Abstract
Predictions of rates of inbreeding (ΔF), based on the concept of longterm genetic contributions assuming the infinitesimal model, are developed for populations with discrete or overlapping generations undergoing mass selection. Phenotypes of individuals are assumed to be recorded prior to reproductive age and to remain constant over time. The prediction method accounts for inheritance of selective advantage both within and between age classes and for changing selection intensities with age. Terms corresponding to previous methods that assume constant selection intensity with age are identified. Predictions are accurate (relative errors ≤8%), except for cases with extreme selection intensities in females in combination with high heritability. With overlapping generations ΔF reaches a maximum when parents are equally distributed over age classes, which is mainly due to selection of the same individuals in consecutive years. ΔF/year decreases much more slowly compared to ΔF/generation as the number of younger individuals increases, whereas the decrease is more similar as the number of older individuals increases. The minimum ΔF (per year or per generation) is obtained when most parents were in the later age classes, which is mainly due to an increased number of parents per generation. With overlapping generations, the relationship between heritability and ΔF is dependent on the age structure of the population.
IN the absence of selection and with a Poisson distribution of family size, expected rates of inbreeding are related directly to the number of parents: E(ΔF) ≈ 1/8N_{m} + 1/8N_{f} (Wright 1969, p. 212). In selected populations, however, superior families contribute more offspring to the next generation than average families. This increases the rate of inbreeding of a selected population compared to an unselected population. Prediction of rates of inbreeding in selected populations is difficult, because selection decisions are correlated over generations due to the inheritance of selective advantage. Methods accounting for only one or two generations of selection (e.g., Burrows 1984a,b) therefore generally underestimate the rate of inbreeding (Wrayet al. 1990; see Caballero 1994 for a review).
Two approaches to prediction of rates of inbreeding for selected populations can be distinguished. First, rates of inbreeding can be predicted on the basis of the variance of allele frequency, using the idea of accumulation of selective advantages over generations (Robertson 1961). Using this approach and equilibrium genetic variances, Santiago and Caballero (1995) obtained accurate predictions for populations with discrete generations under mass selection. Nomura (1996) extended that method to populations with overlapping generations and equal numbers of parents per sex in every age class. Second, rates of inbreeding can be predicted using the concept of longterm genetic contributions. Rates of inbreeding are proportional to the sum of squared longterm genetic contributions of ancestors (Wray and Thompson 1990). Wray and Thompson (1990) obtained accurate predictions of rates of inbreeding for populations with discrete generations under mass selection, using iterative regression methods. For discrete generations and mass selection a closed form expression was obtained by Woolliams et al. (1993). For more complicated selection schemes, however, predictions became unmanageable due to the recursive nature of the procedure and the need for predicting the variance of longterm genetic contributions (Wrayet al. 1994).
Recently, Woolliams and Bijma (2000) showed that the variance of longterm genetic contributions is related to their squared expectation, making a separate prediction of the variance redundant. Furthermore, Woolliams et al. (1999) obtained general predictions of expected genetic contributions using equilibrium genetic variances instead of second generation genetic variances (Woolliamset al. 1993). Using the approach of Woolliams et al. (1999), Bijma and Woolliams (1999) obtained accurate predictions of genetic contributions for populations with overlapping generations under mass or sibindex selection. However, they did not develop predictions for rates of inbreeding for those schemes.
The aim of this article is twofold. First, explicit prediction equations for rates of inbreeding in populations with discrete or overlapping generations under mass selection are developed, on the basis of the theory of Woolliams et al. (1999) and Woolliams and Bijma (2000). These predictions are valid for any distribution of parents across age classes, overcoming the restriction of Nomura (1996), to give a general and practical method for mass selection with overlapping generations. These methods are compared to methods of Santiago and Caballero (1995) for discrete generations, and to methods of Nomura (1996) for the special case of equal numbers of parents per age class with overlapping generations. The accuracy of predictions is examined using simulation. Second, relationships between rates of inbreeding and genetic or population parameters are examined, and differences between populations with discrete and overlapping generations are presented and discussed.
DERIVATION OF EXPRESSIONS
Population model: This section describes the population and the selection procedures for which rates of inbreeding are predicted. This model is also used in the simulation. The trait considered is assumed to be determined by an infinite number of additive loci, each having an infinitesimal effect (infinitesimal model; Fisher 1918). Phenotypic values are the sum of additive genetic values (breeding values) and environmental values, P = A + E. A population with either discrete or overlapping generations under mass selection is modeled. With parents up to a maximum age of c_{max} there are 2c_{max} categories, one for each sex and age of parent. Categories are indexed by k or by l, so k = 1 … c_{max} are males, and k = c_{max}+1 … 2c_{max} are females. With discrete generations, there are only two categories: males and females that are indexed by s = m or f. Before reproductive age, phenotypes of individuals are recorded and remain unchanged over time, so that ranking of individuals within categories is constant over time. Within categories, individuals are ranked on their phenotype and each year the highestranking n_{k} individuals are selected from the kth category, to produce the next cohort. The total number of male and female parents of each cohort is
General: The prediction of ΔF is based on the concept of longterm genetic contributions (James and McBride 1958). The longterm genetic contribution (r_{i}) of ancestor i in cohort t_{1} is defined as the proportion of genes present in individuals in cohort t_{2} deriving by descent from i, where (t_{2} − t_{1}) → ∞ (Woolliamset al. 1993). In the remainder of the current article, longterm genetic contributions of ancestors are referred to as “genetic contributions,” or simply as “contributions.”
Rates of inbreeding are predicted from Woolliams and Bijma (2000),
To calculate
Rates of inbreeding are predicted in three steps. First, expected genetic contributions are predicted using the method of Woolliams et al. (1999). Second,
The difference between the current prediction and the method of Woolliams et al. (1993) is (1) the current prediction is based on equilibrium genetic variances, which simplifies the prediction of u_{i,s} (Woolliamset al. 1999); (2) the variance of genetic contributions is not predicted separately, since it is related to the mean (Woolliams and Bijma 2000).
Discrete generations
Step 1: prediction of expected longterm genetic contributions: Expected genetic contributions of ancestors are obtained from the linear model (Bijma and Woolliams 1999),
Step 2: derivation of
Step 3: Correction of E(ΔF) for deviations of V_{n} from
Poisson variances: With fixed n_{o}, family size follows a hypergeometric distribution (Burrows 1984b) and a correction is required according to the second term of Equation 1. In this article, the hypergeometric variance is approximated by a binomial variance, which simplifies the prediction. For more complicated selection strategies, e.g., index selection, a hypergeometric variance may be required (Woolliams and Bijma 2000).
With discrete generations, the second term of Equation 1 reduces to 1/8[N_{m}Δ_{m} + N_{f}Δ_{f}], where Δ_{s} = α^{T}V_{n(s),dev}α, α^{T} = (α_{m} α_{f}) and V_{n(s),dev} is the 2 × 2 matrix of deviations of the (co)variance of family size from Poisson variances for sex s (Woolliams and Bijma 2000). Diagonal elements of V_{n(s),dev} are obtained as V_{n(s),dev} = V_{n(s)− Vn(s),Poisson}, which are of the form np(1 − p) − np = −np^{2}, where n is the number of candidates and p is the selected proportion. Offdiagonal elements of V_{n(s),dev} are zero. For discrete generations the total correction (appendix a) equals
Relation to Santiago and Caballero (1995): The prediction equation of Santiago and Caballero (1995) can be related directly to the current prediction. With random mating and assuming α_{I,s} = α_{O} = 0 (see Santiago and Caballero 1995 for notation), Equations 21 and 36 of Santiago and Caballero (1995) reduce to
The correction for deviations of V_{n} from Poisson variances can also be related to Equation 36 of Santiago and Caballero (1995). They use V_{n(s)}(s′,s′) = N_{s′}/N_{s}[1 − N_{s′}/ñ_{s}N_{s}] (see Santiago and Caballero 1995, Equation 30 and ignore the term
Overlapping generations
Step 1: prediction of expected longterm genetic contributions: Genetic contributions are predicted using Equation 2 again, but now categories refer to sexage class combinations, which are indexed by k instead of s, so that k = 1 … 2c_{max} and u_{i}_{,}_{k} is the expected genetic contribution of individual i originating from its selection in category k. Solutions for α_{k} and β_{k} are obtained from Woolliams et al. (1999),
Contributions predicted from Equations 9 and 10 are per year; i.e., they are the longterm contribution of a single cohort, not of a total generation. Rates of inbreeding predicted from these contributions therefore are also per year.
Step 2: derivation of
The scalar equivalent of the first term of Equation 1 is
As with discrete generations,
Crossproducts in Equations 11 and 12 arise only from the individuals selected in both categories, which are all the individuals selected from the smallest category [i.e., min(n_{k},n_{l})]. Crossproducts are therefore
Step 3: correction of E(ΔF_{y}) for deviations of V_{n} from Poisson variances: The second term of Equation 1 is ⅛1^{T}NΔ, where Δ is a 2c_{max} vector of elements Δ_{k} = α^{T}V_{n(k),dev}α, and V_{n(k),dev} is a 2c_{max} × 2c_{max} matrix with deviations from Poisson variances (Woolliams and Bijma 2000). Similar to the discrete generation case, V_{n(k)} is approximated by a binomial variance. Elements of V_{n(k),dev} and a numerical example are in appendix b.
Relation to Nomura (1996): Nomura (1996) developed predictions for the special case of equal numbers of parents per sex selected from every age class (denoted n_{m} and n_{f}), i.e., for constant selection intensity with age. With those schemes every parent is selected in every category (except for categories with zero parents) and there are only two exclusive categories; i.e., males selected always and females selected always. In this respect, schemes with equal numbers of parents selected from every age class are like discrete generations, i.e., only two categories that do not compete for being selected. Bijma et al. (1999) show that Equations 30 and 31 of Nomura (1996) reduce to
Furthermore, Nomura (1996) calculated Q using (I − P)^{−1} [P is a gene flow matrix identifying the contribution of parental age groups to selected offspring multiplied by the proportion of genetic variance remaining after selection; Nomura (1996, Appendix)], which, for his special case, is equivalent to our (I − G^{T}*π^{T})^{−1} (see Woolliamset al. 1999, Equation 10). Analogous to Santiago and Caballero (1995), Nomura (1996) calculated
Stochastic simulation: To examine the accuracy of the prediction equations, the breeding scheme described in the “population model” section was simulated and rates of inbreeding were calculated from simulated data. The simulation procedure is described in Bijma and Woolliams (1999). In the simulated data, an ancestor cohort t_{1} and a descendent cohort t_{2} were chosen (Bijma and Woolliams 1999). Inbreeding coefficients of individuals in cohorts t_{1} and t_{2} were calculated from the simulated pedigree, using the algorithm of Meuwissen and Luo (1992). Rates of inbreeding per year were calculated as
RESULTS
Discrete generations: For examination of the accuracy of predictions and to identify the origin of prediction errors, Table 1 shows simulated and predicted ΔF. Two types of predictions are in Table 1: ΔF_{pred}* is the prediction using α and β estimated from simulation, and ΔF_{pred} is the full deterministic prediction using α and β from Equations 9 and 10. Differences between ΔF_{pred} and ΔF_{pred}* reflect prediction errors originating from the prediction of β [in discrete generations, α_{s} = 1/(2N_{s}) is known]. Differences between ΔF_{sim} and ΔF_{pred}* reflect errors in Equation 1.
Generally, errors of the full prediction in Table 1 are small, most errors are below 5%, maximum errors are up to 8.1%, and trends agree well between simulations and predictions. Though errors are small, some trends can be observed. Most errors are positive and errors tend to be highest for N_{m} = 10, but errors tend to be negative for n_{o} = 8 and N_{m} = 100. Prediction errors are partly due to errors in the prediction of β; i.e., ΔF_{pred}* is generally more accurate than ΔF_{pred}. Because we have approximated the hypergeometric variance of family size by a binomial variance, positive errors for small numbers of parents were expected. The correction for hypergeometric variances becomes larger with fewer parents (Burrows 1984b), whereas a binomial correction is unaffected by the number of parents. Because the correction is a negative value, a binomial correction results in an overprediction for small numbers of parents. The current prediction was compared to the prediction of Santiago and Caballero (1995). As expected from the close agreement between equations of both methods, both methods gave very similar results (Bijmaet al. 1999).
Figure 1 shows the relationship between ΔF and heritability
Figure 1 shows that ΔF has a maximum for intermediate heritabilities (except for n_{o} = 2), and changes in ΔF are more pronounced with greater selection intensity. The maximum of ΔF for intermediate h^{2} is due to the Bulmer effect. When the Bulmer effect is ignored in Equation 7 (i.e., κ = 0) the rate of inbreeding increases with h^{2} over the whole range. The logic behind this is that with increasing h^{2} the reduction of betweenfamily variance increases, reducing the importance of the family component in the phenotype. [Note also that the intraclass correlation between full sibs [ρ = ½ h^{2} (1 − ρh^{2})] has a maximum for
With n_{o} = 2, one male and one female offspring are selected from every pair of parents, which gives zero variance of family size, β = 0, and minimal inbreeding. Expected longterm genetic contributions are equal for all parents and the variance of the contributions is zero; i.e., expected and realized contributions are equal. The absence of variance of family size with n_{o} = 2 is taken into account by the correction of ΔF for deviations of V_{n}
from Poisson variances. Without this correction, ΔF_{pred} is equal to a situation with
With higher selection intensities (n_{o} = 8 or 32), ΔF increases considerably with heritability. For example, for h^{2} = 0.6, ΔF increases by 54% compared to random selection (i.e., h^{2} = 0) for n_{o} = 8, and by 105% for n_{o} = 32. The large increase of ΔF with selection intensity originates from the regression of the number of selected offspring on the breeding value of the parent, which is linear in i
For practical selection intensities (n_{o} = 2, 8), there is close agreement between ΔF_{pred} and ΔF_{sim}. For large selection intensities errors are larger (e.g., for n_{o} = 200, N_{m} = N_{f} = 40 and h^{2} = 0.4, an error of −18% was found). Large errors for extreme selection intensities do not undermine the general theory, i.e., Equation 1 is still valid, but the linear model (Equation 2) may be insufficient to predict expected genetic contributions (Woolliams and Bijma 2000).
Overlapping generations: Table 2 shows simulated and predicted rates of inbreeding per generation and generation intervals. Predictions of ΔF using α and β from simulation (such as ΔF_{pred}* in Table 1) are not included, because standard errors on β were too large to draw conclusive inferences. Because the potential number of alternative schemes is very large with overlapping generations, a wide range of schemes was evaluated. Only schemes 1, 3, 5, 6, and 7 are within the scope of Nomura (1996). Schemes 1–5 represent a situation with two age classes with gradually increasing ages of females. Scheme 6 is similar to scheme 5 but with a mating ratio of two. Scheme 7 has equal numbers of parents in all categories. With schemes 8 and 9, parents were ranked on estimated breeding value
Generation intervals are systematically underpredicted in Table 2 (except for schemes with only one reproductive category per sex in which case L is fixed; schemes 1, 5, and 6). The underprediction is entirely explained by the way L_{sim} is calculated, i.e.,
Results from the current prediction were compared to results from the prediction of Nomura (1996) for the special case of equal numbers of parents selected from every age class. [A comparison was made for all schemes presented by Nomura (1996).] As expected from theory, results from both methods were similar (Bijmaet al. 1999).
Relationship between ΔF and distribution of parents over age classes: Figure 2 shows the relationship between the rate of inbreeding (per year and per generation) and the proportion of parents selected from the second age class (p_{2}), for a population with two age classes, N_{m} = N_{f} = 20,
For random selection, Hill (1979) showed that the rate of inbreeding in overlapping generations is related to the lifetime variance of family size and the number of parents entering the population per generation. The same pattern can be observed in Figure 2, which shows that ΔF_{L} has a maximum when parents are equally distributed over age classes, i.e., for N = diag{10, 10, 10, 10}, where the 10 parents selected in age class 1 the first year are the same as the 10 parents selected in age class 2 the next year. Thus only 10 distinct parents are selected from every cohort for this scheme, and with L = 1.41 the number of parents entering the population per generation equals only 14.1. For N = diag{20, 0, 20, 0}, 20 distinct parents are selected from every cohort and with L = 1, 20 parents enter the population per generation. The rate of inbreeding per generation reaches a minimum for p_{2} = 0.95 (N = diag{1, 19, 1, 19}). At first glance, this result is counterintuitive; i.e., one might expect approximately equal rates of inbreeding per generation for N = diag{19, 1, 19, 1} and for N = diag{1, 19, 1, 19}. However, for N = diag{1, 19, 1, 19}, 19 distinct individuals are selected from every cohort and, with L = 1.90, the number of parents per generation equals 36.1.
Line subdivision and migration: As mentioned earlier, the scheme with N = diag{0, 20, 0, 20} has two nonmixing lines. Changing this scheme to N = diag{1, 19, 1, 19} is equivalent to allowing some migration between both lines. Figure 3 shows a comparison between full line subdivision, line subdivision with migration, and one single line for schemes with 2 or 3 age classes. Note that the total number of parents per year is equal per comparison. The comparison shows that allowing some migration between lines substantially reduces ΔF_{L} (i.e., 0.0104 vs. 0.0141 and 0.0075 vs. 0.0141). The smallest ΔF is obtained when lines are joined together ({40, 40} with a cohort interval of 2 years and {60, 60} with a cohort interval of 3 years). When comparing these rates of inbreeding, it must be realized, however, that the schemes with full line subdivision accumulate a betweenline genetic variance equal to
Relationship between ΔF and heritability: Figure 4 shows the relationship between
Rates of inbreeding per year can be obtained from
Figure 4 as ΔF_{Y} = ΔF_{L}/L, which shows the same trends with
DISCUSSION
Explicit prediction equations for rates of inbreeding in populations with either discrete or overlapping generations under mass selection were developed, on the basis of the approach of Woolliams and Bijma (2000) and Woolliams et al. (1999). Except for extreme selection intensities in females, predictions were accurate for discrete as well as for overlapping generations. Though based on a different approach, the current method extends the method of Nomura (1996) to populations with overlapping generations and an arbitrary distribution of parents across age classes, removing the stringent restriction of Nomura (1996). Relationships between rates of inbreeding and genetic and population parameters were also presented. General relationships apparent in discrete generations could not be extended to overlapping generations. For the prediction of rates of inbreeding in overlapping generations it is crucial to account for the inheritance of selective advantage both between and within categories. For discrete generations with only two categories (males and females), which do not compete for selection, only competition between selection candidates within categories is relevant.
The current method was compared to methods based upon the proportion of genetic variance transmitted to the offspring, which showed that with random mating, the equations of both Santiago and Caballero (1995) and Nomura (1996) can be reduced to simple expressions in terms of expected genetic contributions. Santiago and Caballero (1995) suggested that the differences between their results using the reduced genetic variance and those of Woolliams et al. (1993) using longterm contributions were due to the difference in approach. The present results show that the differences obtained previously were most likely due to errors in the derivations involving complex pathways over multiple generations that were needed by Woolliams et al. (1993). These complexities were avoided by Santiago and Caballero (1995). However, Woolliams and Bijma (2000) were able to derive the present results using longterm contributions by modeling the transfer of selective advantages in a single generation by assuming an equilibrium. The idea of basing the prediction on Bulmer's equilibrium variances was introduced by Santiago and Caballero (1995). However, their approach to modeling the inheritance of selected advantage by the proportion of genetic variance retained is correct only for mass selection [see Woolliams et al. (1999) for a general approach].
Prediction errors became large when the number of selection candidates per dam became extremely large (Figure 2), but these situations are out of the range of most artificial selection programs. Certain species (e.g., fish or chicken) are able to produce many offspring per dam, but the number of selection candidates per dam is generally lower. High selection intensities in males can easily be obtained with a limited number of selection candidates per dam when the mating ratio is large. For these situations predictions were accurate (see Table 1, schemes with d = 5, n_{o} = 8 → i = 2.063). The errors with large n_{o} were not present for low h^{2} (results not shown), which indicates that the current method is also applicable to species with a large number of offspring when natural directional selection acts on a trait with low heritability.
In this article, equations for predicting rates of inbreeding were developed assuming a model of truncation selection on a normally distributed trait controlled by an infinitesimal model of gene effects. The predicted rate of inbreeding relates to homozygosity (by descent) at a neutral locus, unlinked to genes affecting the trait under selection (Woolliams and Bijma 2000). When the infinitesimal model does not hold, and the number of genes affecting the trait is large, or when the number of chromosomes is small, it is questionable whether neutral and unlinked loci exist at all. When loci are nonneutral, or linked to nonneutral loci, predicted rates of inbreeding cannot be related directly to the homozygosity at the locus, because a covariance between the genetic contribution and the gene frequency will arise due to selection (Woolliams and Bijma 1999). However, the rate of inbreeding can still be related to rates of inbreeding obtained by analyzing pedigrees using Wright's (1922) path coefficient method, or Malecot's (1948) coefficient of kinship, and also to estimates of inbreeding depression based on inbreeding levels calculated from the pedigree. Recently, Santiago and Caballero (1998) extended prediction methods for effective population size to populations with linked loci undergoing mass selection but for discrete generations only.
In general, to obtain accurate predictions of ΔF one needs to account for more than one generation of inheritance of selective advantage between categories. It was sufficient for Nomura (1996) to account for only a single generation because of the special case of equal numbers of parents per age class. In that case, shifting contributions between age classes has only a minor effect on ΔF because the contributions will remain with the same individuals with the same relative fitness, because every individual is selected in every category. Therefore the lifetime contribution will not be affected. For schemes where the number of parents differs between age classes, shifting of contributions between categories means shifting to other individuals (at least partly), which will affect the lifetime contribution. Consider, for example, scheme 10 in Table 2 with
The use of the concept of longterm genetic contributions to predict rates of inbreeding has several appealing properties. First, the derivation of the relationship between rates of inbreeding and genetic contributions is based directly on the probability of alleles being identical by descent, which enhances the intuitive understanding (Woolliams and Bijma 2000). Furthermore, rates of genetic gain can easily be obtained from the covariance between the genetic contribution and the Mendelian sampling component of the breeding value (Woolliams and Thompson 1994; Woolliamset al. 1999), which integrates methods for predicting genetic gain and rates of inbreeding. Finally, the prediction procedure for genetic contributions describes mechanisms determining the impact of current individuals on future populations and the turnover rate of genes and gives therefore an understanding of the mechanisms determining the development of the pedigree (Woolliamset al. 1999; Bijma and Woolliams 1999). Because the approach is general, it is clear how prediction equations can be extended to other situations.
With a fixed total number of parents selected per year, populations showed maximum rates of inbreeding (per year and per generation) when the number of parents entering the populations per generation was least, which occurred with an equal number of parents in every age class. Rates of inbreeding were smallest when most parents were in the older age classes, because those schemes had the largest number of parents entering the population per generation. This result broadly resembles the results of Hill (1974) for random selection in overlapping generations, although selected populations have an additional component of inbreeding arising from the expected differential contributions within age classes, which will modify this relationship. Schemes with most parents in the later age classes resembled population subdivision with some migration between lines. Because the selective advantage of categories depends on heritability, genetic contributions of categories are strongly affected when heritability changes (Bijma and Woolliams 1999); i.e., contributions generally shifted to the younger age classes when heritability increased. Therefore, no general relationship between heritability and rate of inbreeding could be observed with overlapping generations.
In this article, equations were developed to predict rates of inbreeding for diploid populations with two sexes under controlled selection. The results are therefore primarily relevant for populations under artificial selection, for example, in animal breeding or in selection experiments. Though this article focuses on mass selection within age classes, results for mass selection across age classes can easily be accomplished by choosing the appropriate N, as in schemes 8 and 9 in Table 2. An extension to a situation where individuals in older age classes have more information, e.g., progeny information, only requires the calculation of probabilities of selecting the same individual on different ages, which can be done using standard index theory. The method can also be extended to other selection strategies and modes of inheritance (e.g., index selection and imprinting), using the key results of Woolliams and Bijma (2000) and Woolliams et al. (1999).
In animal breeding, optimization of breeding programs has focused for a long time on the maximization of genetic gain for the short term, partly because methods to predict longterm response were not available. When rates of inbreeding in selected populations can be predicted, predictions of longterm response under the infinitesimal model become available. This article enables methods for the optimization of breeding schemes on the long term (e.g., Villanuevaet al. 1996; Villanueva and Woolliams 1997) to be extended to populations with overlapping generations and mass selection.
Acknowledgments
J.A.W. gratefully acknowledges the Ministry of Agriculture, Fisheries and Food (United Kingdom) for financial support. Ab Groen is acknowledged for giving useful comments on the manuscript, and Tetsuro Nomura for generously sending us a copy of his programs. This research was financially supported by the Netherlands Technology Foundation (STW) and was coordinated by the Earth and Life Science Foundation (ALW).
APPENDIX A: DISCRETE GENERATIONS
Derivation of Equation 6: Starting from Equations 3 and 4, and substituting
Derivation of Equation 8: With a binomial distribution of family size, the deviation from a Poisson variance equals np(1 − p) − np = −np^{2}, where n is the number of candidates (½n_{o}d for sires and ½n_{o} for dams) and p is the selected proportion [1/(½n_{o}d) for male offspring and 1/(½n_{o}) for female offspring]. Elements of V_{n(s)dev} are therefore V_{n(m)dev} =[−1/(½n_{o}d), 0; 0, −d/(½n_{o})] and V_{n(f)dev} =[−1/(½n_{o}d^{2}), 0; 0, −1/(½n_{o})]. From Δ_{s} = α^{T}V_{n(s)dev}α it follows that
Example. For N_{m} = 20, N_{f} = 60, n_{o} = 8, and h^{2} = 0.4, selected proportions, selection intensities, and variance reduction coefficients are p_{m} = 0.083, p_{f} = 0.250, i_{m} = 1.839, i_{f} = 1.271, i = 1.555, κ_{m} = 0.839, κ_{f} = 0.759, κ = 0.799. Bulmer's (1971) equilibrium genetic variance and heritability are
APPENDIX B: OVERLAPPING GENERATIONS
Corrections for deviations of V_{n(k)} from Poisson variances: From Equation 1, the correction equals 1/81^{T}NΔ, where Δ is a 2c_{max} vector of elements Δ_{k} = α^{T}V_{n(k)dev}α, and where V_{n}_{(}_{k}_{)dev} is a 2c_{max} × 2c_{max} matrix with deviations of V_{n}_{(}_{k}_{)} from Poisson variances (Woolliams and Bijma 2000). Similar to the case of discrete generations, deviations from Poisson variances are −np^{2}, where n is the number of candidates (½n_{o}d for sires and ½n_{o} for dams) and p is the selected proportion. The selected proportion in subclass kl, i.e., among offspring in category k descending from parents in category l, equals
Example. For N = diag{12, 8, 15, 25},
For N = diag{12, 8, 15, 25} there are four exclusive categories: (1) males selected both at 1 and 2 years of age (i.e., the eight highestranking males), for which E[u^{2}_{i,s=1}] = E[u^{2}_{i,k=1} + u_{i,k=2})^{2}]; (2) males selected only at 1 year of age (i.e., males ranking 9–12) for which
From Equation 14,
Using Equation 13 (note the reordering) with N_{o} = diag{12, 8, 25, 15} and
The correction for deviations of V_{n(s)} from Poisson variances, for sires in age class one to selected male offspring in age class two, is
Footnotes

Communicating editor: R. G. Shaw
 Received March 30, 1999.
 Accepted December 6, 1999.
 Copyright © 2000 by the Genetics Society of America