The Matched Pairs Problem with Exponential Variates

Two classes of tests for the hypothesis of bivariate symmetry are studied. For paired exponential survival times (t_1j, t_2j), the classes of tests are those based on t_1j-t_2j and those based on log t_1j–log t_2j. For each class the sign, signed ranks, t and likelihood ratio tests are compared via Pitman's criterion of asymptotic relative efficiency (ARE). For tests based on t_1j — t_2j, it is found that: (1) the efficacy of the paired t depends on the coefficient of variation (CV) of the pair means, (2) the signed rank test has the same ARE to the sign test as for the usual location problem. For tests based on log t_1j — log t_2j, the ARE comparisons reduce to the well-known results for the one-sample location problem for samples from a logistic density. Hence, the signed rank test is asymptotically efficient. Furthermore, analyses based on log t_1j — log t_2j are not complicated by the underlying pairing mechanism.     

On the Use of the Shapiro‐Wilk Test in Two‐Stage Adaptive Inference for Paired Data from Moderate to Very Heavy Tailed Distributions

Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t‐test is optimum. On the other hand, if the differences are not normal, the t‐test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non‐parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro‐Wilk test of normality to decide whether to use the t‐test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t‐tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy‐tailed, than currently used methods.     

Exact,Distribution Free Confidence Intervals for Late Effects in Censored Matched Pairs

When comparing censored survival times for matched treated and control subjects, a late effect on survival is one that does not begin to appear until some time has passed. In a study of provider specialty in the treatment of ovarian cancer, a late divergence in the Kaplan–Meier survival curves hinted at superior survival among patients of gynecological oncologists, who employ chemotherapy less intensively, when compared to patients of medical oncologists, who employ chemotherapy more intensively; we ask whether this late divergence should be taken seriously. Specifically, we develop exact, permutation tests, and exact confidence intervals formed by inverting the tests, for late effects in matched pairs subject to random but heterogeneous censoring. Unlike other exact confidence intervals with censored data, the proposed intervals do not require knowledge of censoring times for patients who die. Exact distributions are consequences of two results about signs, signed ranks, and their conditional independence properties. One test, the late effects sign test, has the binomial distribution; the other, the late effects signed rank test, uses nonstandard ranks but nonetheless has the same exact distribution as Wilcoxon's signed rank test. A simulation shows that the late effects signed rank test has substantially more power to detect late effects than do conventional tests. The confidence statement provides information about both the timing and magnitude of late effects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tests for profile analysis of paired curves based on friedman ranking methods

Two nonparametric tests are proposed for the comparison of a paired sample of response curves with T congruent time points. The first procedure rank transforms each curve and tests the homogeneity of the resulting pair of averaged rank vectors. The second procedure rank transforms each pair of curves and tests the homogeneity of the related pair of averaged rank vectors. The first test detects only pure interactions; the second test checks if any difference exists between the rank curves. Both tests are presented in finite and asymptotic as well as in combined (by T singular tests) and multivariate form.     

A Finite Stage Decision Procedure in the Analysis of Variance

In this paper we have developed a finite stage Bayes test in the analysis of variance. Determining such a decision rule the losses of erroneously accepting the hypotheses as well as the observation costs and the a priori probabilities were considered. We have given a method of constructing the continuation intervals (a_t, b_t) (t = 1, …, r) of a r-stage test and have demonstrated how the decision process operates. Furthermore it has been investigated in which way the Bayes risk depends on the several parameters. Especially we have shown that the Bayes risk of an r-stage test is much smaller than the Bayes risk of a corresponding test with fixed sample size as a rule.     

Testing Equivalence Simultaneously for Location and Dispersion of two Normally Distributed Populations

In clinical trials with an active control usually therapeutical equivalence of a new treatment is investigated by looking at a location parameter of the distributions of the primary efficacy variable. But even if the location parameters are close to each other existing differences in variability may be connected with different risks for under or over treatment in an individual patient. Assuming normally distributed responses a multiple test procedure applying two shifted one-sided t-tests for the mean and accordingly two one-sided F-tests for the variances is proposed. Equivalence in location and variability is established if all four tests lead to a rejection at the (one-sided) level α. A conservative procedure “correcting” the t-tests for heteroscedasticity is derived. The choice of a design in terms of the global level α, the global power, the relevant deviations in the population means and variances, as well as the sample size is outlined. Numerical calculations of the actual level and power for the proposed designs show, that for balanced sample sizes the classical uncorrected one-sided t-tests can be used safely without exaggerating the global type I error probability. Finally an example is given.     

Analysis of Profiles Based on Ordinal Classification Functions and Rank Tests

In this article a test for the comparison of K groups of independent profiles is suggested. It is based on rank tests, in which the criterion is obtained by using a suitably chosen ordinal classification function. This function represents a specific view on the profiles. The test, which is easy to use, can be regarded as a generalisation of other well known tests and can be applied within a variety of situations. The relationships to other procedures for the analysis of profiles are described. Some properties are discussed.     

An Approximate Solution to the Behrens-Fisher Problem

An approximate and practical solution is proposed for the Behrens-Fisher problem. This solution is compared to the solutions considered by Mehta and Srinivasan (1970) and Welch's (1937) approximate t-test in terms of the stability of the size and magnitude of the power. It is shown that the stability of the size of the new test is better than that of Welch's t when at least one of the sample sizes is small. When the sample sizes are moderately large or large the sizes and powers of all the recommended tests are almost the same.     

Robustness of Statistical Tests in the Two-Sample Location Problem

A simulation study was performed to compare three statistical tests with respect to their performances in the two-sample location problem for contaminated normal distributions. The three tests were: the t-test, the rank-transformed t-test, and the Wilcoxon rank-sum test. The results showed the t-test to be inferior to the other two tests.     

Bivariate Sign Tests Sensitive to Homeo- and Heteropoetic Treatment Effects

Starting from N paired differences of a continuous variable, BENNETT 's (1962) bivariate sign test is identified as a test against so called homeopoetic treatment effects (or location shifts in one or both variables X and Y). As a complement, a new sign test is identified to be sensitive to so called heteropoetic treatment effects (shifts in dispersion). Both tests are shown to be important in biomedical and psychosocial research. A numerical example is given from psychopharmacology using the new bivariate sign test.     

Trimmed Weighted Simes' Test for Two One‐Sided Hypotheses With Arbitrarily Correlated Test Statistics

The two‐sided Simes test is known to control the type I error rate with bivariate normal test statistics. For one‐sided hypotheses, control of the type I error rate requires that the correlation between the bivariate normal test statistics is non‐negative. In this article, we introduce a trimmed version of the one‐sided weighted Simes test for two hypotheses which rejects if (i) the one‐sided weighted Simes test rejects and (ii) both p‐values are below one minus the respective weighted Bonferroni adjusted level. We show that the trimmed version controls the type I error rate at nominal significance level α if (i) the common distribution of test statistics is point symmetric and (ii) the two‐sided weighted Simes test at level 2α controls the level. These assumptions apply, for instance, to bivariate normal test statistics with arbitrary correlation. In a simulation study, we compare the power of the trimmed weighted Simes test with the power of the weighted Bonferroni test and the untrimmed weighted Simes test. An additional result of this article ensures type I error rate control of the usual weighted Simes test under a weak version of the positive regression dependence condition for the case of two hypotheses. This condition is shown to apply to the two‐sided p‐values of one‐ or two‐sample t‐tests for bivariate normal endpoints with arbitrary correlation and to the corresponding one‐sided p‐values if the correlation is non‐negative. The Simes test for such types of bivariate t‐tests has not been considered before. According to our main result, the trimmed version of the weighted Simes test then also applies to the one‐sided bivariate t‐test with arbitrary correlation.     

Properties and Application of the Location-Test V (n,k)

The present paper is concerned with the properties of a test statistic V(n, k) to test location differences in the one-sample case with known hypothetical distribution G(x). The test is similar to the WILCOXON two-sample statistic after replacement of the second sample by quantiles of the hypothetical distribution. A comparison with the exact distribution of V(n, k) shows that an approximation by means of the normal distribution provides good results even for small sample sizes. The V-test is unbiased against one-tailed alternatives and it is consistent with a restriction which is hardly relevant in practical applications. With regard to the application we are interested especially in the power and robustness against extreme observations for small sample size n. It is shown that in a normal distribution with known standard deviation V(n, k) is more powerful than STUDENT's t for small n and more robust in the sense considered here. The test statistic is based on grouping of the observations into classes of equal expected frequency. A generalization to arbitrary classes provides an essential extension of applicability such as to discrete distributions and to situations where only relative frequencies of G(x) in fixed classes are known.     

An enhanced quantile approach for assessing differential gene expressions

Summary .   Due to the small number of replicates in typical gene microarray experiments, the performance of statistical inference is often unsatisfactory without some form of information-sharing across genes. In this article, we propose an enhanced quantile rank score test (EQRS) for detecting differential expression in GeneChip studies by analyzing the quantiles of gene intensity distributions through probe-level measurements. A measure of sign correlation, δ, plays an important role in the rank score tests. By sharing information across genes, we develop a calibrated estimate of δ, which reduces the variability at small sample sizes. We compare the EQRS test with four other approaches for determining differential expression: the gene-specific quantile rank score test, the quantile rank score test assuming a common δ, a modified t -test using summarized probe-set-level intensities, and the Mack–Skillings rank test on probe-level data. The proposed EQRS is shown to be favorable for preserving false discovery rates and for being robust against outlying arrays. In addition, we demonstrate the merits of the proposed approach using a GeneChip study comparing gene expression in the livers of mice exposed to chronic intermittent hypoxia and of those exposed to intermittent room air.     

Sample Sizes for Comparisons of k Treatments with a Control Based on Different Definitions of the Power

The determination of sample sizes for the comparison of k treatments against a control by means of the test of Dunnett (1955, 1964) as well as by means of the multiple t-test will be considered. The power in multiple comparisons can be defined in different ways, see Hochberg and Tamhane (1987). We will derive formulas for the per-pair power, the any-pair power and the all-pairs power for both one- and two-sided comparisons. Tables will be provided that allow sample sizes to be determined for preassigned values of the power.     

UMP(U)‐Tests for a Binomial Parameter: A Paradox

We consider uniformly most powerful (UMP) as well as uniformly most powerful unbiased (UMPU) tests and their non‐randomized versions for certain hypotheses concerning a binomial parameter. It will be shown that the power function of a UMP(U)‐test based on sample size n can coincide on the entire parameter space with the power function of the corresponding test based on sample size n + 1. A complete characterization of this paradox will be derived. Apart some exceptional cases for two‐sided tests and equivalence tests the paradox appears if and only if a test based on sample size n is non‐randomized.     

Size and Power of Test Statistics for Gene Correlation in 2 × 2 Contingency Tables

Monte Carlo simulation of size and power of two proposed tests for linkage disequilibrium between two genes each with two alleles were investigated. Results were compared with two commonly used statistics, the correlation coefficient r and the log-odds ratio tests. Depending on the sign of the linkage disequilibrium, the new tests were found to be more powerful than either of the correlation or log-odds ratio tests. However, on average (positive and negative linkage disequilibrium) the Chi-square test using the correlation coefficient was to a small extent more powerful than the other tests.     

Removal of Viable Airborne Fungi from Indoor Environments by Benzalkonium Chloride-Based Aerosol Disinfectants

Fungi are ubiquitous in indoor environments, and some taxa can cause clinical symptoms in humans. Thus, from the viewpoint of public health, methods to reduce indoor airborne fungi are needed. The goal of this study was to examine the efficacies of benzalkonium chloride (BAC)–based aerosol disinfectants to remove airborne viable fungi from indoor environments. The laboratory- and field-based experiments were conducted to compare airborne culturable fungal concentrations before and after the disinfectant aerosol applications. The laboratory-based experiments showed the greater efficacies by the BAC-based disinfectant aerosol than by pure-water control aerosol (p <.05, t-test). In the field study using the BAC-based disinfectant aerosols, on average a 58% reduction of total airborne culturable fungal concentrations were observed. Additionally, the significant reduction was found for a group of airborne yeasts or yeast-like organisms (p <.05, Wilcoxon signed rank test). The BAC-based aerosol disinfectants are effective when used to reduce the numbers of airborne culturable fungi, in particular yeasts or yeast-like organisms, from indoor environments.     

An Exact Paired Rank Test

An exact rank test for two dependent samples based on overall mid‐ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data.     

Tests for Homogeneity of Variances Using Robust Weighted Likelihood Estimates

The classical normal-theory tests for testing the null hypothesis of common variance and the classical estimates of scale have long been known to be quite nonrobust to even mild deviations from normality assumptions for moderate sample sizes. Levene (1960) suggested a one-way ANOVA type statistic as a robust test. Brown and Forsythe (1974) considered a modified version of Levene's test by replacing the sample means with sample medians as estimates of population locations, and their test is computationally the simplest among the three tests recommended by Conover , Johnson , and Johnson (1981) in terms of robustness and power. In this paper a new robust and powerful test for homogeneity of variances is proposed based on a modification of Levene's test using the weighted likelihood estimates (Markatou , Basu , and Lindsay , 1996) of the population means. For two and three populations the proposed test using the Hellinger distance based weighted likelihood estimates is observed to achieve better empirical level and power than Brown-Forsythe's test in symmetric distributions having a thicker tail than the normal, and higher empirical power in skew distributions under the use of F distribution critical values.     

Obesity‐Related Knowledge,Attitudes, and Behaviors in Obese and Non‐obese Urban Philadelphia Female Adolescents

Objectives: To examine relationships between knowledge, attitudinal and behavioral factors, and obesity and to determine how these factors influence obesity status in west Philadelphia female adolescents. Research Methods and Procedures: A matched‐pairs study was conducted with 32 stature‐ and age‐matched pairs of obese (body mass index and triceps skinfold ≥95th percentile of National Health and Nutrition Examination Survey I) and non‐obese (body mass index and triceps skinfold between the 15th and 85th percentiles of National Health and Nutrition Examination Survey I) female African American adolescents (aged 11 to 15 years), selected from a school‐based study sample, based on obesity status and matching criteria. Adolescents were compared on the following measures: physical activity, inactivity, dietary intake, eating attitudes, health behavior knowledge, body image, self‐esteem, and maturation status. Differences between obese and non‐obese females were tested using paired t tests and Wilcoxon matched‐pairs signed‐rank tests. Results: Physical activity, inactivity, and perception of ideal body size emerged as the most important contributory factors to obesity status. There were no statistically significant matched‐pair differences in macronutrient and micronutrient intakes, self‐esteem, eating attitudes, health behavior knowledge, or maturation status of these adolescents. Obese adolescents had significantly lower levels of physical activity, higher inactivity, and a larger perception of ideal body size than non‐obese adolescents. Discussion: Knowledge and attitudinal factors (with the exception of perception of ideal body size) had far less association with obesity than activity‐related behavioral factors. These findings suggest that future intervention strategies should pay particular attention to physical activity, inactivity, and body image attitudes.