The optimal size of choice sets in choice experiments

In this paper, we establish the optimal size of the choice sets in generic choice experiments for asymmetric attributes when estimating main effects only. We give an upper bound for the determinant of the information matrix when estimating main effects and all two-factor interactions for binary attributes. We also derive the information matrix for a choice experiment in which the choice sets are of different sizes and use this to determine the optimal sizes for the choice sets.     

Considerations for Effective Simulation Study Analysis

This article discusses estimation of several percentiles simultaneously, develops a simple test to compare the sizes of two test statistics, and considers the use of logit models to adjust power curves to have the same null hypothesis level.     

A sample size selection procedure for pairwise comparisons of means with unequal variances

A sample size selection procedure for paired comparisons of means is presented which controls the half width of the confidence intervals while allowing for unequal variances of treatment means.     

Choosing the best pairwise comparisons of means from non-normal populations,with unequal variances,but equal sample sizes

A Monte Carlo simulation evaluated five pairwise multiple comparison procedures for controlling Type I error rates, any-pair power, and all-pairs power. Realistic conditions of non-normality were based on a previous survey. Variance ratios were varied from 1:1 to 64:1. Procedures evaluated included Tukey's honestly significant difference (HSD) preceded by an F test, the Hayter–Fisher, the Games–Howell preceded by an F test, the Pertiz with F tests, and the Peritz with Alexander–Govern tests. Tukey's procedure shows the greatest robustness in Type I error control. Any-pair power is generally best with one of the Peritz procedures. All-pairs power is best with the Pertiz F test procedure. However, Tukey's HSD preceded by the Alexander–Govern F test may provide the best combination for controlling Type I and power rates in a variety of conditions of non-normality and variance heterogeneity.     

Expanded confidence intervals

Confidence intervals for location parameters are expanded (in either direction) to some “crucial” points and the resulting increase in the confidence coefficient investigated.Particaular crucial points are chosen to illuminate some hypothesis testing problems.Special results are dervied for the normal distribution with estimated variance and, in particular, for the problem of classifiying treatments as better or worse than a control.For this problem the usual two-sided Dunnett procedure is seen to be inefficient.Suggestions are made for the use of already published tables for this problem.Mention is made of the use of expanded confidence intervals for all pairwise comparisons of treatments using an “honest ordering difference” rather than Tukey's “honest siginificant difference”.     

Measuring cross-country technological catch-up through variable-parameter FDH

In this paper, we estimate indices of technological catch-up for a sample of 52 countries through the FDH approach with variable scaling parameters proposed in Kerstens and Vanden Eeckaut (1999). We show that this technique has significant advantages for the international comparisons of productivity and, more generally, wherever the assumption of convexity for the production set is not likely to be appropriate. We want to thank Laurens Cherchye, Hal Fried, Kristian Kerstens and an anonymous referee for useful comments on preceding versions of this paper. The usual disclaimer applies. Financial support from the MURST, Italy, is gratefully acknowledged.     

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.     

Non-parametric predictive inference for future order statistics

This article presents non-parametric predictive inference for future order statistics. Given the data consisting of n real-valued observations, m future observations are considered and predictive probabilities are presented for the rth-ordered future observation. In addition, joint and conditional probabilities for events involving multiple future order statistics are presented. The article further presents the use of such predictive probabilities for order statistics in statistical inference, in particular considering pairwise and multiple comparisons based on two or more independent groups of data.     

Optimal designs for stated choice experiments that incorporate ties

In 1970 Davidson generalised the Bradley–Terry model to allow respondents to say that the two options presented in a choice task were equally attractive. In this paper we extend this idea to the MNL model with m options in each choice set and we show that the optimal designs for the MNL model are also optimal in this setting.     

Use of tail areas for testing goodness-of-fit

Characterizing a set of data as a random sample from a specified distribution is often a precursor to statistical inference or hypothesis testing involving the extremes of the distribution -precisely the regions of greatest uncertainty. It seems reasonable then to exploit as best we can our limited knowledge of this region. Toward this end we investigate here the areas in the tails of the distribution as determined by the extreme order statistics as a criterion for testing goodness-of-fit.     

A Nonparametric Test for the Ordered Alternative Based on Kendall's Correlation Coefficient

Several methods have been developed for testing the ordered alternative. These include the Jonckheere–Terpstra (JT) test (Jonckheere, 1954 Jonckheere , A. R. ( 1954 ). A distribution free k-sample test against ordered alternatives . Biometrika 41 : 133 – 145 .[Crossref], [Web of Science ®] , [Google Scholar]; Terpstra, 1952 Terpstra , T. ( 1952 ). The asymptotic normality and consistency of Kendall's test against trend when ties are present in one ranking . Indigationes Mathematicae 14 : 327 – 333 . [Google Scholar]), a modified JT test (MJT) (Tryon and Hettmansperger, 1987 Tryon , V. P. , Hettmansperger , T. P. ( 1987 ). A class of nonparametric tests for homogeneity against ordered alternatives . Annals of Statistics 1 : 1061 – 1070 . [Google Scholar]), and a test proposed by Terpstra and Magel (TM) (Terpstra and Magel, 2003 Terpstra , J. T. , Magel , R. C. ( 2003 ). A new nonparametric test for the ordered alternative problem . Journal of Nonparametric Statistics 15 : 289 – 301 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), among others. This article proposes a new method for testing the ordered alternative. The proposed test is based on Kendall's tau statistic. The asymptotic distribution of the test statistic is given. A Monte Carlo simulation study is conducted comparing the estimated powers of the proposed test with existing tests under a variety of sample sizes and distributions.     

Multiple comparisons of mean vectors with large dimension under general conditions

ABSTRACT

Multiple comparisons for two or more mean vectors are considered when the dimension of the vectors may exceed the sample size, the design may be unbalanced, populations need not be normal, and the true covariance matrices may be unequal. Pairwise comparisons, including comparisons with a control, and their linear combinations are considered. Under fairly general conditions, the asymptotic multivariate distribution of the vector of test statistics is derived whose quantiles can be used in multiple testing. Simulations are used to show the accuracy of the tests. Real data applications are also demonstrated.     

A catalogue of designs for partial profiles in paired comparison experiments with three groups of factors

A common strategy for avoiding information overload in multi-factor paired comparison experiments is to employ pairs of options which have different levels for only some of the factors in a study. For the practically important case where the factors fall into three groups such that all factors within a group have the same number of levels and where one is only interested in estimating the main effects, a comprehensive catalogue of D-optimal approximate designs is presented. These optimal designs use at most three different types of pairs and have a block diagonal information matrix.     

Designs for first-order interactions in paired comparison experiments with two-level factors

For paired comparison experiments involving options described by a common set of two-level factors a new method for generating exact designs is presented. These designs allow the efficient estimation of main effects and first-order interactions and perform better than alternative designs available in the literature.     

Tables for pairwise multiple comparisons using shaffer's modifiedsequentially-rejective procedure

A sequentially rejective (SR) testing procedure introduced by Holm (1979) and modified (MSR) by Shaffer (1986) is considered for testing all pairwise mean comparisons.For such comparisons, both the SR and MSR methods require that the observed test statistics be ordered and compared, each in turn, to appropriate percentiles on Student's t distribution.For the MSR method these percentiles are based on the maximum number of true null hypotheses remaining at each stage of the sequential procedure, given prior significance at previous stages, A function is developed for determining this number from the number of means being tested and the stage of the test.For a test of all pairwise comparisons, the logical implications which follow the rejection of a null hypothesis renders the MSR procedure uniformly more powerful than the SR procedure.Tables of percentiles for comparing K means, 3 < K < 6, using the MSR method are presented.These tables use Sidak's (1967) multiplicative inequality and simplify the use of t he MSR procedure.Several modifications to the MSR are suggested as a means of further increasing the power for testing the pairwise comparisons.General use of the MSR and the corresponding function for testing other parameters besides the mean is discussed.     

Pairwise Comparisons Associated with the K Independent Sample Median Test

In this article, a Tukey-type method is proposed that will allow simultaneous pairwise comparisons among all pairs of samples associated with Mood's procedure. An example is also provided for illustrative purposes.     

Generalized confidence intervals compatible with the Min test for simultaneous comparisons of one subpopulation to several other subpopulations

A problem where one subpopulation is compared with several other subpopulations in terms of means with the goal of estimating the smallest difference between the means commonly arises in biology, medicine, and many other scientific fields. A generalization of Strass-burger-Bretz-Hochberg approach for two comparisons is presented for cases with three and more comparisons. The method allows constructing an interval estimator for the smallest mean difference, which is compatible with the Min test. An application to a fluency-disorder study is illustrated. Simulations confirmed adequate probability coverage for normally distributed outcomes for a number of designs.     

A method of unconstrained multiple comparisons with the best

There are three types of multiple comparisons: all-pairwise multiple comparisons (MCA), multiple comparisons with the best (MCB), and multiple comparisons with a control (MCC). There are also three levels of multiple comparisons inference: confidence sets, subset comparisons, test of homogeneity. In current practice, MCA procedures dominate. In correct attempts at more efficient comparisons, in the form of employing lower level MCA procedures for higher level inference, account for the most frequent abuses in multiple comparisons. A better strategy is to choose the correct type of inference at the level of inference desired. In particular, very often the simulataneous comparisons of each treatment with the best of the other treatments (MCB) suffice. Hsu (1984b) gave simultaneous confidence intervals for θ_i ? max_j≠iθ_j having the simple form [? (Y_i ?max_j≠i Y_j ? C) (Y_i?max_j≠i Y_j + C)⁺]. Those intervals were constrained, sothat even if a treatment is inferred to be the best, no positive bound on how much it is better thatn the rest is given, a somewhat undesirable property. In this article it is shown that by employing a slightly larger critical value, the nonpositivity constraint on the lower bound is removed.