Singularities in Mixed Model

Definition

A singularity is reported in ASReml when the diagonal element of the mixed model equations is effectively zero (see the !TOLERANCE qualifier) during absorption. It indicates there is either

no data for that fixed effect, or

a linear dependence in the design matrix means there is no information left to estimate the effect.

Control

ASReml handles singularities by using a generalized inverse in which the singular row/column is zero and the associated fixed effect is zero. Which equations are singular depends on the order the equations are processed. This is controlled by ASReml for the sparse terms but by the user for the dense terms. They should be specified with main effects before interactions so that the ANOVA table has correct marginalization. Since ASReml processes the dense terms from the bottom up, the first level (the last level processed) is often singular.

Reporting

The number of singularities is reported in the .asr file immediately prior to the REML log-likelihood ( LogL) line for that iteration. The effects (and associated standard or prediction error) which correspond to these singularities are zero in the .sln file.

Singularities in the sparse_fixed terms of the model may change with changes in the random terms included in the model. If this happens it will mean that changes in the REML log-likelihood are not valid for testing the changes made to the random model. This situation is not easily detected as the only evidence will be in the .sln file where different fixed effects are singular. A likelihood ratio test is not valid if the fixed model has changed.

Ordering of terms

The order in which estimates for the fixed and random effects in linear mixed model are reported will usually differ from the order the model terms are specified. Solutions to the mixed model equations are obtained using the methods outlined Gilmour et al., 1995. ASReml orders the equations in the sparse part to maintain as much sparsity as it can during the solution. After absorbing them, it absorbs the model terms associated with the dense equations in the order specified.

Example

A sequence of models is presented to facilitate an understanding of over-parameterised models. It is assumed that var is a factor with 4 levels, trt with 3 levels and rep with 3 levels and that all var.trt combinations are present in the data.

model	number of singularities	order of fitting
yield ~ var !r rep	has no singularities.	rep is absorbed before var.
yield ~ mu var !r rep	has 1 singularity;	rep is absorbed before mu which is absorbed before var;
		first level of var is aliassed and set to zero.
yield ~ var trt !r rep	has 1 singularity;	rep is absorbed before var which is absorbed before trt;
		first level of trt is aliassed and set to zero.
yield ~ mu var trt var.trt !r rep	has 8 singularities;	rep is absorbed first, then mu, then var, then trt and then var.trt;
		first levels of both var and trt are aliassed and set to zero,
		together with subsequent interactions.
yield ~ mu var trt !r rep !f var.trt	has 8 singularities;	rep and var.trt are absorbed first in some order which maintains
		high sparsity, then mu, then var and then trt.
		Consequently there are no singularities in var.trt;
		mu, var and trt, are completely singular and set to zero.
		The order within [ var.trt rep ] is determined internally.