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6 Command file: Specifying the terms in the mixed model 102the !DENSE data line qualifier or mv is included before !r, see Table 5.5. Thespecial term mv is always included in sparse. Thus random and sparse terms areestimated using sparse matrix methods which result in faster processing. Theinverse coefficient matrix is fully formed for the terms in the dense set. Theinverse coefficient matrix is only partially formed for terms in the sparse set.Typically, the sparse set is large and sparse storage results in savings in memoryand computing. A consequence is that the variance matrix for estimates is onlyavailable for equations in the dense portion.Ordering of terms in ASRemlThe order in which estimates for the fixed and random effects in linear mixedmodel are reported will usually differ from the order the model terms are specified.Solutions to the mixed model equations are obtained using the methods outlinedGilmour et al., 1995. ASReml orders the equations in the sparse part to maintainas much sparsity as it can during the solution. After absorbing them, it absorbsthe model terms associated with the dense equations in the order specified.Aliassing and singularitiesA singularity is reported in ASReml when the diagonal element of the mixed modelequations is effectively zero (see the !TOLERANCE qualifier) during absorption. Itindicates there is either• no data for that fixed effect, or• a linear dependence in the design matrix means there is no information left toestimate the effect.ASReml handles singularities by using a generalized inverse in which the singularrow/column is zero and the associated fixed effect is zero. Which equations aresingular depends on the order the equations are processed. This is controlled byASReml for the sparse terms but by the user for the dense terms. They shouldbe specified with main effects before interactions so that the ANOVA table hascorrect marginalization. Since ASReml processes the dense terms from the bottomup, the first level (the last level processed) is often singular.The number of singularities is reported in the .asr file immediately prior tothe REML log-likelihood (LogL) line for that iteration (see Section 13.3). Theeffects (and associated standard or prediction error) which correspond to thesesingularities are zero in the .sln file.