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2 Some theory 22the terms A, B, C, A.B, A.C and B.C are marginal to A.B.C. Nelder (1977, 1994)argues that meaningful and interesting tests for terms in such models can onlybe conducted for those tests which respect marginality relations. This philosophyunderpins the following description of the second Wald statistic availablein ASReml, the so-called “conditional” Wald statistic. This method is invokedby placing !FCON on the datafile line. ASReml attempts to construct conditionalWald statistics for each term in the fixed dense linear model so that marginalityrelations are respected. As a simple example, for the three way factorial modelthe conditional Wald statistics would be computed asTerm Sums of Squares M code1 R(1) .A R(A | 1,B,C,B.C) = R(1,A,B,C,B.C) - R(1,B,C,B.C) AB R(B | 1,A,C,A.C) = R(1,A,B,C,A.C) - R(1,A,C,A.C) AC R(C | 1,A,B,A.B) = R(1,A,B,C,A.B) - R(1,A,B,A.B) AA.B R(A.B | 1,A,B,C,A.C,B.C) = R(1,A,B,C,A.B,A.C,B.C) - R(1,A,B,C,A.C,B.C) BA.C R(A.C | 1,A,B,C,A.B,B.C) = R(1,A,B,C,A.B,A.C,B.C) - R(1,A,B,C,A.B,B.C) BB.C R(B.C | 1,A,B,C,A.B,A.C) = R(1,A,B,C,A.B,A.C,B.C) - R(1,A,B,C,A.B,A.C) BA.B.C R(A.B.C | 1,A,B,C,A.B,A.C,B.C) = R(1,A,B,C,A.B,A.C,B.C,A.B.C) -R(1,A,B,C,A.B,A.C,B.C)COf these the conditional Wald statistic for the 1, B.C and A.B.C terms would bethe same as the incremental Wald statistics produced using the linear modely ∼ 1 + A + B + C + A.B + A.C + B.C + A.B.CThe preceeding table includes a so-called M (marginality) code reported by ASRemlwhen conditional Wald statistics are presented. All terms with the highest M codeletter are tested conditionally on all other terms in the model, i.e. by droppingthe term from the maximum model. All terms with the preceeding M code letter,are marginal to at least one term in a higher group, and so forth. For example,in the table, model term A.B has M code B because it is marginal to model termA.B.C and model term A has M code A because it is marginal to A.B, A.C andA.B.C. Model term mu (M code .) is a special case in that its test is conditionalon all covariates but no factors. Following is some ASReml output from the .aovtable which reports the terms in the conditional statistics.Marginality pattern for F-con calculation-- Model terms --Model Term DF 1 2 3 4 5 6 7 81 mu 1 * . . . . . . .2 water 1 I * C C . . c .3 variety 7 I I * C . c . .4 sow 2 I I I * C . . .5 water.variety 7 I I I I * C C .