Hierarchical models for the probabilities of conception

In the last thirty years, there has been considerable interest in finding better models to fit data for probabilities of conception. An important early model was proposed by Barrett and Marshall (1969) and extended by Schwartz, MacDonald and Heuchel (1980). Recently, researchers have further extended these models by adding covariates. However, the increasingly complicated models are challenging to analyze with frequentist methods such as the EM algorithm. Bayesian models are more feasible, and the computation can be done via Markov chain Monte Carlo (MCMC). We consider a Bayesian model with an effect for protected intercourse to analyze data from the California Women's Reproductive Health Study and assess the effects of water contaminants and hormones. There are two main contributions in the paper. (1) For protected intercourse, we propose modeling the ratios of daily conception probabilities with protected intercourse to corresponding daily conception probabilities with unprotected intercourse. Due to the small sample size of our data set, we assume the ratios are the same for each day but unknown. (2) We consider Bayesian analysis under a unimodality assumption where the probabilities of conception increase before ovulation and decrease after ovulation. Gibbs sampling is used for finding the Bayesian estimates. There is some evidence that the two covariates affect fecundability.