Linear selection indices for non-linear profit functions

Summary Two methods of deriving linear selection indices for non-linear profit functions have been proposed. One is by linear approximation of profit, and another is the graphical method of Moav and Hill (1966). When profit is defined as the function of population means, the graphical method is optimal. In this paper, profit is defined as the function of the phenotypic values of individual animals; it is then shown that the graphical method is not generally optimal. We propose new methods for constructing selection indices. First, a numerical method equivalent to the graphical method is proposed. Furthermore, we propose two other methods using quadratic approximation of profit: one is based on Taylor series about means before selection, and the other is based on Tayler series about means after selection. Among these different methods, it is shown that the method using quadratic approximation based on Taylor series about means after selection is the most efficient.     

Selection indices for desired relative genetic gains with inequality constraints

Summary A selection index which maximizes genetic gains in a desired direction has been previously suggested. We extend this index to the case where desired relative genetic gains are constrained to be not less or not greater than pre-specified levels. Further, we suggest an index for desired relative genetic gains constrained to be between certain levels. All these indices are obtained using quadratic programming techniques.     

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_X²=0.25 and h_Y²=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_X²=0.60, h_Y²=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_X²= h_Y²=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.     

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     

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: economic weights for maximum simultaneous genetic gain

Summary Selection indices that maximize the correlation between an individual organism's index score and its breeding value frequently require a priori known economic weights before the optimum phenotypic weights can be estimated. The long generation intervals and economic uncertainty that surround forest tree breeding can make the choice of weights arbitrary. In this paper an algorithm is introduced for finding economic weights that will ensure maximum simultaneous progress in all index traits. At the outset the traits are assumed to be of equal preference. The solutions are functions of the eigenvalues and eigenvectors of a quadratic form of the additive genetic and phenotypic covariance matrices. Examples of applications in tree breeding emphasize the practical aspects of the method.     

通用选择指数原理

本文在对各类种畜评定方法探讨的基础上,提出了通用选择指数概念,对各类评定方法在理论上加以综合,推导出统一的计算公式,并编制出相应的计算程序。为充分利用种畜各种信息,特别是为在多信息来源需作约束、最宜选择时的种畜评定,提供了行之有效的计算方法。避免了在实际应用时对各类评定方法的独立探讨以及因多种计算体系导致应用上的混乱。此外,本文还得出了有关选择理论的三个结论,并改正了文献[2]中的失误之处。     

Construction of Selection Indexes with Restrictions

In this paper the problem of unistage selection with inequality constraints is formulated. If the predictor and criterion variables are all normally distributed, this problem can be written as a convex programming problem, with a linear objective function and with linear constraints and a quadratic constraint. Using the duality theory, for convex nonlinear programming it is proved, that its dual problem can be transformed into a convex minimization problem with non-negativity conditions. Good computational methods are known for solving this problem. By the help of the dual problem sufficient conditions for a solution of the original primal problem are derived and illustrated by an example of practical interest.     

Robustness of Selection Procedures

In the article Bechhofers Indifference-zone formulation for selecting the t populations with the t highest means is considered in a set of non-normal distributions. Selection rules based on the sample mean, the 10% and the 20% trimmed means, two estimators proposed by Tiku (1981) for valuating the smallest and highest accepted sample values higher, the sample median and a linear combination of quantile estimators, two adaptive procedures and a ranksum procedure are investigated in a large scale simulation experiment in respect of their robustness against deviations from an assumed distribution. Robustness is understood as a small percentage of the difference β_A-β between the actual probability of incorrect selection β_A and the nominal β-value. We obtained a relatively good robustness for the classical sample mean selection rule and useful derivations for the employment of other selection rules in an area of practical importance.     

Subset Selection for Exponential Populations: Determination of the Selection Constant

Given are k(≧2) exponential populations differing only in their location parameter. One wishes to choose the best one, that is the population with the largest value of the location parameter. A possible method for solving this problem is to select a subset of the k populations of size at least one which includes the best population with a required confidence P*(k^?1 ≤ P* ≤1). In this paper the required selection constant is determined for different values of k and P*. Also an approximation for the selection constant is derived. A comparison with the exact results is made.     

Index selection for genetic improvement of quantitative characters

Summary This paper reviews the basic theory and summarizes various modifications of the selection index. The limitations of selection index are discussed in four parts: (1) changes of parameters due to selection. (2) sampling errors of parameter estimation. (3) evaluation of relative economic weights and (4) internal deterrents to index selection.     

Subset Selection Procedures for Randomized Designs

Starting from the principle that there exists a randomization procedure that assigns treatments to experimental units, four subset selection rules for the problem of selecting the best treatment from a set of different treatments are proposed. Two of these are extensions of already existing subset selection procedures, which were defined for unbalanced designs, and need a separate selection constant for each individual treatment. The other two rules proposed are new and need only one selection constant for all treatments. The various procedures are compared, and illustrated by application to a plant breeding variety trial.     

Selection theory for selfed progenies

Summary The purpose of this article was to extend the model used to predict selection response with selfed progeny from 2 alleles per locus to a model which is general for number and frequency of alleles at loci. To accomplish this, 4 areas had to be dealt with: 1) simplification of the derivation and calculation of the condensed coefficients of identity; 2) presentation of the genetic variances expressed among and within selfed progenies as linear function of 5 population parameters; 3) presentation of selection response equations for selfed progenies as functions of these 5 population parameters; and 4) to identify a set of progeny to evaluate, such that one might be able to estimate these 5 population parameters.The five population parameters used in predicting gains were the additive genetic variance, the dominance variance, the covariance of additive and homozygous dominance deviations, the variance of the homozygous dominance deviations and a squared inbreeding depression term.Contribution from the Missouri Agricultural Experiment Station. Journal Series No. 9971     

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.     

Selection for efficiency of feed utilization in growing mice

Summary Selection was practised for improved feed efficiency (gain/feed intake) of mice on two alternative feeding regimes. In one set of lines animals were fed ad libitum, in the other set they were individually fed a fixed amount of feed (about 10% below the control ad libitum intake) which was not changed over generations. For each treatment, a pair of replicate lines (E) were selected on efficiency from 3–5 weeks of age for 8 generations and another pair (L) from 5–7 weeks for 7 generations. A control line was maintained for both E and L lines. In terminal generations mice from each line were tested on each feeding regime, and carcasses of ad libitum fed mice were analysed.The realized heritability (within families) for efficiency averaged 13%, without much variation over treatments. In the E lines efficiency increased by about 18% of the control mean and in the L lines by about 60%, although absolute changes were small, and responses were similar on the two feeding regimes. Weights at the start of test decreased in the E lines and increased in the L lines; weights at the end of test increased in both.When tested on the alternative regimes, no interactions were detected for live weights, weight gains or efficiency; selection under fixed intake led to the same increase in appetite as did that under ad libitum.There were no interactions for carcass composition. Selection for efficiency led to an increase in fatness on both selection regimes and both weight ranges.     

Selection for improved yield in inter-specific mixtures or intercrops

Summary A simple recurrent selection scheme using randomly constructed intercrop or mixture treatments for the mutual improvement of two or more species is described. Several possible selection criteria are available for each species, including its own yield, those of each of its associates, and combinations of these. Statistical expressions are developed to describe the expected gain in economic yield in each species given selection for one or more species on the basis of any common criterion, and it is shown that the gains from all responding species and applied selection criteria are additive. Negative correlations between the direct and associate effects of genotypes favour the selection of whole mixtures. Selection indices of several species yields can be applied either on an individual species basis or to whole mixtures, but only in the latter case can the statistics required for the calculation of the optimum index be estimated from the data provided by the trial. Some general properties and possible long term effects of different selection methods are discussed.     

Selection indices for quality evaluation in wheat breeding

Summary From multilocation trials involving 125 cultivars of wheat of mainly French and European origin four tests — protein content, Pelshenke, modified Zeleny and the mixograph — were used to establish six selection indices. Three of these indices — IW₁, IW₂ and IW₃ — were calculated in order to evaluate the genetic potentiality of the lines for dough strength as given by the Chopin alveograph. The indices IV₁, IV₂ and IV₃ were established to evaluate loaf volume as measured by the French bread-making standard. A quality index IQ was calculated from the allelic effects of the high-molecular-weight (HMW) subunits of glutenin from 195 cultivars assessed by the Chopin alveograph and the Pelshenke test. Comparison of the relative efficiency of each of the six indices to the individual tests revealed the superiority of the indices over one or several technological parameters. The six selection indices and the quality index were compared using 30 very diverse F₄ lines. Their ability to retain the good quality lines is discussed in particular.     

Selection for increased length of reproductive life in mice

Summary An experiment was conducted in mice to examine whether selection can increase reproductive life and lifetime production of progeny. Mice in two lines with litter size standardized at birth and in two lines without standardization were pair-mated at 7 weeks of age and maintained as long as they produced litters up to 382 days. Progeny from the sixth litters were used to maintain the four selected lines, while progeny from the first litters were bred to maintain unselected control lines. Selected and control lines were compared at five and six generations of the selected lines. Contemporary comparisons revealed that the length of reproductive life and most lifetime production traits were significantly greater in the selected than in control lines. Realized heritability of the length of reproductive life ranged from 0.08 to 0.13. It was concluded that the length of reproductive life and lifetime production in mice can be increased by selection.Animal Research Centre Publication No. 1618     

Ponderal indices at birth

At birth, the index weight/height is the highest correlated with weight, height and cephalic perimeter, for the index of Quételet (BMI) the correlations are also significant although lower. In both cases, the variation with age is significant. For the Rohrer index, the situation is different, the changes with age being almost absent. In front of these results, can we decide which index would be adequate to establish the nutritional status at birth? For Cole (1986), for such a ponderal index the correlation with height would have to be absent. Rolland-Cachera (1982) proposed also that the correlations with height would be absent but would be high with weight. In the case of the newborns of our study the only index being not correlated to height is the Rohrer index with low correlation also with the cephalic perimeter and high correlation with weight (although here the correlation is even higher with BMI and W/H). During the utero-placentary disfunction the foetal weight is affected but not height and cephalic perimeter: in this case, the advantage is also to an index correlated with weight but not with height.     

Spike-train spectra and network response functions for non-linear integrate-and-fire neurons

Reduced models have long been used as a tool for the analysis of the complex activity taking place in neurons and their coupled networks. Recent advances in experimental and theoretical techniques have further demonstrated the usefulness of this approach. Despite the often gross simplification of the underlying biophysical properties, reduced models can still present significant difficulties in their analysis, with the majority of exact and perturbative results available only for the leaky integrate-and-fire model. Here an elementary numerical scheme is demonstrated which can be used to calculate a number of biologically important properties of the general class of non-linear integrate-and-fire models. Exact results for the first-passage-time density and spike-train spectrum are derived, as well as the linear response properties and emergent states of recurrent networks. Given that the exponential integrate-fire model has recently been shown to agree closely with the experimentally measured response of pyramidal cells, the methodology presented here promises to provide a convenient tool to facilitate the analysis of cortical-network dynamics.