Homogeneity of Variance

Comparing Distributions Among Groups

Consider a simple comparison between two groups (male and female) on some dependent variable like time spent a week online using Facebook. Each group generates data used to estimate a mean and standard deviation for the separate groups. The test of the difference between groups, using an independent samples t-test, creates a numerator that is the difference between the means of the two groups (M1 – M2). The denominator contains a term that is the sample weighted average of the two standard deviations. An assumption of this test, similar to that of any analysis of variance (ANOVA) or the other family of associated statistics (e.g., ANCOVA, MANOVA, MANCOVA, multiple regression), involves the same set of assumptions that the comparison of group variances/standard deviations are not dissimilar.

Often that assumption remains untested or unreported when considered. For example, SPSS employs Levene’s test for the homogeneity of distribution that uses the mean as the measure of centrality. Another test, the Brown–Forscythe, uses essentially the same formula but employs the median instead of the mean for the particular distribution. While statistician R. A. Fisher developed an F-test for homogeneity of variance, the test is extremely sensitive to any departure from nonnormality, creating a very high standard for data sets to meet in achieving homogeneity.

The impact of lack of homogeneity may not involve a serious risk of generating an incorrect conclusion. The F-test is extremely robust to the violation of equality of variances, particularly if the distribution of cases is relatively balanced and the probability level for significance is not extreme [Page 667](e.g., p < .05). The F-test begins to lose fidelity as the distribution of observations in the cells becomes more and more extreme or smaller and smaller values for p are selected, such as p < .005 or p < .001. Essentially, the more the extreme set of assumptions or tests, the greater the need to ensure homogeneity when comparing the cell distributions.

The impact of violation would require consideration of the investigator employing a nondistributional or nonparametric test for the comparison of groups such as Kruskal–Wallis. The nonparametric tests may compare favorably in terms of power, reducing the probability of Type II or false negative error as compared to the use of a parametric test such as F or t. Similarly, an investigator may consider choosing to transform the underlying data to logs or some other distribution. However, the impact of such transformations may not improve the ability to conduct a significance test because the impact on the level of power and Type I error may even be more pronounced after such manipulation.

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