Homoscedasticity

Homoscedasticity as a Statistical Assumption

Homoscedasticity is one of three major assumptions underlying parametric statistical analyses. In univariate analyses, such as the analysis of variance (ANOVA), with one quantitative dependent variable (Y) and one or more categorical independent variables (X), the homoscedasticity assumption is known as homogeneity of variance. In this context, it is assumed that equal variances of the dependent variable exist across levels of the independent variables.

In multivariate analyses, homoscedasticity means all pairwise combinations of variables (X and Y) are normally distributed. In regression contexts, homoscedasticity refers to constant variance of the residuals (i.e., the difference between the actual and the predicted value of a data point), or conditional variance, regardless of changes in X:

Heteroscedasticity

Violation of the homoscedasticity assumption results in heteroscedasticity when values of the dependent variable seem to increase or decrease as a function of the independent variables. Typically, homoscedasticity violations occur when one or more of the variables under investigation are not normally distributed. Sometimes heteroscedasticity might occur from a few discrepant values (atypical data points) that might reflect actual extreme observations or recording or measurement error.

Scholars and statisticians have different views on the implications of heteroscedasticity in parametric analyses. Some have argued that heteroscedasticity in ANOVA might not be problematic if there are equal numbers of observations across all cells. More recent research contradicts this view and argues that, even in designs with relatively equal cell sizes, heteroscedasticity increases the Type I error rate (i.e., error of rejecting a correct null hypothesis). Still others have persuasively argued that heteroscedasticity might be substantively interesting to some researchers.

Regardless, homoscedasticity violations result in biased statistical results and inaccurate inferences about the population. Therefore, before conducting parametric analyses, it is critical to evaluate and address normality violations and examine data for outlying observations. Detection of homoscedasticity violations in multivariate analyses is often made post hoc, that is, by examining the variation of residuals values.

Exploratory Data Analysis