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

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.

Plane Answers to Complex Questions

“…Tests for lack of fit using near replicates are reviewed by Neill and Johnson (1984). The theory behind such tests is beautifully explained in Christensen (1989Christensen ( , 1991. (OK, so I'm biased in favor of this particular author.)…”

“…Sherfey (1998, 1999) provide a theoretical basis for chosing near replicate clusters. Christensen (1991) suggests that a very good all-purpose near replicate lack of fit test was introduced by Shillington (1979). Shillington's test involves writing the regression model in terms of c clusters of near replicates with the ith cluster containing Ni cases, say .. , c, j = 1, ... , N i . Note that at this point we have done nothing to the model except play with the subscripts; model (1) is just the original model.…”

“…Christensen (1989Christensen ( , 1991 gives details and explains why this should be a good all-purpose test -even though it is not the optimal test for either of the alternatives developed by Christensen. The near replicate lack of fit test proposed in Christensen (1989) is the test of model (1) against model (4). The test proposed in Christensen (1991) uses the same numerator as Shillington's test, but a denominator sum of squares that is the SSE from model (1) minus the numerator sum of squares from Shillington's test. Both of Christensen's tests are optimal for certain types of lack of fit.…”

“…Christensen (1991) shows that the tests can be quite different and that for the single most interesting type of lack of fit, the 91 test will typically be more powerful than Shillington's test, which is typically better than the 89 test.…”

Lack-of-fit Testing for Polynomial Regression Models Without Replications

“…Christensen (2003) noted that a small value of the F-statistic does not indicate a lack of fit in the proposed model, but in the combination of these two pure types, and this can be extremely difficult to detect. Miller et al (1998Miller et al ( , 1999, Miller and Neil (2007) presented a lack-of-fit test based on families of groupings of the observation, using the test given by Christensen (1989Christensen ( , 1991, and those given by Khuri (1985) and Levy and Neill (1990).…”