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

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  • Hal Caswell

    (Universiteit van Amsterdam)

Abstract

Background: Although the matrix model for kinship networks includes many demographic processes, it is deterministic. It provides values of age-stage distributions of kin, but no information on (co)variances. Because kin populations are small, demographic stochasticity is expected to create appreciable inter-individual variation. Objective: To develop a stochastic kinship model that includes demographic stochasticity and projects (co)variances of kin age distributions, and functions thereof. 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 nonlinear 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 presentations of the kinship model have acknowledged the limitation to expected values. That limitation is now removed; both means and variances are easily calculated with minimal modification of code.

Suggested Citation

  • Hal Caswell, 2024. "The formal demography of kinship VI: Demographic stochasticity and variance in the kinship network," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 51(39), pages 1201-1256.
    • Handle: RePEc:dem:demres:v:51:y:2024:i:39
    DOI: 10.4054/DemRes.2024.51.39
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    References listed on IDEAS

    as
    1. Hal Caswell, 2022. "The formal demography of kinship IV: Two-sex models and their approximations," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 47(13), pages 359-396.
    2. Xi Song & Robert D. Mare, 2019. "Shared Lifetimes, Multigenerational Exposure, and Educational Mobility," Demography, Springer;Population Association of America (PAA), vol. 56(3), pages 891-916, June.
    3. Thomas Pullum, 1982. "The Eventual Frequencies of Kin in a Stable Population," Demography, Springer;Population Association of America (PAA), vol. 19(4), pages 549-565, November.
    4. Hal Caswell & Xi Song, 2021. "The formal demography of kinship III: Kinship dynamics with time-varying demographic rates," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 45(16), pages 517-546.
    5. Thomas Leopold & Charlotte Clara Becker & Zafer Buyukkececi & Marcel Raab & Beyda Çineli, 2024. "KINMATRIX: A new data resource for studies of families and kinship," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 51(25), pages 789-808.
    6. Thomas Pullum & Douglas Wolf, 1991. "Correlations between frequencies of kin," Demography, Springer;Population Association of America (PAA), vol. 28(3), pages 391-409, August.
    7. Hal Caswell & Rachel Margolis & Ashton Verdery, 2023. "The formal demography of kinship V: Kin loss, bereavement, and causes of death," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 49(41), pages 1163-1200.
    8. Sha Jiang & Shripad Tuljapurkar & Hal Caswell & Zhen Guo & Wenyun Zuo, 2023. "How does the demographic transition affect kinship networks?," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 48(32), pages 899-930.
    9. Hal Caswell, 2020. "The formal demography of kinship II: Multistate models, parity, and sibship," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 42(38), pages 1097-1146.
    10. Alexia Prskawetz & Jože Sambt, 2014. "Economic support ratios and the demographic dividend in Europe," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 30(34), pages 963-1010.
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    More about this item

    Keywords

    kinship; matrix models; stochastic models; prevalence; dependency ratios; multitype branching processes;
    All these keywords.

    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics
    • Z0 - Other Special Topics - - General

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