Simultaneous inference in generalized linear model settings

Abstract

Generalized Linear Models (GLM's) are utilized in a variety of statistical applications. Many times the estimated quantities from the models are of primary interest. These estimated quantities may include the mean response, odds ratio, relative risk, or attributable proportion. In these cases overall conclusions about these quantities may be desirable. Currently few sophisticated methods exist to simultaneously estimate these quantities from a GLM. I propose several methods of estimating these quantities simultaneously and compare them to the existing methods. Intervals for the expected response of the GLM and any set of linear combinations of the GLM are explored. Most existing methods emphasis the simultaneous estimation of the expected response; few consider estimation of the sets of regression parameters, and hence quantities such as the odds ratio or relative risk. Additionally, almost all intervals employ maximum likelihood estimators (MLEs) for the model parameters. MLEs are often biased estimators for GLMs, particularly at small sample sizes. Thus, another set of intervals is proposed that utilize an alternative estimator for the parameters, the penalized maximum likelihood estimator (pMLE). This estimator is very similar to the usual MLE, but it is shifted in order to account for the bias typically present in the MLE for GLMs. Various critical values of the simultaneous intervals are explored for both the MLE and pMLE based intervals. Emphasis is placed on scenarios where the sample size is small relative to the number of parameters being estimated. Simulation studies compare the various intervals and suggest general recommendations. The pMLE based intervals proposed exhibit superior performance, particularly at small and moderate sample sizes. While usual MLE based intervals typically do not attain the desired level of confidence at the small sample sizes, the pMLE based intervals do. Additionally, at moderate to large sample sizes the pMLE based intervals are, in many cases, less conservative than the usual MLE based intervals.