Application of ‘Delete = Replace’ to Deletion Diagnostics for Variance Component Estimation in the Linear Mixed Model

Abstract

Summary. ‘Delete = replace’ is a powerful and intuitive modelling identity. This paper extends previous work by stating and proving the identity in more general terms and extending its application to deletion diagnostics for estimates of variance components obtained by restricted maximum likelihood estimation for the linear mixed model. We present a new, fast, transparent and approximate computational procedure, arising as a by‐product of the fitting process. We illustrate the effect of the deletion of individual observations, of ‘subjects’ and of arbitrary subsets. Central to the identity and its application is the conditional residual.

Case deletion diagnostics in multilevel models

“…, r , are random, independent and zero mean vectors with cov(γ j ) = σ 2 j I q j , Z j are N × q j design matrices for the random effects, and e is a random error vector (independent of γ j ) with cov(e) = σ 2 0 I N . Christensen et al [6] and Haslett and Dillane [13] studied the deletion diagnostic for restricted maximum likelihood estimation (REMLE) of variance components in model (6.1). We can apply the results of Section 3 to study the case deletion diagnostic for the MLE of the variance components because MLE is equivalent to IGLS under the normality assumption.…”

“…Haslett [12] suggested a simpler case-deletion measure using marginal and conditional residuals. Hodges [14] studied case influence in hierarchical models based on a reformulation of the model and an approximate case deletion formula, and Haslett and Dillane [13] developed a case deletion diagnostic for estimates of the variance components in linear mixed models. We note however that no influence diagnostic measures have been studied for multilevel models.…”

Influence diagnostics and outlier tests for varying coefficient mixed models

“…For example, [7,8] discussed LMs; [9,10] considered linear mixed models (LMMs). Haslett and Dillane [11] proved a 'delete = replace' identity in LMs and applied it to deletion diagnostics for estimators of variance components. Xiang, Tse and Lee [12] investigated generalized linear mixed models.…”

Local influence in multilevel models

“…A simpler casedeletion measure based on marginal and conditional residuals was suggested by Haslett (1999). Haslett & Dillane (2004) developed a deletion diagnostic for estimates of variance components obtained by the restricted maximum likelihood method for linear mixed models.…”