An F-type Multiple Testing Approach for Assessing Randomness of Linear Mixed Models: Scientific Explanation

In linear mixed models the assessing of the signi?cance of all or a subset of the random effects is often of primary interest. Many techniques have been proposed for this purpose but none of them is completely satisfactory. One of the oldest methods for testing randomness is the F-test but it is often overlooked in modern applications due to poor statistical power and non-applicability in some important situations. In this work a two-step procedure is developed for generalizing an F-test and improving its statistical power. In the ?rst step, by comparing two covariance matrices of a least squares statistics, we obtain a “repeatable” F-type test. In the second step, by changing the projected matrix which de?nes the least squares statistic we apply the test repeteadly to the same data in order to have a set of correlated statistics analyzed within a multiple testing approach. The resulting test is suf?ciently general, easy to compute, with an exact distribution under the null and alternative hypothesis and, perhaps more importantly, with a strong increase of statistical power with respect to the F-test. In the light of these results we believe that our two-stage approach based on a combinnation of a ”repeatable” F-type test with a multiple testing approach may suggest a procedure for improving statistical power in linear mixed models.