ASReml-S reference manual - VSN International

2.6.2 Incremental and Conditional Wald Statistics2.6 Inference for fixed effects 17The basic tool for inference is the Wald statistic defined in equation 14.1. However, thereare several ways L can be defined to construct a test for a particular model term, two ofwhich are available in asreml. An F-statistic is obtained by dividing the Wald statistic byr, the numerator degrees of freedom. In this form it is possible to perform an approximateF test if we can deduce the denominator degrees of freedom. For balanced designs, theseWald F statistics are numerically identical to the F-tests obtained from the standardanalysis of variance.The first method for computing Wald statistics (for each term) is the incremental form.For this method, Wald statistics are computed from an incremental sum of squares inthe spirit of the approach used in classical regression analysis [see Searle, 1971]. Forexample, if we consider a very simple model with terms relating to the main effects oftwo qualitative factors A and B, given symbolically byy ∼ 1 + A + Bwhere 1 represents the constant term (µ), then the incremental sums of squares for thismodel can be written as the sequenceR(1)R(A|1) = R(1, A) − R(1)R(B|1, A) = R(1, A, B) − R(1, A)where the R(·) operator denotes the reduction in the total sums of squares due to a modelcontaining its argument and R(·|·) denotes the difference between the reduction in thesums of squares for any pair of (nested) models. Thus R(B|1, A) represents the differencebetween the reduction in sums of squares between the maximal modely ∼ 1 + A + BandImplicit in these calculations is thaty ∼ 1 + A• we only compute Wald statistics for estimable functions [Searle, 1971, p 408]• all variance parameters are held fixed at the current REML estimates from the maximalmodelIn this example, it is clear that the incremental Wald statistics may not produce thedesired test for the main effect of A, as in many cases we would like to produce a Waldstatistic for A based onR(A|1, B) = R(1, A, B) − R(1, B)The issue is further complicated when we invoke marginality considerations. The issue ofmarginality between terms in a linear (mixed) model has been discussed in much detailby Nelder [1977]. In this paper Nelder defines marginality for terms in a factorial linearmodel with qualitative factors, but later [Nelder, 1994] extended this concept to functionalmarginality for terms involving quantitative covariates and for mixed terms which involvean interaction between quantitative covariates and qualitative factors. Referring to oursimple illustrative example above, with a full factorial linear model given symbolicallybyy ∼ 1 + A + B + A.B