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A Comprehensive Study of Regression Analysis and the Existing Techniques

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• Published: 01 December 2015

Points of Significance

Multiple linear regression

• Martin Krzywinski 2 &
• Naomi Altman 1  

Nature Methods volume  12 ,  pages 1103–1104 ( 2015 ) Cite this article

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When multiple variables are associated with a response, the interpretation of a prediction equation is seldom simple.

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Last month we explored how to model a simple relationship between two variables, such as the dependence of weight on height 1 . In the more realistic scenario of dependence on several variables, we can use multiple linear regression (MLR). Although MLR is similar to linear regression, the interpretation of MLR correlation coefficients is confounded by the way in which the predictor variables relate to one another.

In simple linear regression 1 , we model how the mean of variable Y depends linearly on the value of a predictor variable X ; this relationship is expressed as the conditional expectation E( Y | X ) = β 0 + β 1 X . For more than one predictor variable X 1 , . . ., X p , this becomes β 0 + Σ β j X j . As for simple linear regression, one can use the least-squares estimator (LSE) to determine estimates b j of the β j regression parameters by minimizing the residual sum of squares, SSE = Σ( y i − ŷ i ) 2 , where ŷ i = b 0 + Σ j b j xij . When we use the regression sum of squares, SSR = Σ( ŷ i − Y − ) 2 , the ratio R 2 = SSR/(SSR + SSE) is the amount of variation explained by the regression model and in multiple regression is called the coefficient of determination.

The slope β j is the change in Y if predictor j is changed by one unit and others are held constant. When normality and independence assumptions are fulfilled, we can test whether any (or all) of the slopes are zero using a t -test (or regression F -test). Although the interpretation of β j seems to be identical to its interpretation in the simple linear regression model, the innocuous phrase “and others are held constant” turns out to have profound implications.

To illustrate MLR—and some of its perils—here we simulate predicting the weight ( W , in kilograms) of adult males from their height ( H , in centimeters) and their maximum jump height ( J , in centimeters). We use a model similar to that presented in our previous column 1 , but we now include the effect of J as E( W | H , J ) = β H H + β J J + β 0 + ε, with β H = 0.7, β J = −0.08, β 0 = −46.5 and normally distributed noise ε with zero mean and σ = 1 ( Table 1 ). We set β J negative because we expect a negative correlation between W and J when height is held constant (i.e., among men of the same height, lighter men will tend to jump higher). For this example we simulated a sample of size n = 40 with H and J normally distributed with means of 165 cm (σ = 3) and 50 cm (σ = 12.5), respectively.

Although the statistical theory for MLR seems similar to that for simple linear regression, the interpretation of the results is much more complex. Problems in interpretation arise entirely as a result of the sample correlation 2 among the predictors. We do, in fact, expect a positive correlation between H and J —tall men will tend to jump higher than short ones. To illustrate how this correlation can affect the results, we generated values using the model for weight with samples of J and H with different amounts of correlation.

Let's look first at the regression coefficients estimated when the predictors are uncorrelated, r ( H , J ) = 0, as evidenced by the zero slope in association between H and J ( Fig. 1a ). Here r is the Pearson correlation coefficient 2 . If we ignore the effect of J and regress W on H , we find Ŵ = 0.71 H − 51.7 ( R 2 = 0.66) ( Table 1 and Fig. 1b ). Ignoring H , we find Ŵ = −0.088 J + 69.3 ( R 2 = 0.19). If both predictors are fitted in the regression, we obtain Ŵ = 0.71 H − 0.088 J − 47.3 ( R 2 = 0.85). This regression fit is a plane in three dimensions ( H , J , W ) and is not shown in Figure 1 . In all three cases, the results of the F -test for zero slopes show high significance ( P ≤ 0.005).

( a ) Simulated values of uncorrelated predictors, r ( H , J ) = 0. The thick gray line is the regression line, and thin gray lines show the 95% confidence interval of the fit. ( b ) Regression of weight ( W ) on height ( H ) and of weight on jump height ( J ) for uncorrelated predictors shown in a . Regression slopes are shown ( b H = 0.71, b J = −0.088). ( c ) Simulated values of correlated predictors, r ( H , J ) = 0.9. Regression and 95% confidence interval are denoted as in a . ( d ) Regression (red lines) using correlated predictors shown in c . Light red lines denote the 95% confidence interval. Notice that b J = 0.097 is now positive. The regression line from b is shown in blue. In all graphs, horizontal and vertical dotted lines show average values.

When the sample correlations of the predictors are exactly zero, the regression slopes ( b H and b J ) for the “one predictor at a time” regressions and the multiple regression are identical, and the simple regression R 2 sums to multiple regression R 2 (0.66 + 0.19 = 0.85; Fig. 2 ). The intercept changes when we add a predictor with a nonzero mean to satisfy the constraint that the least-squares regression line goes through the sample means, which is always true when the regression model includes an intercept.

Shown are the values of regression coefficient estimates ( b H , b J , b 0 ) and R 2 and the significance of the test used to determine whether the coefficient is zero from 250 simulations at each value of predictor sample correlation −1 < r ( H , J ) < 1 for each scenario where either H or J or both H and J predictors are fitted in the regression. Thick and thin black curves show the coefficient estimate median and the boundaries of the 10th–90th percentile range, respectively. Histograms show the fraction of estimated P values in different significance ranges, and correlation intervals are highlighted in red where >20% of the P values are >0.01. Actual regression coefficients ( β H , β J , β 0 ) are marked on vertical axes. The decrease in significance for b J when jump height is the only predictor and r ( H , J ) is moderate (red arrow) is due to insufficient statistical power ( b J is close to zero). When predictors are uncorrelated, r ( H , J ) = 0, R 2 of individual regressions sum to R 2 of multiple regression (0.66 + 0.19 = 0.85). Panels are organized to correspond to Table 1 , which shows estimates of a single trial at two different predictor correlations.

Balanced factorial experiments show a sample correlation of zero among the predictors when their levels have been fixed. For example, we might fix three heights and three jump heights and select two men representative of each combination, for a total of 18 subjects to be weighed. But if we select the samples and then measure the predictors and response, the predictors are unlikely to have zero correlation.

When we simulate highly correlated predictors r ( H , J ) = 0.9 ( Fig. 1c ), we find that the regression parameters change depending on whether we use one or both predictors ( Table 1 and Fig. 1d ). If we consider only the effect of H , the coefficient β H = 0.7 is inaccurately estimated as b H = 0.44. If we include only J , we estimate β J = −0.08 inaccurately, and even with the wrong sign ( b J = 0.097). When we use both predictors, the estimates are quite close to the actual coefficients ( b H = 0.63, b J = −0.056).

In fact, as the correlation between predictors r ( H , J ) changes, the estimates of the slopes ( b H , b J ) and intercept ( b 0 ) vary greatly when only one predictor is fitted. We show the effects of this variation for all values of predictor correlation (both positive and negative) across 250 trials at each value ( Fig. 2 ). We include negative correlation because although J and H are likely to be positively correlated, other scenarios might use negatively correlated predictors (e.g., lung capacity and smoking habits). For example, if we include only H in the regression and ignore the effect of J , b H steadily decreases from about 1 to 0.35 as r ( H , J ) increases. Why is this? For a given height, larger values of J (an indicator of fitness) are associated with lower weight. If J and H are negatively correlated, as J increases, H decreases, and both changes result in a lower value of W . Conversely, as J decreases, H increases, and thus W increases. If we use only H as a predictor, J is lurking in the background, depressing W at low values of H and enhancing W at high levels of H , so that the effect of H is overestimated ( b H increases). The opposite effect occurs when J and H are positively correlated. A similar effect occurs for b J , which increases in magnitude (becomes more negative) when J and H are negatively correlated. Supplementary Figure 1 shows the effect of correlation when both regression coefficients are positive.

When both predictors are fitted ( Fig. 2 ), the regression coefficient estimates ( b H , b J , b 0 ) are centered at the actual coefficients ( β H , β J , β 0 ) with the correct sign and magnitude regardless of the correlation of the predictors. However, the standard error in the estimates steadily increases as the absolute value of the predictor correlation increases.

Neglecting important predictors has implications not only for R 2 , which is a measure of the predictive power of the regression, but also for interpretation of the regression coefficients. Unconsidered variables that may have a strong effect on the estimated regression coefficients are sometimes called 'lurking variables'. For example, muscle mass might be a lurking variable with a causal effect on both body weight and jump height. The results and interpretation of the regression will also change if other predictors are added.

Given that missing predictors can affect the regression, should we try to include as many predictors as possible? No, for three reasons. First, any correlation among predictors will increase the standard error of the estimated regression coefficients. Second, having more slope parameters in our model will reduce interpretability and cause problems with multiple testing. Third, the model may suffer from overfitting. As the number of predictors approaches the sample size, we begin fitting the model to the noise. As a result, we may seem to have a very good fit to the data but still make poor predictions.

MLR is powerful for incorporating many predictors and for estimating the effects of a predictor on the response in the presence of other covariates. However, the estimated regression coefficients depend on the predictors in the model, and they can be quite variable when the predictors are correlated. Accurate prediction of the response is not an indication that regression slopes reflect the true relationship between the predictors and the response.

Altman, N. & Krzywinski, M. Nat. Methods 12 , 999–1000 (2015).

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Altman, N. & Krzywinski, M. Nat. Methods 12 , 899–900 (2015).

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Martin Krzywinski is a staff scientist at Canada's Michael Smith Genome Sciences Centre.,

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Supplementary figure 1 regression coefficients and r 2.

The significance and value of regression coefficients and R 2 for a model with both regression coefficients positive, E( W | H,J ) = 0.7 H + 0.08 J - 46.5 + ε. The format of the figure is the same as that of Figure 2 .

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Krzywinski, M., Altman, N. Multiple linear regression. Nat Methods 12 , 1103–1104 (2015). https://doi.org/10.1038/nmeth.3665

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Abstract: Least squares linear regression is one of the oldest and widely used data analysis tools. Although the theoretical analysis of the ordinary least squares (OLS) estimator is as old, several fundamental questions are yet to be answered. Suppose regression observations $(X_1,Y_1),\ldots,(X_n,Y_n)\in\mathbb{R}^d\times\mathbb{R}$ (not necessarily independent) are available. Some of the questions we deal with are as follows: under what conditions, does the OLS estimator converge and what is the limit? What happens if the dimension is allowed to grow with $n$? What happens if the observations are dependent with dependence possibly strengthening with $n$? How to do statistical inference under these kinds of misspecification? What happens to the OLS estimator under variable selection? How to do inference under misspecification and variable selection? We answer all the questions raised above with one simple deterministic inequality which holds for any set of observations and any sample size. This implies that all our results are a finite sample (non-asymptotic) in nature. In the end, one only needs to bound certain random quantities under specific settings of interest to get concrete rates and we derive these bounds for the case of independent observations. In particular, the problem of inference after variable selection is studied, for the first time, when $d$, the number of covariates increases (almost exponentially) with sample size $n$. We provide comments on the ``right'' statistic to consider for inference under variable selection and efficient computation of quantiles.

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Linear Regression in Medical Research

Patrick schober.

From the * Department of Anesthesiology, Amsterdam UMC, Vrije Universiteit Amsterdam, Amsterdam, the Netherlands

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† Department of Surgery and Perioperative Care, Dell Medical School at the University of Texas at Austin, Austin, Texas.

Related Article, see p 110

Linear regression is used to quantify the relationship between ≥1 independent (predictor) variables and a continuous dependent (outcome) variable.

In this issue of Anesthesia & Analgesia , Müller-Wirtz et al 1 report results of a study in which they used linear regression to assess the relationship in a rat model between tissue propofol concentrations and exhaled propofol concentrations (Figure.

Table 2 given in Müller-Wirtz et al, 1 showing the estimated relationships between tissue (or plasma) propofol concentrations and exhaled propofol concentrations. The authors appropriately report the 95% confidence intervals as a measure of the precision of their estimates, as well as the coefficient of determination ( R 2 ). The presented values indicate, for example, that (1) the exhaled propofol concentrations are estimated to increase on average by 4.6 units, equal to the slope (regression) coefficient, for each 1-unit increase of plasma propofol concentration; (2) the “true” mean increase could plausibly be expected to lie anywhere between 3.6 and 5.7 units as indicated by the slope coefficient’s confidence interval; and (3) the R 2 suggests that about 71% of the variability in the exhaled concentration can be explained by its relationship with plasma propofol concentrations.

Linear regression is used to estimate the association of ≥1 independent (predictor) variables with a continuous dependent (outcome) variable. 2 In the most simple case, thus referred to as “simple linear regression,” there is only one independent variable. Simple linear regression fits a straight line to the data points that best characterizes the relationship

between the dependent ( Y ) variable and the independent ( X ) variable, with the y -axis intercept ( b 0 ), and the regression coefficient being the slope ( b 1 ) of this line:

A model that includes several independent variables is referred to as “multiple linear regression” or “multivariable linear regression.” Even though the term linear regression suggests otherwise, it can also be used to model curved relationships.

Linear regression is an extremely versatile technique that can be used to address a variety of research questions and study aims. Researchers may want to test whether there is evidence for a relationship between a categorical (grouping) variable (eg, treatment group or patient sex) and a quantitative outcome (eg, blood pressure). The 2-sample t test and analysis of variance, 3 which are commonly used for this purpose, are essentially special cases of linear regression. However, linear regression is more flexible, allowing for >1 independent variable and allowing for continuous independent variables. Moreover, when there is >1 independent variable, researchers can also test for the interaction of variables—in other words, whether the effect of 1 independent variable depends on the value or level of another independent variable.

Linear regression not only tests for relationships but also quantifies their direction and strength. The regression coefficient describes the average (expected) change in the dependent variable for each 1-unit change in the independent variable for continuous independent variables or the expected difference versus a reference category for categorical independent variables. The coefficient of determination, commonly referred to as R 2 , describes the proportion of the variability in the outcome variable that can be explained by the independent variables. With simple linear regression, the coefficient of determination is also equal to the square of the Pearson correlation between the x and y values.

When including several independent variables, the regression model estimates the effect of each independent variable while holding the values of all other independent variables constant. 4 Thus, linear regression is useful (1) to distinguish the effects of different variables on the outcome and (2) to control for other variables—like systematic confounding in observational studies or baseline imbalances due to chance in a randomized controlled trial. Ultimately, linear regression can be used to predict the value of the dependent outcome variable based on the value(s) of the independent predictor variable(s).

Valid inferences from linear regression rely on its assumptions being met, including

• the residuals are the differences between the observed values and the values predicted by the regression model, and the residuals must be approximately normally distributed and have approximately the same variance over the range of predicted values;
• the residuals are also assumed to be uncorrelated. In simple language, the observations must be independent of each other; for example, there must not be repeated measurements within the same subjects. Other techniques like linear mixed-effects models are required for correlated data 5 ; and
• the model must be correctly specified, as explained in more detail in the next paragraph.

Whereas Müller-Wirtz et al 1 used simple linear regression to address their research question, researchers often need to specify a multivariable model and make choices on which independent variables to include and on how to model the functional relationship between variables (eg, straight line versus curve; inclusion of interaction terms).

Variable selection is a much-debated topic, and the details are beyond the scope of this Statistical Minute. Basically, variable selection depends on whether the purpose of the model is to understand the relationship between variables or to make predictions. This is also predicated on whether there is informed a priori theory to guide variable selection and on whether the model needs to control for variables that are not of primary interest but are confounders that could distort the relationship between other variables.

Omitting important variables or interactions can lead to biased estimates and a model that poorly describes the true underlying relationships, whereas including too many variables leads to modeling the noise (sampling error) in the data and reduces the precision of the estimates. Various statistics and plots, including adjusted R 2 , Mallows C p , and residual plots are available to assess the goodness of fit of the chosen linear regression model.

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Linear Regression is a method for modelling a relationship between a dependent variable and independent variables. These models can be fit with numerous approaches. The most common is least squares , where we minimize the mean square error between the predicted values $\hat{y} = \textbf{X}\hat{\beta}$ and actual values $y$: $\left(y-\textbf{X}\beta\right)^{2}$.

We can also define the problem in probabilistic terms as a generalized linear model (GLM) where the pdf is a Gaussian distribution, and then perform maximum likelihood estimation to estimate $\hat{\beta}$.

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This research was conducted to see the effect of service quality, promotion and price on customer satisfaction. This research was conducted at the Besuki branch of JNE. Sampling was done by random sampling technique where all the population was taken at random to be the research sample. This is done to increase customer satisfaction at JNE Besuki branch through service quality, promotion and price. The analytical tool used is multiple linear regression to determine service quality, promotion and price on customer satisfaction. The results show that service quality affects customer satisfaction, promotion affects customer satisfaction, price affects customer satisfaction. Keyword : service quality, promotion, price, customer satisfaction

PENGARUH KUALITAS PRODUK, HARGA DAN INFLUENCER MARKETING TERHADAP KEPUTUSAN PEMBELIAN SCARLETT BODY WHITENING

This research aimed to figure out the influence between product quality, price and marketing influencer with the purchasing decision of Scarlett Body Whitening in East Java. The research instrument employed questionnaire to collect data from Scarlett Body Whitening consumers in East Java. Since there was no valid data for number of the consumers, the research used Roscoe method to take the sample. Data analyzed using multiple linear regression test. Product quality and price have a positive and significant effect on purchasing decisions. Meanwhile, the marketing influencer had no significant effect on purchase decision for Scarlett Body Whitening. Need further research to ensure that marketing influencer had an effect on purchase decision.   Keywords: Product quality, price, marketing influencer, buying decision

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