The Beta-Geometric Distribution Applied to Comparative Fecundability Studies

A convenient measure of fecundability is time (number of menstrual cycles) required to achieve pregnancy. Couples attempting pregnancy are heterogeneous in their per-cycle probability of success. If success probabilities vary among couples according to a beta distribution, then cycles to pregnancy will have a beta-geometric distribution. Under this model, the inverse of the cycle-specific conception rate is a linear function of time. Data on cycles to pregnancy can be used to estimate the beta parameters by maximum likelihood in a straightforward manner with a package such as GLIM. The likelihood ratio test can thus be employed in studies of exposures that may impair fecundability. Covariates are incorporated in a natural way. The model is illustrated by applying it to data on cycles to pregnancy in smokers and nonsmokers, with adjustment for covariates. For a cross-sectional study, when length-biased sampling is taken into account, the pre-interview attempt time is shown to follow a beta-geometric distribution, so that the same methods of analysis can be applied even though all of the available data are right-censored. For a cohort followed prospectively, there will be some couples enrolled whose fecundability is effectively 0, and for such applications, the beta could be considered to be contaminated by a distribution degenerate at 0. The mixing parameter (proportion sterile) can be estimated by application of the expectation-maximization (EM) algorithm. This, too, can be carried out using GLIM.