The formal demography of kinship II: Multistate models, parity, and sibship

Background Recent kinship models focus on the age structures of kin as a function of the age of the focal individual. However, variables in addition to age have important impacts. Generalizing age-specific models to multistate models including other variables is an important and hitherto unsolved problem. Objectives Our aim is to develop a multistate kinship model, classifying individuals jointly by age and other criteria (generically, “stages”). Methods We use the vec-permutation method to create multistate projection matrices including age- and stage-dependent survival, fertility, and transitions. These matrices operate on block-structured population vectors that describe the age×stage structure of each kind of kin, at each age of a focal individual. Results The new matrix formulation is directly comparable to, and greatly extends, the recent age-classified kinship model of Caswell (2019a). As an application, we derive a model that includes age and parity. We obtain, for all types of kin, the joint age×parity structure, the marginal age and parity structures, and the (normalized) parity distributions, at every age of the focal individual. We show how to use the age×parity distributions to calculate the distributions of sibship sizes of kin. As an example, we apply the model to Slovakia (1960–2014). The results include a dramatic shift in the parity distribution as the frequency of low-parity kin increased and that of high-parity kin decreased. Contribution The new model extends the formal demographic analysis of kinship to age×stage-classified models. In addition to parity, other stage classifications, including marital status, maternal age effects, and sex are now open to analysis.