Pearson residual and efficiency of parameter estimates in generalized linear model

Abstract

We demonstrate that the efficiency of regression parameter estimates in the generalized linear model can be expressed as a function of Pearson residuals and likelihood based information. The relationship provides an easy way to derive sandwich variance estimators on β ^ for a specific distribution within the exponential family. In generalized linear models, the correlation between Pearson residual and Fisher information can be used to predict the error ratio of quasi-likelihood variance versus sandwich variance when the sample size is sufficiently large. The derived theory can help to determine which conventional approach to use in the generalized linear model for certain types of data analysis, such as analyzing heteroscedastic data in linear regression; or to analyze over-dispersed data for single parameter families of distributions. The results from re-analysis of a clinical trial data set are used to illustrate issues explored in the paper.