Fecundability studies, where couples attempting pregnancy subject to 'failure' (conception) one time in each menstrual cycle, present a natural discrete failure-time scenario. Because the biologic capacity to conceive varies among couples in the population, a complication arises in choosing a method of analysis, related to the fact that the maximum follow-up time can vary from study to study, and follow-up time could potentially have different effects on parameters based on different approaches to modelling. Traditional development in fertility studies has been based on an implicit assumption that binary outcomes for different menstrual cycles are mutually independent. We contrast traditional models to a random effects model where cycle viability is modelled as subject-specific. We clarify the interpretations for different parameters from different models. We show that the traditional approach yields some regression parameters that depend on follow-up time, limiting the generalizability of inferences based on this analytic approach. By contrast, the subject-specific model consistently estimates parameters of interest, if the underlying distribution is properly specified. Data from a fecundability study carried out in North Carolina serves to illustrate these points.