A general framework for constraint approaches to adjusted risk differences.

The risk difference is an intelligible measure for comparing disease incidence in two exposure or treatment groups. Despite its convenience in interpretation, it is less prevalent in epidemiological and clinical areas where regression models are required in order to adjust for confounding. One major barrier to its popularity is that standard linear binomial or Poisson regression models can provide estimated probabilities out of the range of (0,1), resulting in possible convergence issues. For estimating adjusted risk differences, we propose a general framework covering various constraint approaches based on binomial and Poisson regression models. The proposed methods span the areas of ordinary least squares, maximum likelihood estimation, and Bayesian inference. Compared to existing approaches, our methods prevent estimates and confidence intervals of predicted probabilities from falling out of the valid range. Through extensive simulation studies, we demonstrate that the proposed methods solve the issue of having estimates or confidence limits of predicted probabilities out of (0,1), while offering performance comparable to its alternative in terms of the bias, variability, and coverage rates in point and interval estimation of the risk difference. An application study is performed using data from the Prospective Registry Evaluating Myocardial Infarction: Event and Recovery (PREMIER) study.

Constraint approaches to the estimation of relative risk.

In medical and epidemiologic studies, relative risk is usually the parameter of interest. However, calculating relative risk using standard log-Binomial regression approach often encounters non-convergence. A modified Poisson regression, which uses robust variance, was proposed by Zou in 2004. Although the modified Poisson regression with sandwich variance estimator is valid for the estimation of relative risk, the predicted probability of the outcome may be greater than the natural boundary 1 for the unobserved but plausible covariate combinations. Moreover, the lower and upper bounds of confidence intervals for predicted probabilities could fall out of (0, 1). Chu and Cole, in 2010, proposed a Bayesian approach to overcome this issue. Posterior median was used to get the parameter estimation. However, the Bayesian approach may provide biased estimation, especially when the probability of outcome is high. In this article, we propose an alternative constraint optimization approach for estimating relative risk. Our approach can reach similar or better performance than Bayesian approach in terms of bias, root mean square error, coverage rate, and predictive probabilities. Simulation studies are conducted to demonstrate the usefulness of this approach. Our method is also illustrated by Prospective Registry Evaluating Myocardial Infarction: Event and Recovery data.

Bayesian penalized log-likelihood ratio approach for dose response clinical trial studies.

In literature, there are a few unified approaches to test proof of concept and estimate a target dose, including the multiple comparison procedure using modeling approach, and the permutation approach proposed by Klingenberg. We discuss and compare the operating characteristics of these unified approaches and further develop an alternative approach in a Bayesian framework based on the posterior distribution of a penalized log-likelihood ratio test statistic. Our Bayesian approach is much more flexible to handle linear or nonlinear dose-response relationships and is more efficient than the permutation approach. The operating characteristics of our Bayesian approach are comparable to and sometimes better than both approaches in a wide range of dose-response relationships. It yields credible intervals as well as predictive distribution for the response rate at a specific dose level for the target dose estimation. Our Bayesian approach can be easily extended to continuous, categorical, and time-to-event responses. We illustrate the performance of our proposed method with extensive simulations and Phase II clinical trial data examples.