \(\gamma \) acts trivially on the two terms on the right, this implies that \([\Lambda \otimes _{\mathbb {Z}[G]} {{\,\textrm{Ar}\,}}_{F}]\) is trivial in \({{\,\textrm{G}\,}}_{0}(\Lambda ).\) By the first equality of Proposition  5.3 we have

in \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}_{(P)}[G]).\) Since \(\textrm{j}\) is the automorphism of \({{\,\textrm{G}\,}}_{0}(\mathbb {Z}_{(P)}[G])\) induced by the involution \(\sigma \mapsto \sigma ^{-1}\) on \(\mathbb {Z}[G],\) and since we have \(\gamma ^{-1}=\gamma ,\) we conclude that

in \({{\,\textrm{G}\,}}_{0}(\Lambda ),\) as desired. \(\square \)

Let \(p > 19\) be a prime number satisfying \(p\equiv \pm 3 \bmod 8\) and let \(\zeta _{p}\) denote a primitive p -th root of unity. Then \({{\,\textrm{Cl}\,}}(\mathbb {Z}[\zeta _{p}, \frac{1}{2p}])\) is non-trivial.

Let C be the class group of \(\mathbb {Q}(\zeta _p)\) and let \(C^{+}\) be the class group of the maximal real subfield \(\mathbb {Q}(\zeta _p)^{+}.\) By [ 26 , Theorem 4.14], the natural map \(C^{+} \rightarrow C\) is an injection, and so we can and do view \(C^{+}\) as a subgroup of C . By [ 26 , Theorem 2.13], 2 splits into \((p-1)/f\) distinct primes in \(\mathbb {Q}(\zeta _{p}),\) where f is the multiplicative order of \(2 \bmod p.\) The condition \(p\equiv \pm 3 \bmod 8\) is equivalent to 2 being a quadratic non-residue \(\bmod \) p ,  and thus f must be even. Since \(\mathbb {Q}(\zeta _p)/\mathbb {Q}\) is cyclic, we conclude that any prime of \(\mathbb {Q}(\zeta _p)^+\) lying above 2 must be inert in \(\mathbb {Q}(\zeta _p)/\mathbb {Q}(\zeta _p)^+.\) Thus the class in C of any prime of \(\mathbb {Q}(\zeta _p)\) above 2 must in fact lie in \(C^+.\) Since \(p>19,\) we have that \(\#C/\#(C^{+}) > 1\) by [ 26 , Corollary 11.18]. Moreover, the unique prime of \(\mathbb {Q}(\zeta _{p})\) above p is principal (generated by \(1-\zeta _{p}).\) Therefore the quotient of C by the subgroup generated by the classes of primes above 2 and p is non-trivial, which gives the desired result. \(\square \)

Proposition 8.7

Let \(p>19\) be a prime number satisfying \(p\equiv \pm 3 \bmod 8.\) Then \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}\) is non-trivial.

Since \(G/G'\) is cyclic of order 2 p ,  we have that

Let \(\Lambda '\) be the image of \(\Lambda \) under the projection \(\mathbb {Z}_{(P)}[G] \rightarrow \mathbb {Z}_{(P)}[G/G'].\) Then \(\Lambda '\) is a direct factor of \(\Lambda \) and \(\Lambda ' \cong \mathbb {Z}[\frac{1}{2p}] \times \mathbb {Z}[\zeta _{p}, \frac{1}{2p}].\) Thus \({{\,\textrm{G}\,}}_{0}(\Lambda ')_{{{\,\textrm{tors}\,}}}\) is a direct summand of \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}\) and \({{\,\textrm{G}\,}}_{0}(\Lambda ')_{{{\,\textrm{tors}\,}}} \cong {{\,\textrm{Cl}\,}}(\mathbb {Z}[\zeta _{p}, \frac{1}{2p}])\) by [13, Theorem (38.67)]. Therefore the result follows from Lemma  8.6 . \(\square \)

The conclusions of Lemma  8.6 and Proposition  8.7 hold under the weaker assumption on p that \(p>19\) be a prime number such that 2 generates a group of even cardinality in \((\mathbb {Z}/p\mathbb {Z})^\times ,\) and the proofs carry over almost verbatim. The set of all such primes has been investigated, and Hasse [ 14 ] has computed its Dirichlet density to be 17/24.

Proof of Theorem 1.3

Let p be a prime number satisfying the hypothesis of Proposition  8.7 , and let G and \(\Lambda \) be as in Notation  8.1 . Then \({{\,\textrm{G}\,}}_0(\Lambda )_{{{\,\textrm{tors}\,}}}\) is non-trivial. The family \(\mathcal {F}\) as in Notation  8.1 is infinite by Lemma  8.4 , and for all \((F,i)\in \mathcal {F}\) the class of \(\Lambda \otimes _{\mathbb {Z}[G]}{{\,\textrm{Cl}\,}}_F\) in \({{\,\textrm{G}\,}}_0(\Lambda )\) is trivial by Proposition  8.5 . \(\square \)

Finally, the next result shows that Heuristic  3.2 is false for \(\mathcal {F}\) and \(\Lambda \) as above and for some natural functions f .

Proposition 8.9

Suppose that for all homomorphisms \(\phi :{{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}\rightarrow \mathbb {C}^\times ,\) Heuristic  3.2 holds with \(K=\mathbb {Q}\) and with f being the function that assigns to a finite \(\Lambda \) -module M the value of \(\phi \) on the class of M in \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}.\) Then as ( F ,  i ) ranges over \(\mathcal {F},\) the class of \(\Lambda \otimes _{\mathbb {Z}[G]}{{\,\textrm{Cl}\,}}_F\) in \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}}\) is equidistributed.

By Lemma  7.5 there is a direct product decomposition

Note that the group \(G/G'\) is cyclic of order 2 p . Let \(\bar{\gamma }\) be the image of \(\gamma \) under the projection map \(G \rightarrow G/G',\) and let \(\bar{\Lambda }\) be the quotient of \(\mathbb {Z}_{(P)}[G/G']\) by the two-sided ideal generated by \(1+\bar{\gamma }.\) Then \(\Lambda \cong \bar{\Lambda } \times T,\) where T is a direct factor of \(\prod _{\chi \in {{\,\textrm{Irr}\,}}_{{{\,\textrm{na}\,}}}(G)/\sim }R_\chi .\) Moreover, for \(\chi \in {{\,\textrm{Irr}\,}}_{{{\,\textrm{na}\,}}}(G),\) we have that \(\mathbb {Q}(\chi )=\mathbb {Q}\) by Proposition  8.2 , and thus \({{\,\textrm{G}\,}}_{0}(R_{\chi })_{{{\,\textrm{tors}\,}}}\) is trivial by Lemma  7.8 . Hence there is a canonical isomorphism \({{\,\textrm{G}\,}}_{0}(\Lambda )_{{{\,\textrm{tors}\,}}} \cong {{\,\textrm{G}\,}}_{0}(\bar{\Lambda })_{{{\,\textrm{tors}\,}}}.\) The set \(\mathcal {F}\) is infinite by Lemma  8.4 . Note that \(F^{G'}\) is an imaginary abelian number field that is a quadratic extension of a real field, and \(\bar{\Lambda }\otimes _{\mathbb {Z}[G]}{{\,\textrm{Cl}\,}}_F\) is the maximal quotient of \(\mathbb {Z}_{(P)}\otimes _{\mathbb {Z}}{{\,\textrm{Cl}\,}}_{F^{G'}}\) on which complex conjugation acts by \(-1.\) Equidistribution of minus parts of class groups in the corresponding Grothendieck group was shown in [ 4 , Proposition 4.4] to follow from the Cohen–Lenstra–Martinet heuristic for families of imaginary extensions with Galois group isomorphic to \(G/G'.\) However, it is easy to see that the prediction is the same in the present situation, despite the fields being ordered differently; see Remark  3.1 . \(\square \)

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Acknowledgements

We would like to thank Ted Chinburg, Aurel Page, and Peter Stevenhagen for useful conversations, and Andreas Nickel and an anonymous referee for helpful comments on earlier drafts of the manuscript. The first named author gratefully acknowledges financial support through EPSRC Fellowship EP/P019188/1, ‘Cohen–Lenstra heuristics, Brauer relations, and low-dimensional manifolds’. The second named author gratefully acknowledges financial support through EPSRC First Grant EP/N005716/1 ‘Equivariant Conjectures in Arithmetic’.

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Bartel, A., Johnston, H. & Lenstra, H.W. Arakelov class groups of random number fields. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02862-4

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The process creates a payroll action representing the payroll and a payroll relationship action for each relationship processed.

The process loads into memory the element entries for the payroll relationship action.

The process identifies and determines any formulas to run for calculating the element entries.

At the end of the process, you find one run result value for each element entry value. If the element entry involves currency conversion, the payroll calculation uses the current conversion rate and rounds the monetary result based on the formula rules.

For each run result, the process determines which balances the result should feed. The process then writes and updates the balances to the database.

Example 1: Calculation Based on Annual Salary Basis

Your payroll provider might require you to pass values for gross earnings periodically, based on the payroll frequency of each employee. If you use an annual salary basis to store the values, run the Calculate Gross Earnings process to calculate the values by payroll period.

The formula attached to the annual salary calculates the periodic value and feeds the value to a run result during the payroll run. You can extract the run result value using a payroll interface report.

Example 2: Calculation Based on an Element Group

To avoid processing all the regular and supplemental earnings in a calculation process, the process considers only the earnings elements that you associate with an element group. Specify the value of the element group as a parameter while you submit the Calculate Gross Earnings process.