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. Objective: The aim is to develop a multistate kinship model, classifying individuals jointly by age and other criteria (generically, “stages”). Methods: The vec-permutation method is used 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, a model is derived including age and parity. It provides, 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. The age×parity distributions provide the distributions of sibship sizes of kin. As an example, the model is applied to Slovakia (1960–2014). The results show a dramatic shift in the parity distribution as the frequency of low-parity kin increased and that of high-parity kin decreased. Contribution: This 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.

中文翻译：

---

亲属关系的正式人口统计：多状态模型，奇偶校验和同居关系

背景：最近的亲属关系模型关注亲属的年龄结构，其是对象个体年龄的函数。但是，除年龄外，变量也有重要影响。将针对年龄的模型推广到包括其他变量在内的多状态模型是一个重要且迄今尚未解决的问题。目标：目的是建立一个多州血统模型，根据年龄和其他标准（通常称为“阶段”）对个人进行分类。方法：vec-permutation方法用于创建多状态投影矩阵，包括年龄和阶段相关的生存，生育力和过渡。这些矩阵在块结构的种群矢量上运行，该种群矢量描述了焦点个体在每个年龄下每种亲属的年龄×阶段结构。结果：新的基质配方可直接与之媲美并大大扩展，Caswell（2019a）的最新的按年龄分类的血缘关系模型。作为应用，可以推导出包括年龄和奇偶校验的模型。它为所有类型的亲属提供了焦点个体每个年龄的联合年龄×奇偶校验结构，边缘年龄和奇偶校验结构以及（规范化的）奇偶校验分布。年龄×胎次分布提供了近亲同胞大小的分布。例如，该模型应用于斯洛伐克（1960–2014）。结果显示，随着低奇偶校验亲属的频率增加而高奇偶校验亲属的频率降低，奇偶校验分布发生了显着变化。贡献：该模型将对亲属关系的正式人口统计学分析扩展到了年龄×阶段分类模型。除平价外，其他阶段的分类，包括婚姻状况，产妇年龄影响和性别，现在也可以分析。