Selection indices for non-linear profit functions

Summary Conventional selection index theory assumes that the total merit or profitability of animals is a linear function of measurable traits. However, in many cases merit may be a non-linear function of these traits. A linear selection index can still be used in this situation but the optimum index depends on the selection intensity to be used and on the number of generation over which the selection response is to be maximized. Nonlinear selection indices have been suggested but these result in a lower selection response than the best linear index. Linear selection indices suggested in the past are shown to correspond to the optimum linear index for either a very small selection response or, in the case of restricted indices, a very large selection response. The economic value of a trait may depend on management decisions taken by the farmer. In this situation the economic values should be calculated assuming that the management decisions taken maximize profit given the present genetic value of the animals.     

Index versus tandem selection after repeated generations of selection

Summary A theoretical comparison between two multiple-trait selection methods, index and tandem selection, after several generations of selection was carried out. An infinite number of loci determining the traits, directional and truncation selection, discrete generations and infinite population size were assumed. Under these assumptions, changes in genetic parameters over generations are due to linkage disequilibrium generated by selection. Changes continue for several generations until equilibrium is approached. Algebraic expressions for asymptotic responses from index selection can be derived if index weights are maintained constant across generations. Expressions at equilibrium for genetic parameters and responses are given for the index and its component traits. The loss in response by using initial index weights throughout all generations, instead of updating them to account for changes in genetic parameters, was analyzed. The benefit of using optimum weights was very small ranging from 0% to about 1.5% for all cases studied. Recurrence formulae to predict genetic parameters and responses at each generation of selection are given for both index and tandem selection. A comparison between expected response in the aggregate genotype at equilibrium from index and tandem selection is made considering two traits of economic importance. The results indicate that although index selection is more efficient for improving the aggregate breeding value, its relative efficiency with respect to tandem selection decreases after repeated cycles of selection. The reduction in relative efficiency is highest with the highest selection intensity and heritabilities and with negative correlations between the two traits. The advantage of index over tandem selection might be further reduced if changes in genetic parameters due to gene frequency changes produced by selection, random fluctuations due to the finite size of the population, and errors in estimation of parameters, were also considered.     

以提高饲料效率为目标的选择指数制订方法

将构成饲料效率的两个组份性状通过价格定义成一个效益性状,探讨了以提高饲料效率为目标的选择指数制订方法,该方法避开了直接选择饲料效率在统计方面所遇到的诸多麻烦相应地解决了一类比值性状的综合选择问题。     

Selection index updating

Summary When traits become evident at different ages or there are large differences in the costs of measuring various traits, selection by independent culling levels may give a higher aggregate economic return than index selection because not all traits need to be measured on all individuals. The problems with optimum independent culling selection is that general solutions are not possible and numerical integration is needed for specific cases. Recently, Xu and Muir (1991) developed a new independent culling level procedure by use of orthogonal transformation of the original characters. With their procedure, explicit solutions for optimum truncation points are possible without numerical integration. As such, the procedure is proficient for any number of stages, and generalized theoretical comparisons of alternative breeding strategies are possible. However, their procedure was limited to the case where selection is for one character at each stage. In this paper, our previous results are extended to the general case of multi-stage index selection, called selection index updating. This procedure is called selection index updating because as traits become available in latter stages, each subsequent index contains all of the traits available up to that stage.The procedure is to develop sequential indices for each stage such that correlations among indices at different stages are zero. Optimum culling points are obtained for the updating procedure by using Xu and Muir's (1991) iterative equations. Due to the property of orthogonality of the updated indices, aggregate gain can be partitioned into gains due to various stages of selection. Partitioning of aggregate economic gain is useful to breeders who desire to adjust individual trait selection intensity based on facilities available at that stage. Methods are discussed to modify the procedure to obtain maximum aggregate economic return per unit of cost associated with obtaining measures on each trait. An application of multi-stage selection is demonstrated using a set of data for Rhode Island Red layer type chickens. A second example demonstrates the use of multi-stage selection optimized with respect to aggregate economic gain and costs associated with obtaining measurements.Journal Paper No. 12813 of the Purdue University Agricultural Experiment Station     

Index selection with nonlinear profit function as a tool to achieve simultaneous genetic gain

Summary Simultaneous improvement of several, and often negatively correlated, traits is frequently a desired objective in forest tree breeding. A profit function that includes a combination of both linear weights and weights for the cross-products of trait combinations facilitates the construction of a linear index, with an attractive response in all traits. A detailed algorithm for finding the index coefficients is provided, along with three examples of applications in tree breeding. The index is also a powerful tool in optimizing the selection for a ratio of two traits. It is argued that a more equal progress in several traits provides a safetey net when faced with economic uncertainties. The provided algorithm eliminates the need for direct search techniques. Existence of a dual set of linear weights means that the statistical properties of the index based on nonlinear profit functions are identical to those of the classical Smith-Hazel type of index.     

Selection strategies and artificial evolution

Summary Artificial selection results in biolgical changes, creating artificial evolution. When using selection indexes, the artificial evolution depends on the relative economic (or other) weight of traits in the breeding objective, and on the phenotypic and genetic variances and covariances among these traits and the traits recorded in the selection index. As shown here, the selection strategy (in this case, individual selection versus progeny test selection) can also have marked effects on the kind of artificial evolution produced. Thus, where economic weights are uncertain, choice between alternative selection strategies might take into account the different types of animal or plant resulting.     

A comparison of restricted selection index and linear programming in sire selection

Summary The objective of restricted selection index is to enhance genetic change in one trait while restricting to zero change in a second trait. Linear programming is another, yet conceptually different, technique to maximize one function while enforcing limits on others. The objective of this research was to compare restricted selection index and linear programming in ability to maximize performance in one trait while limiting change in a second trait to zero. Results of a numerical study demonstrate that linear programming is a more effective method to limit correlated response than restricted selection index. On average, both methods limited response in a correlated trait to zero. However, the squared deviation of actual response in the restricted trait from zero was smaller with linear programming than with restricted selection index. Response to selection in the unrestricted trait is greater with restricted selection index than with linear programming.     

Moments,cumulants, and polygenic dynamics

A new approach for describing the evolution of polygenic traits subject to selection and mutation is presented. Differential equations for the change of cumulants of the allelic frequency distribution at a particular locus and for the cumulants of the distributions of genotypic and phenotypic values are derived. The derivation is based on the assumptions of random mating, no sex differences, absence of random drift, additive gene action, linkage equilibrium, and Hardy-Weinberg proportions. Cumulants are a set of parameters that, like moments, describe the shape of a probability density. Compared with moments, however, they have properties that make them a much more convenient tool for investigating polygenic traits. Applications to directional and stabilizing selection are given.     

Evolutionary genetics of sex-limited traits under fluctuating selection

Ultimately based on the different gamete size and the resulting sex roles, most animal species have acquired traits that occur only in one of the sexes. For those sex-limited traits the impact of a single selection event is reduced, because the genes coding for these traits also occur unexpressed in the other sex and are therefore partly hidden to selection. All sex-limited traits thus show a storage effect analogous to a seed bank. This storage effect has to be considered when measuring the effect of fluctuating selection on sex-limited traits. Here, I develop appropriate equations to measure the ability of an allele to invade a population when fluctuating selection acts on sex-limited traits. A comparison of the equations for X-chromosomal and autosomal traits gives the following result: the evolution of traits limited to the heterogametic sex should preferentially involve X-chromosomal genes and the evolution of traits limited to the homogametic sex should preferentially involve autosomal genes. This difference between the sexes may contribute to explain (1) the large effect of X-chromosomal genes in causing hybrid sterility in the heterogametic sex and (2) the large effect of X-chromosomal genes on sexually selected traits in heterogametic males.     

Restricted selection index in mice designed to change body fat without changing body weight: correlated responses

Summary Correlated responses were studied in lines of mice selected for eight generations based on the criterion of a restricted selection index. Two replicate lines were selected in each treatment as follows: HE, high epididymal fat pad weight (EF) with zero change in body weight (BW) at 12 weeks of age; LE; low EF with zero change in BW; and RS, randomly. Correlated responses showed considerable variation between replicates, suggesting that genetic drift was important. Further, correlated responses for most traits were relatively small, probably because of low selection intensity. The HE line responded as expected in component traits of the restricted index. Associated compositional traits in HE responded as predicted since traits correlated with adiposity increased and hind carcass weight did not change significantly. Feed intake increased and feed efficiency (weight gain/feed intake) decreased in HE, as predicted. In contrast, the LE line did not respond in component traits as predicted since EF did not decrease and BW increased. Consequently, LE exhibited little change in traits associated with adiposity, but hind carcass weight, feed intake and feed efficiency increased. Of the correlated responses scored for fitness traits (littering rate, number of days from pairing of mate to littering, litter size and preweaning pup survival rate), significant effects were found for decreased littering rate in LE and increased prenatal survival rate in HE. In summary, correlated responses to restricted index selection generally agreed with expectation when responses in component traits of the index were considered.The research reported in this publication was funded by the North Carolina Agricultural Research Service (NCARS), Raleigh, NC 27695-7643, USA. The use of trade names in this publication does not imply endorsement by the NCARS, nor criticism of similar ones not mentioned     

Effects of mating system in Japanese quail

Summary A 17-generation selection experiment was conducted to study direct and correlated responses to mass selection under a mating system with alternating generations of full-sib inbreeding and wide outbreeding (population I) as compared with a mass selected, randomly mated population (population II). The selection criterion was an index of total egg mass to 78 days divided by adult female body weight. Estimates of heritabilities and genetic correlations are reported. Estimated heritabilities for the index were 0.38±0.04 and 0.29±0.05 in population I and II, respectively. Realized heritabilites were 0.10±0.05 and 0.12±0.03. For most traits studied the mean phenotypic values in the cyclic mated population decreased for inbred generations. Increased inbreeding levels also caused outbred generation means of population I to decrease through the first six or seven generations. After this period of adaptation to inbreeding selection response was positive for the index and positively correlated traits. Total response to selection under the cyclic inbred-outbred mating system did not exceed selection response made under random mating. However, the rate of response in the cyclically mated population exceeded that in the randomly mated population in later generations when the cyclically mated population had apparently adapted to inbreeding.Published with approval of the Director of the Montana Agricultural Experiment Station, Journal Series No. 1221     

A JOINT INDEX FOR THE INTENSITY OF SEX‐SPECIFIC SELECTION

A growing body of experimental and field data shows that selective pressures often differ between males and females. Surprisingly, to date, little attempt has been made to formalize a metric expressing the relative behavior of directional selection in the two sexes. We propose an index that describes the extent to which concordant or antagonistic selection is operating between the sexes for a given trait. This joint index could prove especially useful for the study of intralocus sexual conflict and the evolution of sexual dimorphism, providing a common scale to directly compare different traits within or among taxonomic levels, and allowing an assessment on how common sexually antagonistic selection might be in extant populations.     

Selection response for part period egg number and egg mass in chickens — a comparison

Summary Two sublines from each of four flocks of White Leghorn were subjected to two cycles of selection for part year egg number and part year egg mass. A family index (combined selection) was the selection criterion for each of the two traits. The generation means of an unselected pedigreed randombred control, bred and maintained along with the selected sub-lines, were used for correcting short term environmental trends. The sublines selected for a common selection criterion were considered as replicates. The control corrected generation means, when averaged over replicates, gave the mean direct and correlated responses for the selected and different unselected traits. The average genetic change per generation was 2.16 eggs in egg number selected sublines and 146 g of egg mass in egg mass selected sublines. The direct response marginally exceeded the correlated response for both part period egg number and egg mass seen in the corresponding sublines. Except for egg weight the correlated responses for different unselected traits were in the same direction in both egg number and egg mass selected sublines. While the egg weight in egg number lines did not change, its response was positive in egg mass selected lines. A comparison between the lines revealed that the egg mass selected lines matured later and laid heavier but slightly less numerous eggs than the egg number selected sub-lines.     

Evaluation of selection index: application to the choice of an indirect multitrait selection index for soybean breeding

Summary Selection indices can be used to predict one trait from information available on several traits in order to improve the prediction accuracy. Plant or animal breeders are interested in selecting only the best individuals, and need to compare the efficiency of different trait combinations in order to choose the index ensuring the best prediction quality for individual values. As the usual tools for index evaluation do not remain unbiased in all cases, we propose a robust way of evaluation by means of an estimator of the mean-square error of prediction (EMSEP). This estimator remains valid even when parameters are not known, as usually assumed, but are estimated. EMSEP is applied to the choice of an indirect multitrait selection index at the F₅ generation of a classical breeding scheme for soybeans. Best predictions for precocity are obtained by means of indices using only part of the available information.     

Variations of flower size and reproductive traits in self-incompatible <Emphasis Type="Italic">Trollius ranunculoides</Emphasis> (Ranunculaceae) among local habitats at Alpine Meadow

For plants that rely on animals for pollination, the ability to attract the animals to their flowers can be a crucial component of fitness. A large number of studies have documented pollinators to be important selective agents driving the evolution of flower size and correlated traits on a large scale. In this paper, we studied variations of reproductive traits in self-incompatible Trollius ranunculoides (Ranunculaceae) among local habitats at Alpine Meadow. The results showed significant variations of floral size, seed mass per fruit and sex allocation (male/female mass ratio) between different habitats, where floral size and seed mass was not explained fully by variation of plant size among habitats. It suggested that other factors unrelated to plant size might also influence floral variation. However, in our manipulated experiment, it showed no effects of manipulated floral size not only on visit rate of effective pollinators (bees and flies) but also on female success (seed set, seed mass per fruit), irrespective of flower density. Consequently, we could not conclude that the variation of floral size in T. ranunculoides was due to phenotypic plasticity, or natural selection. But if selection occurred, it should not be mediated by pollinators. It was likely that variation of sex allocation between habitats lead to changes of flower or corolla size, because plant invested much less to male function (female-biased sex allocation and larger single seed mass) in shade habitat (bottom of bush) than other exposed habitats, to gain higher fitness. In addition, high-floral density in T. ranunculoides had a negative effect on service of main pollinator (bees) and female success. This situation would influence the strength of selection on floral size.     

Asymmetrical correlated responses to selection under an infinitesimal genetic model

Summary Asymmetry in correlated responses to selection is expected when more than one cycle of selection is practised due to changes in genetic parameters produced by selection. In large populations, under the infinitesimal model these changes are due to linkage disequilibrium generated by selection and not to gene frequency changes. This study examines the conditions under which asymmetrical correlated responses are to be expected when an infinitesimal model is considered. Asymmetrical correlated responses in two traits in respect to which trait is selected are expected if the two traits have different heritabilities. Predicted asymmetry increases with the absolute value of the genetic correlation between the two traits, the difference between the two heritabilities, the intensity of selection and the number of generations of selection. Linkage disequilibrium generated by selection should be taken into account in explaining asymmetrical correlated responses observed in selection experiments.     

Rules for assortative mating in relation to selection for linear merit functions

Summary This study examined how assortative mating (without selection) based on linear combinations of two traits could be used to change genetic parameters so as to increase efficiency of selection. The efficiency of the Smith-Hazel index for improvement of multiple traits is a function of phenotypic and genetic variances and covariances, and of the relative economic values of the traits involved. Assortative mating is known to change genetic variances and covariances. Recursive formulae were derived to obtain these variances and covariances after t generations of assortative mating on linear combinations (mating rules) of phenotypic values for two traits, with a given correlation between mates. Selection efficiency after t generations of assortative mating without selection was expressed as a function of random mating genetic parameters, economic values, the mating rule, and the correlation between mates. Selection efficiency was maximized with respect to the coefficients in the mating rule. Because the objective function was nonlinear, a computer routine was used for maximizing it. Two cases were considered. When random mating heritabilities for the two traits were h _X ² =0.25 and h _Y ² =0.50, the genetic correlation r_XY=-0.60, and the economic values were a_X=3 and a_Y=1, continued assortative mating based on the optimal mating rule for 31 generations (with a correlation of 0.80 between mates) increased selection efficiency by 29%. Heritabilities changed to 0.38 and 0.66, respectively, and the genetic correlation became – 0.79. When h _X ² =0.60, h _Y ² =0.60, r_XY=– 0.20, a₁=1 and a₂=1, 36 generations of continued assortative mating with the optimal mating rule increased the efficiency of selection by 17%, heritabilities became h _X ² = h _Y ² =0.71, and the genetic correlation changed to 0.25. Only three generations of assortative mating were required to change the sign of the genetic correlation.     

Including genetic relationships in selection decisions: alternative methodologies

Investigations are made of variations in an iterative methodology previously introduced for reducing inbreeding by including genetic relationships in selection decisions, using adjusted estimated breeding values (EBV). An alternative computing strategy for maximising the value of the population selection criterion is shown to involve less computation, which results in function values as great or greater than the original method. Alteration of weights for different types of relationships in the adjusted EBV has no detectable effect on genetic gain at a given level of inbreeding. Selection using the adjusted EBV method in one sex and truncation on EBV in the other sex results in less genetic gain at a given level of inbreeding than using adjusted EBV in both sexes, but results in more gain at a given level of inbreeding than three selection strategies that do not include genetic relationships in selection decisions. The advantage of the adjusted EBV method over these three methods is retained when selection is for a sexlimited trait.     

Selection on floral and carbon uptake traits of Lobelia siphilitica is similar in females and hermaphrodites

Sexual dimorphism is common in plants and animals. Although this dimorphism is often assumed to be adaptive, natural selection has rarely been measured on sexually dimorphic traits of plants. We measured phenotypic selection via seed set on two floral and four carbon uptake traits of female and hermaphrodite Lobelia siphilitica. Because females can reproduce only via seeds, which are costlier than pollen, we predicted that females with smaller flowers and enhanced carbon uptake would have higher fitness, resulting in either sex morph-specific directional selection or stabilizing selection for different optimal trait values in females and hermaphrodites. We found that directional selection on one carbon uptake trait differed between females and hermaphrodites. We did not detect significant stabilizing selection on traits of either sex morph. Our results provide little support for the hypothesis that sexual dimorphism in gynodioecious plants evolved in response to sex morph-specific selection.     

多个主成分性状的综合优化研究

考虑育种目标的要求,对多个独立的主成分性状进行综合优化,建立了综合主成分性状的数学模型．综合主成分性状是S个表型方差（遗传方差）最大的主成分性状在育种目标约束下表型方差（或遗传方差）极大化下的线性组合,作为育种指标,选择效果好,预见性强．