A note on covariance decomposition in linear models with nested-error structure: new and alternative derivations of the F-test

This article aims at utilizing unexploited decompositions of the covariance matrix of the onefold and twofold nested error regression models to derive F-tests for the fixed effects as well as the variance components. Under each model, the decomposition yields symmetric idempotent matrices that are mutually orthogonal. Transforming the response vector of the working model using such matrices permits new derivations of the classical F-test for zero variance components. Importantly, new exact tests are derived as convenient alternatives to the invalid least squares F-test for linear hypothesis on the fixed effects in both models.