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.     

Simultaneous Selection of Extreme Populations: A Subset Selection Approach

The problem of selecting a “best” (largest mean, or smallest mean) population from a collection of k independent populations was formulated and solved by Bechhofer (1954). Gupta (1965) solved another important problem, that of selecting a subset of populations containing the “best” population from the original collection of populations. Since then many variations of the problem have been considered. Tong (1969) and Lewis (1980) have investigated the problem of selecting extreme populations (populations with a largest, and populations with a smallest, mean) with respect to one and two standard populations, respectively. In this paper we study the selection of extreme populations in absence of any standard population. We formulate subset-selection procedures when variances are known and equal, and also in the most general case when they are unknown and unequal. Nonexistence of a single-stage procedure is noted for this latter case (even if variances are equal). A two-stage procedure and some of its associated properties are discussed. Tables needed for application are provided, as is a worked example.     

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 of an Almost Best Treatment

A generalized goal using subset selection is discussed for the location parameter case. This goal is to select a non-empty subset from a set of k (k ≥ 2) treatments that contains at least one ε-best treatment with confidence level P*. For a set of treatments an ε-best treatment is defined as a treatment with location parameter on a distance less than or equal to ε(ε ≥o) from the best treatment, where best is defined as largest value of the location parameter. The efficiency of subset selection of an ε-best treatment relative to subset selection of the best treatment is investigated and is computed for some values of k and the confidence level for the Normal case as well as for the Logistic case.     

Selection system efficiencies for computer simulated progeny test field designs in loblolly pine

Summary Six simulated progeny test field designs in combination with three within-family selection systems were tested on three loblolly pine (Pinus taeda L.) progeny test sites in southeastern Oklahoma and southwestern Arkansas, to compare genetic gains for the single trait, height. Residual deviations obtained by subtraction of family and plantation mean effects for each plantation were combined with simulated genetic effects with known family variance structure. The simulated genetic populations, arranged in the following progeny test field designs — large square or almost square plots, five- and ten-tree row plots, five-tree noncontiguous plots, two tree row plots, and single-tree plots — were superimposed on the residual data for each plantation. Within-family selection methods based on deviations from block means, deviations from neighborhood means and deviations from plot means were built into the model. Realized genetic gain attained by each design — selection system combination was compared with the genetic gain theoretically possible if selection accuracy were perfect, and with expected gain estimated using the general linear model. In general, average realized genetic gain compared well with expected gain. Differences between designs with large versus small plots were generally lower than expected, although the single-tree plot design always yielded highest realized gain. Realized gain was generally higher than expected when within-family selection was based on deviations from block or neighborhood means, but equal to or lower than expected when selection was based on deviations from plot means.     

Selection Problems in Balanced Block Designs

If we have independent random variables the solution of BECHHOFER's selection problem in special cases is easily applied because we can use tables of a standardized multivariate normal distribution. The aim of this note is to show how such tables are also applicable in a case of correlated random variables like the estimates of treatment effects in BIB and BCB. Formulae for a choice of minimal sample sizes are also given.     

Further Investigations on the Robustness of the Standardized Selection Difference

The robustness of the standardized selection difference is investigated for the family of central chi-squared-distributions. The results are compared with those for the uniform, a family of triangular distributions and the exponential distribution. Beside these exact results and some simulation results are considered.     

On Indifference Zone Selection with a Preference Threshold

This paper discusses a selection criterion that generalizes the well-known concept of indifference zone selection through a preference threshold. A population is preferred to another population if the difference in the sums of observed values exceeds a given nonnegative threshold value. We present an argument for this selection rule by modelling preference by imprecise previsions. We aim at guidelines to design a selection experiment, which is characterized by two numbers: the number of necessary observations per population, and the preference threshold. Next to the probability of correct selection we also need a second specification. In this paper we consider a probability of false selection that is strongly related to the minimum probability of correct selection. Based on this model the outcome of an experiment may be ‘no selection’, at least not based on strong preference of a single population. The ideas are presented through a simple selection problem for normal populations with common known variance. Although the theory has a frequentist nature, the derivation and justification of the selection rule through imprecise previsions relies on Bayesian foundations, and via this route we gain more insight into the selection criterion.     

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     

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.     

Model Building and Analysis for Block Designs with Nested Rows and Columns

The paper deals with two main problems: it gives a unified approach to model building for observations obtained in a block design with nested rows and columns and it provides the analysis of variance for such data. Also, some statistical properties of the design are examined.     

Effect of Non-normality on Response to Selection in small Populations

The expressions for the r-th largest standardised deviate have been derived for beta, gamma, exponential and log-normal distributions. Using these expressions the intensities of selection have been computed for different parametric values of the distributions and of p, the proportion saved for samples of size upto 10 and compared with the corresponding values for normal distribution. The normal approximation to beta and gamma distributions is seen to be quite robust in predicting response to selection under moderately heavy and low cullings.     

Selection for increased percentage phaseolin in common bean

Summary Two selection methods were compared to determine which was more efficient for increasing percentage phaseolin in the common bean (Phaseolus vulgaris L.). A base population consisting of families segregating for six seed protein alleles (Phas^S, Phas^C, Phas^T, phas^-, lec^-, and Arcl⁺), all of which have measurable effects on percentage phaseolin, was subjected to either three cycles of S₁ family recurrent selection for increased percentage phaseolin (PPS), or one cycle of selection for combinations of the protein alleles (PAS) known to have positive effects on phaseolin accumulation. One cycle of PAS resulted in an increase in percentage phaseolin that was equivalent to three cycles of PPS. Selection under both methods produced increases in several correlated traits including percentage total protein, phaseolin as a percent of total protein, mg protein/seed, and mg phaseolin/seed. The amount of nonphaseolin protein per seed decreased, while seed yield was unaffected by either selection procedure. By selecting for favorable seed protein alleles identified by electrophoresis, it was possible to rapidly increase percentage phaseolin without the need for field evaluation.     

Two-Stage Procedures for Comparing Treatments with a Control: Elimination at the First Stage and Estimation at the Second Stage

We consider the problem of comparing a set of p₁ test treatments with a control treatment. This is to be accomplished in two stages as follows: In the first stage, N₁ observations are allocated among the p₁ treatments and the control, and the subset selection procedure of Gupta and Sobel (1958) is employed to eliminate “inferior” treatments. In the second stage, N₂ observations are allocated among the (randomly) selected subset of p₂(≤p₁) treatments and the control, and joint confidence interval estimates of the treatment versus control differences are calculated using Dunnett's (1955) procedure. Here both N₁ and N₂ are assumed to be fixed in advance, and the so-called square root rule is used to allocate observations among the treatments and the control in each stage. Dunnett's procedure is applied using two different types of estimates of the treatment versus control mean differences: The unpooled estimates are based on only the data obtained in the second stage, while the pooled estimates are based on the data obtained in both stages. The procedure based on unpooled estimates uses the critical point from a p₂-variate Student t-distribution, while that based on pooled estimates uses the critical point from a p₁-variate Student t-distribution. The two procedures and a composite of the two are compared via Monte Carlo simulation. It is shown that the expected value of p₂ determines which procedure yields shorter confidence intervals on the average. Extensions of the procedures to the case of unequal sample sizes are given. Applicability of the proposed two-stage procedures to a drug screening problem is discussed.     

Sequentially Rejective Test Procedures for Detecting Outlying Cells in One- and Two-Sample Multinomial Experiments

For multiple testing of multinomial models in the case of one or two samples we propose using test procedures based on the principle described by MARCUS, PERITZ and GABRIEL (1976). These methods are based in each step of the sequentially rejective strategy on tests which exhaust the full α level (i.e. which are not conservative). The tests can be performed in a finite or asymptotic version.     

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     

Three Multiple Comparison Procedures for Trimmed Means

Two common goals when choosing a method for performing all pairwise comparisons of J independent groups are controlling experiment wise Type I error and maximizing power. Typically groups are compared in terms of their means, but it has been known for over 30 years that the power of these methods becomes highly unsatisfactory under slight departures from normality toward heavy-tailed distributions. An approach to this problem, well-known in the statistical literature, is to replace the sample mean with a measure of location having a standard error that is relatively unaffected by heavy tails and outliers. One possibility is to use the trimmed mean. This paper describes three such multiple comparison procedures and compares them to two methods for comparing means.