Jason W. Osborne and Elaine Waters
North Carolina State University and University of Oklahoma

Most statistical tests rely upon certain assumptions about the variables used in the analysis.  When these assumptions are not met the results may not be trustworthy, resulting in a Type I or Type II error, or over- or under-estimation of significance or effect size(s).  As Pedhazur (1997, p. 33) notes, "Knowledge and understanding of the situations when violations of assumptions lead to serious biases, and when they are of little consequence, are essential to meaningful data analysis".  However, as Osborne, Christensen, and Gunter (2001) observe, few articles report having tested assumptions of the statistical tests they rely on for drawing their conclusions.  This creates a situation where we have a rich literature in education and social science, but we are forced to call into question the validity of many of these results, conclusions, and assertions, as we have no idea whether the assumptions of the statistical tests were met.  Our goal for this paper is to present a discussion of the assumptions of multiple regression tailored toward the practicing researcher.

Several assumptions of multiple regression are “robust” to violation (e.g., normal distribution of errors), and others are fulfilled in the proper design of a study (e.g., independence of observations).  Therefore, we will focus on the assumptions of multiple regression that are not robust to violation, and that researchers can deal with if violated.   Specifically, we will discuss the assumptions of linearity, reliability of measurement, homoscedasticity, and normality. 

VARIABLES ARE NORMALLY DISTRIBUTED. 

Regression assumes that variables have normal distributions.  Non-normally distributed variables (highly skewed or kurtotic variables, or variables with substantial outliers) can distort relationships and significance tests.  There are several pieces of information that are useful to the researcher in testing this assumption:  visual inspection of data plots, skew, kurtosis, and P-P plots give researchers information about normality, and Kolmogorov-Smirnov tests provide inferential statistics on normality.  Outliers can be identified either through visual inspection of histograms or frequency distributions, or by converting data to z-scores. 

Bivariate/multivariate data cleaning can also be important (Tabachnick & Fidell, p 139) in multiple regression.  Most regression or multivariate statistics texts (e.g., Pedhazur, 1997; Tabachnick & Fidell, 2000) discuss the examination of standardized or studentized residuals, or indices of leverage.  Analyses by Osborne (2001) show that removal of univariate and bivariate outliers can reduce the probability of Type I and Type II errors, and improve accuracy of estimates. 

Outlier (univariate or bivariate) removal is straightforward in most statistical software.  However, it is not always desirable to remove outliers.  In this case transformations (e.g., square root, log, or inverse), can improve normality, but complicate the interpretation of the results, and should be used deliberately and in an informed manner.  A full treatment of transformations is beyond the scope of this article, but is discussed in many popular statistical textbooks. 

ASSUMPTION OF A LINEAR RELATIONSHIP BETWEEN THE INDEPENDENT AND DEPENDENT VARIABLE(S).

Standard multiple regression can only accurately estimate the relationship between dependent and independent variables if the relationships are linear in nature.  As there are many instances in the social sciences where non-linear relationships occur (e.g., anxiety), it is essential to examine analyses for non-linearity.  If the relationship between independent variables (IV) and the dependent variable (DV) is not linear, the results of the regression analysis will under-estimate the true relationship.  This under-estimation carries two risks:  increased chance of a Type II error for that IV, and in the case of multiple regression, an increased risk of Type I errors (over-estimation) for other IVs that share variance with that IV.

Authors such as Pedhazur (1997), Cohen and Cohen (1983), and Berry and Feldman (1985) suggest three primary ways to detect non-linearity.  The first method is the use of theory or previous research to inform current analyses.  However, as many prior researchers have probably overlooked the possibility of non-linear relationships, this method is not foolproof.  A preferable method of detection is examination of residual plots (plots of the standardized residuals as a function of standardized predicted values, readily available in most statistical software).  Figure 1 shows scatterplots of residuals that indicate curvilinear and linear relationships.

Figure 1.  Example of curvilinear and linear relationships with standardized residuals by standardized predicted values.
Curvilinear Relationship	Linear Relationship

VARIABLES ARE MEASURED WITHOUT ERROR (RELIABLY)

The nature of our educational and social science research means that many variables we are interested in are also difficult to measure, making measurement error a particular concern.  In simple correlation and regression, unreliable measurement causes relationships to be under-estimated increasing the risk of Type II errors.  In the case of multiple regression or partial correlation, effect sizes of other variables can be over-estimated if the covariate is not reliably measured, as the full effect of the covariate(s) would not be removed.  This is a significant concern if the goal of research is to accurately model the “real” relationships evident in the population.  Although most authors assume that reliability estimates (Cronbach alphas) of .7-.8 are acceptable (e.g., Nunnally, 1978) and Osborne, Christensen, and Gunter (2001) reported that the average alpha reported in top Educational Psychology journals was .83, measurement of this quality still contains enough measurement error to make correction worthwhile, as illustrated below.

Correction for low reliability is simple, and widely disseminated in most texts on regression, but rarely seen in the literature.  We argue that authors should correct for low reliability to obtain a more accurate picture of the “true” relationship in the population, and, in the case of multiple regression or partial correlation, to avoid over-estimating the effect of another variable.

Reliability and simple regression

Since “the presence of measurement errors in behavioral research is the rule rather than the exception” and “reliabilities of many measures used in the behavioral sciences are, at best, moderate” (Pedhazur, 1997, p. 172); it is important that researchers be aware of accepted methods of dealing with this issue.  For simple regression, Equation #1 provides an estimate of the “true” relationship between the IV and DV in the population:

(1)

In this equation, r₁₂ is the observed correlation, and r₁₁ and r₂₂ are the reliability estimates of the variables.  Table 1 and Figure 2 presents examples of the results of such a correction. 

Figure 2: Change in variance accounted for as correlations are corrected for low reliability

 As Table 1 illustrates, even in cases where reliability is .80, correction for attenuation substantially changes the effect size (increasing variance accounted for by about 50%).  When reliability drops to .70 or below this correction yields a substantially different picture of the “true” nature of the relationship, and potentially avoids a Type II error.

Reliability and Multiple Regression

With each independent variable added to the regression equation, the effects of less than perfect reliability on the strength of the relationship becomes more complex and the results of the analysis more questionable.  With the addition of one independent variable with less than perfect reliability each succeeding variable entered has the opportunity to claim part of the error variance left over by the unreliable variable(s).  The apportionment of the explained variance among the independent variables will thus be incorrect.  The more independent variables added to the equation with low levels of reliability the greater the likelihood that the variance accounted for is not apportioned correctly.  This can lead to erroneous findings and increased potential for Type II errors for the variables with poor reliability, and Type I errors for the other variables in the equation.  Obviously, this gets increasingly complex as the number of variables in the equation grows. 

A simple example, drawing heavily from Pedhazur (1997), is a case where one is attempting to assess the relationship between two variables controlling for a third variable (r_12.3).  When one is correcting for low reliability in all three variables Equation #2 is used:

(3)

Table 2 shows some of the many possible combinations of reliabilities, correlations, and the effects of correcting for only the covariate or all variables.  Some points of interest:  (a) as in Table 1, even small correlations see substantial effect size (r²) changes when corrected for low reliability, in this case often toward reduced effect sizes (b) in some cases the corrected correlation is not only substantially different in magnitude, but also in direction of the relationship, and (c) as expected, the most dramatic changes occur when the covariate has a substantial relationship with the other variables. 

ASSUMPTION OF HOMOSCEDASTICITY 

Homoscedasticity means that the variance of errors is the same across all levels of the IV.  When the variance of errors differs at different values of the IV, heteroscedasticity is indicated.  According to Berry and Feldman (1985) and Tabachnick and Fidell (1996) slight heteroscedasticity has little effect on significance tests; however, when heteroscedasticity is marked it can lead to serious distortion of findings and seriously weaken the analysis thus increasing the possibility of a Type I error.

This assumption can be checked by visual examination of a plot of the standardized residuals (the errors) by the regression standardized predicted value.  Most modern statistical packages include this as an option.  Figure 3 show examples of plots that might result from homoscedastic and heteroscedastic data.

Figure 3. Examples of homoscedasticity and heteroscedasticity

Ideally, residuals are randomly scattered around 0 (the horizontal line) providing a relatively even distribution.  Heteroscedasticity is indicated when the residuals are not evenly scattered around the line.  There are many forms heteroscedasticity can take, such as a bow-tie or fan shape.  When the plot of residuals appears to deviate substantially from normal, more formal tests for heteroscedasticity should be performed.  Possible tests for this are the Goldfeld-Quandt test when the error term either decreases or increases consistently as the value of the DV increases as shown in the fan shaped plot or the Glejser tests for heteroscedasticity when the error term has small variances at central observations and larger variance at the extremes of the observations as in the bowtie shaped plot (Berry & Feldman, 1985).  In cases where skew is present in the IVs, transformation of variables can reduce the heteroscedasticity. 

CONCLUSION

The goal of this article was to raise awareness of the importance of checking assumptions in simple and multiple regression.  We focused on four assumptions that were not highly robust to violations, or easily dealt with through design of the study, that researchers could easily check and deal with, and that, in our opinion, appear to carry substantial benefits. 

We believe that checking these assumptions carry significant benefits for the researcher. Making sure an analysis meets the associated assumptions helps avoid Type I and II errors.  Attending to issues such as attenuation due to low reliability, curvilinearity, and non-normality often boosts effect sizes, usually a desirable outcome.

Finally, there are many non-parametric statistical techniques available to researchers when the assumptions of a parametric statistical technique is not met.  Although these often are somewhat lower in power than parametric techniques, they provide valuable alternatives, and researchers should be familiar with them.

References 

Berry, W. D., & Feldman, S. (1985).  Multiple Regression in Practice.  Sage University Paper Series on Quantitative Applications in the Social Sciences, series no. 07-050). Newbury Park, CA:  Sage.

Cohen, J., & Cohen, P. (1983).  Applied multiple regression/correlation analysis for the behavioral sciences.  Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Nunnally, J. C. (1978).  Psychometric Theory (2^nd ed.).  New York: McGraw Hill.

Osborne, J. W., Christensen, W. R., & Gunter, J. (April, 2001).  Educational Psychology from a Statistician’s Perspective:  A Review of the Power and Goodness of Educational Psychology Research.  Paper presented at the national meeting of the American Education Research Association (AERA), Seattle, WA.

Osborne, J. W. (2001). A new look at outliers and fringeliers: Their effects on statistic accuracy and Type I and Type II error rates. Unpublished manuscript, Department of Educational Research and Leadership and Counselor Education, North Carolina State University.

Pedhazur, E. J., (1997). Multiple Regression in Behavioral Research (3^rd ed.). Orlando, FL:Harcourt Brace.

Tabachnick, B. G., Fidell, L. S. (1996).  Using Multivariate Statistics (3rd ed.).  New York: Harper Collins College Publishers

Tabachnick, B. G., Fidell, L. S. (2001).  Using Multivariate Statistics (4th ed.).  Needham Heights, MA:  Allyn and Bacon

Contact Information:

Jason W. Osborne, Ph.D
ERLCE, Campus Box 7801
Poe Hall 608,
North Carolina State University
Raleigh NC 27695-7801

(919) 515-1714

Jason_Osborne@ncsu.edu