Subset selection of the best treatment via many-one tests

The method of Gupta (1956, 1965) was developed to select a subset from k normal populations that contains the best populations with given probability. This paper shows a duality between the general goal of selecting a subset for the best population and many-one tests. A population should be regarded as ‘candidate’ for the best population and thus retained in the subset if the samples from the other populations are not significantly better. Based on this ‘idea’ a general selection procedure is proposed using many-one tests for the comparison of each population against the remaining ones.     

On selecting a subset which contains all populations better than a control

Let π₁…, π_k denote k(≥ 2) populations with unknown means μ₁ , …, μ_k and variances σ₁ ² , …, σ_k ² , respectively and let π_o denote the control population having mean μ_o and variance σ_o ² . It is assumed that these populations are normally distributed with correlation matrix {ρ_ij}. The goal is to select a subset, of populations of π₁ , …, π_k which contains all the populations with means larger than or equal to the mean of the control one. Procedures are given for selecting such a subset so that the probability that all the populations with means larger than or equal to the mean of the control one are included in the selected subset is at least equal to a predetermined value P?(l/k < P? < 1). The goal treated here is a first step screening procedure that allows the experimenter to choose a subset and withhold judgement about which one has the largest mean. Then, if the one with the largest mean is desired it can be chosen from the selected subset on the basis of cost and other considerations. Percentage points are also included.     

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.     

Combining two classical approaches for statistical selection

This paper presents a selection procedure that combines Bechhofer's indifference zone selection and Gupta's subset selection approaches, by using a preference threshold. For normal populations with common known variance, a subset is selected of all populations that have sample sums within the distance of this threshold from the largest sample sum. We derive the minimal necessary sample size and the value for the preference threshold, in order to satisfy two probability requirements for correct selection, one related to indifference zone selection, the other to subset selection. Simulation studies are used to illustrate the method.     

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.     

A restricted subset selection procedure for Weibull populations

Consider k(k ≥ 2) two-parameter Weibull populations. We want to select a subset of the populations not exceeding m in size such that the subset contains at least ? of the t best populations. We have proposed a procedure which uses either the maximum likelihood estimators or ‘simplified’ linear estimators of the parameters. The estimators are based on type II censored data. The ranking of the populations is done by comparing their reliabilities at a certain fixed time. In selected cases the constants for the procedure are tabulated using Monte Carlo methods.     

Selecting logistic populations using the sample medians

This paper is concerned primarily with subset selection procedures based on the sample mediansof logistic populations. A procedure is given which chooses a nonempty subset from among kindependent logistic populations, having a common known variance, so that the populations with thelargest location parameter is contained in the subset with a pre‐specified probability. Theconstants required to apply the median procedure with small sample sizes (≤= 19) are tabulated and can also be used to construct simultaneous confidence intervals. Asymptotic formulae are provided for application with larger sample sizes. It is shown that, under certain situations, rules based on the median are substantially more efficient than analogous procedures based either on sample means or on the sum of joint ranks.     

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.     

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.     

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.     

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.     

Conditional probability of correct selection under the continuum partition with applications

The problem of selecting the best population from among a finite number of populations in the presence of uncertainty is a problem one faces in many scientific investigations, and has been studied extensively, Many selection procedures have been derived for different selection goals. However, most of these selection procedures, being frequentist in nature, don't tell how to incorporate the information in a particular sample to give a data-dependent measure of correct selection achieved for this particular sample. They often assign the same decision and probability of correct selection for two different sample values, one of which actually seems intuitively much more conclusive than the other. The methodology of conditional inference offers an approach which achieves both frequentist interpret ability and a data-dependent measure of conclusiveness. By partitioning the sample space into a family of subsets, the achieved probability of correct selection is computed by conditioning on which subset the sample falls in. In this paper, the partition considered is the so called continuum partition, while the selection rules are both the fixed-size and random-size subset selection rules. Under the distributional assumption of being monotone likelihood ratio, results on least favourable configuration and alpha-correct selection are established. These re-sults are not only useful in themselves, but also are used to design a new sequential procedure with elimination for selecting the best of k Binomial populations. Comparisons between this new procedure and some other se-quential selection procedures with regard to total expected sample size and some risk functions are carried out by simulations.     

Screening for Multiple Target Detection

We apply statistical selection theory to multiple target detection problems by analyzing the Mahalanobis distances between multivariate normal populations and a desired standard (a known characteristic of a target). We want to achieve the goal of selecting a subset that contains no non target (negative) sites, which entails screening out all non targets. Correct selection (CS) is defined according to this goal. We consider two cases: (1) that all covariance matrices are known; and (2) that all covariance matrices are unknown, including both heteroscedastic and homoscedastic cases. Optimal selection procedures are proposed in order to reach the selection goal. The least favorable configurations (LFC) are found. Tables and figures are presented to illustrate the properties of our proposed procedures. Simulation examples are given to show that our procedures work well. The log-concavity results of the operating characteristic functions are also given.     

SELECTING THE t BEST POPULATIONS — SCALE PARAMETER CASE

Selection from k independent populations of the t (< k) populations with the smallest scale parameters has been considered under the Indifference Zone approach by Bechhofer k Sobel (1954). The same problem has been considered under the Subset Selection approach by Gupta & Sobel (1962a) for the normal variances case and by Carroll, Gupta & Huang (1975) for the more general case of stochastically increasing distributions. This paper uses the Subset Selection approach to place confidence bounds on the probability of selecting all “good” populations, or only “good” populations, for the Case of scale parameters, where a “good” population is defined to have one of the t smallest scale parameters. This is an extension of the location parameter results obtained by Bofinger & Mengersen (1986). Special results are obtained for the case of selecting normal populations based on variances and the necessary tables are presented.     

Subset selection for normal means in multi-factor experiments

The procedure of Gupta [1956], [1965] for selecting a random sized subset of k ≧ 2 normal populations which contains the population with the largest population mean when the populations have a common variance is generalized to multi-factor experiments. Two-factor experiments with equal replication on each factor-level combination are discussed in detail. The cases of zero and non-zero interactions between factor levels are considered. For the two-factor, zero interaction case with a common number of observations at each factor-level combination, a table of constants necessary to implement the procedure is provided for experiments having selected levels per factor; the constants are equi-coordinate upper percentage points of a multivariate Student t distribution.     

Extension of two-sided test to multiple treatment trials

In a two-treatment trial, a two-sided test is often used to reach a conclusion, Usually we are interested in doing a two-sided test because of no prior preference between the two treatments and we want a three-decision framework. When a standard control is just as good as the new experimental treatment (which has the same toxicity and cost), then we will accept both treatments. Only when the standard control is clearly worse or better than the new experimental treatment, then we choose only one treatment. In this paper, we extend the concept of a two-sided test to the multiple treatment trial where three or more treatments are involved. The procedure turns out to be a subset selection procedure; however, the theoretical framework and performance requirement are different from the existing subset selection procedures. Two procedures (exclusion or inclusion) are developed here for the case of normal data with equal known variance. If the sample size is large, they can be applied with unknown variance and with the binomial data or survival data with random censoring.     

Selecting all treatments better than a control using existing tables

In this paper subset selection procedures for selecting all treatment populations with means larger than a control population are proposed. The treatments and control are assumed to have a multivariate normal distribution. Various covariance structures are considered. All of the proposed procedures are easily implemented using existing tables of the multivariate normal and multivariate t distributions. Some other procedures which have been proposed require extensive and unavailable tables for their implementation     

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     

Some subset selection problems

Consider that we have a collection of k populations π₁, π₂…,π_k. The quality of the ith population is characterized by a real parameter θ_i and the population is to be designated as superior or inferior depending on how much the θ_i differs from θ_max = max{θ₁, θ₂,…,θ_k}. From the set {π₁, π₂,…,π_k}, we wish to select the subset of superior populations. In this paper we devise rules of selection which have the property that their selected set excludes all the inferior populations with probability at least 1?α, where a is a specified number.