A global test for effects in 2^kfactorial design without replicates

Abstract

In this paper we consider the problem of testing the effects in 2k factorial designs when there are no replicates and an independent estimate of the variance is not available. First we consider the use of the so-called scale-ratio tests for testing the null hypothesis of no significant effects; these tests are similar in spirit to the ‘global’ F-test commonly used in the classical ANOVA setup. The class of Pitman efficient scale-ratio tests is derived and a new Pitman efficient test based on the ratio of two M-estimates of scale is proposed. The new global test is compared via Monte Carlo with those proposed Paulson (1952) and Ferguson (1961) and shown to represent a substantial improvement when the fraction of significant effects is large. Next, we consider the problem of identifying the significant effects. A forward stepwise procedure based on the new test is proposed and shown (via Monte Carlo) to perform fairly well. Unlike other forward procedures, the present one is unaffected by masking because the test statistic is based on robust estimates of location and scale.