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

Abstract

BackgroundRecent 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.ObjectivesOur aim is to develop a multistate kinship model, classifying individuals jointly by age and other criteria (generically, “stages”).MethodsWe 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.ResultsThe 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.ContributionThe 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.

The formal demography of kinship III: Kinship dynamics with time-varying demographic rates

“…The analysis here retains the Focal-centric approach of Goodman, Keyfitz, and Pullum (1974) and follows Caswell (2019, 2020) in treating the kin of Focal as a population. Whereas in the time-invariant model dynamics unfold in relation to the age of Focal, in the model here, both Focal’s age and calendar time appear in the dynamics.…”

“…For example, Caswell (2019) extended the matrices and vectors to include the age structures of dead as well as living kin, and calculated Focal’s lifetime experience of the death of kin. Caswell (2020) extended the model to include, in addition to age, arbitrary stage structures with the possibility of movement among those stages, with an age×parity-dependent analysis as an example. In these multistate formulations, the model equations are unchanged; only the contents and block structures of the matrices change.…”

“…These models provide a tool for demographers to examine how different components of population structure are connected to one another and to identify the effect of changing one component on population processes. Goodman, Keyfitz, and Pullum (1974) and Caswell (2019, 2020) have emphasized repeatedly that their models are not intended to predict the results of census data on the numbers of kin in a population. Such data will reflect processes operating in the real population but not included in the models; the models provide a background against which to compare the effect of those processes.…”

“…Time-invariant models are a powerful way to explore a set of conditions by asking what would happen if those conditions were to be maintained (Keyfitz, 1972; Cohen, 1979; Caswell, 2001). The early kinship models (Lotka, 1931; Coale, 1965; Goodman, Keyfitz, and Pullum, 1974, 1975) and the more recent alternative by Caswell (2019, 2020) are all time-invariant. Every individual in the kinship network experiences a fixed set of demographic rates.…”

“…This classical approach is challenging to implement, gives only the numbers, not age distributions of kin, and is difficult to extend beyond the simplest case. A new approach was introduced by Caswell (2019, 2020), using matrix operations to project the population of each type of kin from one age to the next and account for the production of new kin of one type (e.g., nieces) by the reproduction of a different type (e.g., sisters). The first of these papers provided the general theory for the dynamics of the age structure of any kind of kin; the second extended the theory to multistate models that project the age × stage structure of all types of kin and applied the theory to analyze the age × parity structure of kin.…”