A comparison of three invariant tests of additivity in two-way classifications with no replications

Abstract

Tusell (1990) proposed an innovative test of additivity in non-replicated two-way classifications. Tusell's procedure consists of employing the well-known likelihood ratio (LR) test of sphericity as a test of additivity. Ordinarily, additivity is assumed when testing sphericity. Tusell deduced that if sphericity is assumed, then application of the LR criterion results in a test of additivity. This article gives the limiting distribution of the LR sphericity criterion under nonadditivity. Simulation studies that compare Tusell's test with Johnson and Graybill's (1972) LR test against a rank-1 alternative and with Boik's (1990) locally best invariant test of addivity are reported. The LR sphericity test is the most powerful test (among the three) when the noncentrality matrix has multiple non-zero eigenvalues which are comparable in magnitude. For most other conditions, the locally best invariant test is the most powerful test. In practice, prior information about the structure of the noncentrality matrix is generally not available. For these cases, the locally best invariant test is recommended.