Simultaneous Estimation of Parameter Sets in Generalized Linear Models
Epidemiological and medical research routinely employ generalized linear models (GLM) to quantify the relationships between the incidence of disease, such as cervical cancer, and particular risk factors, such as infection with a particular type of the HPV virus. After model estimation, interest usually focuses on estimation of odds ratios or relative risks, both functions of the model parameters. Often simultaneous estimation of these quantities, or particular subsets of these quantities, are warranted. For example, if the incidence of cervical cancer is of interest, then researchers may wish to order the odds ratios for cervical cancer with regard to infection of particular types of the HPV virus, a process requiring simultaneous estimation. Current simultaneous estimation methods only perform well when there are a very small number of comparisons and/or the sample size is relatively large. Additionally, the estimated quantities can have significant bias even when the sample size is large. We propose simultaneous bounds that: (1) perform well for a small or large number of comparisons, (2) exhibit improved performance for small to moderate sample sizes, (3) provide bias adjustment not reliant on asymptotics, and (4) avoids the infinite parameter estimates that can occur with maximum likelihood estimators. These bounds adapt tube-formula based simultaneous confidence regions (SCRs) for use with the pMLE. Simulations demonstrate that the proposed bounds achieve the desired level of confidence at smaller sample sizes than previous methods.