A multivariate solution of the multivariate ranking and selection problem

The problem of selection of the best multivariate population is given a new formulation which does not involve reducing the populations to univariate quantities. This formulation's solution is developed for known, and (using the Heteroscedastic Method) also for unknown, variance-covariance matrices. Preference reversals and arbitrary nonlinear preference functions are explicitly allowed in this new theory     

On Calculations Involving the Maximum Cell Frequency

We describe a method of computing the cumulative distribution function of the maximum and minimum cell frequencies in sampling distributions commonly encountered in the analysis of categorical data.The procedure is efficient for exact or approximate calculation in both homogeneous and non-homogeneous cases, is non-recursive, and does not require Dirichlet integrals.Some related statistical problems are also discussed.     

Nonparametric prediction analysis for binary data

A nonparametric inference algorithm developed by Davis and Geman (1983) is extended problem. The algorithm and applied to a medical prediction employs an estimation procedure for acquiring pairwise statistics among variables of a binary data set, allows for the data-driven creation of interaction terms among the variables, and employs a decision rule which asymptotically gives the minimum expected error. The inference procedure was designed for large data sets but has been extended via the method of cross-validation to encompass smaller data sets.     

Some efficient random imputation methods

Imputation methods that assign a selection of respondents’ values for missing i tern nonresponses give rise to an addd,tional source of sampling variation, which we term imputation varLance , We examine the effect of imputation variance on the precision of the mean, and propose four procedures for sampling the rEespondents that reduce this additional variance. Two of the procedures employ improved sample designs through selection of respc,ndents by sampling without replacement and by stratified sampl;lng. The other two increase the sample base by the use of multiple imputations.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

On f-tests and linear hypothesis in a general linear model

A simple method of setting linear hypotheses testable by F-tests in a general linear model when the covariance matrix has a general form and is completely unknown, is provided. With some additional conditions imposed on the covariance matrix, there exist the UMP invariant tests of certain linear hypotheses. We derive them to compare the powers with those of F-tests obtained under no restrictions on the covariance matrix. The results are illustrated in a multiple regression model with some examples.     

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.     

Sensitivity analysis for two control groups

In many experiments where data have been collected at two points in time (pre-treatment and post-treatment), investigators wish to determine if there is a difference between two treatment groups. In recent years it has been proposed that an appropriate statistical analysis to determine if treatment differences exist is to use the post-treatment values as the primary comparison variables and the pre-treatment values as covariates. When there are several outcome variables, we propose new tests based on residuals as alternatives to existing methods and investigate how the powers of the new and existing tests are affected by various choices of covariates. The limiting distribution of the test statistic of the new test based on residuals is given. Monte Carlo simulations are employed in the power comparisons.     

Optimal design of clinical trials comparing several treatments with a control

Clinical trials are often designed to compare several treatments with a common control arm in pairwise fashion. In this paper we study optimal designs for such studies, based on minimizing the total number of patients required to achieve a given level of power. A common approach when designing studies to compare several treatments with a control is to achieve the desired power for each individual pairwise treatment comparison. However, it is often more appropriate to characterize power in terms of the family of null hypotheses being tested, and to control the probability of rejecting all, or alternatively any, of these individual hypotheses. While all approaches lead to unbalanced designs with more patients allocated to the control arm, it is found that the optimal design and required number of patients can vary substantially depending on the chosen characterization of power. The methods make allowance for both continuous and binary outcomes and are illustrated with reference to two clinical trials, one involving multiple doses compared to placebo and the other involving combination therapy compared to mono-therapies. In one example a 55% reduction in sample size is achieved through an optimal design combined with the appropriate characterization of power.