Small-Sample Characterizations of Near Replicate Lack-of-Fit Tests

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Abstract
Several ad hoc tests based on near replicates have been proposed for testing lack of fit in regression analysis. Christensen characterized lack of fit as existing between clusters of near replicates, within clusters, or as a combination of these pure types. Of these, the between-cluster variety is the type commonly associated with the idea of lack of fit. Christensen examined a test that was new to the normal theory regression literature and established uniformly most powerful invariant (UMPI) properties of the test. In particular, the test is UMPI for orthogonal lack of fit within clusters. In this article, a new test is proposed that is UMPI for orthogonal lack of fit between clusters. The relationship of this optimal test to other proposed tests is examined, giving the first small-sample theoretical justification for these tests. The power of the new test is compared to that of others in the literature, and consistency results are discussed.

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