The formal demography of kinship VI: Demographic stochasticity, variance, and covariance in the kinship network

Background The matrix model for kinship networks includes many demographic processes but is deterministic, projecting expected values of age-stage distributions of kin. It provides no information on (co)variances. Because kin populations are small, demographic stochasticity is expected to create appreciable inter-individual variation. Objectives To develop a stochastic kinship model to project (co)variances of kin age-stage distributions, and functions thereof, including demographic stochasticity. Methods Kin populations are described by multitype branching processes. Means and covariances are projected using matrices that are generalizations of the deterministic model. The analysis requires only an age-specific mortality and fertility schedule. Both linear and non-linear transformations of the kin age distribution are treated as outputs accompanying the state equations. Results The stochastic model follows the same mathematical framework as the deterministic model, modified to treat initial conditions as mixture distributions. Variances in numbers of most kin are compatible with Poisson distributions. Variances for parents and ancestors are compatible with binomial distributions. Prediction intervals are provided, as are probabilities of having at least one or two kin of each type. Prevalences of conditions are treated either as fixed or random proportions. Dependency ratios and their variances are calculated for any desired group of kin types. An example compares Japan under 1947 rates (high mortality, high fertility) and 2019 rates (low mortality, low fertility). Contribution Previous versions of the kinship model have acknowledged their limitation to expected values. That limitation is now removed; means and variances are easily and quickly calculated with minimal modification of code.