Weighted nonparametric maximum likelihood estimate of a mixing distribution in nonrandomized clinical trials

Hierarchical models have a variety of applications, including multi-center clinical trials, local estimation of disease rates, longitudinal studies, risk assessment, and meta-analysis. In a hierarchical model, observations are sampled conditional on individual unit-specific parameters and these parameters are sampled from a mixing distribution. In observational studies or nonrandomized clinical trails, observations may be biased samples from a population and heterogeneous with respect to some confounding factors. Without controlling the heterogeneity in the sample, the standard estimation of the mixing distribution may lead to inaccurate statistical inferences. In this article, we propose a weighted nonparametric maximum likelihood estimate (NPMLE) of the mixing distribution and its smoothed version via weighted smoothing by roughening. The proposed estimator reduces bias by assigning a weight to each subject in the sample. The weighted NPMLE is shown to be weighted self-consistent and therefore can be easily calculated through a recursive approach. Simulation studies were conducted to evaluate the performance of the proposed estimator. We applied this method to clinical trial data evaluating a new treatment for stress urinary incontinence.