Assumptions Underlying Multiple Regression

• Level of measurement. For multiple regression, both the predictor variables and the criterion variable should be assessed at the interval or ratio level of measurement. Nominal-level predictor variables can be used if they have been appropriately dummy- or effect-coded.

• Random sampling. Each participant in the sample contributes one score on each predictor variable, and one score on the criterion variable. These sets of scores should represent a random sample drawn from the population of interest.

• Normal distribution of the criterion variable. For any combination of values on the predictor variables, the criterion variable should be normally distributed.

• Homogeneity of variance. For any combination of values of the predictor variables, the criterion variable should demonstrate a constant variance.

• Independent observations. A given observation should not be affected by (or related to) any other observation in the sample. For example, this assumption would be violated if the various observations represented repeated measurements taken from a single participant. It would also be violated if the study included multiple participants, some of whom contributed more that one observation to the dataset (i.e., some participants contributed more than one set of scores on the criterion variable and predictor variables).

• Linearity. The relationship between the criterion variable and each predictor variable should be linear. This means that the mean criterion scores at each value of a given predictor should fall on a straight line.

• Errors of prediction. Errors of prediction should be normally distributed and the distribution of errors should be centered at zero. Error of prediction associated with a given observation should not be correlated with those associated with the other observations. Errors of prediction should demonstrate consistent variance. Errors of prediction should not be correlated with the predictor variables.

• Absence of measurement error. The predictor variables should be measured without error. Pronounced violations of this assumption lead to underestimation of the regression coefficient for the corresponding predictor.

• Absence of specification errors. The term specification error generally refers to situations in which the model represented by the regression equation is not theoretically tenable. In multiple regression, specification errors most frequently result from omitting relevant predictor variables from the equation or including irrelevant predictor variables in the equation. Specification errors also result when researchers posit a linear relationship among variables that are actually curvilinear.

It is infrequent that all of these assumptions will be fully satisfied in applied research. Fortunately, regression analysis is generally robust against minor violations of most of these assumptions. However, it is less robust against violations of the assumptions involving independent observations, measurement error, or specification errors (Pedhazur, 1982).

In addition to considering the preceding assumptions, researchers are also advised to inspect their data for possible problems involving outliers or multicollinearity. An outlier is an unusual observation that does not fit the regression model well. Outliers are often the result of mistakes made when entering data, and can profoundly bias parameter estimates (such as regression coefficients). As previously noted, multicollinearity exists when two or more predictor variables demonstrate a high degree of correlation with one another (i.e., r > |.90|). Multicollinearity can cause regression coefficient estimates to fail to demonstrate statistical significance, be biased, or even demonstrate the incorrect sign. Cohen, Cohen, West, and Aiken (2003) discuss these problems, and show how to detect outliers, multicollinearity, and other problems sometimes encountered in regression analysis.