Prediction of genetic gain from quadratic optimisation with constrained rates of inbreeding

There are selection methods available that allow the optimisation of genetic contributions of selection candidates for maximising the rate of genetic gain while restricting the rate of inbreeding. These methods imply selection on quadratic indices as the selection merit of a particular individual is a quadratic function of its estimated breeding value. This study provides deterministic predictions of genetic gain from selection on quadratic indices for a given set of resources (the number of candidates), heritability, and target rate of inbreeding. The rate of gain was obtained as a function of the accuracy of the Mendelian sampling term at the time of convergence of long-term contributions of selected candidates and the theoretical ideal rate of gain for a given rate of inbreeding after an exact allocation of long-term contributions to Mendelian sampling terms. The expected benefits from quadratic indices over traditional linear indices (i.e. truncation selection), both using BLUP breeding values, were quantified. The results clearly indicate higher gains from quadratic optimisation than from truncation selection. With constant rate of inbreeding and number of candidates, the benefits were generally largest for intermediate heritabilities but evident over the entire range. The advantage of quadratic indices was not highly sensitive to the rate of inbreeding for the constraints considered.


INTRODUCTION
Quadratic optimisation [7,8,10,11] provides a solution to the problem of optimising selection decisions in breeding schemes for maximising genetic gain (ΔG) with constrained rates of inbreeding (ΔF). The selection merit of individual candidates is a quadratic function of their estimated breeding values and therefore selection is based on quadratic indices. Stochastic simulation studies have shown that the extra response from quadratic indices over traditional linear indices (i.e. truncation selection) at the same ΔF is large, ranging from 20% to 60% (e.g. [10]). Also, substantial increases in ΔG are expected when applying quadratic indices to real livestock populations [1,9].
Although the benefits in terms of gain from quadratic optimisation are clear from empirical studies, there is not yet a framework available for obtaining deterministic predictions of ΔG when using this approach. Grundy et al. [7] showed that the ideal optimal solution for a given constraint on ΔF could be obtained after an exact linear allocation of long-term genetic contributions of selected candidates (r) to their Mendelian sampling terms (a). Using stochastic simulations, Avendaño et al. [2] empirically confirmed that quadratic optimisation allocates contributions of selected candidates according to the best information on their Mendelian sampling terms and not on their breeding values. This provided the link between the optimisation of breeding schemes using quadratic indices and the maximisation of the covariance between r and a implicit in the definition of genetic gain of Woolliams and Thompson [18] (i.e. E [ΔG] = r i a i ).
Grundy et al. [7] also provided a deterministic prediction for the ideal rate of genetic gain (ΔG ideal ) but the empirical gains obtained after applying quadratic optimisation were consistently lower than ΔG ideal . The lack of knowledge about the true Mendelian sampling term at selection time was one of the reasons of the inability of quadratic optimisation to attain the ideal optimal solution [20]. Recently, Avendaño et al. [3] have provided deterministic predictions of the accuracy of Mendelian sampling terms both at selection time and at convergence of long-term contributions that can prove to be useful for filling the gap in the prediction of gain from selection on quadratic indices.
The aim of this study was to develop a deterministic framework for predicting the rate of genetic gain under quadratic optimisation when BLUP is used to estimate breeding values. The approach combined predictions of the accuracy of the estimated Mendelian sampling term at the time of convergence of longterm contributions [3] with predictions of the ideal rate of genetic gain [7]. Predictions of gain from quadratic optimisation were then compared to those from traditional truncation selection and the expected benefits from using quadratic indices were quantified.

Ideal and upper bound genetic gain under a constrained rate of inbreeding
Assuming that Mendelian sampling terms are normally distributed with standard deviation equal to one, Grundy et al. [7] showed that the ideal theoretical rate of genetic gain (ΔG ideal ) can be defined in terms of the standardised truncation point (x) and the selection intensity (i) as where k = i(i − x) and i and x are the solution for being T the total number of candidates per generation (T/2 males and T/2 females), ΔF the desired rate of inbreeding, and p the proportion of selected candidates. Values for p, and hence x and i, can be found using the Newton-Rapson method for defined values of T ΔF (App. A). The variance of the Mendelian sampling terms is not reduced by selection through gametic phase disequilibrium (i.e. the Bulmer effect). Hence, ΔG ideal from expression (1) can be conveniently expressed in terms of the trait additive genetic variance in the base population (σ 2 A 0 ), resources available (T ) and the target ΔF as This assumes that breeding values and Mendelian sampling terms are known without error (i.e. the initial heritability, h 2 0 , is equal to one and the environmental variance equals to zero). It also assumes a constant Mendelian variance such that losses from inbreeding are restored by mutational variance. In other circumstances this will provide a medium term approximation.
It is valid to ask how the expression for ΔG ideal which implies an exact allocation of long-term contributions to Mendelian sampling terms compares to conventional formulae for predicting the rate of gain under standard truncation selection under the same circumstances of known breeding values and Mendelian sampling terms. The equilibrium rate of gain after accounting for gametic phase disequilibrium generated by selection [6] for mass truncation selection can be approximated as where σ 2 P and h 2 are respectively the phenotypic variance and heritability, both at equilibrium. With the assumption of h 2 0 = h 2 = 1 (i.e. breeding values are assumed known without error) then σ P = σ A , σ A 0 /σ A = √ 1 + k and the equation (4) can be re-written as Equation (5) shows that the form of ΔG ideal in equation (3) is not unexpected and that the expressions for the rate of genetic gain after linear allocation of Mendelian sampling terms and long-term contributions for constrained ΔF, and that for truncation selection are closely related. Importantly, it arises that for the same σ A 0 , ΔG m_tru < ΔG ideal . The validity of this inequality is shown in Figure 1, where both expressions for the rate of gain are compared at the same x and k (i.e. not at the same ΔF). Figure 1 shows that, assuming an exact allocation of long-term contributions to Mendelian sampling terms, and known Mendelian sampling terms for each selected candidate, ΔG ideal is always greater than ΔG m_tru when compared at the same truncation point. As expected, the difference between ΔG ideal and ΔG m_tru was reduced as x and k increased, approaching zero at the limit (k = 1). Typical values for x in practical livestock breeding programmes could range from −1.0 to 3.5, giving a corresponding range for k of 0.34 to 0.95. As pointed out by Grundy et al. [7] and Woolliams et al. [20], one of the factors for which the ΔG ideal is unattainable is that the true Mendelian sampling terms are unknown, so contributions of selected candidates are allocated according to initial estimates at the time of selection. Grundy et al. [7] suggested an intuitive expression for predicting the upper bound genetic gain obtained under quadratic optimisation (ΔG quad ) that accounts for this lack of knowledge of the true Mendelian sampling terms: where ρ conv is the accuracy of the Mendelian sampling term at convergence of long-term contributions.

Predicting the effective number of parents at the time of selection
The population structure at the time of selection is required for obtaining the predictions for ρ conv from the extended pseudo-BLUP index of Avendaño et al. [3]. In their study, the 'effective number of parents' of equal contributions [12] at selection time was calculated empirically (i.e. from simulations) as N c = c 2 −1 , where the sum of squares of contributions was taken over all selected candidates resulting from the optimisation. The population structure at selection time was derived as n s = n d = 1 2 N c and n o = T/n d , where n s , n d and n o are respectively the number of sires, the number of dams and the number of offspring per dam. A mating ratio equal to one was therefore assumed.
Since the primary objective here was to obtain a completely deterministic prediction for ΔG quad , an approach was derived for predicting N c . The approach was based on predictions for the ratio N c /N r , where N r is the 'effective number of ancestors'. The ratio N c /N r measures the change in the equivalent number of parents from initial selection that are still represented when their contributions converge. The effective number of ancestors can be derived directly from the inbreeding constraint as N r = [4ΔF] −1 given that E(ΔF) = 1 4 r 2 i [22]. Therefore, N c could be obtained from a prediction of N c /N r and the ΔF constraint. The prediction approach for the ratio N c /N r is described in Appendix B.

Predicting the accuracy of the Mendelian sampling term
Predictions for the accuracy of the Mendelian sampling term needed to predict ΔG quad from equation (6) were obtained using the approach of Avendaño et al. [3]. Briefly, a selection index theory framework was used where the original pseudo-BLUP index of Wray and Hill [21], which assumed a nested mating structure with m dams mated to each sire, was extended to accommodate three extra sources of information related to the Mendelian sampling term.
Here a mating ratio m = 1 was assumed. The three sources were the estimated Mendelian sampling term of the dam, the mean estimated Mendelian sampling term of all dams mated to the sire, and the estimated Mendelian sampling term of the sire. This extension allowed the accuracy of the Mendelian sampling term to be predicted for selected candidates just after being selected (ρ sel ), and after becoming sires (ρ sir ) or dams (ρ dam ) with recorded offspring. The latter was considered to be the accuracy at the convergence of long-term contributions. With a mating ratio equal to one, ρ sir = ρ dam = ρ conv .

Simulations
The rate of gain ΔG quad predicted from equation (6) was compared to the empirical responses obtained from stochastic simulations (ΔG sim ) over several generations. An additive infinitesimal model [6] was assumed for the trait under selection. The true breeding values of base population individuals were obtained from a normal distribution with mean zero and variance equal to the initial heritability (h 2 0 ) so the phenotypic variance was assumed equal to one. In subsequent generations, the true breeding value of the progeny was obtained as half the sum of the true breeding values of their parents plus a random Mendelian sampling term. The Mendelian sampling term of an individual was sampled from a normal distribution with mean zero and variance 1 2 h 2 0 (i.e. the effects of inbreeding on the genetic variance were ignored). The phenotypic value for any individual at any generation was obtained by adding an environmental component sampled from a normal distribution with mean zero and variance 1−h 2 0 to the true breeding value. Populations with discrete generations were evaluated over 10 generations of selection. In the base generation (t = 0), T individuals (T/2 males and T/2 females) with family full-sib structure were generated. The first generation of selection was obtained from the mating of animals selected at t = 0.
Selection decisions were optimised for each generation by using the optimisation algorithm described by Meuwissen [10] which gives the maximum genetic gain while constraining the inbreeding rate to a specific value. A BLUP animal model was used to estimate breeding values. The constant rate of inbreeding in the long-term was achieved by setting the constraint on the average coancestry of selected candidates (c T Ac/2) to 1 -(1-ΔF) t , where A is the average relationship matrix, c is a vector of projected use of candidates (i.e. mating proportions), ΔF is the desired rate and t is the generation number [7]. Mating among selected candidates was at random. Populations of T = 100, 200 or 300 candidates per generation were simulated. A broad range of trait h 2 0 (0.01, 0.25, 0.50, 0.75 and 0.99) was studied. ΔF was constrained to 0.05, 0.025, 0.0125 and 0.01 per generation. The parameter values go beyond those commonly encountered in practice which correspond to the low to intermediate values for h 2 0 , and the lower values of ΔF. One hundred replicates were performed for each scenario.

Comparison of deterministically predicted genetic gains from optimised selection and truncation selection at the same rate of inbreeding
The rate of gain (ΔG quad ) predicted from equation (6) was also compared to the predicted rate of gain obtained from truncation selection (ΔG tru ) at the same ΔF for a range of scenarios all using BLUP genetic evaluation. The predictions of genetic gain and rate of inbreeding for truncation selection schemes were obtained using the SelAction software [4,5,13]. A mating ratio equal to one was assumed for all schemes under truncation selection (i.e., n s = n d = T/2 and n o = T /n d across generations).
Comparisons between truncation and optimised selection at the same ΔF were carried out in two ways. Firstly, ΔG quad and ΔG tru were compared for an extensive range of ΔF at two fixed levels of h 2 0 (0.10 and 0.35). Comparisons across ΔF were carried out with T = 100 or 300 candidates. In order to obtain an extensive range of ΔF (from 0.1% up to 4.0%), the population structure was varied by changing the proportion of selected individuals at regular intervals. For each population structure, SelAction [13] was used to predict ΔG tru and the corresponding ΔF. For each level of ΔF predicted under truncation selection, N c was obtained from T , ΔF and h 2 0 using the approach described above. Then ΔG quad under optimised selection was calculated after predicting ρ conv and ΔG ideal from the modified pseudo-BLUP index [3] and equation (3), respectively.
Secondly, ΔG quad and ΔG tru were compared for an extensive range of h 2 0 and two fixed levels of ΔF(0.01 and 0.025). Comparisons across h 2 0 (with h 2 0 ranging from 0.05 to 0.85) were carried out with T = 100, 300 and 1000. For a given combination of T and h 2 0 , the population structure that gave the desired ΔF under truncation selection was obtained using SelAction. A maximum deviation of 0.05 with respect to the target ΔF was allowed. Then, as before the corresponding N c was obtained from T , ΔF and h 2 0 , and ΔG quad was calculated from ρ conv and ΔG ideal . Table I shows a comparison between ΔG obs (simulations) and ΔG quad (predictions) for the broad range of h 2 0 and ΔF constraints studied. The % error ranged from -0.7% (h 2 0 = 0.5 and ΔF = 0.0250) to 15.0% (h 2 = 0.99 and ΔF = 0.0100). Over-prediction was the norm for the most relaxed ΔF constraint (i.e. ΔF = 0.0500) and for h 2 0 = 0.99. However, under-prediction was typically observed for the most stringent ΔF constraints with h 2 0 < 0.99. It should be noted that accurate (% error less than 10%) predictions were obtained for the range of ΔF constraints most likely to be applied in practice (e.g. from 0.0100 to 0.0250) in selection programmes aimed to improve typical traits of economic importance (i.e. with h 2 0 levels of up to 0.50). In Figure 2, predictions of ΔG quad (using predicted N c ) are compared with the rates of gain from simulations for two scheme sizes (T = 100 and T = 300), a broad range of h 2 0 and two ΔF constraints (ΔF = 0.0100 and ΔF = 0.0250). Standard errors for simulated rates of gain were always less than 0.02 and ranged from 0.001 to 0.014. Predictions for ΔG quad were very similar to those using empirical N c for both scheme sizes (not shown). Excluding the extreme h 2 0 = 0.99, Figure 2a shows a good agreement between ΔG quad and ΔG obs for the smaller scheme (T = 100) at both ΔF constraints. For ΔF = 0.0100, ΔG quad under-predicted ΔG obs on average by -5.9%, but the differences between predicted and observed values were non-significant (p > 0.05) for h 2 0 ranging from 0.01 to 0.7. Similarly, for ΔF = 0.0250, the difference between ΔG quad and ΔG obs was non-significant (p > 0.05) for h 2 0 ≤ 0.6.  For the largest scheme (T = 300), ΔG quad still provided good predictions of gain for the most stringent constraint. For ΔF = 0.0100, the difference between ΔG quad and ΔG obs was non-significant (p > 0.05) for h 2 0 < 0.9 (Fig. 2b). For h 2 0 ≤ 0.7, ΔG quad over-predicted ΔG obs by about 3.9%, and for h 2 0 ≥ 0.6 ΔG quad under-predicted ΔG obs by about -3.7%. Predictions were less accurate for ΔF = 0.0250 and ΔG quad over-predicted ΔG obs by on average 11.8% in the whole range of h 2 0 up to 0.8. The difference between ΔG quad and ΔG obs was significant (p < 0.05) for h 2 0 ranging from 0.1 to 0.6.

Comparison of predicted rates of genetic gain from truncation and quadratic optimisation at the same rate of inbreeding
The rates of gain under truncation and optimised selection for different ΔF levels are presented in Figure 3. At a given ΔF, the gain from optimised selection (ΔG quad ) was always higher than that from truncation selection (ΔG tru ) for both heritabilities and scheme sizes. The advantage of ΔG quad over ΔG tru increased as ΔF increased (i.e. as the constraint in quadratic optimisation becomes less stringent) and was the greatest for the highest h 2 0 and the largest breeding scheme. For instance, for ΔF = 0.0100 and h 2 0 = 0.35, the advantage of ΔG quad over ΔG tru was 26.5% (0.286 versus 0.226) for T = 100 and 32.5%  T was increased up to 1000 candidates (results not shown). In this scenario, the superiority of ΔG quad over ΔG tru was about 40% for ΔF = 0.0100. Figure 4 shows the change of ΔG quad /ΔG tru with h 2 0 for three population sizes and two fixed values of ΔF (0.0100 and 0.0250). Two situations can be identified according to ranges of h 2 0 below or above 0.55. For h 2 0 up to 0.55, ΔG quad /ΔG tru was greater for ΔF = 0.0250 than for ΔF = 0.0100 and the maximum ratio increased with T . For ΔF = 0.0100, the maximum ΔG quad /ΔG tru was 1.41 for T = 1000, 1.32 for T = 300 and 1.27 for T = 100, and it

DISCUSSION
This study presents deterministic predictions of the maximum potential rate of genetic gain that can be obtained in schemes using selection on quadratic indices where the rate of inbreeding is restricted to pre-defined levels. The two key components required for the prediction are the following: (i) a prediction of the ideal rate of gain after a perfect allocation of long-term contributions to Mendelian sampling terms for a given ΔF, and (ii) a prediction of the ultimate accuracy of the Mendelian sampling term at the time of convergence of the long-term contributions of selected candidates. The first component (i.e. prediction of ΔG ideal ) was available from Grundy et al. [7] and the second component (i.e. prediction of the accuracy) from Avendaño et al. [3]. The approach provided accurate predictions of ΔG quad for realistic h 2 0 and target ΔF values in breeding schemes of contrasting size. However, ΔG quad was less satisfactory and over-predicted gain with the extreme values of ΔF (5%) or h 2 0 (0.99). The deterministic method provides a solution to the gap in prediction tools for the design of breeding schemes under constrained rates of inbreeding and gives the necessary accompanying tool to the current available operational dynamic selection algorithms.
Deterministic methods for the strategic optimisation of breeding schemes, that is, the maximisation of genetic gain for a pre-defined ΔF were previously developed for schemes under mass [15,16], and index selection [15]. However, these methods dealt with truncation selection, and were unlinked to dynamic selection algorithms based upon quadratic indices [7,10]. For truncation selection, Villanueva et al. [17] and Bijma et al. [5] used predictions of ΔG and/or ΔF based on lifetime contributions (u i ) of selected candidates. The basic framework for predicting u i was laid down by Woolliams et al. [19] and is based on u i = E(r i |s i ) where s i is the selective advantage of the individual i. However, although this framework is relevant for truncation selection it may not be appropriate for quadratic indices [19]. Under truncation selection, selective advantages are defined as functions of the breeding value of the candidates whereas under quadratic optimisation the selective advantage is related to the Mendelian sampling term [2]. The main limitation for extending the prediction approach based on u i to constrained optimisation under BLUP selection is the difficulty of predicting the selection intensity for the group of selected candidates for which the average coancestry has been restricted [20].
Here, the predicted accuracy of the Mendelian sampling term was the key parameter for obtaining predictions of genetic gain under constrained ΔF. The lack of knowledge of the Mendelian sampling term (the selective advantage in quadratic indices) and the availability of only initial estimates at the time of selection is one of the central factors that prevent attaining the ideal optimal solution of an exact allocation of long-term contributions to Mendelian sampling terms [7,20]. Therefore, by using predictions of the ultimate accuracy, the over-prediction represented by ΔG ideal is directly adjusted. The inability to set the contributions of selected candidates to their desired values (i.e. independently from the contributions of future descendants) is the second factor that drives the observed outcome of the constrained optimisation away from the ideal solution [7,20]. Not accounting for this effect when adjusting ΔG ideal does not seem to represent a strong limitation since predictions ignoring it were reasonably accurate for a broad range of parameters.
Although the deterministic framework presented is self-contained and can be used by the specification of solely T , h 2 0 and ΔF, it should be recognised that it has an element of empirical nature and so is limited. This arises from the regression approach for predicting the ratio N c /N r that implied the empirical relationship between the optimisation outcome at selection time ( c 2 ) and at convergence ( r 2 ) for different h 2 0 , ΔF restrictions and population sizes. Apart from the method of inferring equilibrium structure, further sources of error may arise from the use of standard selection intensities that assume infinite population sizes, particularly in the smaller schemes where indices among relatives may be highly correlated, and from the use of linear indices in the method to calculate ρ conv for quadratic optimisation. The relatively poor performance of the prediction for the extreme heritability h 2 0 = 0.99 is not unexpected since a very high accuracy at the start of the optimisation process implies that the contributions assigned initially will differ greatly among individuals and there will be no scope to change them over time e.g. to accommodate desired contributions from their offspring. In contrast, with lower values of h 2 0 , the assigned initial contributions will be less variable and these can be adjusted more easily, if necessary, in subsequent generations as information accumulates.
Predictions for ΔG quad ignore the effects of inbreeding on the genetic variance and thus refer to either (i) a near-asymptotic response to selection in the medium term after accounting for the Bulmer effect, or (ii) where losses from inbreeding are balanced by mutational variance entering the population. Villanueva and Woolliams [16] accounted for the effect of inbreeding on genetic variance and found for sib-indices that the optimum schemes for maximising gain at different generations (e.g. t = 5 or t = 20) under restricted inbreeding were similar. In essence, by applying a restriction on the accumulation of inbreeding, the loss of variance is also restricted. In any case, comparisons of predicted and simulated gains were free of bias since the rate of gain obtained from simulations also ignored reductions of genetic variance due to inbreeding.
A novel result of this study was the deterministic comparison of the rate of gain for schemes under BLUP optimised and truncation selection at the same ΔF. The results in Figure 4 suggest that, for realistic target ΔF values in commercial breeding populations, the maximum advantage of quadratic optimisation (ΔG quad ) over truncation selection (ΔG tru ) occurs at h 2 0 ranging from 0.15 to 0.35. In addition, the results indicate that for any level of ΔF, ΔG is higher for optimised than for truncation selection and that for h 2 0 up to 0.55 the superiority of ΔG quad over ΔG tru increases as the population size increases. Benchmark maximum values found for the ratio ΔG quad /ΔG tru were 1.40 and 1.45 for the largest scheme (i.e. T = 1, 000), and 1.27 and 1.32 for the smallest scheme (i.e. T = 100) for ΔF = 0.0100 and ΔF = 0.0250, respectively. The predicted superiority of quadratic optimisation over truncation selection agreed with empirical evidence from real livestock populations of Aberdeen Angus (beef cattle) and Meatlinc (sheep) [1]. At the observed ΔF in these populations, ΔG quad / ΔG tru was 1.30 and 1.17 for Aberdeen Angus (for ΔF = 0.0020) and Meatlinc (for ΔF = 0.0100), respectively.
A number of points should be addressed in relation to the predicted superiority of ΔG quad over ΔG tru . First, it might be counter intuitive that ΔG quad / ΔG tru increased with the size of the scheme. At first sight it might be expected that the benefits would be the greatest when the constraint on ΔF is hardest to meet (i.e. when T is small). However, large resources enable larger family sizes to be available, consequently increasing the accuracy of the evaluation of all terms required, and also allowing for selection and use more closely aligned to the optimum indicated by the quadratic index. Second, it should be noted that values for ΔG quad /ΔG tru might be overestimated for two reasons. The benefits of the quadratic index over truncation selection shown in Figures 3 and 4 include the benefits of factorial mating (assumed with quadratic indices) over hierarchical mating with a mating ratio equal to one (assumed with truncation selection). Nevertheless, the effect of the family structure on ΔG quad / ΔG tru is expected to become less important as the scheme size increases [14]. Also, following the approach of Villanueva and Woolliams [16], higher responses could be obtained from truncation selection if the weights given to the different sources of information in the pseudo-BLUP index were optimised for maximising ΔG tru for a given constraint on ΔF. In any case, the results are indicative of the potential benefits of quadratic indices and the potential trends in the benefits for the key parameters involved.
As it stands today, the prediction framework can be readily applied in species with no restriction in reproductive rates (e.g. fish and trees) to improve traits with phenotypes available in both sexes. Extensions of this framework, particularly to accommodate reproductive limitations would be relevant to allow its application in livestock populations, particularly of cattle and sheep. Although a mating ratio of one has been assumed here, the current predictions of the Mendelian sampling accuracy allow any mating ratio and are thus not a limitation. In contrast, an expression for the ideal optimal ΔG when reproductive limitations exist needs to be developed.
These results are critical from the point of view of the strategic design of practical breeding schemes, since with this framework breeders will be able to set their risk preference (i.e. the target ΔF) and predict the benefit from changing from their current selection practices (namely, truncation selection) to optimised selection. In a subsequent stage, available operational tools [7,10] for making selection decisions on a day-to day basis can be applied. Both the design tool (a priori) and the operational (a posteriori) tool have the same underlying definition of genetic gain that is expressed in terms of long-term genetic contributions and Mendelian sampling terms [18]. Thus, a prediction framework for ΔG quad is a necessary accompanying tool for the available operational dynamic selection algorithms. B.V. acknowledges financial support from the Secretaría de Estado de Educación y Universidades (Ministerio de Educación, Cultura y Deporte, Spain). Valentin Kremer is acknowledged for advise on the use of SelAction.

APPENDIX A: IMPLEMENTATION OF THE NEWTON-RAPHSON METHOD FOR FINDING THE TRUNCATION POINT AND THE SELECTION INTENSITY FOR A GIVEN ΔF CONSTRAINT
In order to find the values of i and x that satisfy (4T ΔF) −1 = p(i − x) 2 (1 − ix + x 2 ) −1 for given values of T and ΔF, a starting value of zero was given to x. Then, iterations were performed such as 2 , and p and i are taken from the truncated normal distribution for the normal deviate x t−1.

APPENDIX B: PREDICTION OF THE RATIO OF EFFECTIVE NUMBER OF PARENTS AT THE TIME OF SELECTION (N c ) TO EFFECTIVE NUMBER OF PARENTS AT THE TIME OF CONVERGENCE OF CONTRIBUTIONS (N r )
The prediction of N c /N r was carried out in several steps. Firstly, the empirical ratios N c /N r observed from the simulation were plotted against 1 − h 2 0 for − Figure B1. The correlation between the ratio N c /N r resulting from the simulation (empirical) and the predicted N c /N r was 0.92. This correlation was obtained using different values for h 2 0 and T ΔF resulting from T = 100, 200 and 300, and ΔF = 0.025, 0.0125 and 0.010.
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