Probability models for family building: An analytical review

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Abstract
Considerable literature now exists on stochastic models for the reproductive history of a cohort of couples. These models are of varying complexity and the relationships between separate treatments are not always clear. A classification system for such models is proposed, followed by a historical review of models for family building and for logically related processes. Models, differing only in treatment of time as discrete or continuous, are presented in detail for the simple case where the prob ability of conception is constant, and all conceptions lead to live births which are associated with a fixed nonsusceptible period. Analysis of different treatments is facilitated by introducing the notion of the time when a conception is recorded. Emphasis is placed on results for the probability of a recording at a specified time t, the probability of r recordings by time t, and the expected number of recordings in time t. Differences between the discrete and continuous time models are made explicit. It is shown that results for these models can be derived using renewal theory techniques, which are presented. More complex models based on renewal theory and allowing for several pregnancy outcomes or for variability in parameters are briefly described, followed by generalized models which allow parameters to vary with time. Applications of family building models are summarized.

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