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2 Some theory 21squares for this model 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 dueto a model containing its argument and R(·|·) denotes the difference between thereduction in the sums of squares for any pair of (nested) models. Thus R(B|1, A)represents the difference between the reduction in sums of squares between theso-called maximal “model”y ∼ 1 + A + BandImplicit in these calculations is thaty ∼ 1 + A• we only compute Wald statistics for estimable functions (Searle, 1971, page408),• all variance parameters are held fixed at the current REML estimates from themaximal modelIn this example, it is clear that the incremental Wald statistics may not producethe desired test for the main effect of A, as in many cases we would like to producea Wald statistic 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 of marginality between terms in a linear (mixed) model has been discussedin much detail by Nelder (1977). In this paper Nelder defines marginalityfor terms in a factorial linear model with qualitative factors, but later Nelder(1994) extended this concept to functional marginality for terms involving quantitativecovariates and for mixed terms which involve an interaction betweenquantitative covariates and qualitative factors. Referring to our simple illustrativeexample above, with a full factorial linear model given symbolically byy ∼ 1 + A + B + A.Bthen A and B are said to be marginal to A.B, and 1 is marginal to A and B. In athree way factorial model given byy ∼ 1 + A + B + C + A.B + A.C + B.C + A.B.C