Influence Diagnostics for Correlated Data

Correlated or clustered data are frequently encountered in medical and dental research. Although there has been a large amount of research on analytic methods for clustered data, there has been only a limited amount of work in the area of influence diagnostics for these methods. We present a simple approach to influence diagnostics for clustered data that is practical even for very large data sets. By applying a method of partitioning the fixed effects in a linear model with clustered data and exchangeable correlation structure (Scott and Holt, 1982), ordinary least-squares estimation can be used to obtain influence diagnostics with balanced data. Using a series of simulations, we evaluate the performance of the influence diagnostics for a partitioned linear model where data are unbalanced and where cluster deletion is performed as opposed to point deletion. When the correlation structure is exchangeable and the data are unbalanced, these influence measures provide a reasonably high degree of sensitivity for identifying influential observations. This technique may offer a less cumbersome approach to influence diagnostics with clustered data. Introduction The study of influence is an important aspect in any type of regression analysis because not all cases in a set of data are equal participants in determining regression coefficient estimates, statistical testing, and other statistics. In fact, the character of a regression model may be determined by only a few observations within a set of data while the majority of the data are largely ignored. A large body of literature related to influence diagnostics for linear models now exists, and there are several measures now available for detecting various types of influential observations, including leverage points and outliers as well as groups of influential observations within a set of data. Case-deletion diagnostics have reached such popularity in standard linear regression analysis that they are included in most statistical software packages. In addition, several books have been written to further disseminate both the theory and applications of methods for detecting influential observations, including books by Cook and Weisberg (1982), Belsley, Kuh, and Welsch (1983), and Chatterjee and Hadi (1987). Only in recent years have these methods been extended to analysis of correlated data. Local influence methods for influential groups of observations first described by Cook (1986) have been extended to mixed analysis of variance models by Beckman, Nachtsheim, and Cook (1987). Case deletion diagnostics were extended to linear mixed models by Christensen, Pearson, and Johnson (1992). Further extension of case deletion diagnostics to generalized estimating equations was carried out by Preisser and Qaqish (1996), where both cluster-deletion and point-deletion diagnostics were investigated. Banerjee and Frees (1997) investigated case-deletion diagnostics for both generalized linear mixed models and marginal models (GEE). In this paper, we present a simple method for casedeletion diagnostics for linear models with clustered data and an exchangeable working correlation structure. By partitioning the fixed effects of a linear model into within-cluster variation and between-cluster variation, we are able to obtain influence diagnostics based on cluster deletion alone that can be used to obtain information about both influential clusters as well as clusters with influential points within those clusters. In addition, when an exchangeable working correlation is appropriate, we show that the use of ordinary least squares estimators can be used to obtain clusterdeletion diagnostics with reasonable accuracy compared to using generalized least squares estimators. Partitioning Fixed Effects in Linear Models Let us consider the general linear model defined by 0 α α = + + g g Y X e where Xg is a matrix of unpartitioned covariates of dimension N × p − 1, Y is a vector of responses, e is a vector of random errors, 0 α is the intercept, α g is the vector of coefficients for the matrix of unpartitioned covariates, and N is the total number of observations. By partitioning the fixed effects of this general linear model into between-cluster effects and within-cluster effects, the partitioned model can be rewritten as 0 α α α = + + + w w b b Y X X e where Xw is a matrix corresponding to the within-cluster variation, Xb is a matrix corresponding to the betweencluster variation, Y is the vector of responses, e is a vector of random errors, α0 is the intercept, αw is the vector of coefficients for within-cluster variation, and Joint Statistical Meetings Biometrics Section-to include ENAR & WNAR