The generalized linear mixed model (GLMM) is one of many extensions of the usual linear model. In a GLMM, observations are not necessarily independent, but it is assumed that data can be clustered so that observations in different clusters are independent. A random component is then added to account for the dependencies within clusters, and for the “over-dispersion” often found in longitudinal studies.

In this paper, the authors first describe the details of the GLMM model, and then cite several published methods for getting maximum likelihood estimates for the parameters of the linear model, as well as the (non-linear) parameters of the added random component. For the purposes of this paper, they adopt a Gauss-Newton type of iteration.

The influence diagnostics mentioned in the title of this paper describe the influence of individual clusters on the parameter values. The first is a Cook’s distance measure of the effect of deleting a single cluster, and the second addresses the effect of pair-wise deletions. This second measure is used to detect cases where a pair of clusters has greater joint influence than the sum of their individual influences. The second measure can also be used to detect cases where one cluster masks the influence of another.

To avoid repetition of the iterative method for each single and pairwise deletion, the authors develop a first-order approximation to a generalized Cook’s distance measure. This approximation requires about the same amount of computation as a single iteration of the full solution method. The authors illustrate the method with data from a published study on the length of hospital stays for patients at various hospitals in Western Australia. The length of stay is assumed to be independent between hospitals, but there are certain correlations within each hospital. Using this data, the authors present examples of influential clusters, for example hospitals, as well as of masking clusters.