Some multiple decision problems in analysis of variance

In most practical situations to which the analysis of variance tests are applied, they do not supply the information that the experimenter aims at. If, for example, in one-way ANOVA the hypothesis is rejected in actual application of the F-test, the resulting conclusion that the true means θ₁,…,θ_k are not all equal, would by itself usually be insufficient to satisfy the experimenter. In fact his problems would begin at this stage. The experimenter may desire to select the “best” population or a subset of the “good” populations; he may like to rank the populations in order of “goodness” or he may like to draw some other inferences about the parameters of interest.

The extensive literature on selection and ranking procedures depends heavily on the use of independence between populations (block, treatments, etc.) in the analysis of variance. In practical applications, it is desirable to drop this assumption or independence and consider cases more general than the normal.

In the present paper, we derive a method to construct optimal (in some sense) selection procedures to select a nonempty subset of the k populations containing the best population as ranked in terms of θ_i’s which control the size of the selected subset and which maximizes the minimum average probability of selecting the best. We also consider the usual selection procedures in one-way ANOVA based on the generalized least squares estimates and apply the method to two-way layout case. Some examples are discussed and some results on comparisons with other procedures are also obtained.     

Selecting a Good Stochastic System for the Large Number of Alternatives

In this article, we present the problem of selecting a good stochastic system with high probability and minimum total simulation cost when the number of alternatives is very large. We propose a sequential approach that starts with the Ordinal Optimization procedure to select a subset that overlaps with the set of the actual best m% systems with high probability. Then we use Optimal Computing Budget Allocation to allocate the available computing budget in a way that maximizes the Probability of Correct Selection. This is followed by a Subset Selection procedure to get a smaller subset that contains the best system among the subset that is selected before. Finally, the Indifference-Zone procedure is used to select the best system among the survivors in the previous stage. The numerical test involved with all these procedures shows the results for selecting a good stochastic system with high probability and a minimum number of simulation samples, when the number of alternatives is large. The results also show that the proposed approach is able to identify a good system in a very short simulation time.     

Distribution-free subset selection for incompletely ranked data

In market research and some other areas, it is common that a sample of n judges (consumers, evaluators, etc.) are asked to independently rank a series of k objects or candidates. It is usually difficult to obtain the judges' full cooperation to completely rank all k objects. A practical way to overcome this difficulty is to give each judge the freedom to choose the number of top candidates he is willing to rank. A frequently encountered question in this type of survey is how to select the best object or candidate from the incompletely ranked data. This paper proposes a subset selection procedure which constructs a random subset of all the k objects involved in the survey such that the best object is included in the subset with a prespecified confidence. It is shown that the proposed subset selection procedure is distribution-free over a very broad class of underlying distributions. An example from a market research study is used to illustrate the proposed procedure.     

Conditionally unbiased estimation in the normal setting with unknown variances

To efficiently and completely correct for selection bias in adaptive two-stage trials, uniformly minimum variance conditionally unbiased estimators (UMVCUEs) have been derived for trial designs with normally distributed data. However, a common assumption is that the variances are known exactly, which is unlikely to be the case in practice. We extend the work of Cohen and Sackrowitz (Statistics & Probability Letters, 8(3):273-278, 1989), who proposed an UMVCUE for the best performing candidate in the normal setting with a common unknown variance. Our extension allows for multiple selected candidates, as well as unequal stage one and two sample sizes.     

A Restricted Subset Selection Rule for Selecting at Least One of the t Best Normal Populations in Terms of Their Means When Their Common Variance is Known,Case II

Consider k( ? 2) normal populations with unknown means μ₁, …, μ_k, and a common known variance σ². Let μ_[1] ? ??? ? μ_[k] denote the ordered μ_i.The populations associated with the t(1 ? t ? k ? 1) largest means are called the t best populations. Hsu and Panchapakesan (2004) proposed and investigated a procedure R_HPfor selecting a non empty subset of the k populations whose size is at most m(1 ? m ? k ? t) so that at least one of the t best populations is included in the selected subset with a minimum guaranteed probability P* whenever μ_{[k ? t + 1]} ? μ_{[k ? t]} ? δ*, where P*?and?δ* are specified in advance of the experiment. This probability requirement is known as the indifference-zone probability requirement. In the present article, we investigate the same procedure R_HP for the same goal as before but when k ? t < m ? k ? 1 so that at least one of the t best populations is included in the selected subset with a minimum guaranteed probability P* whatever be the configuration of the unknown μ_i. The probability requirement in this latter case is termed the subset selection probability requirement. Santner (1976) proposed and investigated a different procedure (R_S) based on samples of size n from each of the populations, considering both cases, 1 ? m ? k ? t and k ? t < m ? k. The special case of t = 1 was earlier studied by Gupta and Santner (1973) and Hsu and Panchapakesan (2002) for their respective procedures.     

Selection and screening procedures to determine optimal product designs

To compare several promising product designs, manufacturers must measure their performance under multiple environmental conditions. In many applications, a product design is considered to be seriously flawed if its performance is poor for any level of the environmental factor. For example, if a particular automobile battery design does not function well under temperature extremes, then a manufacturer may not want to put this design into production. Thus, this paper considers the measure of a product's quality to be its worst performance over the levels of the environmental factor. We develop statistical procedures to identify (a near) optimal product design among a given set of product designs, i.e., the manufacturing design that maximizes the worst product performance over the levels of the environmental variable. We accomplish this by intuitive procedures based on the split-plot experimental design (and the randomized complete block design as a special case); split-plot designs have the essential structure of a product array and the practical convenience of local randomization. Two classes of statistical procedures are provided. In the first, the δ-best formulation of selection problems, we determine the number of replications of the basic split-plot design that are needed to guarantee, with a given confidence level, the selection of a product design whose minimum performance is within a specified amount, δ, of the performance of the optimal product design. In particular, if the difference between the quality of the best and second best manufacturing designs is δ or more, then the procedure guarantees that the best design will be selected with specified probability. For applications where a split-plot experiment that involves several product designs has been completed without the planning required of the δ-best formulation, we provide procedures to construct a ‘confidence subset’ of the manufacturing designs; the selected subset contains the optimal product design with a prespecified confidence level. The latter is called the subset selection formulation of selection problems. Examples are provided to illustrate the procedures.     

On some optimal selection procedures for weibull populations

Consider k (k >(>)2) Weibull populations. We shall derive a method of constructing optimal selection procedures to select a subset of the k populations containing the best population which control the size of the selected subset and which maximises the minimum probability of making a correct selection. Procedures and results are derived for the case when sample sizes are unequal. Some tables and figures are given at the end of this paper.     

Average worth estimation of the selected subset of Poisson populations

Let Π₁, …, Π _p be p(p≥2) independent Poisson populations with unknown parameters θ₁, …, θ _p , respectively. Let X _i denote an observation from the population Π _i , 1≤i≤p. Suppose a subset of random size, which includes the best population corresponding to the largest (smallest) θ _i , is selected using Gupta and Huang [On subset selection procedures for Poisson populations and some applications to the multinomial selection problems, in Applied Statistics, R.P. Gupta, ed., North-Holland, Amsterdam, 1975, pp. 97–109] and (Gupta et al. [On subset selection procedures for Poisson populations, Bull. Malaysian Math. Soc. 2 (1979), pp. 89–110]) selection rule. In this paper, the problem of estimating the average worth of the selected subset is considered under the squared error loss function. The natural estimator is shown to be biased and the UMVUE is obtained using Robbins [The UV method of estimation, in Statistical Decision Theory and Related Topics-IV, S.S. Gupta and J.O. Berger, eds., Springer, New York, vol. 1, 1988, pp. 265–270] UV method of estimation. The natural estimator is shown to be inadmissible, by constructing a class of dominating estimators. Using Monte Carlo simulations, the bias and risk of the natural, dominated and UMVU estimators are computed and compared.     

Selecting the normal population with the smallest variance: A restricted subset selection rule

Consider k( ? 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ²_[1] ? ??? ? σ_[k]² denote the ordered variances. Our goal is to select a non empty subset of the k populations whose size is at most m(1 ? m ? k ? 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probability P* whenever σ²_[2]/σ_[1]² ? δ* > 1, where P* and δ* are specified in advance of the experiment. Based on samples of size n from each of the populations, we propose and investigate a procedure called R_BCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example.     

Subset selection procedures for ΔP-superior populations

In some ranking and selection problems it is reasonable to consider any population which is inferior but sufficiently close to the best (t-th best) as acceptable. Under this assumption, this paper studies classes of procedures to meet two possible goals. A and B. Goal A is to select a subset which contains only good populations, while Goal B is of a screening nature and requires selection of a subset of size not exceeding m (1 ≤ m ≤ k) and containing at least one good population. In each case results loading to the determination of the sample size required to attain the goals above with prespecified probability are obtained. Properties of the procedures are discussed.     

Control-variate methods for comparison with a standard

Comparison with a standard is a general multiple comparison problem, where each system is required to be compared with a single system, referred to as a ‘standard’, as well as with other alternative systems. Screening procedures specially designed to be used for comparison with a standard have been proposed to find a subset that includes all the systems better than the standard in terms of the expected performance. Selection procedures are derived to determine the best system among a number of systems that are better than the standard, or to select the standard when it is equal to or better than the other alternatives. We develop new procedures for screening and selection through the use of two variance reduction techniques, common random numbers and control variates, which are particularly useful in the context of simulation experiments. Empirical results and a realistic example are also provided to compare our procedures with the existing ones.     

Criterion and selection of optimal subsets in the linear regression,two stage procedures

In many practical situation the regression analysis with stochastic regressors is used. The estimations of this model are often influenced by a high degree of multicollinearity. For avoidance of this fact a criterion and a procedure for the selection of an optimal subset for regression will be derived on the base of the partition of the moments of the conditional normal distribution of the regressand under the condition of the regressors. Further two stage procedures improving the result of the subset regression. based also on the partition of the conditional moments will be given.     

A combination multiple comparisons and subset selection procedure to identify treatments that are strictly inferior to the best

This paper considers the problem of identifying which treatments are strictly worse than the best treatment or treatments in a one-way layout, which has many important applications in screening trials for new product development. A procedure is proposed that selects a subset of the treatments containing only treatments that are known to be strictly worse than the best treatment or treatments. In addition, simultaneous confidence intervals are obtained which provide upper bounds on how inferior the treatments are compared with these best treatments. In this way, the new procedure shares the characteristics of both subset selection procedures and multiple comparison procedures. Some tables of critical points are provided for implementing the new procedure, and some examples of its use are given.     

A two stage procedure for selecting δ∗-optimal means in the normal model

Suppose π₁,…,π_k are k normal populations with π_i having unknown mean μ_i and unknown variance σ². The population π_i will be called δ^?-optimal (or good) if μ_i is within a specified amountδ? of the largest mean. A two stage procedure is proposed which selects a subset of the k populations and guarantees with probability at least P^? that the selected subset contains only δ?-optimal π_i ’s. In addition to screening out non-good populations the rule guarantees a high proportion of sufficiently good π_i’S will be selected.     

Confidence bounds and selection of the t best populations

Confidence statements about location (or scale) parameters associated with K populations, which may be used in making selection decisions about those populations, are investigated. When a subset of fixed size t is selected from the K populations a lower bound is obtained for the minimum selected parameter as a function of the maximum non-selected parameter. Tables are produced for the normal means case when the variance is common but unknown. It is pointed out that these tables may be used to find confidence intervals discussed by Hsu (1984     

A subset selection procedure for multinomial distributions

A subset selection procedure is developed for selecting a subset containing the multinomial population that has the highest value of a certain linear combination of the multinomial cell probabilities; such population is called the ‘best’. The multivariate normal large sample approximation to the multinomial distribution is used to derive expressions for the probability of a correct selection, and for the threshold constant involved in the procedure. The procedure guarantees that the probability of a correct selection is at least at a pre-assigned level. The proposed procedure is an extension of Gupta and Sobel's [14] selection procedure for binomials and of Bakir's [2] restrictive selection procedure for multinomials. One illustration of the procedure concerns population income mobility in four countries: Peru, Russia, South Africa and the USA. Analysis indicates that Russia and Peru fall in the selected subset containing the best population with respect to income mobility from poverty to a higher-income status. The procedure is also applied to data concerning grade distribution for students in a certain freshman class.     

SNP selection for predicting a quantitative trait

Molecular markers combined with powerful statistical tools have made it possible to detect and analyze multiple loci on the genome that are responsible for the phenotypic variation in quantitative traits. The objectives of the study presented in this paper are to identify a subset of single nucleotide polymorphism (SNP) markers that are associated with a particular trait and to construct a model that can best predict the value of the trait given the genotypic information of the SNPs using a three-step strategy. In the first step, a genome-wide association test is performed to screen SNPs that are associated with the quantitative trait of interest. SNPs with p-values of less than 5% are then analyzed in the second step. In the second step, a large number of randomly selected models, each consisting of a fixed number of randomly selected SNPs, are analyzed using the least angle regression method. This step will further remove redundant SNPs due to the complicated association among SNPs. A subset of SNPs that are shown to have a significant effect on the response trait more often than by chance are considered for the third step. In the third step, two alternative methods are considered: the least angle shrinkage and selection operation and sparse partial least squares regression. For both methods, the predictive ability of the fitted model is evaluated by an independent test set. The performance of the proposed method is illustrated by the analysis of a real data set on Canadian Holstein cattle.     

On the performance of some subset selection procedures

In this paper, we restrict attention to the problem of subset selection of normal populations. The approaches and results of some previous comparison studies of subset selection procedures are discussed briefly. And then the result of a new Monte Carlo study comparing the performance of two classical procedures and the Bayes procedure is presented.     

SELECTION PROCEDURES FOR SCALE PARAMETERS USING TWO-SAMPLE U-STATISTICS

Let be k independent populations having the same known quantile of order p (0 p 1) and let F(x)=F(x/_i) be the absolutely continuous cumulative distribution function of the ith population indexed by the scale parameter ₁, i = 1,…, k. We propose subset selection procedures based on two-sample U-statistics for selecting a subset of k populations containing the one associated with the smallest scale parameter. These procedures are compared with the subset selection procedures based on two-sample linear rank statistics given by Gill & Mehta (1989) in the sense of Pitman asymptotic relative efficiency, with interesting results.     

On selecting the treatment with the largest probability of survival

Suppose there are k(>= 2) treatments and each treatment is a Bernoulli process with binomial sampling. The problem of selecting a random-sized subset which contains the treatment with the largest survival probability (reliability or probability of success) is considered. Based on the ideas from both classical approaches and general Bayesian statistical decision approach, a new subset selection procedure is proposed to solve this kind of problem in both balanced and unbalanced designs. Comparing with the classical procedures, the proposed procedure has a significantly smaller selected subset. The optimal properties and performance of it were examined. The methods of selecting and fitting the priors and the results of Monte Carlo simulations on selected important cases are also studied.