The preceding analysis can be adapted to random-mating, constant-size, monogamous populations as follows. With Poisson-distributed reproductive success, approximately 20% of each generation’s lineages become extinct, with a mean time to extinction of slightly more than 1.5 generations [10 - 14]. Thus, all present-day individuals will be descended from only 80% of an ancestral population that existed two or more generations in the past. In terms of the population modeled by simulation, a “census” size of N = 20,000 (or 10,000 monogamous pairs) means that approximately 8000 generation 0 pairs gave rise to persistent lineages. To avoid confusion, this “effective monogamous pair population size” will be referred to as N_m. An individual in generation G is descended from 2^G−1 ancestor pairs in generation 0, and that is the size of the ancestor-pair sample represented by that individual. Let C_G be the probability that a random pair of individuals share at least one pair of ancestors who lived G generations previously; that is, the probability that they are (G − 1)^th cousins. C_G is obtained from Equation (2) by substituting N_m for N and 2^G−1 for a:

C G = 1 − ( N m − 1 N m ) ( 2 G − 1 ) 2 (3)

In the case of promiscuous (non-monogamous) reproduction, each offspring is produced by a newly created random pairing of male and female parents. As long as N is reasonably large, almost all sibs will be half-sibs and almost all 1^st cousins, half-1^st cousins, etc. For simplicity, I will generally not use the “half” modifier for the remainder of this paper, with the understanding that it is implied when discussing results for promiscuous reproduction. With promiscuous reproduction, every individual is descended from 2^G ancestors who lived G generations previously. Approximately 80% of each generation’s members give rise to persistent lineages. Thus, for a total population size N = 20,000, the “effective promiscuous population size” is N_p = 16,000. Let D_G be the probability that a random pair of individuals share at least one ancestor who lived G generations in the past. D_G is obtained from Equation (2) by substituting N_p for N and 2^G for a:

D G = 1 − ( N p − 1 N p ) ( 2 G ) 2 (4)

If the relationship of interest is sibs, rather than cousins, Equations (3) and (4) are approximations to the probability of pairwise shared ancestry, given the suggested values for N_m and N_p. That is because sibs share ancestors who lived only one generation previously. Therefore, in that case, N_m and N_p should be about 86.5% of the census population size—that is, the “effective size” of the immediately preceding generation. This special case is of little consequence for the present investigation and is, therefore, ignored in the analytical calculations shown in the Results.

The analytical model is readily extended to consideration of relationships among individuals in samples larger than pairs. In a sample of s individuals, drawn without replacement, there are s(s − 1)/2 possible pairwise comparisons. For monogamous reproduction, the probability that any one pair in the sample will not be (G − 1)^th cousins is (1 − C_G). The probability that all of the pairwise comparisons are not cousins (or sibs if G = 1) is:

Pr ( nopairsare   ( G − 1 ) th cousins ) = ( 1 − C G ) s ( s − 1 ) / 2 (5)

Therefore, the probability that there is at least one pair of (G − 1)^th cousins in the sample is:

Pr ( atleastonepairof   ( G − 1 ) th cousins ) = 1 − ( 1 − C G ) s ( s − 1 ) / 2 (6)

As an example of a prediction by Equation (5) and Equation (6), consider the case of sampling 20 individuals from a monogamous population of size N = 20,000. There are 190 possible pairwise comparisons. The probability that any one pair will not share an ancestor who lived three generations previously (i.e., will not be 2^nd cousins) is (1 − C₃) = 0.998. The probability that none of the 190 possible pairs will be 2^nd cousins is 0.998¹⁹⁰ ≈ 0.684 (Equation (5)). Or, conversely, there is about a 32% chance that the sample will contain at least one pair of 2^nd cousins (Equation (6)); and by similar calculation, there is a nearly 100% chance that the same sample will contain at least one pair of 4^th cousins. If reproduction is promiscuous, Equations (5) and (6) are modified by substituting D_G for C_G. Then, with N = 20,000 and s = 20, the probability that the sample will contain at least one pair of (half) 2^nd cousins is about 53.2%, and there is about a 95% chance that a sample will contain at least one pair of 3^rd cousins.

Equation (3) gives the probability that a random pair of individuals will be (G − 1)^th cousins (in the monogamous case). As already noted, in a sample of s individuals, there are s(s − 1)/2 possible pairwise comparisons. Thus, the expected number of pairs of cousins in the sample is simply the product of (1) the probability that any one pair are cousins and (2) the number of possible pairs. Let P_G be the number of pairs of (G − 1)^th cousins in a sample from a population with monogamous reproduction; and Q_G the number of pairs of (mostly half) cousins if reproduction is promiscuous. Then:

P G = C G ( s ( s − 1 ) 2 ) (7)

and

Q G = D G ( s ( s − 1 ) 2 ) (8)

In a sample of 20 individuals taken from a monogamous population of 20,000, the expected number of pairs of 2^nd cousins is 0.38, which is consistent with the 32% probability that such a sample will contain at least one pair of 2^nd cousins. That same sample would be expected to contain 5.98 pairs of 4^th cousins.

Shchur and Nielsen [9] derive expectations for the quantities, U_p and V_p, which are the number of individuals in a sample which do not have (p − 1)^th cousins in that same sample. U_p applies to a monogamous population and V_p to a non-monogamous (promiscuous reproduction) population. They depend upon both sample size and effective population size, as expected from the preceding discussion. U_p, for example, and P_G (Equation (7)) convey related information. 1 − U_P is the expected number of individuals in a sample who are (p − 1)^th cousins of at least one other member of the sample. That is not the same as the number of pairs of (p − 1)^th cousins, but it sets upper and lower bounds on the number of pairs. For example, if there are r related individuals, the minimum number of pairs is r/2, and the maximum number of cousin pairs is r(r − 1)/2. Thus, there is an approximate test for consistency between the Shchur-Nielsen expectations for the number of related individuals in a sample and the expected number of cousin pairs in a sample (Equations (7) and (8)). Given N = 20,000 and monogamy, the expected number of 4^th cousins, (1 − U₅), in a sample of 20 individuals is 9.45. The minimum number of pairs expected by Shchur-Nielsen is, therefore, 4.73, which is consistent with the 5.98 pairs obtained with Equation (7). (As explained below, Sec. 3.2.3, the effective population size used for the Shchur-Nielsen calculations in this example is 8000.)

There is very close agreement between the analytical model (Equation (5)) and the simulation results (Figure 2). It is noteworthy that, at least for panmictic populations, a very small sampling fraction is sufficient to guarantee that even closely related individuals will be included in a sample. For example, with N = 20,000 and s = 200, there is an approximately 100% probability that at least one pair of 1^st cousins will be included (Figure 2(b), Figure 2(d)). Furthermore, required sample sizes increase much more slowly than the population size. As noted above, with N = 20,000, s = 20, and monogamy, there is about a 32% chance that a sample will contain one or more pairs of 2^nd cousins (Figure 2(a)). Similar calculations can be performed for any population and sample size. If N is one million, a sample of 140 individuals would also have about a 32% chance of including at least one pair of 2^nd cousins (from Equation (6)). In other words, a 50× increase in population size requires only a 7× increase in sample size to produce a similar probability that a sample will include one or more pairs of (G − 1)^th cousins.

Very small sample sizes are sufficient to guarantee that a sample will include at least one pair of relatively distantly related cousins. For N = 20,000 and s = 20, there is an essentially 100% probability that the sample will contain at least one pair of 4^th, and more distantly related, cousins (Figure 2(a), Figure 2(c)).

There is close agreement between the simulation results and analytical expectation (Figure 3). With

monogamous reproduction, and N = 20,000, every member of a sample is related to every other member of a sample as 8^th cousins (G = 9, Figure 3(a), Figure 3(b)). That result is independent of sample size, and depends only on the population size (effective number of monogamous pairs, N_m). It reflects the fact that random pairs of individuals, drawn from the population, will be 8^th cousins with probability approaching 1.0 (Figure 1(a)). To be sure, the actual number of pairs of cousins of a given degree in a sample does depend on sample size, because there are more possible pairwise combinations with larger samples. The analytical plots in Figure 3(a) and Figure 3(b) are identical except that every point in Figure 3(b) is 104.74× higher than the corresponding point in Figure 3(a). With promiscuous reproduction the number of cousin pairs of given degree is greater than for monogamous reproduction, as expected (Figure 3(c) and Figure 3(d)). Almost all possible pairs in a sample are 7^th cousins (G = 8).

For consistency with the preceding, I will replace U_p and V_p of Shchur and Nielsen [9] with U_G and V_G, which are the expected number of individuals in a sample that do not have (G − 1)^th cousins, where G explicitly references generations. In order to compare expectations for different sample sizes, they divide U_G and V_G by sample size to obtain the expected proportion of unrelated individuals in the sample. A possible source of confusion in this analysis has to do with “effective population size”. Shchur and Nielsen

refer to N as effective population size and later state “we assume that there are exactly N male and N female individuals” ([9], p. 1281). From that, it would appear that the appropriate value for N in their expressions for the expectations of U_G and V_G would be N = 8000, the number of generation 0 males or females with persistent genealogies, given a total population size of 20,000. Results for the calculated Shchur-Nielsen expectation and the simulations are shown in Figure 4. The correspondence between expectation and simulation is high, especially given that there is some stochasticity in the simulations. The correspondence appears to confirm that the interpretation of N in the Shchur-Nielsen formulas was correct.