Why is anova called an omnibus test

It may seem odd that the technique is called "Analysis of Variance" rather than "Analysis of Means. ANOVA is used to test general rather than specific differences among means. This can be seen best by example. In the case study " Smiles and Leniency ," the effect of different types of smiles on the leniency shown to a person was investigated. Four different types of smiles neutral, false, felt, miserable were investigated. Thus, when the omnibus test is significant, but no post-hoc between-group comparison shows significant difference, one is bewildered at what is going on and wondering how to interpret the results.

At the end of the spectrum, when the omnibus test is not significant, one wonders if all post-hoc tests will be non-significant as well so that stopping after a nonsignificant omnibus test will not lead to any missed opportunity of finding group difference. In this report, we investigate this perplexing phenomenon and discuss how to interpret such results. Comparison of groups is a common issue of interest in most biomedical and psychosocial research studies.

In many studies, there are more than two groups, in which case the popular t-test for two independent groups no longer applies and models for comparing more than two groups must be used, such as the analysis of variance, ANOVA, model. Under this approach, one first performs an omnibus test, which tests the null hypothesis of no difference across groups, i.

If this test is not significant, there is no evidence in the data to reject the null and one then concludes that there is no evidence to suggest that the group means are different.

Otherwise, post-hoc tests are performed to find sources of difference. During post-hoc analysis, one compares pairs of groups and finds all pairs that show significant difference. This hierarchical procedure is predicated upon the premise that if the omnibus test is significant, there must exist at least two groups that are significantly different and vice versa.

The hierarchical procedure is taught in basic as well as advanced statistics courses and built into many popular statistical packages. For example, when performing the analysis of variance ANOVA model for comparing multiple groups, the omnibus test is carried out by the F-statistic.

In practice, however, it seems quite often that none of the post-hoc tests are significant, while the omnibus test is significant. The reverse seems to occur often as well; when the omnibus test is not significant, although some of the post-hoc tests are significant. To the best of our knowledge, there does not appear a general, commonly accepted approach to handle such a situation.

In this report, we examine this hierarchical approach and see how well it performs using simulated data. We want to know if a significant omnibus test guarantees at least one post-hoc test and vice versa.

Although the statistical problem of comparing multiple groups is relevant to all statistical models, we focus on the relatively simpler analysis of variance ANOVA model and start with a brief overview of this popular model for comparing more than two groups. The analysis of variance ANOVA model is widely used in research studies for comparing multiple groups. This model extends the popular t-test for comparing two independent groups to the general setting of more than two groups.

Consider a continuous outcome of interest, Y , and let I denote the number of groups. We are interested in comparing the population mean of Y across the I groups.

With the statistical model in Equation 1 , the primary objective of group comparison can be stated in terms of statistical hypotheses as follows. First, we want to know if all the groups have the same mean. Under the ANOVA above, the null and alternative hypothesis for this comparison of interest is stated as:.

Thus, under the null H 0 , all groups have the same mean. If H 0 is rejected in favor of the alternative H a , there are at least two groups that have different means. If this omnibus test is not rejected, then one concludes that there is evidence to indicate different means across the groups.

Otherwise, there is evidence against the null in favor of the alternative and one then proceeds to the next step to identify the groups that have different means from each other. Thus, performing all such tests can potentially increase type I errors. The popular t-test for comparing two independent groups is inappropriate and specially designed tests must be used to account for accumulated type I errors due to multiple testing to ensure correct type I errors.

Next we review the omnibus and some post-hoc tests, which will later be used in our simulation studies. To set up this table, let us define the following quantities:. These sums of squares characterize variability of all the groups 1 when ignoring differences in the group means SS Total , and 2 after accounting for such differences SS R. For example, it can be shown that. Thus, if the group means help explain a large amount of variability in group differences, SS R will be close to SS Total , resulting in small SS E , in which case groups means are likely to be different.

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