Effective number of breeders from sibship reconstruction: empirical evaluations using hatchery steelhead

Abstract Effective population size (N e) is among the most important metrics in evolutionary biology. In natural populations, it is often difficult to collect adequate demographic data to calculate N e directly. Consequently, genetic methods to estimate N e have been developed. Two N e estimators based on sibship reconstruction using multilocus genotype data have been developed in recent years: sibship assignment and parentage analysis without parents. In this study, we evaluated the accuracy of sibship reconstruction using a large empirical dataset from five hatchery steelhead populations with known pedigrees and using 95 single nucleotide polymorphism (SNP) markers. We challenged the software COLONY with 2,599,961 known relationships and demonstrated that reconstruction of full‐sib and unrelated pairs was greater than 95% and 99% accurate, respectively. However, reconstruction of half‐sib pairs was poor (<5% accurate). Despite poor half‐sib reconstruction, both estimators provided accurate estimates of the effective number of breeders (N b) when sample sizes were near or greater than the true N b and when assuming a monogamous mating system. We further demonstrated that both methods provide roughly equivalent estimates of N b. Our results indicate that sibship reconstruction and current SNP panels provide promise for estimating N b in steelhead populations in the region.


Introduction
Two methods for estimating Ne from sets of sibling assignments (PWOP: Waples 2011, andCOLONY: Wang 2009) are compared. Here we discuss how they are conceptually related and show that the PWOP method is a special case of COPLONY method. The COLONY method can explicitly incorporate deviations from Hardy-Weinberg equilibrium and differences in sex ratio, while the PWOP method relies only on sibling relationships. This difference may not large in practice, as often these values are unknown and assigned default values by researchers, but see Wang (2009) and Wang (2016) for useful discussion.
In Supplemental Figures 1 and 2, using empirical data from the current study, we show that both the above methods give essentially the same estimates when applied to the same set of sibling assignment inferred from genetic data. Sibling assignments were estimated with COLONY2 (Jones and Wang 2010), assuming either a monogamous or polygamous mating system.
Definitions N e = Inbreeding effective population size, a measure of how the average inbreeding coefficient changes from one generation to the next.  S = Number of (observed) offspring. Each offspring has two parents, so S = k i /2. p same = Chance that two random gametes in the offspring generation come from the same parent. In an ideal population, p same = 1 N . The chance that two random gametes are identical by descent in the previous generation is P same /2 if the contribution of alleles by a parent is random and independent. Crow and Denniston (1988):

Equations
In equation (1) k i contains all potential parents, even those with zero offspring. This equation assumes an equal sex ratio and progeny distribution.
Waples and Waples (2011) noted that (1) holds even when excluding individuals from the parental population that do not contribute offspring. Waples and Waples (2011) equation 2a: They also note that this implies a sample of offspring can be used to estimate N e by estimation of k i (and crucially also k 2 i ). As k i = 2S, the above equation can also be written as (Waples 2011, equation 2b): Starting from the same vector of k i values, we can equivalently calculate the chance that two gametes sampled at random without replacement share the same parent.
where 2S(2S − 1) is the number of gamete pairs and k i (k i − 1) is the number of gamete pairs sharing a parent, equal to −2S + k 2 i . This allows equation 4 to be rewritten as: which is simply the reciprocal of 3, providing the simple relationship: This connects the equations of Crow and Denniston (1988) and Waples (2011) to the chance that a random pair of gametes share a parent.
Wang (2009) addresses a very similar situation: "Equations for the effective size (N e ) of a population were derived in terms of the frequencies of a pair of offspring taken at random from the population being sibs sharing the same one or two parents".
Wang (2009) equation 10: This equation directly addresses two potential departures from an ideal population; sex ratio and Hardy-Weinberg proportions. α is a measure of departure from Hardy-Weinberg equilibrium (i.e. F IS ), Q 1 , Q 2 , and Q 3 are the probabilities of a pair of offspring being paternal half-siblings, maternal half-siblings, and full-siblings, respectively. N 1 and N 2 are the number of male and female parents.
If we assume Hardy-Weinberg equilibrium (α = 0), (7) becomes: Wang (2009) equation 8: if we assume that the sex ratio is equal, this becomes: and therefore: Substitution into (8) leads to: Which is the same as 6 and 2.
For a detailed discussion of the implications of these simplifying assumptions, see Wang (2009) and Wang (2016).
For an empirical example of the similarity of these methods, see the notebook at: http://rwaples.github.io/PWOP-vs-SA/ Figure 1: Scatterplot of N e estimates with the PWOP and COLONY2 methods for sibling assignments assuming a monogamous mating structure. The Pearson correlation coefficient and p-value (as computed by scipy.stats.pearsonr) are reported. Figure 2: Scatterplot of N e estimates with the PWOP and COLONY2 methods for sibling assignments assuming a polygamous mating structure. The Pearson correlation coefficient and p-value (as computed by scipy.stats.pearsonr) are reported.