Omnibus Test - Omnibus Tests in One Way Analysis of Variance

Omnibus Tests in One Way Analysis of Variance

The F-test in ANOVA is an example of an omnibus test, which tests the overall significance of the model. Significant F test means that among the tested means, at least two of the means are significantly different, but this result doesn't specify exactly what means are different one from the other. Actually, testing means' differences is been made by the quadratic rational F statistic ( F=MSB/MSW). In order to determine which mean differ from another mean or which contrast of means are significantly different, Post Hoc tests (Multiple Comparison tests) or planned tests should be conducted after obtaining a significant omnibus F test. It may be consider using the simple Bonferroni correction or other suitable correction. Another omnibus test we can find in ANOVA is the F test for testing one of the ANOVA assumptions: the equality of variance between groups. In One-Way ANOVA, for example, the hypotheses tested omnibus F test are:

H0: µ₁=µ₂=….= µ_k

H1: µ₁≠µ₂≠….≠ µ_k

These hypotheses examine model fit of the most common model: y_ij = µ_j + ε_ij, where y_ij is the dependant variable, µ_j is the j-th independent variable's expectancy, which usually is referred as "group expectancy" or "factor expectancy" ; and ε_ij are the errors results on using the model.

The F statistics of the omnibus test is: Whereas, \bar y is the overall sample mean, \bar y_j is the group j sample mean, k is the number of groups and n_j is sample size of group j .

The F statistic is distributed F_{(k-1,n-k),(α)} under assuming of null hypothesis and normality assumption. F test is considered robust in some situations, even when the normality assumption isn't met.