A New Approach to Rank Several Multivariate Normal Populations with Application to Life Cycle Assessment

The need to establish the relative superiority of each treatment when compared to all the others, i.e., ordering the underlying populations according to some pre-specified criteria, often occurs in many applied research studies and technical/business problems. When populations are multivariate in nature, the problem may become quite difficult to deal with especially in case of small sample sizes or unreplicated designs. The purpose of this work is to propose a new approach for the problem of ranking several multivariate normal populations. It will be theoretically argued and numerically proved that our method controls the risk of false ranking classification under the hypothesis of population homogeneity while under the nonhomogeneity alternatives we expect that the true rank can be estimated with satisfactory accuracy, especially for the “best” populations. Our simulation study proved also that the method is robust in the case of moderate deviations from multivariate normality. Finally, an application to a real case study in the field of life cycle assessment is proposed to highlight the practical relevance of the proposed methodology.     

Robust designs for first-order multiple design multivariate models

This paper deals with the methodology of obtaining designs that are robust to nonnormality of the error distribution for tests associated with first-order multiple design multivariate linear models.A sequential procedure is given for the generation of the proposed designs.     

An Optimality Probability Index for Correlated Multiple Responses

In the real problems, there are many cases which have correlated quality characteristics so multiple response optimization can be more realistic if we can consider correlation structure of responses. In this study we propose a new method which uses multivariate normal probability to find the optimal treatment in an experimental design. Moreover, a heuristic method is used to find better factors’ level in all possible combinations in the designs with large number of controllable factors and their levels. Some simulated numerical examples and a real case were studied by the proposed approach and the comparison of the results with previous methods show efficiency of the proposed method.     

The construction of optimal designs for dose-escalation studies

Methods for the construction of A-, MV-, D- and E-optimal designs for dose-escalation studies are presented. Algebraic results proved elusive and explicit expressions for the requisite optimal designs are only given for a restricted class of traditional designs. Recourse to numerical procedures and heuristics is therefore made. Complete enumeration of all possible designs is discussed but is, as expected, highly computer intensive. Two exchange algorithms, one based on block exchanges and termed the Block Exchange Algorithm and the other a candidate-set-free algorithm based on individual exchanges and termed the Best Move Algorithm, are therefore introduced. Of these the latter is the most computationally effective. The methodology is illustrated by means of a range of carefully selected examples.     

ON OPTIMIZING MULTI-LEVEL DESIGNS: POWER UNDER BUDGET CONSTRAINTS

This paper derives a procedure for efficiently allocating the number of units in multi‐level designs given prespecified power levels. The derivation of the procedure is based on a constrained optimization problem that maximizes a general form of a ratio of expected mean squares subject to a budget constraint. The procedure makes use of variance component estimates to optimize designs during the budget formulating stages. The method provides more general closed form solutions than other currently available formulae. As such, the proposed procedure allows for the determination of the optimal numbers of units for studies that involve more complex designs. A method is also described for optimizing designs when variance component estimates are not available. Case studies are provided to demonstrate the method.     

Adaptive graph‐based multiple testing procedures

Multiple testing procedures defined by directed, weighted graphs have recently been proposed as an intuitive visual tool for constructing multiple testing strategies that reflect the often complex contextual relations between hypotheses in clinical trials. Many well‐known sequentially rejective tests, such as (parallel) gatekeeping tests or hierarchical testing procedures are special cases of the graph based tests. We generalize these graph‐based multiple testing procedures to adaptive trial designs with an interim analysis. These designs permit mid‐trial design modifications based on unblinded interim data as well as external information, while providing strong family wise error rate control. To maintain the familywise error rate, it is not required to prespecify the adaption rule in detail. Because the adaptive test does not require knowledge of the multivariate distribution of test statistics, it is applicable in a wide range of scenarios including trials with multiple treatment comparisons, endpoints or subgroups, or combinations thereof. Examples of adaptations are dropping of treatment arms, selection of subpopulations, and sample size reassessment. If, in the interim analysis, it is decided to continue the trial as planned, the adaptive test reduces to the originally planned multiple testing procedure. Only if adaptations are actually implemented, an adjusted test needs to be applied. The procedure is illustrated with a case study and its operating characteristics are investigated by simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

A general procedure for deriving distributions

A general procedure for deriving the exact and asymptotic distributions of a certain class of test statistics in multivariate analysis is proposed. The method is based on an asymptotic expansion of gamma ratios in terms of generalized Bernoulli polynomials. The exact and asymptotic results are obtained and the method is illustrated in the problem of testing linear hypotheses in the multinomial case. In this problem the method yields Box's (1949) expansion as a special case.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

A randomized response procedure for multiple-sensitive questions

The Randomized Response (RR) technique is a well-established interview procedure which guarantees privacy protection in social surveys dealing with sensitive items. The RR method assumes a stochastic mechanism to create uncertainty about the true status of the respondents in order to ensure privacy protection and to avoid tendencies to dissimulate or respond in a socially desirable direction. A very general model for the RR method was introduced by Franklin (Commun Stat Theory Methods 18:489?C505, 1989) when a single-sensitive question is under study. However, since social surveys are often based on questionnaires containing more than a single-sensitive question, the analysis of multivariate RR data is of considerable interest. This paper focuses on the generalization of the Franklin model in a multiple-sensitive question setting and on related inferential issues.     

Bayesian adaptive linearization method for phase I drug combination trials with dimension reduction

Many phase I drug combination designs have been proposed to find the maximum tolerated combination (MTC). Due to the two‐dimension nature of drug combination trials, these designs typically require complicated statistical modeling and estimation, which limit their use in practice. In this article, we propose an easy‐to‐implement Bayesian phase I combination design, called Bayesian adaptive linearization method (BALM), to simplify the dose finding for drug combination trials. BALM takes the dimension reduction approach. It selects a subset of combinations, through a procedure called linearization, to convert the two‐dimensional dose matrix into a string of combinations that are fully ordered in toxicity. As a result, existing single‐agent dose‐finding methods can be directly used to find the MTC. In case that the selected linear path does not contain the MTC, a dose‐insertion procedure is performed to add new doses whose expected toxicity rate is equal to the target toxicity rate. Our simulation studies show that the proposed BALM design performs better than competing, more complicated combination designs.     

Optimal generalized case–cohort sampling design under the additive hazard model

Generalized case–cohort designs have been proved to be a cost-effective way to enhance effectiveness in large epidemiological cohort. In generalized case–cohort design, we first select a subcohort from the underlying cohort by simple random sampling, and then sample a subset of the failures in the remaining subjects. In this article, we propose the inference procedure for the unknown regression parameters in the additive hazards model and develop an optimal sample size allocations to achieve maximum power at a given budget in generalized case–cohort design. The finite sample performance of the proposed method is evaluated through simulation studies. The proposed method is applied to a real data set from the National Wilm's Tumor Study Group.     

An estimated‐score approach for dealing with missing covariate data in matched case–control studies

Matched case–control designs are commonly used in epidemiological studies for estimating the effect of exposure variables on the risk of a disease by controlling the effect of confounding variables. Due to retrospective nature of the study, information on a covariate could be missing for some subjects. A straightforward application of the conditional logistic likelihood for analyzing matched case–control data with the partially missing covariate may yield inefficient estimators of the parameters. A robust method has been proposed to handle this problem using an estimated conditional score approach when the missingness mechanism does not depend on the disease status. Within the conditional logistic likelihood framework, an empirical procedure is used to estimate the odds of the disease for the subjects with missing covariate values. The asymptotic distribution and the asymptotic variance of the estimator when the matching variables and the completely observed covariates are categorical. The finite sample performance of the proposed estimator is assessed through a simulation study. Finally, the proposed method has been applied to analyze two matched case–control studies. The Canadian Journal of Statistics 38: 680–697; 2010 © 2010 Statistical Society of Canada     

Block-circulant matrices for constructing optimal Latin hypercube designs

Computer simulations are usually needed to study a complex physical process. In this paper, we propose new procedures for constructing orthogonal or low-correlation block-circulant Latin hypercube designs. The basic concept of these methods is to use vectors with a constant periodic autocorrelation function to obtain suitable block-circulant Latin hypercube designs. A general procedure for constructing orthogonal Latin hypercube designs with favorable properties and allowing run sizes being different from a power of 2 (or a power of 2 plus 1), is presented here for the first time. In addition, an expansion of the method is given for constructing Latin hypercube designs with low correlation. This expansion is useful when orthogonal Latin hypercube designs do not exist. The properties of the generated designs are further investigated. Some examples of the new designs, as generated by the proposed procedures, are tabulated. In addition, a brief comparison with the designs that appear in the literature is given.     

Some inference results in several symmetric multivariate exponential models

In this paper, we consider multivariate exponential models with identical marginals. We obtain MLEs of the parameters and large sample tests for mutual independence of k components in the multivariate exponential models proposed by Weinman (1966), Marshall-Olkin (1967) and Block (1975).     

A Variable Selection Method for Analyzing Supersaturated Designs

Supersaturated designs are a large class of factorial designs which can be used for screening out the important factors from a large set of potentially active variables. The huge advantage of these designs is that they reduce the experimental cost drastically, but their critical disadvantage is the confounding involved in the statistical analysis. In this article, we propose a method for analyzing data using several types of supersaturated designs. Modifications of widely used information criteria are given and applied to the variable selection procedure for the identification of the active factors. The effectiveness of the proposed method is depicted via simulated experiments and comparisons.     

The glm log-rank test:general linear modeling of log-rank scores as a method of analysis for survival data

The use of general linear modeling (GLM) procedures based on log-rank scores is proposed for the analysis of survival data and compared to standard survival analysis procedures. For the comparison of two groups, this approach performed similarly to the traditional log-rank test. In the case of more complicated designs - without ties in the survival times - the approach was only marginally less powerful than tests from proportional hazards models, and clearly less powerful than a likelihood ratio test for a fully parametric model; however, with ties in the survival time, the approach proved more powerful than tests from Cox's semi-parametric proportional hazards procedure. The method appears to provide a reasonably powerful alternative for the analysis of survival data, is easily used in complicated study designs, avoids (semi-)parametric assumptions, and is quite computationally easy and inexpensive to employ.     

Regression analysis of multivariate current status data under a varying coefficients additive hazards frailty model

This article discusses regression analysis of multivariate current status failure time data for which the observation time may be related to the underlying survival time. A local partial likelihood technique is used to estimate the varying coefficient covariate effect functions under the additive hazards frailty model. The asymptotic properties of the proposed estimators are established. An extensive simulation study is conducted for the evaluation of the proposed procedure, the results of which indicate that the proposed method works well in practice. Also, a real data study is provided to illustrate the performance of the proposed method.     

Construction of incomplete Sudoku square and partially balanced incomplete block designs

Abstract

The present article deals with the study of association among the elements of a Sudoku square. In this direction, we have defined an association scheme and constructed incomplete Sudoku square designs which are capable of studying four explanatory variables and also happen to be the designs for two-way elimination of heterogeneity. Some series of Partially Balanced Incomplete Block (PBIB) designs have also been obtained.     

An investigation of Bayesian inference approach to model validation with non-normal data

Quantitative model validation is playing an increasingly important role in performance and reliability assessment of a complex system whenever computer modelling and simulation are involved. The foci of this paper are to pursue a Bayesian probabilistic approach to quantitative model validation with non-normality data, considering data uncertainty and to investigate the impact of normality assumption on validation accuracy. The Box–Cox transformation method is employed to convert the non-normality data, with the purpose of facilitating the overall validation assessment of computational models with higher accuracy. Explicit expressions for the interval hypothesis testing-based Bayes factor are derived for the transformed data in the context of univariate and multivariate cases. Bayesian confidence measure is presented based on the Bayes factor metric. A generalized procedure is proposed to implement the proposed probabilistic methodology for model validation of complicated systems. Classic hypothesis testing method is employed to conduct a comparison study. The impact of data normality assumption and decision threshold variation on model assessment accuracy is investigated by using both classical and Bayesian approaches. The proposed methodology and procedure are demonstrated with a univariate stochastic damage accumulation model, a multivariate heat conduction problem and a multivariate dynamic system.     

ON SELECTING VARIABLES AND ASSESSING THEIR PERFORMANCE IN LINEAR DISCRIMINANT ANALYSIS

Linear discriminant analysis between two populations is considered in this paper. Error rate is reviewed as a criterion for selection of variables, and a stepwise procedure is outlined that selects variables on the basis of empirical estimates of error. Problems with assessment of the selected variables are highlighted. A leave-one-out method is proposed for estimating the true error rate of the selected variables, or alternatively of the selection procedure itself. Monte Carlo simulations, of multivariate binary as well as multivariate normal data, demonstrate the feasibility of the proposed method and indicate its much greater accuracy relative to that of other available methods.