Accuracy of marker-assisted selection with auxiliary traits

Genetic information on molecular markers is increasingly being used in plant and animal improvement programmes particularly as indirect means to improve a metric trait by selection either on an individual basis or on the basis of an index incorporating such information. This paper examines the utility of an index of selection that not only combines phenotypic and molecular information on the trait under improvement but also combines similar information on one or more auxiliary traits. The accuracy of such a selection procedure has been theoretically studied for sufficiently large populations so that the effects of detected quantitative trait loci can be perfectly estimated. The theory is illustrated numerically by considering one auxiliary trait. It is shown that the use of an auxiliary trait improves the selection accuracy; and, hence, the relative efficiency of index selection compared to individual selection which is based on the same intensity of selection. This is particularly so for higher magnitudes of residual genetic correlation and environmental correlation having opposite signs, lower values of the proportion of genetic variation in the main trait associated with the markers, negligible proportion of genetic variation in the auxiliary trait associated with the markers, and lower values of the heritability of the main trait but higher values of the heritability of the auxiliary trait.     

The value of cows in reference populations for genomic selection of new functional traits

Today, almost all reference populations consist of progeny tested bulls. However, older progeny tested bulls do not have reliable estimated breeding values (EBV) for new traits. Thus, to be able to select for these new traits, it is necessary to build a reference population. We used a deterministic prediction model to test the hypothesis that the value of cows in reference populations depends on the availability of phenotypic records. To test the hypothesis, we investigated different strategies of building a reference population for a new functional trait over a 10-year period. The trait was either recorded on a large scale (30 000 cows per year) or on a small scale (2000 cows per year). For large-scale recording, we compared four scenarios where the reference population consisted of 30 sires; 30 sires and 170 test bulls; 30 sires and 2000 cows; or 30 sires, 2000 cows and 170 test bulls in the first year with measurements of the new functional trait. In addition to varying the make-up of the reference population, we also varied the heritability of the trait (h2 = 0.05 v. 0.15). The results showed that a reference population of test bulls, cows and sires results in the highest accuracy of the direct genomic values (DGV) for a new functional trait, regardless of its heritability. For small-scale recording, we compared two scenarios where the reference population consisted of the 2000 cows with phenotypic records or the 30 sires of these cows in the first year with measurements of the new functional trait. The results showed that a reference population of cows results in the highest accuracy of the DGV whether the heritability is 0.05 or 0.15, because variation is lost when phenotypic data on cows are summarized in EBV of their sires. The main conclusions from this study are: (i) the fewer phenotypic records, the larger effect of including cows in the reference population; (ii) for small-scale recording, the accuracy of the DGV will continue to increase for several years, whereas the increases in the accuracy of the DGV quickly decrease with large-scale recording; (iii) it is possible to achieve accuracies of the DGV that enable selection for new functional traits recorded on a large scale within 3 years from commencement of recording; and (iv) a higher heritability benefits a reference population of cows more than a reference population of bulls.     

Resource allocation optimization with multi-trait genomic prediction for bread wheat (<Emphasis Type="Italic">Triticum aestivum</Emphasis> L.) baking quality

Key Message

Multi-trait genomic prediction models are useful to allocate available resources in breeding programs by targeted phenotyping of correlated traits when predicting expensive and labor-intensive quality parameters.

Abstract

Multi-trait genomic prediction models can be used to predict labor-intensive or expensive correlated traits where phenotyping depth of correlated traits could be larger than phenotyping depth of targeted traits, reducing resources and improving prediction accuracy. This is particularly important in the context of allocating phenotyping resource in plant breeding programs. The objective of this work was to evaluate multi-trait models predictive ability with different depth of phenotypic information from correlated traits. We evaluated 495 wheat advanced breeding lines for eight baking quality traits which were genotyped with genotyping-by-sequencing. Through different approaches for cross-validation, we evaluated the predictive ability of a single-trait model and a multi-trait model. Moreover, we evaluated different sizes of the training population (from 50 to 396 individuals) for the trait of interest, different depth of phenotypic information for correlated traits (50 and 100%) and the number of correlated traits to be used (one to three). There was no loss in the predictive ability by reducing the training population up to a 30% (149 individuals) when using correlated traits. A multi-trait model with one highly correlated trait phenotyped for both the training and testing sets was the best model considering phenotyping resources and the gain in predictive ability. The inclusion of correlated traits in the training and testing lines is a strategic approach to replace phenotyping of labor-intensive and high cost traits in a breeding program.

     

Heritability of ram mating success in multi-sire breeding situations

Multi-sire mating of a mob of ewes is commonly used in commercial sheep production systems. However, ram mating success (defined as the number of lambs sired by an individual) can vary between rams in the mating group. If this trait was repeatable and heritable, selection of rams capable of siring larger numbers of lambs could reduce the number of rams required for mating and ultimately lead to increased genetic gain. However, genetic correlations with other productive traits, such as growth and female fertility, could influence the potential for ram mating success to be used as a selection trait. In order to investigate this trait, parentage records (including accuracy of sire assignment) from 15 commercial ram breeding flocks of various breeds were utilised to examine the repeatability and heritability of ram mating success in multi-sire mating groups. In addition, genetic and phenotypic correlations with growth and female fertility traits were estimated using ASReml. The final model used for the ram mating success traits included age of the ram and mating group as fixed effects. Older rams (3+years old) had 15% to 20% greater mating success than younger rams (1 or 2 years of age). Increasing the stringency of the criteria for inclusion of both an individual lamb, based on accuracy of sire assignment, or a whole mating group, based on how many lambs had an assigned sire, increased repeatability and heritability estimates of the ram mating success traits examined. With the most stringent criteria employed, where assignment of sire accuracy was >0.95 and the total number of lambs in the progeny group that failed to have a sire assigned was<0.05, repeatability and heritability for log_e(number of lambs) was 0.40±0.09 and 0.26±0.12, respectively. For proportion of lambs sired, repeatability and heritability were both 0.30±0.09. The two ram mating traits (log_e(nlamb) and proportion) were highly correlated, both phenotypically and genetically (0.88±0.01 and 0.94±0.06, respectively). Both phenotypic and genetic correlations between ram mating success and growth and other female fertility traits were low and non-significant. In conclusion, there is scope to select rams capable of producing high numbers of progeny and thus increase selection pressure on rams to increase genetic gain.     

QTLs for fertility in table grape (<Emphasis Type="Italic">Vitis vinifera</Emphasis> L.)

Fertility (number of inflorescences per shoot) is a trait of major importance for grapevine breeding. We present and compare the results of quantitative trait locus (QTL) detection for fertility in two table grape progenies, MTP3140 and MTP3234. Novel parental and consensus maps were built for MTP3140. We found a main QTL on linkage group (LG) 5 in both progenies, which was also the most stable one across years. It explained up to 18.5% of the total intra-cross phenotypic variance. Three other QTLs, one on LG 5 and two on LG 14, were repeated over years but were found in a single progeny. Our results suggest that QTLs effect was mainly additive. The prospects for testing candidate genes are discussed as well as the potential interest of these results for marker-assisted selection.     

Response to genomic selection: The Bulmer effect and the potential of genomic selection when the number of phenotypic records is limiting

Background

Over the last ten years, genomic selection has developed enormously. Simulations and results on real data suggest that breeding values can be predicted with high accuracy using genetic markers alone. However, to reach high accuracies, large reference populations are needed. In many livestock populations or even species, such populations cannot be established when traits are difficult or expensive to record, or when the population size is small. The value of genomic selection is then questionable.

Methods

In this study, we compare traditional breeding schemes based on own performance or progeny information to genomic selection schemes, for which the number of phenotypic records is limiting. Deterministic simulations were performed using selection index theory. Our focus was on the equilibrium response obtained after a few generations of selection. Therefore, we first investigated the magnitude of the Bulmer effect with genomic selection.

Results

Results showed that the reduction in response due to the Bulmer effect is the same for genomic selection as for selection based on traditional BLUP estimated breeding values, and is independent of the accuracy of selection. The reduction in response with genomic selection is greater than with selection based directly on phenotypes without the use of pedigree information, such as mass selection. To maximize the accuracy of genomic estimated breeding values when the number of phenotypic records is limiting, the same individuals should be phenotyped and genotyped, rather than genotyping parents and phenotyping their progeny. When the generation interval cannot be reduced with genomic selection, large reference populations are required to obtain a similar response to that with selection based on BLUP estimated breeding values based on own performance or progeny information. However, when a genomic selection scheme has a moderate decrease in generation interval, relatively small reference population sizes are needed to obtain a similar response to that with selection on traditional BLUP estimated breeding values.

Conclusions

When the trait of interest cannot be recorded on the selection candidate, genomic selection schemes are very attractive even when the number of phenotypic records is limited, because traditional breeding requires progeny testing schemes with long generation intervals in those cases.     

Maternal environmental effects on the phenotypic responses of the twining vine Ipomoea purpurea to support availability

The presence of physical support elicits a number of morphological changes in the shoot of the common morning glory Ipomoea purpurea , including a shortening of internodes and petioles and a thickening of the main stem. Working with experimentally supported and non-supported plants of I . purpurea , I tested the existence of maternal environmental effects of physical support in this twining vine. I evaluated whether the offspring of supported plants differed from the offspring of non-supported plants in a number of morphological and reproductive characters. Stem diameter was the only shoot trait that showed a significant effect of the maternal environment (support). The stems of the progeny of supported mother plants were thicker than the stems of the progeny of non-supported mother plants. This was true for both supported and non-supported progeny. Stem diameter, however, was not significantly related to plant fitness (seed number) in supported or non-supported progeny plants. The maternal environment did not affect the magnitude of the phenotypic responses to support of shoot traits in the progeny plants. There were no maternal environmental effects of physical support on the proportion of flowering plants at week 10 in both non-supported and supported progeny. Likewise, the presence of physical support in the maternal environment did not affect reproductive traits (seed number, seed size, percentage of reproductive biomass) in supported and non-supported progeny plants. Finally, the relationship between seed size and seed number was affected by the maternal support environment. The progeny of supported mother plants showed a significantly negative association (trade-off) between seed size and number, whereas such a trade-off was not observed in the progeny of non-supported plants. This was true for both supported and non-supported progeny.     

Mapping of quantitative trait loci controlling adaptive traits in coastal Douglas fir. III. Quantitative trait loci-by-environment interactions

Quantitative trait loci (QTL) were mapped in the woody perennial Douglas fir (Pseudotsuga menziesii var. menziesii [Mirb.] Franco) for complex traits controlling the timing of growth initiation and growth cessation. QTL were estimated under controlled environmental conditions to identify QTL interactions with photoperiod, moisture stress, winter chilling, and spring temperatures. A three-generation mapping population of 460 cloned progeny was used for genetic mapping and phenotypic evaluations. An all-marker interval mapping method was used for scanning the genome for the presence of QTL and single-factor ANOVA was used for estimating QTL-by-environment interactions. A modest number of QTL were detected per trait, with individual QTL explaining up to 9.5% of the phenotypic variation. Two QTL-by-treatment interactions were found for growth initiation, whereas several QTL-by-treatment interactions were detected among growth cessation traits. This is the first report of QTL interactions with specific environmental signals in forest trees and will assist in the identification of candidate genes controlling these important adaptive traits in perennial plants.     

Analyses of muscular dystrophy and Con A deficiency traits in testcross progeny of chickens

Hereditary muscular dystrophic chickens of the Storrs strain possess two genetic disorders, muscular dystrophy (MD) and a deficient concanavalin A (Con A), a T-cell mitogen, mediated splenic lymphocyte blastogenic response. A possible amelioration of the MD phenotype in MD chickens expressing normal Con A was postulated on the basis of progeny segregating for these two traits in F2 genetic analyses. To test this possibility, testcross progeny were examined for segregation of MD and Con A deficiency traits, and for the degree of muscle destruction and Con A deficiency. The data show both traits to be inherited independently as autosomal recessive traits, and do not support any phenotypic modifications occurring in chickens expressing MD with normal Con A. In the testcross progeny, the Con A deficiency disorder is equally deficient in normal and MD progeny, and the degree of muscle destruction as measured by serum creatine phosphokinase levels is equally great in MD chickens with or without the Con A deficiency trait. The reduced numbers of MD chickens in the testcross progeny can be accounted for by chance and probably reflect losses during in ovo development.     

Genetic markers linked to quantitative traits in poultry

This study utilized DNA fingerprints and crosses of two genetically distinct lines of layer-type chickens to identify genetic markers linked to quantitative trait loci (QTL). In phase I, back-cross (BC¹) hens were separately ranked for each of eight traits and then blood pools were produced in groups along each phenotypic distribution. The DNA was isolated from the blood pools and used in a gradient analysis to screen for DNA fingerprint bands that exhibited intensity gradients associated with the phenotypic traits. To identify linkage of bands with QTL and to estimate band effects, F₂ progeny were produced in phase II from the phase I BC, population. A single-trait animal model was used for analysis of associations of all individual DNA fingerprint bands of sires and their progeny phenotypic performance. Twenty fingerprint bands, only two of which had shown trait-associated gradients in phase I, were identified by the animal model analysis of the progeny test as QTL linked (P≤005) to specific traits of growth, reproduction and egg quality. These 20 bands warrant further study as potentially valuable molecular markers for QTL.     

Molecular diversity, structure and domestication of grasses

Map-based cloning has been considered problematic for isolating quantitative trait loci (QTLs) due to the confounding phenotypic effects of environment and other QTLs. However, five recent studies, all in plants, have succeeded in cloning QTLs using map-based methods. We review the important features of these studies and evaluate the prospects for broader application of the techniques. Successful map-based cloning requires that QTLs represent single genes that can be isolated in near-isogenic lines, and that genotypes can be unambiguously inferred by progeny testing. In plants or animals for which map-based cloning of genes with discrete phenotypes is feasible, the modified procedures required for QTLs should not be limiting in most cases. The choice between map-based cloning and alternative methods will depend on details of the species and traits being studied.     

大豆遗传图谱的构建和若干农艺性状的QTL定位分析

大豆许多重要农艺性状都是由微效多基因控制的数量性状,对这些数量性状进行QTL定位是大豆数量性状遗传研究领域的一个重要内容.本研究利用栽培大豆科新3号为父本、中黄20为母本杂交得到含192个单株的F2分离群体,构建了含122 个SSR标记、覆盖1719.6cM、由33个连锁群组成的连锁遗传图谱.利用复合区间作图法,对该群体的株高、主茎节数、单株粒重和蛋白质含量等农艺性状的调查数据进行QTL分析,共找到两个株高QTL,贡献率分别为9.15%和6.08%;两个主茎节数QTL,贡献率分别为10. 1%和8.6%;一个蛋白质含量QTL,贡献率为9.8%;一个单株粒重QTL,贡献率为11.4% .通过遗传作图共找到与所定位的4个农艺性状QTL连锁的6个SSR标记,这些标记可以应用于大豆种质资源的分子标记辅助选择,从而为大豆分子标记辅助育种提供理论依据.     

THE ASSOCIATION BETWEEN FLUCTUATING ASYMMETRY,TRAIT VARIABILITY,TRAIT HERITABILITY,AND STRESS: A MULTIPLY REPLICATED EXPERIMENT ON COMBINED STRESSES IN DROSOPHILA MELANOGASTER

A number of hypotheses have been proposed about the association between developmental stability phenotypic variability, heritability, and environmental stress. Stress is often considered to increase both the asymmetry and phenotypic variability of bilateral traits, although this may depend on trait heritability. Empirical studies of such associations often yield inconsistent results. This may reflect the diversity of traits and conditions used or a low repeatability of any associations. To test for repeatable associations between these variables, multiply replicated experiments were undertaken on Drosophila melanogaster using a combination stress at the egg, larval and adult stages of reduced protein, ethanol in the medium, and a cold shock. Both metric and meristic traits were measured and levels of heritable variation for each trait estimated by maximum likelihood and parent-offspring regression over three generations. Trait means were reduced by stress, whereas among-individual variation increased Fluctuating asymmetry (FA) was increased by stress in some cases, but few comparisons were significant. Only one trait orbital bristle, showed consistent increases in FA. Changes in trait means, trait phenotypic variability, and developmental stability as a result of stress were not correlated. Extreme phenotypes tended to have higher levels of FA but only the results for orbital bristles were significant. All traits had low to intermediate heritabilities except orbital bristle, which showed no heritable variation. Only traits with low heritability and high levels of phenotypic variability may show consistent increases in FA under stress. Overall, the independence of phenotypic variability, plasticity, and the developmental stability of traits extend to changes in these measures under stressful conditions.     

AFLP mapping and detection of quantitative trait loci (QTLs) for economically important traits in Pinus sylvestris: a preliminary study

We have applied a two-way pseudo-testcross strategy in an analysis of Pinus sylvestris for genetic mapping and detection of quantitative trait loci (QTLs) associated with economically important traits targeted in the Swedish tree-breeding programme. Based on 94 full-sib progeny of a cross between two plus-trees from northern Sweden we generated two parental maps using AFLP markers. The female map was comprised of 94 markers assigned to 15 linkage groups giving a size of 796 cM. On the male map 155 markers were assigned to 15 linkage groups, giving a total size of 1335 cM. The recombination frequency was found to be sex-dependent, being 29.3% higher in male than in female gametes. On the female map, 12 QTLs were detected (but none for branch diameter or wood density). Three QTLs for tree height accounted for 25.8% of the total phenotypic variation of this trait. When the QTLs detected for all the traits were taken independently, the percentages of phenotypic variance ranged from 9.3% to 22.7%. The highest value was observed for frost hardiness, an important trait in northern Sweden for which a major gene seemed to be involved. A cluster of QTLs for tree height, trunk diameter and volume was located on one linkage group. On the male map, four QTLs for trunk diameter and volume were detected. Due to the reduced number of individuals under study, the results are preliminary and have to be validated on more trees.     

Extent of QTL Reuse During Repeated Phenotypic Divergence of Sympatric Threespine Stickleback

How predictable is the genetic basis of phenotypic adaptation? Answering this question begins by estimating the repeatability of adaptation at the genetic level. Here, we provide a comprehensive estimate of the repeatability of the genetic basis of adaptive phenotypic evolution in a natural system. We used quantitative trait locus (QTL) mapping to discover genomic regions controlling a large number of morphological traits that have diverged in parallel between pairs of threespine stickleback (Gasterosteus aculeatus species complex) in Paxton and Priest lakes, British Columbia. We found that nearly half of QTL affected the same traits in the same direction in both species pairs. Another 40% influenced a parallel phenotypic trait in one lake but not the other. The remaining 10% of QTL had phenotypic effects in opposite directions in the two species pairs. Similarity in the proportional contributions of all QTL to parallel trait differences was about 0.4. Surprisingly, QTL reuse was unrelated to phenotypic effect size. Our results indicate that repeated use of the same genomic regions is a pervasive feature of parallel phenotypic adaptation, at least in sticklebacks. Identifying the causes of this pattern would aid prediction of the genetic basis of phenotypic evolution.     

Genetic architecture of growth and body composition in unique chicken populations

A resource population was established by crossing one modern broiler sire from a commercial broiler breeder male line with dams from two unrelated highly inbred lines; F1 birds were intercrossed to produce two F2 populations. A variety of phe notypic measurements related to growth, muscling, internal organs, and skeleton were recorded for the F2 populations and contemporary pure inbred and broiler birds. Based on the means and phenotypic distributions of the F2 populations com pared to their parental lines, the effective number of genes affecting each trait and heterosis were estimated and discussed relative to the known genetic selection history for each trait. The results suggest that a high number of genes with small epistatic effects are involved in determining the phenotype for traits that broilers were traditionally selected for, and a lower number of genes with major effects are involved in determining the phenotype for traits related to fitness. The estimated number of genes and the phenotypic distributions of the different traits suggest that a quantitative trait loci (QTL) search might be more effectively applied for traits with a low number of involved genes and a high phenotypic distribution among the F2 birds than for traits that show a lower phenotypic distribution and a high number of genes.     

A multi-marker test based on family data in genome-wide association study

Background

Requirements for successful implementation of multivariate animal threshold models including phenotypic and genotypic information are not known yet. Here simulated horse data were used to investigate the properties of multivariate estimators of genetic parameters for categorical, continuous and molecular genetic data in the context of important radiological health traits using mixed linear-threshold animal models via Gibbs sampling. The simulated pedigree comprised 7 generations and 40000 animals per generation. Additive genetic values, residuals and fixed effects for one continuous trait and liabilities of four binary traits were simulated, resembling situations encountered in the Warmblood horse. Quantitative trait locus (QTL) effects and genetic marker information were simulated for one of the liabilities. Different scenarios with respect to recombination rate between genetic markers and QTL and polymorphism information content of genetic markers were studied. For each scenario ten replicates were sampled from the simulated population, and within each replicate six different datasets differing in number and distribution of animals with trait records and availability of genetic marker information were generated. (Co)Variance components were estimated using a Bayesian mixed linear-threshold animal model via Gibbs sampling. Residual variances were fixed to zero and a proper prior was used for the genetic covariance matrix.

Results

Effective sample sizes (ESS) and biases of genetic parameters differed significantly between datasets. Bias of heritability estimates was -6% to +6% for the continuous trait, -6% to +10% for the binary traits of moderate heritability, and -21% to +25% for the binary traits of low heritability. Additive genetic correlations were mostly underestimated between the continuous trait and binary traits of low heritability, under- or overestimated between the continuous trait and binary traits of moderate heritability, and overestimated between two binary traits. Use of trait information on two subsequent generations of animals increased ESS and reduced bias of parameter estimates more than mere increase of the number of informative animals from one generation. Consideration of genotype information as a fixed effect in the model resulted in overestimation of polygenic heritability of the QTL trait, but increased accuracy of estimated additive genetic correlations of the QTL trait.

Conclusion

Combined use of phenotype and genotype information on parents and offspring will help to identify agonistic and antagonistic genetic correlations between traits of interests, facilitating design of effective multiple trait selection schemes.     

QTL detection and allelic effects for growth and fat traits in outbred pig populations

Quantitative trait loci (QTL) for growth and fatness traits have previously been identified on chromosomes 4 and 7 in several experimental pig populations. The segregation of these QTL in commercial pigs was studied in a sample of 2713 animals from five different populations. Variance component analysis (VCA) using a marker-based identity by descent (IBD) matrix was applied. The IBD coefficient was estimated with simple deterministic (SMD) and Markov chain Monte Carlo (MCMC) methods. Data for two growth traits, average daily gain on test and whole life daily gain, and back fat thickness were analysed. With both methods, seven out of 26 combinations of population, chromosome and trait, were significant. Additionally, QTL genotypic and allelic effects were estimated when the QTL effect was significant. The range of QTL genotypic effects in a population varied from 4.8% to 10.9% of the phenotypic mean for growth traits and 7.9% to 19.5% for back fat trait. Heritabilities of the QTL genotypic values ranged from 8.6% to 18.2% for growth traits, and 14.5% to 19.2% for back fat. Very similar results were obtained with both SMD and MCMC. However, the MCMC method required a large number of iterations, and hence computation time, especially when the QTL test position was close to the marker.     

The reaction norm for abdominal pigmentation and its curve in Drosophila mediopunctata depend on the mean phenotypic value

The idea of a general independence between the phenotypic plasticity and the mean value of a trait is, presently, a consensus. Here, we use the reaction norm of abdominal pigmentation (number of dark spots) of Drosophila mediopunctata in response to temperature, to test this idea. We raised eight strains, bearing two different chromosomal inversions and with varying mean phenotypic values, under 11 temperatures in a thermal gradient to test for predictions concerning mean phenotypic values, chromosomal inversions, and reaction norms. Our results revealed a strong effect of different phenotypic groups and no effect of different karyotypes on reaction norms. Moreover, we found a significant negative correlation between mean phenotypic value and the curvature of the reaction norms, revealing a high dependency of the reaction norm shape on mean phenotypic value. These results clearly reject the idea of genetic independence between mean value and phenotypic plasticity, and may indicate a pattern of correlation, which may include results from other traits and species, with an importance that has not been fully appreciated.     

An approximate method for optimum independent culling level selection for n stages of selection with explicit solutions

Summary An approximate method with explicit solutions to apply independent culling levels for multiple traits in n-stages of selection was developed. An approximate solution was found for sequentially selected traits. Two assumptions were necessary. The first was to assume that subsequent selection would not appreciably change the mean of traits already selected, and the second was to approximate the variance of a correlated trait in a selected population with an upward biased projection. The procedure was shown to give near optimal results regardless of selection intensity or genetic correlations if phenotypic correlations among traits were low. The procedure gave poor results only for certain sequences of selection when phenotypic correlations were high. However, in those cases good results were obtained using a different sequence of selection. With high correlations, the procedure is recommended only after comparing solutions and expected genetic gain for all sequences of selection. If the expected aggregate gain for the sequence of selection desired is less than that of another order, culling points associated with the optimal ordering must be determined. Genetic gain from use of culling points is independent of order of selection. The procedure is recommended for use with computer programs that attempt to find optimal culling points to reduce computational time and to check results.Journal Paper No. 12448 of the Purdue University Agricultural Experiment Station