O'brien Test for Homogeneity of Variance

Motivation and Method

Several tests are available for detecting whether several samples come from populations having the same variances. In the case of two samples, the ratio of the population estimates (computed from the samples) is distributed as a Fisher distribution under the usual assumptions. Unfortunately, there is no straightforward extension of this approach to designs involving more than two samples. By contrast, the O'Brien test is designed to test the homogeneity of variance assumption for several samples at once and with the versatility for ANOVA designs, including contrast analysis and analysis of subdesigns.

The main idea behind the O'Brien test is to transform the original scores so that the transformed scores reflect the variation of the original scores. An ANOVA on the transformed scores will then reveal differences in the variability (i.e., variance) of the original scores, and therefore this analysis will test the homogeneity of variance assumption. A straightforward application of this idea will be to replace the original scores with the absolute value of their deviation to the mean of their experimental group. So, if we denote by Yas the score of participant s in experimental condition a whose mean is denoted by Ma., this first idea amounts to transforming Yas into vas as follows:

This transformation has the advantage of being simple and easy to understand, but unfortunately, it creates some statistical problems (i.e., the F distribution does not model the probability distribution under the null hypothesis), and in particular, it leads to an excess of Type I errors (i.e., we reject the null hypothesis more often than the α level indicates).

A better approach is to replace each score by its absolute distance to the median of its group. Specifically, each score is replaced by

where Mda. = median of Group a. This transformation gives very satisfactory results for an omnibus testing of the homogeneity of variance assumption. However, in order to implement more-sophisticated statistical procedures (e.g., contrast analyses, multiples comparisons), an even better transformation was proposed by O'Brien. Here, the scores are transformed as

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