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.     

Comparison of Two Distribution-Free Procedures for Multiple Comparisons With a Control

Chakraborti and Desu (1988) presented a distribution-free procedure for testing that k (≥1) distributions are equal to a control distribution. They compared their procedure, a generalization of the test proposed by Mathisen (1943), to the procedure proposed by Slivka (1970). They asserted that their procedure has shorter expected duration than Slivka's procedure in life-testing experiments where observations become available in an ordered manner. Here it is proven that, in fact, Slivka's procedure has shorter duration in such circumstances. Normal approximations are presented which indicate that their procedure requires a smaller sample size to guarantee a specified power for Lehmann alternatives and proportional hazard alternatives when all observations are to be observed.     

Simultaneous Confidence Intervals for the Mean of Multivariate Poisson Distribution: A Comparison

Haibing (2009) proposed a procedure for successive comparisons between ordered treatment effects in one-way layout and showed that the proposed procedure has greater power than the procedure proposed by Lee and Spurrier (1995). Critical constants required for the proposed procedure were estimated using Monte Carlo simulation and few values of the constants were tabulated which limit the applications of the proposed procedure. In this article, a numerical method, using recursive integration methodology, is discussed to compute the critical constants which work efficiently for a large number of treatments and extensive values of critical constants are tabulated for the use of practitioners. Power comparisons of Haibing's and Lee and Spurrier's procedure is also discussed.     

Non-parametric testing for the number of change points in a sequence of independent random variables

A non-parametric procedure is derived for testing for the number of change points in a sequence of independent continuously distributed variables when there is no prior information available. The procedure is based on the Kruskal–Wallis test, which is maximized as a function of all possible places of the change points. The procedure consists of a sequence of non-parametric tests of nested hypotheses corresponding to a decreasing number of change points. The properties of this procedure are analyzed by Monte Carlo methods and compared to a parametric procedure for the case that the variables are exponentially distributed. The critical values are given for sample sizes up to 200.     

Multiple comparison procedures for finding non-maximum normal means

Abstract

In this study, we discuss multiple comparison procedures for finding normal means which are not maximum among several normal means. Specifically, we propose the single step procedure, the sequentially rejective step down procedure and the step up procedure. For the single step procedure we determine the critical value for a specified significance level. For the sequentially rejective step down procedure and the step up procedure we determine the critical value at each step of the test for a specified significance level. For three procedures we formulate the power of the test under a specified alternative hypothesis. We give some numerical examples regarding critical values and power of the test intended to compare three procedures.     

MULTIPLE TESTING VIA FDR FOR LARGE SCALE IMAGING DATA

The multiple testing procedure plays an important role in detecting the presence of spatial signals for large scale imaging data. Typically, the spatial signals are sparse but clustered. This paper provides empirical evidence that for a range of commonly used control levels, the conventional FDR procedure can lack the ability to detect statistical significance, even if the p-values under the true null hypotheses are independent and uniformly distributed; more generally, ignoring the neighboring information of spatially structured data will tend to diminish the detection effectiveness of the FDR procedure. This paper first introduces a scalar quantity to characterize the extent to which the "lack of identification phenomenon" (LIP) of the FDR procedure occurs. Second, we propose a new multiple comparison procedure, called FDR(L), to accommodate the spatial information of neighboring p-values, via a local aggregation of p-values. Theoretical properties of the FDR(L) procedure are investigated under weak dependence of p-values. It is shown that the FDR(L) procedure alleviates the LIP of the FDR procedure, thus substantially facilitating the selection of more stringent control levels. Simulation evaluations indicate that the FDR(L) procedure improves the detection sensitivity of the FDR procedure with little loss in detection specificity. The computational simplicity and detection effectiveness of the FDR(L) procedure are illustrated through a real brain fMRI dataset.     

On the conservative simultaneous confidence procedures for multiple comparisons among mean vectors

In this paper, conservative simultaneous confidence intervals for multiple comparisons among mean vectors in multivariate normal distributions are considered. Some properties of the multivariate Tukey–Kramer procedure for pairwise comparisons and the conservative simultaneous confidence procedure for comparisons with a control are presented. Particularly, the upper bound for the conservativeness of the simultaneous confidence procedure for comparisons with a control is obtained. Finally, numerical results by Monte Carlo simulations and an example to illustrate the procedure are given.     

Analysis of means using ranks for the randomized complete block design

A nonparametric test procedure is proposed for the analysis of randomized complete block designs. Such a procedure may be carried out graphically in the form of a Shewhart control chart. Exact and asymptotic critical values are given for the implementation of the proposed procedure. A Monte Carlo study is made to compare the powers of the proposed procedure to those of analysis of variance, the analysis of means, and the Friedman procedures. Results of the study indicate that the proposed procedure has superior power performance when testing against slippage alternative hypotheses under heavy-tailed distributions such as the Cauchy distribution. However, when testing against symmetric alternatives under light-tailed distributions, the proposed procedure does not perform well     

Admissibility of a test procedure in a conditionally specified random model

Admissibility of a test procedure is a desirable property though not a compelling one; while lack of it often results in discarding a given test procedure. In this paper we have proved necessary conditions for admissibility for a test procedure.     

A unified sequential test procedure for simultaneous testing the equality of several binomial proportions to a specified standard

In this article, we propose a unified sequentially rejective test procedure for testing simultaneously the equality of several independent binomial proportions to a specified standard. The proposed test procedure is general enough to include some well-known multiple testing procedures such as the Ordinary Bonferroni procedure, Hochberg procedure and Rom procedure. It involves multiple tests of significance based on the simple binomial tests (exact or approximate) which can be easily found in many elementary standard statistics textbooks. Unlike the traditional Chi-square test of the overall hypothesis, the procedure can identify the subset of the binomial proportions, which are different from the prespecified standard with the control of the familywise type I error rate. Moreover, the power computation of the procedure is provided and the procedure is illustrated by two real examples from an ecological study and a carcinogenicity study.     

Max-min multiple comparison procedure for comparing several dose levels with a zero dose control

The comparison of increasing doses of a compound to a zero dose control is of interest in medical and toxicological studies. Assume that the mean dose effects are non-decreasing among the non-zero doses of the compound. A simple procedure that modifies Dunnett's procedure is proposed to construct simultaneous confidence intervals for pairwise comparisons of each dose group with the zero dose control by utilizing the ordering of the means. The simultaneous lower bounds and upper bounds by the new procedure are monotone, which is not the case with Dunnett's procedure. This is useful to categorize dose levels. The expected gains of the new procedure over Dunnett's procedure are studied. The procedure is shown by real data to compare well with its predecessor.     

An asymptotic distribution-free selection procedure for a two-way layout problem

This paper deals with an asymptotic distribution-free subset selection procedure for a two-way layout problem. The treatment effect with the largest unknown value is of interest to us. The block effect is a nuisance parameter in this problem. The proposed procedure is based on the Hodges-Lehmann estimators of location parameters. The asymptotic relative efficiency of the proposed procedure with the normal means procedure is evaluated. It is shown that the proposed procedure has a high efficiency.     

A New Look at Half-normal Plots for Assessing the Significance of Contrasts for Unreplicated Factorials

The visual inspection of a half-normal plot is a popular procedure for interpreting data from unreplicated factorial experiments. As this technique is somewhat subjective, the plot should preferably be supplemented with a more formal procedure. However, despite the improvements proposed by Zahn, the formal method using the half-normal plot suggested by Daniel seems to be rarely used. We investigate this procedure by using simulations and real data and suggest a modification to improve its acceptability. We also present corrected critical values for a procedure proposed by Lenth which was found to be miscalibrated.     

Best Linear Classification Rule for Multivariate Interval Screened Data

In classification analysis, the target variable is often in practice defined by an underlying multivariate interval screening scheme. This engenders the problem of properly characterizing the screened populations as well as that of obtaining a classification procedure. Such problems paved the way for the development of yet another linear classification procedure and the incorporation of a class of skew-elliptical distributions for describing evolutions in the populations. To render the linear procedure effective, this article considers derivation and properties of the classification procedure as well as efficient estimation. The procedure is illustrated in applications to real and simulation data.     

Nonparametric approaches to testing and multiple comparisons in a one-way anova

Two questions of interest involving nonparametric multiple comparisons are considered. The first question concerns whether it is appropriate to use a multiple comparison procedure as a test of the equality of k treatments, and if it is, which procedure performs best as a test. Our results show that for smaller k values some multiple comparison procedures perform well as tests. The second question concerns whether a joint ranking or a separate ranking multiple comparison procedure performs better as a test and as a device for treatment separation. We find that the joint ranking procedure does slightly better as a test, but for treatment separation the answer depends on the situation.     

A nonparametric coherent confidence procedure

ABSTRACT

Holm's step-down testing procedure starts with the smallest p-value and sequentially screens larger p-values without any information on confidence intervals. This article changes the conventional step-down testing framework by presenting a nonparametric procedure that starts with the largest p-value and sequentially screens smaller p-values in a step-by-step manner to construct a set of simultaneous confidence sets. We use a partitioning approach to prove that the new procedure controls the simultaneous confidence level (thus strongly controlling the familywise error rate). Discernible features of the new stepwise procedure include consistency with individual inference, coherence, and confidence estimations for follow-up investigations. In a simple simulation study, the proposed procedure (treated as a testing procedure), is more powerful than Holm's procedure when the correlation coefficient is large, and vice versa when it is small. In the data analysis of a medical study, the new procedure is able to detect the efficacy of Aspirin as a cardiovascular prophylaxis in a nonparametric setting.     

A comparison of two procedures to select the best binomial population with sequential elimination of inferior populations

We compare the selection procedure of Levin and Robbins [1981. Selecting the highest probability in binomial or multinomial trials. Proc. Nat. Acad. Sci. USA 78, 4663–4666.] with the procedure of Paulson [1994. Sequential procedures for selecting the best one of k Koopman–Darmois populations. Sequential Analysis 13, 207–220.] to identify the best of several binomial populations with sequential elimination of unlikely candidates. We point out situations in which the Levin–Robbins procedure dominates the Paulson procedure in terms of the duration of the experiment, the expected total number of observations, and the expected number of failures. Because the Levin–Robbins procedure is also easier to implement than Paulson's procedure and gives a tighter guarantee for the probability of correct selection, we conclude that it holds a competitive edge over Paulson's procedure.     

A Baseline-free Procedure for Transformation Models Under Interval Censorship

An important property of Cox regression model is that the estimation of regression parameters using the partial likelihood procedure does not depend on its baseline survival function. We call such a procedure baseline-free. Using marginal likelihood, we show that an baseline-free procedure can be derived for a class of general transformation models under interval censoring framework. The baseline-free procedure results a simplified and stable computation algorithm for some complicated and important semiparametric models, such as frailty models and heteroscedastic hazard/rank regression models, where the estimation procedures so far available involve estimation of the infinite dimensional baseline function. A detailed computational algorithm using Markov Chain Monte Carlo stochastic approximation is presented. The proposed procedure is demonstrated through extensive simulation studies, showing the validity of asymptotic consistency and normality. We also illustrate the procedure with a real data set from a study of breast cancer. A heuristic argument showing that the score function is a mean zero martingale is provided.     

Exact sample-size determination in testing non-inferiority under a simple crossover trial

For testing the non-inferiority (or equivalence) of an experimental treatment to a standard treatment, the odds ratio (OR) of patient response rates has been recommended to measure the relative treatment efficacy. On the basis of an exact test procedure proposed elsewhere for a simple crossover design, we develop an exact sample-size calculation procedure with respect to the OR of patient response rates for a desired power of detecting non-inferiority at a given nominal type I error. We note that the sample size calculated for a desired power based on an asymptotic test procedure can be much smaller than that based on the exact test procedure under a given situation. We further discuss the advantage and disadvantage of sample-size calculation using the exact test and the asymptotic test procedures. We employ an example by studying two inhalation devices for asthmatics to illustrate the use of sample-size calculation procedure developed here.