Modeling conception as an aggregated Bernoulli outcome with latent variables via the EM algorithm

We describe a general method for analyzing aggregated Bernoulli outcomes. The research is motivated by an epidemiological reproductive study where the outcome was whether pregnancy was detected in a woman's particular menstrual cycle and the Bernoulli "trials" corresponded to days with intercourse during the cycle. Each cycle is either "viable" or not, i.e., is or is not susceptible to conception. We develop an EM algorithm approach to maximizing the observed-data pseudo-likelihood, based on a set of unobservable latent outcomes linked to the specific days with intercourse. This method is flexible in that it allows one to model effect of covariates on the susceptibility factor and on the latent outcomes. Application of the method to fertility studies enables one to investigate covariates with a long-term or transient effect on the daily conception probability. A complication is that most couples contribute more than one cycle in a prospective study. A generalized estimating equation approach adjusts for the dependency among outcomes within individual couples. The method can be applied in any setting where dependency among Bernoulli trials is induced by a susceptibility factor and the trial outcomes are only observable in the aggregate.