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.     

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.     

A sampling study of the size and power of tests for paired survival data

The paired-t, sign, and signed rank tests were compared for samples from a bivariate exponential distribution. Each is a valid α-level test. One test was not uniformly more powerful than the others for all sample sizes, α levels, correlations, and alternative hypotheses considered, but the signed rank test did well consistently. It was always preferable to the sign test and never was appreciably worse than the paired-t test. The relative performance of the tests depends on α as well as the sample size.     

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.     

A New Powerful Nonparametric Rank Test for Ordered Alternative Problem

We propose a new nonparametric test for ordered alternative problem based on the rank difference between two observations from different groups. These groups are assumed to be independent from each other. The exact mean and variance of the test statistic under the null distribution are derived, and its asymptotic distribution is proven to be normal. Furthermore, an extensive power comparison between the new test and other commonly used tests shows that the new test is generally more powerful than others under various conditions, including the same type of distribution, and mixed distributions. A real example from an anti-hypertensive drug trial is provided to illustrate the application of the tests. The new test is therefore recommended for use in practice due to easy calculation and substantial power gain.     

Statistical power when testing for genetic differentiation

A variety of statistical procedures are commonly employed when testing for genetic differentiation. In a typical situation two or more samples of individuals have been genotyped at several gene loci by molecular or biochemical means, and in a first step a statistical test for allele frequency homogeneity is performed at each locus separately, using, e.g. the contingency chi-square test, Fisher's exact test, or some modification thereof. In a second step the results from the separate tests are combined for evaluation of the joint null hypothesis that there is no allele frequency difference at any locus, corresponding to the important case where the samples would be regarded as drawn from the same statistical and, hence, biological population. Presently, there are two conceptually different strategies in use for testing the joint null hypothesis of no difference at any locus. One approach is based on the summation of chi-square statistics over loci. Another method is employed by investigators applying the Bonferroni technique (adjusting the P-value required for rejection to account for the elevated alpha errors when performing multiple tests simultaneously) to test if the heterogeneity observed at any particular locus can be regarded significant when considered separately. Under this approach the joint null hypothesis is rejected if one or more of the component single locus tests is considered significant under the Bonferroni criterion. We used computer simulations to evaluate the statistical power and realized alpha errors of these strategies when evaluating the joint hypothesis after scoring multiple loci. We find that the 'extended' Bonferroni approach generally is associated with low statistical power and should not be applied in the current setting. Further, and contrary to what might be expected, we find that 'exact' tests typically behave poorly when combined in existing procedures for joint hypothesis testing. Thus, while exact tests are generally to be preferred over approximate ones when testing each particular locus, approximate tests such as the traditional chi-square seem preferable when addressing the joint hypothesis.     

Linear Rank Statistics for the Simple Tree Alternatives

The problem of comparing several treatments with a control is considered. It is formulated as a test of homogeneity with one-sided alternatives that at least one treatment is better than the control, while no treatment is worse than the control. A class of linear rank tests for the simple tree alternatives is proposed, assuming a location model. The most efficient test of the proposed class for detecting a specified pattern is derived. Two optimal tests for use in the case of vaguely specified tree alternatives are derived. In a simulation study the power of these two tests are compared with the power of Kruskal-Wallis test. Recommendations about the use of the proposed tests are made.     

Statistical tests as inappropriate tools for data analysis performed on non-random samples of plant communities

Many authors apply statistical tests to sets of relevés obtained using non-random methods to investigate phytosociological and ecological relationships. Frequently applied tests include thet-test, ANOVA, Mann-Whitney test, Kruskal-Wallis test, chi-square test (of independence, goodness-of-fit, and homogeneity), Kolmogorov-Smirnov test, concentration analysis, tests of linear correlation and Spearman rank correlation coefficient, computer intensive methods (such as randomization and re-sampling) and others. I examined the extent of reliability of the results of such tests applied to non-random data by examining the tests requirements according to statistical theory. I conclude that when used for such data, the statistical tests do not provide reliable support for the inferences made because non-randomness of samples violated the demand for observations to be independent, and different parts of the investigated communities did not have equal chance to be represented in the sample. Additional requirements, e.g. of normality and homoscedasticity, were also neglected in several cases. The importance of data satisfying the basic requirements set by statistical tests is stressed.     

A New Testing Approach for Comparing the Overall Homogeneity of Survival Curves

The assessment of overall homogeneity of time‐to‐event curves is a key element in survival analysis. The currently commonly used methods, e.g., log‐rank and Wilcoxon tests, may have a significant loss of statistical testing power under certain circumstances. In this paper a new statistical testing approach is developed to compare the overall homogeneity of survival curves. The proposed new method has greater power than the commonly used tests to detect overall differences between crossing survival curves. The small‐sample performance of the new test is investigated under a variety of situations by means of Monte Carlo simulations. Furthermore, the applicability of the proposed testing approach is illustrated by a real data example from a kidney dialysis trial. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparing sire and dam estimates of heritability: jackknife and likelihood approaches

Three estimates of heritability are available from the half-sib pedigree design: the sire, dam and genotypic estimates. Because of its significantly smaller standard error, the genotypic estimate is preferred provided that there are no non-additive effects that inflate the estimate. I present two methods to test for such effects: these are a t-test of the paired sire and dam pseudovalues from the jackknife procedure and the likelihood ratio test from the animal model. Both methods are shown to be valid tests for significant dominance and/or maternal effects. SPLUS coding for the implementation of the jackknife method is provided. Unless sample sizes are very large, the power of the tests is low and hence caution is advised in the use of the genotypic estimate following a nonsignificant test. An approximate power analysis can be done using the data from the jackknife method but the estimated power is typically a substantial underestimate of the true power and its use is not recommended.     

A signed-rank test for clustered data

Summary .   We consider the problem of comparing two outcome measures when the pairs are clustered. Using the general principle of within-cluster resampling, we obtain a novel signed-rank test for clustered paired data. We show by a simple informative cluster size simulation model that only our test maintains the correct size under a null hypothesis of marginal symmetry compared to four other existing signed rank tests; further, our test has adequate power when cluster size is noninformative. In general, cluster size is informative if the distribution of pair-wise differences within a cluster depends on the cluster size. An application of our method to testing radiation toxicity trend is presented.     

Applications of Extensions of Bivariate Rank Sum Statistics to the Crossover Design to Compare Two Treatments Through Four Sequence Groups

Summary This article describes applications of extensions of bivariate rank sum statistics to the crossover design with four sequence groups for two treatments. A randomized clinical trial in ophthalmology provides motivating background for the discussion. The bilateral design for this study has four sequence groups T:T, T:P, P:T, and P:P, respectively, for T as test treatment or P as placebo in the corresponding order for the left and right eyes. This article describes how to use the average of the separate Wilcoxon rank sum statistics for the left and right eyes for the overall comparison between T and P with the correlation between the two eyes taken into account. An extension of this criterion with better sensitivity to potential differences between T and P through reduction of the applicable variance has discussion in terms of a conceptual model with constraints for within‐side homogeneity of groups with the same treatment and between‐side homogeneity of the differences between T and P. Goodness of fit for this model can have assessment with test statistics for its corresponding constraints. Simulation studies for the conceptual model confirm better power for the extended test statistic with its full invocation than other criteria without this property. The methods summarized here are illustrated for the motivating clinical trial in ophthalmology, but they are applicable to other situations with the crossover design with four sequence groups for either two locations for two treatments at the same time for a patient or two successive periods for the assigned treatments for a recurrent disorder. This article also notes that the methods based on its conceptual model can have unsatisfactory power for departures from that model where the difference between T and P via the T:T and P:P groups is not similar to that via the T:P and P:T groups, as might occur when T has a systemic effect in a bilateral trial. For this situation, more robust test statistics have identification, but there is recognition that the parallel groups design with only the T:T and P:P groups may be more useful than the bilateral design with four sequence groups.     

An approximate method for optimum independent culling level selection for n stages of selection with explicit solutions

Summary An approximate method with explicit solutions to apply independent culling levels for multiple traits in n-stages of selection was developed. An approximate solution was found for sequentially selected traits. Two assumptions were necessary. The first was to assume that subsequent selection would not appreciably change the mean of traits already selected, and the second was to approximate the variance of a correlated trait in a selected population with an upward biased projection. The procedure was shown to give near optimal results regardless of selection intensity or genetic correlations if phenotypic correlations among traits were low. The procedure gave poor results only for certain sequences of selection when phenotypic correlations were high. However, in those cases good results were obtained using a different sequence of selection. With high correlations, the procedure is recommended only after comparing solutions and expected genetic gain for all sequences of selection. If the expected aggregate gain for the sequence of selection desired is less than that of another order, culling points associated with the optimal ordering must be determined. Genetic gain from use of culling points is independent of order of selection. The procedure is recommended for use with computer programs that attempt to find optimal culling points to reduce computational time and to check results.Journal Paper No. 12448 of the Purdue University Agricultural Experiment Station     

A mixture-model approach for parallel testing for unequal variances

Testing for unequal variances is usually performed in order to check the validity of the assumptions that underlie standard tests for differences between means (the t-test and anova). However, existing methods for testing for unequal variances (Levene's test and Bartlett's test) are notoriously non-robust to normality assumptions, especially for small sample sizes. Moreover, although these methods were designed to deal with one hypothesis at a time, modern applications (such as to microarrays and fMRI experiments) often involve parallel testing over a large number of levels (genes or voxels). Moreover, in these settings a shift in variance may be biologically relevant, perhaps even more so than a change in the mean. This paper proposes a parsimonious model for parallel testing of the equal variance hypothesis. It is designed to work well when the number of tests is large; typically much larger than the sample sizes. The tests are implemented using an empirical Bayes estimation procedure which `borrows information' across levels. The method is shown to be quite robust to deviations from normality, and to substantially increase the power to detect differences in variance over the more traditional approaches even when the normality assumption is valid.     

Potency control of diphtheria component in adsorbed vaccines by in vitro neutralization tests.

Samples from 20 lots of dT vaccine and from 20 lots of DTP vaccine were used to standardize and validate the Vero cell and the toxin binding inhibition (ToBI) tests for the potency control of diphtheria component. For the Vero cell method, violet crystal solution was used to stain the cells and estimate the endpoint of diluted diphtheria antitoxin. Diphtheria anatoxin was used for performing the ToBI test instead of toxin. The results obtained by both in vitro tests were similar to those obtained by in vivo toxin neutralization test in guinea pigs. The various analysis and the chi(2) test applied to evaluate the reproducibility and homogeneity, respectively, among in vitro tests and in vivo toxin neutralization test did not detect statistical significant difference for both analysed vaccines. An excellent correlation among in vitro tests and in vivo neutralization test was observed by Spearman's correlation coefficient.     

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.     

Morphometric testing of structural hypotheses of the supraorbital region in modern humans

Our understanding of the functional morphology of the primate supraorbital region is based largely on previous morphometric and in vivo mechanical tests of hypotheses in non-human anthropoids. Prior tests of two structural hypotheses explaining morphological variation in the supraorbital region, the craniofacial size hypothesis and the spatial hypothesis, did not fully consider modern humans. We extend these previous findings to include modern humans by conducting morphometric tests of these two hypotheses in a sample of adult Melanesian crania. Morphometric correlates of structural predictions for the craniofacial size and spatial hypotheses were developed and compared to measurements of the supraorbital region via bivariate product-moment correlations. Measurements of the supraorbital region are significantly correlated with a craniofacial size estimate across individuals from this Melanesian sample. This result supports the prediction of the craniofacial size hypothesis that the magnitude of the supraorbital region is proportional to craniofacial size. The predicted link between the degree of neural-orbital disjunction and the magnitude of the supraorbital region, explicated in the spatial hypothesis, receives mixed support in the correlation analysis. These two results agree with previous research indicating that support for the craniofacial size and spatial hypotheses can be found across and within anthropoid primate species, including modern humans. Correlational support for both the craniofacial size and spatial hypotheses suggests multiple factors influence variation in the modern human supraorbital region. Thus, a single hypothesis cannot fully account for modern human variation in this region. The low bivariate correlation coefficients in this study further question whether existing hypotheses can adequately explain morphological variation in the supraorbital region in a primate population sample. Novel functional, structural, behavioral and developmental ideas must be explored if we are to better understand morphological variation in the modern human supraorbital region.     

Comparison of multivariate tests for genetic linkage

OBJECTIVES: Multivariate tests for linkage can provide improved power over univariate tests but the type I error rates and comparative power of commonly used methods have not previously been compared. Here we studied the behavior of bivariate formulations of the variance component (VC) and Haseman-Elston (H-E) approaches. METHODS: We compared through simulation studies the bivariate H-E test with the unconstrained bivariate VC approach and with a VC approach in which the major-gene correlation is constrained to +/-1. We also compared these methods to univariate methods. RESULTS: Bivariate approaches are more powerful than univariate analyses unless the traits are very highly positively correlated. The power of the bivariate H-E test was less than the VC procedures. The constrained test was often less powerful than the unconstrained test. The empirical distributions of the bivariate H-E test and the unconstrained bivariate VC test conformed with asymptotic distributions for samples of 100 or more sibships of size 4. CONCLUSIONS: The unconstrained VC test is valuable for testing for preliminary linkages using multivariate phenotypes. The bivariate H-E test was less powerful than the bivariate VC tests.     

A powerful and robust method for mapping quantitative trait loci in general pedigrees

The variance-components model is the method of choice for mapping quantitative trait loci in general human pedigrees. This model assumes normally distributed trait values and includes a major gene effect, random polygenic and environmental effects, and covariate effects. Violation of the normality assumption has detrimental effects on the type I error and power. One possible way of achieving normality is to transform trait values. The true transformation is unknown in practice, and different transformations may yield conflicting results. In addition, the commonly used transformations are ineffective in dealing with outlying trait values. We propose a novel extension of the variance-components model that allows the true transformation function to be completely unspecified. We present efficient likelihood-based procedures to estimate variance components and to test for genetic linkage. Simulation studies demonstrated that the new method is as powerful as the existing variance-components methods when the normality assumption holds; when the normality assumption fails, the new method still provides accurate control of type I error and is substantially more powerful than the existing methods. We performed a genomewide scan of monoamine oxidase B for the Collaborative Study on the Genetics of Alcoholism. In that study, the results that are based on the existing variance-components method changed dramatically when three outlying trait values were excluded from the analysis, whereas our method yielded essentially the same answers with or without those three outliers. The computer program that implements the new method is freely available.     

The Baumgartner-Weiss-Schindler test for the detection of differentially expressed genes in replicated microarray experiments

MOTIVATION: An important application of microarray experiments is to identify differentially expressed genes. Because microarray data are often not distributed according to a normal distribution nonparametric methods were suggested for their statistical analysis. Here, the Baumgartner-Weiss-Schindler test, a novel and powerful test based on ranks, is investigated and compared with the parametric t-test as well as with two other nonparametric tests (Wilcoxon rank sum test, Fisher-Pitman permutation test) recently recommended for the analysis of gene expression data. RESULTS: Simulation studies show that an exact permutation test based on the Baumgartner-Weiss-Schindler statistic B is preferable to the other three tests. It is less conservative than the Wilcoxon test and more powerful, in particular in case of asymmetric or heavily tailed distributions. When the underlying distribution is symmetric the differences in power between the tests are relatively small. Thus, the Baumgartner-Weiss-Schindler is recommended for the usual situation that the underlying distribution is a priori unknown. AVAILABILITY: SAS code available on request from the authors.