The formal demography of kinship IV: Two-sex models

Background Previous kinship models analyze female kin through female lines of descent, neglecting male kin and male lines of descent. Because males and females differ in mortality and fertility, including both sexes in kinship models is an important unsolved problem. Objectives The objectives are to develop a kinship model including female and male kin through all lines of descent, to explore approximations when full sex-specific rates are unavailable, and to apply the model to several populations as an example. Methods The kin of a focal individual form an age×sex-classified population and are projected as Focal ages using matrix methods, providing expected age-sex structures for every type of kin at every age of Focal. Initial conditions are based on the distribution of ages at maternity and paternity. Results The equations for two-sex kinship dynamics are presented. As an example, the model is applied to populations with large (Senegal), medium (Haiti), and small (France) differences between female and male fertility. Results include numbers and sex ratios of kin as Focal ages. An approximation treating female and male rates as identical provides some insight into kin numbers, even when male and female rates are very different. Contribution Many demographic and sociological parameters (e.g., aspects of health, bereavement, labor force participation) differ markedly between the sexes. This model permits analysis of such parameters in the context of kinship networks. The matrix formulation makes it possible to extend the two-sex analysis to include kin loss, multistate kin demography, and time varying rates.


Introduction
and Keyfitz and Caswell (2005).  The vectors a 1 , a 2 , b 1 , . . . , b 4 are the age structure vectors of each of these types of kin. 118 Our convention is to number females first (1, 3, 5, . . .) and males second (2, 4, 6, . . .). The     Figure 6: An example of reproduction by ancestors (parents, in this case) of Focal. All reproduction by Focal's parents, producing older and younger sisters and brothers, is attributed to Focal's mother.
primes appearing in Figure 2 indicate that those vectors are specific to both sex and line of 120 descent. (We will eliminate them shortly.) 121 We begin by specifying notation. Define The dynamics of grandchildren must account for all four types of grandchildren shown 138 in Figure 2: a 1 a 2 (x) (7) 140 As written, equation (7) permits each of the four types of grandchildren to experience its 141 own survival schedule, and each of the two types of children to contribute according to its 142 own fertility schedule. The corresponding model for great-grandchildren would include eight 143 survival matrices, U 1 , . . . , U 8 and four fertility matrices F 1 , . . . , F 4 . And so on. 144 Let us make the usual assumption that the demographic rates are affected by sex but 145 not by line of descent from an arbitrarily defined Focal individual. Then Under this assumption, the four types of grandchildren can be aggregated into granddaugh-149 ters and grandsons, ignoring their lines of descent from Focal. Define The dynamics then become  Focal has a network of female and male ancestors, as shown in Figure 5. She has two parents 165 (mothers d 1 and fathers d 2 ), four grandparents (maternal g 1 and g 2 ; paternal g 3 and g 4 ). 166 The pattern continues for as many levels as desired, the number of types of ancestors doubling 167 with each level.
168 2 There exist situations in which the full structure of Figure 2 and equation (7) would be of interest. For example, evolutionary calculations based on kin selection depend on the sharing of genes between individuals. Because maternity is more certain than paternity, Focal is more certain that she shares 1/4 of her genes with the children of her daughters, b 1 and b 2 than she is for the children of her sons, b 3 and b 4 . In general, each passage through a male descendant will introduce some uncertainty into the inheritance.
A related issue arises with respect to mitochondrial DNA, which is passed from mothers to offspring, and Y chromosomes, which are transmitted from fathers to sons; calculations involving these forms of inheritance might benefit from being able to distinguish kin by lines of descent. See Tanskanen and Danielsbacka (2019),  especially their Table 2.2, for a summary of these issues.

Consider the grandparents. Their dynamics satisfy
where the recruitment term is zero because Focal accumulates no new biological ancestors 171 after her birth. Combining the maternal and paternal grandparents of each sex yields The reproduction of the ancestors of Focal produces the side chains in Figure 1. In the 175 two-sex model, reproduction must account for the lack of independence of pairs of ancestors.

176
Consider Focal's daughters and sons. They can be assumed to reproduce independently to 177 produce grandchildren, and are treated so in Figure 3 and equation (12). But Focal's mother 178 and father do not reproduce independently to produce her siblings. Therefore, as shown in 179 Figure 6, we credit the reproduction of Focal's parents to her mother, following the female 180 fertility schedule (a kind of female dominance assumption).
Notice the difference between the block-structured fertility matrices in (16) and (12). For some type k of kin, we write 186 where x is the age of Focal. The tilde denotes block structured vectors and matrices composed 187 of female and male parts.

188
The dynamics of k(x) are written as

192
The recruitment subsidy termβ(x) depends on the nature of the kin that provide the re-193 cruitment.

194
• If the subsidy is provided by reproduction of one of the direct ancestors of Focal 195 (parents, grandparents, etc.), then, as in equation (16) 196 198 where k * denotes the source kin (e.g., mothers are the source of the siblings of focal).

199
The matrixF * captures reproduction of both female and male offspring by females.

200
• If the subsidy is provided by any other kin type, then as in equation (12) where k * again denotes the source kin (e.g., children are the source of the grandchildren 204 of focal).

205
• If there is no recruitment subsidy, as in (15), thenβ = 0. The matrices F f and F m contain age-specific fertilities for females and males respectively.

208
The distributions of the ages of mothers and of fathers are obtained by applying these per 209 capita rates to age distributions of women and men, respectively. Let be the age structure of the population. The age distributions of maternity and paternity are The stable sex-age structure is given by the eigenvectorw corresponding to the dominant 222 eigenvalue ofÃ. Without loss of generality, we scalew so that its entries are non-negative, The dynamics of the children of focal are given by The dynamics of the grandchildren of focal are given by The dynamics of the great-grandchildren of focal are given by In each case, the initial condition is zero (Focal has no children, grandchildren, etc. when 246 she is born). The recruitment of each generation of descendants comes from the fertility of 247 the previous generation.

248
The chain of descendants can be extended as far as desired, as is also true of the matrix  Figure 5. We know that, at birth, Focal had exactly one living mother. 256 We will assume that she also has one living father, thus ignoring paternal mortality in the 257 nine months between conception and birth.

258
The ages of Focal's parents at her birth are unknown, so we treat her mother and father 259 as being selected at random from the distributions π f and π m of the ages of mothers and 260 fathers at the birth of children.

261
Under these assumptions, the dynamics of the parents of Focal are given by

264
Focal accumulates no new parents after her birth, so the recruitment term is zero.

265
The dynamics of the grandparents of Focal are given by The initial conditiong(0) is obtained by noting that the grandparents of Focal are the parents 269 of Focal's parents. Thus we could write 270g (0) = mothers of mom fathers of mom (0) + mothers of dad fathers of dad (0) (43) 271 We do not know the ages of Focal's mother or father, but we know their distributions, so The dynamics of the great-grandparents of Focal follow the same pattern. The great-276 grandparents of Focal at her birth are the grandparents of Focal's parents, so As with descendants, the ancestors can be calculated back as far as desired. For this 280 calculation, letk j be the kin vector of the jth generation of ancestors, where parents are 281 generation zero. Then, for j ≥ 1,

285
We turn next to the siblings, nieces, and nephews of Focal. Inconveniently, English seems 286 to have no gender neutral collective term for nieces and nephews, as sibling is for brothers 287 and sisters. The term "nibling" has been suggested, according to the internet. German 288 has such a term, "Geschwisterkind," meaning child of a sibling. There also seems to be no 289 gender-neutral collective term for aunts and uncles (next section).

290
Siblings. The older and younger siblings of Focal are treated separately because they have 291 different dynamics.

292
Focal may have older siblings at her birth, but she can accumulate no more of them after 293 she is born, so the recruitment term is zero. Older siblings at Focal's birth are the children of 294 Focal's mother at Focal's birth. Focal's mother and father do not reproduce independently, 295 so the initial condition is credited to Focal's mother and calculated as an average over π f 296 only. Thus

299
Focal has no younger siblings at her birth, 3 but can accumulate younger siblings through 300 her mother's reproduction. Thus The matrixF * ensures that the recruitment of new younger siblings comes from the repro-304 duction of Focal's mother following the female fertility schedule, as in Figure 6.

305
Niblings. Then niblings of Focal are the children of Focal's older and younger siblings.

306
The recruitment of niblings through older siblings comes from the reproduction of those 307 siblings, both brothers and sisters contributing independently. The initial condition follows 308 from the fact that, at the time of Focal's birth, these niblings are the grandchildren of Focal's 309 mother. Thus, the dynamics of nieces and nephews through older siblings are This line of descent can be continued indefinitely. Each generation receives its recruitment 313 from the generation before (grand-niblings from niblings, etc.). The initial condition for each 314 generation is the corresponding descendant of Focal's mother: niblings are the grandchildren 315 of Focal's mother; grand-niblings are the great-grandchildren, and so on.

316
The recruitment of niblings through younger siblings comes from the reproduction of 317 those younger siblings. The initial condition is zero, because Focal has no younger siblings 318 at birth, and hence can have no nieces or nephews through them. The resulting dynamics

323
The aunts and uncles of Focal (there seems to be no gender-inclusive term) are the siblings 324 of Focal's parents.

325
The aunts and uncles through the older siblings receive no recruitment subsidy because 326 once Focal is born, her parents cannot add any older siblings. The initial condition com-327 bines the older siblings of the mother and father of Focal, following the steps above for the 328 grandparents of focal. The resulting dynamics are Aunts and uncles through younger siblings receive a recruitment subsidy from the re-332 production of the grandmothers of Focal. Grandmothers and grandfathers do not reproduce 333 independently, so only input from grandmothers is counted. The initial condition combines 334 the younger siblings of Focal's mother and of Focal's father, at the time of Focal's birth.

335
The resulting dynamics are nieces/nephews via younger siblings 0Fñ(x) r aunts/uncles older than mother i π fm (i)m(i) 0 s aunts/uncles younger than mother i π fm (i) n(i)F * g (x) t cousins from aunts/uncles older than mother i π fm (i)p(i)Fr(x) v cousins from aunts/uncles younger than mother i π fm (i)q(i)Fs(x)

343
The resulting dynamics are The cousins through the aunts/uncles younger than mother receive a recruitment subsidy 347 from the reproduction of those aunts and uncles. These cousins are the nieces and nephews 348 of Focal's mother through her younger siblings.

349
The resulting dynamics are The chains of descendants through these cousins can be extended indefinitely. Recruit-353 ment into each generation comes from the generation before it, and the initial condition 354 consists of the corresponding level of nieces/nephews of Focal's mother.   TFR and mean age at paternity is much greater than mean age at maternity. In France, the

Kin sex ratios
What is hinted at, but not clearly apparent, in Figure 8 is the shift in the sex composition 404 of the various kin as Focal ages. Figure 9 shows the sex ratio (number of males divided by 405 number of females), for each type of kin, as a function of the age of Focal.

406
The sex ratio declines (i.e., females come to incrasingly outnumber males) with increasing Schoumaker (2019) reports that the difference between male and female fertility is 437 least in countries with low fertility rates; this approximation should, therefore, be 438 most successful in such populations.

439
Model 3. In the absence of male mortality data, an approximation would apply the female 440 mortality schedule to both sexes. In this case, set Model 4. In the absence of male rates of any kind, males and females could be treated as 445 indistinguishable. I refer to this as the androgynous approximation, in which female 446 rates are used for both sexes, leading to In each of these cases, the calculations proceed as prescribed in Section 3, but using the • the survival matrix U is replaced by the block-structured matrixŨ,

502
• the fertility matrix F is replaced with the block structured matricesF andF * , and

503
• the age at maternity distribution π is replaced by the maternity and paternity distri-504 butions π f and π m .

505
The dynamics of the two-sex model reflect the independence of reproduction by some kin 506 (e.g., sons and daughters of Focal produce grandchildren independently) but dependence 507 in others (e.g., Focal's mother and father do not produce siblings of Focal independently).

508
The summaries in Table 1 (two-sex) and Tables

518
This paper has focused on the numbers and sex ratios of kin. But because the model 519 provides the full age×sex structure for each type of kin, many other kinds of weighted num-520 bers (e.g., dependency ratios) are easily computed. Of particular interest are quantities that 521 might, in general, be called "prevalences" -measures of the occurrence of some property, 522 at specified ages, for each sex. The properties are often medical or health conditions (e.g., 523 Caswell, 2019a, for an analysis of dementia lacking. Notice that the differences in numbers of kin between the two-sex model and the 544 androgynous approximation (Figures 10 and 11) are much smaller than the differences in kin 545 numbers among the three countries examined here (Figure 8). More comparative research is 546 warranted.

547
The possible use of model fertility schedules (e.g., Coale and Trussell, 1974) when mea- one is interested in the kinship structure of species whose fertility patterns are very different 555 from ours (Caswell, 2021).

556
The distributions of ages at maternity and paternity of Focal's mother and father deter-557 mine initial conditions for kin. These ages have been treated here as independent; an obvious 558 extension would be to use a joint distribution reflecting the age distribution of couples. Finally, it is important to recall that the projections of the kin populations provide mean 566 age-sex structures, over the distributions produced by the survival and fertility probabilities.

567
Because the kin populations are small, there will be an (as yet unknown) degree of stochastic 568 variation around those means.

569
In summary, the matrix theoretic model makes it possible to expand the demographic 570 detailed included the analysis of a kinship network: from age distributions to multistate 571 age×stage distributions, from time-invariant to time-varying demographic rates, and now 572 from one-sex to two-sex models. It has resulted in an increasingly rich set of demographic 573 outcomes, including many of great demographic interest, including (but not limited to) 574 bereavement, dependency, prevalences, family sizes, and sex ratios. All these are now subject 575 to analysis in terms of demographic rates in a kinship setting. great-grandmothers i π i g(i) 0 m older sisters i π i a(i) 0 n younger sisters 0 Fd(x) p nieces via older sisters i π i b(i) Fm(x) q nieces via younger sisters 0 Fn(x) r aunts older than mother i π i m(i) 0 s aunts younger than mother i π i n(i) Fg(x) t cousins from aunts older than mother i π i p(i) Fr(x) v cousins from aunts younger than mother i π i q(i) Fs(x) 673 Table A-2: Summary of the age×stage-classified kinship model of Caswell (2020). Matrices and vectors bearing tildes (e.g.,ã) age×stage block-structured.

Symbol
Kin initial condition k 0 Subsidy β(x) φ Focalφ 0 0 a daughters 0Fφ(x) b granddaughters 0Fã(x) c great-granddaughters 0Fb(x) d mothersπ 0 g grandmothers i π age id (i) 0 h great-grandmothers i π age ig (i) 0 m older sisters i π age iã (i) 0 n younger sisters 0Fd(i) p nieces via older sisters i π age ib (i)Fm(x) q nieces via younger sisters 0Fñ(i) r aunts older than mother i π age im (x) 0 s aunts younger than mother i π age iñ (i)Fg(x) t cousins: aunts older than mother i π age ip (i)Fr(x) v cousins: aunts younger than mother i π age iq (i)Fs(x) h(x, t) great-grandmothers i π i (t)g(i, t) U t h(x, t) 0 m(x, t) older sisters i π i (t)a(i, t) U t m(x, t) 0 n(x, t) younger sisters 0 U t n(x, t) F t d(x, t) p(x, t) nieces via older sisters i π i (t)b(i, t) U t p(x, t) F t m(x, t) q(x, t) nieces via younger sisters 0 U t q(x, t) F t n(x, t) r(x, t) aunts older than mother i π i (t)m(i, t) U t r(x, t) 0 s(x, t) aunts younger than mother i π i (t)n(i, t) U t s(x, t) F t g(x, t) t(x, t) cousins; aunts older than mother i π i (t)p(i, t) U t t(x, t) F t r(x, t) v(x, t) cousins; aunts younger than mother i π i (t)q(i, t) U t v(x, t) F t s(x, t)