MARKER-BASED SELECTION WITHIN FAMILIES

The situation considered in each family is such that the alleles present at one QTL in the two parents are unequivocally identified by marker genes, and these are located at such a short distance from the QTL that no recombination occurs. Then, if the four genes present in the parents are called (ab) and (cd), four types of offspring are obtained: (ac), (ad), (bc), (bd). The coancestries among those four classes of offspring, conditional on the markers, are easily obtained knowing the number of marker genes in common between any two marker classes. From the standard definition of the coancestry coefficient, when two offspring have received the same two markers (a and c for example), the probability that a random gene from one is identical with a random gene from the other, i.e., their coancestry, is 0.5. Similarly, when two offspring have received only one marker in common (e.g., a), the corresponding coancestry is 0.25. The coancestries obtained can be arranged as shown in Table 1.

Assuming a phenotypic variance of 1 for the trait considered, and otherwise using the notation of Lande and Thompson (1990), we define h², p, r_nh² and t_n to be the heritability of the trait, the proportion of the additive genetic variance due to the QTL, or marked segment, and the genetic and phenotypic variances of family means, respectively. The within-family variance due to the QTL (or marked genetic variance) then is M = (1 − r_n)ph². The variance-covariance matrices of within-family phenotypic and genetic values, P and G, can be written as functions of M and t_n, assuming one individual in each class, i.e., a family size of n = 4, by application of the general expression of the covariance between relatives (e.g., Ollivier 1981, p. 28) P=(1−tn)(10.5M0.5M00.5M100.5M0.5M010.5M00.5M0.5M1)andG=M(10.50.500.5100.50.5010.500.50.51). The above array of coancestries can be extended to any number (m) of dams mated to the same sire, the number of categories of offspring and the size of the P and G matrices being 4m in the general case. These matrices have the same general structure as above, provided t_n and r_n, for full-sib families of size n, be replaced by their corresponding values t_mn and r_mn for full-sib families of size n nested within half-sib families of size mn, with phenotypic correlations t₁ and t₂ among full-sibs and half-sibs, respectively, tmn=[1+(n−1)t1+n(m−1)t2](mn)−1rmn=[1+0.5(n−1)+0.25n(m−1)](mn)−1.

In the more general case of n > 4, given that M is the phenotypic correlation among members of the same marker class and assuming n/4 individuals per class, the phenotypic variance of a class mean is (1 − t_mn)[M + 4 (1 − M)n⁻¹]. The correlation between class means is then either zero or t = 0.5M[M + 4(1 − M)n⁻¹]⁻¹, and general expressions for P and G can be established as given in the appendix (ignoring the negligible predictive value of the deviation from class mean).

The variance-covariance matrix of the predictors of breeding values is obtained, using classical selection index theory, as the product GP⁻¹G, whose diagonal terms yield the variance of the index (I). Applying the rules of inversion of partitioned matrices for obtaining P⁻¹ (Searle 1982), an explicit expression of GP⁻¹G can be obtained as shown in the appendix, and Var(I)=(1−tm)−1Mt[1+(a−b+c)(a+2mb)−1] (1a) where: a=(1−2t)(1−t)−1 (1b) b=t[1+t2(1−t)−1(1−t+2mt)−1] (1c) c=m+(2m+1)t2(1−t)−1(1−t+2mt)−1, (1d) and the accuracy of I is ρ=Var(I)∕h2 .

The particular case of full-sib families implies m = 1, r_mn = r_n and t_mm = t_n. The previous equations then considerably simplify and (1a) becomes Var(I)=(1−tn)−1Mt[1+2(1+2t)−1], (2) which yields the corresponding accuracy ρ₁, the index 1 referring to a within full-sib family selection.

The previous predictions apply when the 4m categories of offspring are equally represented, which requires n ≥ 4. With lower values of n, various structures of the P and G matrices occur, with given probabilities. Only the case of n = 1, which corresponds to a particularly simple situation, will be considered here. When n = 1, each half-sib family is made up of two groups of individuals, receiving either gene present in the sire. The coancestry conditional on the markers is then either 0.25 between offspring belonging to the same group, or zero between offspring belonging to different groups. The predictor is therefore equivalent to a combined selection index, with group size m/2, intragroup phenotypic correlation and genetic covariance both equal to t = 0.5M, and phenotypic variance 1 − t_m. The variance of such an index (e.g., from Ollivier 1981, p. 79) is Var(I)=(1−tm)−1M2[1+(0.5m−1)(0.5−t)2(1−t)−1(1−t+0.5mt)−1] (3) which for large values of m reduces to Var(I)=0.5(1−t2)−1M(1−0.5M)−1. (4)

Table 2 gives values of selection accuracy relative to individual phenotypic selection for various half-sib family sizes. For moderate values of family sizes and p, this type of selection is rather inaccurate. In a typical litter-bearing species, with m = n = 8, it can be seen that p must exceed 0.5 for marker accuracy to exceed individual phenotype accuracy (h). As m increases, less and less is gained from increasing n. For low family sizes, relative accuracy is not much influenced by heritability, whereas for a large number of m or n it is approximately halved when h² increases from 0.10 to 0.50 (see the columns “large” in Table 2). The particular case of full-sib families is presented in Table 3, which shows that low values of relative accuracy are reached, unless extreme and rather unrealistic values of n and p are assumed.

It should be noted that in situations of linkage disequilibrium selection accuracy does not depend on the family structure and, as Table 2 shows, attains values which can never be reached under linkage equilibrium, even with extremely large family sizes. It can be seen that when both m and n become very large in Equation 1a Var(I) → 0.75 ph² (1 − t₂)⁻¹, and when n becomes very large in Equation 2 Var(I) → 0.5 ph² (1 − t₁)⁻¹. The corresponding maximum accuracies, 0.75p(1−t2)−1 , respectively, can be compared to 0.5p(1−t1)−1 applying in situations of linkage disequilibrium.

APPLICATION TO MACS

Once the marker index is defined, a selection index combining phenotypic measurements on the individual and its relatives with this marker index can be established. When only one type of family is considered, the MACS situation of Lande and Thompson (1990, Appendix I) can easily be adapted to the present situation, by replacing the quantity ρ in their notation by ρ²h², as defined from our Equations 1 and 2. The overall accuracy (R) of the MACS index then is RMACS=hρ2h−2+rmn2tmn−1+(1−rmn−ρ2)2D−1 (5a) where D=1−tmn−ρ2h2 (5b) assuming either m = 1 (full-sibs) or n = 1 (half-sibs). In the latter case, an optimal overall index combining marker and phenotypic information can alternatively be derived. The last term in Equation 5a then becomes (1 − r_n)(1 − t_n)⁻¹ .

For evaluating MACS accuracy in the general case of full-sib families nested within half-sib families, i.e., m and n both different from 1, it will be convenient to replace the overall marker index defined by Equation 1a by two independent marker indices, a within- and a between-full-sib family marker index, whose variances are defined by Equation 2 and the difference between 1a and 2, respectively, i.e., (1−rn)σm2 and (ρ² − ρ12h2 )h². By defining the proper trait variances of family means (t_mn and t_n) and their corresponding covariances r_mnh² and r_nh² with additive genetic values, the parameters necessary for obtaining the overall MACS index are Q, the phenotypic variance-covariance matrix of the 5 predictors, and V, the vector of the covariances between the predictors and the individual breeding value Q=(tmntn−tmn(ρ2−ρ12)h2(ρ2−ρ12)h2(ρ2−ρ12)h2(1−tn)ρ12h2ρ12h2ρ12h2) and V=(rmnh2(rn−rmn)h2(ρ2−ρ12)h2(1−rn)h2ρ12h2) The accuracy of the MACS index in the general case then is (V′Q⁻¹V)^0.5h⁻¹, derived by extension of the procedure outlined in Appendix I of Lande and Thompson (1990) for 4 predictors RMACS=hρ2h−2+rmn2tmn−1+(rn−rmn−ρ2+ρ12)2D1−1+(1−rn−ρ12)2D2−1 (6a) where D1=tn−tm−(ρ2−ρ12)h2 (6b) D2=1−tn−ρ1h2. (6c) This can be compared to the classical combined selection (CS) index in the same family structure (as given for instance, in a slightly different form, by Ollivier 1981, p. 80) RCS=hrmn2tmn−1+(rn−rmn)2(tn−tmn)−1+(1−rn)2(1−tn)−1. (7) The relative accuracy of MACS compared to standard CS is given in Table 4 for various genetic and family structure situations. As expected from the accuracies of Tables 2 and 3, large family sizes and high values of p are needed for MACS to be valuable in such situations. Table 4 also shows that for large full-sib families no gain is expected from increasing the number of dams per sire. In fact, Equations 6a and 7 are both independent of m when n is large. Somewhat paradoxically, this maximum relative accuracy can, however, be exceeded by increasing the number of dams, with one progeny for dam. The maximum gain, however, remains around 50% for h² = 0.10, assuming an infinite size of half-sib family and all QTLs marked (see Table 4).