A Robust Test for Multi-Sample Location Problems with Unequal Group Variances and Nonnormality

A robust test (to be referred to as M^* test) is proposed for testing equality of several group means without assuming normality and equality of variances. This test statistic is obtained by combining Tiku's MML robust procedure with the James statistic. Monte Carlo simulation studies indicate that the M^* test is more powerful than the Welch test, the James test, and the tests based on Huber's M-estimators over a wide range of nonnormal universes. It is also more powerful than the Brown and Forsythe test under most of nonnormal distributions and has substantially the same power as the Brown and Forsythe test under normal distribution. Comparing with Tan-Tabatabai test, M^* is almost as powerful as Tan-Tabatabai test.     

A Comparison of the Three Conditional Exact Tests in Two‐way Contingency Tables Using the Unconditional Exact Power

The conditional exact tests of homogeneity of two binomial proportions are often used in small samples, because the exact tests guarantee to keep the size under the nominal level. The Fisher's exact test, the exact chi‐squared test and the exact likelihood ratio test are popular and can be implemented in software StatXact. In this paper we investigate which test is the best in small samples in terms of the unconditional exact power. In equal sample cases it is proved that the three tests produce the same unconditional exact power. A symmetry of the unconditional exact power is also found. In unequal sample cases the unconditional exact powers of the three tests are computed and compared. In most cases the Fisher's exact test turns out to be best, but we characterize some cases in which the exact likelihood ratio test has the highest unconditional exact power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Testing Linear Contrasts of Means in Experimental Design Without Assuming Normality and Homogeneity of Variances

We investigate the efficiencies of TIKU'S (1967, 1980) modified maximum likelihood (MML) estimators of location and scale parameters of symmetric distributions and show that they are remarkably efficient (jointly). We develop test statistics (based on MML estimators), analogous to the classical tests based on sample means and variances, for testing the equality of two means (the population variances not necessarily equal). We show that these tests are remarkably robust to distributional assumptions and generally more powerful than the well-known nonparametric tests (WILCOXON , normal-score, KOLMOGOROV -SMIRNOV ). We generalize the results to testing linear contrasts of means in experimental design (the error variances not necessarily equal). We show that the analogous tests based on ‘adaptive’ robust estimators (wave, bisquare, HAMPEL ,) etc., GROSS (1976, and other ‘adaptive’ robust estimators) give misleading Type I errors.     

A Robust Test for Homoscedasticity in a Two-Way Layout

This paper discusses tests for homogeneity of row-variances in a two-way classification. Parametric tests for this problem are shown to be highly sensitive to departures from the normality assumption. Levene's robust tests tend to be rather liberal when the number of rows exceeds the number of columns. A modification is suggested which improves the approximation for this situation, while retaining the desirable robustness property.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

COMPETITION AND GENOTYPE-BY-ENVIRONMENT INTERACTION IN NATURAL BREEDING SUBSTRATES OF DROSOPHILA

Although empirical studies frequently suggest that genotype-by-environment (G X E) interaction can maintain genetic variation, very few data are available to test for the specific conditions necessary for the existence of a protected polymorphism (i.e., the property of persistence of an allele even when initially rare). Drosophila species live in patchy environments and their local population structure may be characterized to some extent by Levene's migration pattern, namely by a single pool of individuals that presumably mate at random and breed on discrete and ephemeral resources. We present here a field experiment that links Drosophila ecology and population genetics, which used the alcohol dehydrogenase (Adh) and α-glycerophosphate dehydrogenase (αGpdh) polymorphic loci in D. melanogaster flies raised from Opuntia ficus-indica fruits (prickly pears). The results show that there is density-dependent mortality in those fruits with a relatively high number of larvae (i.e., selection is “soft”) and suggest that there is differential viability for αGpdh genotypes. Additionally, a pattern of G X E interaction for fitness values, which is fully compatible with the theoretical conditions required for the existence of a protected polymorphism, was found after weighting the fitness estimates by the relative contribution that each fruit makes to the total adult population. The strong association between Adh^S and αGpdh^F alleles suggests that the occurrence of the common cosmopolitan inversion In(2L)t in the population might be responsible for the negative frequency-dependent selection predicted by Levene's model when genetic variation persists in heterogeneous environments.     

chifish: a computer program testing for genetic heterogeneity at multiple loci using chi‐square and Fisher's exact test

chifish is a 32‐bit Windows/DOS program evaluating divergence at multiple gene loci. It tests the hypothesis of no difference at any locus both by means of Pearson's traditional chi‐square and by using Fisher's method of combining P values obtained by Fisher's exact test. Input data are read from a file formatted for genepop . Commonly used population genetics software do not perform chi‐square tests, and the simultaneous application of both techniques aids in situations where poor power of the ‘exact approach’ may prevent detection of true differentiation (e.g. few populations and few alleles per locus).     

A Robust Genome‐Wide Scan Statistic of the Wellcome Trust Case–Control Consortium

Summary In genome‐wide association (GWA) studies, test statistics that are efficient and robust across various genetic models are preferable, particularly for studying multiple diseases in the Wellcome Trust Case–Control Consortium ( WTCCC, 2007 , Nature 447 , 661–678). A new test statistic, the minimum of the p‐values of the trend test and Pearson's test, was considered by the WTCCC. It is referred to here as MIN2. Because the minimum of two p‐values is no longer a valid p‐value itself, the WTCCC only used it to rank single nucleotide polymorphisms (SNPs) but did not report the p‐values of the associated SNPs when MIN2 was used for ranking. Given its importance in practice, we derive the asymptotic null distribution of MIN2, study some of its analytical properties related to GWA studies, and compare it with existing methods (the trend test, Pearson's test, MAX3, and the constrained likelihood ratio test [CLRT]) by simulations across a wide range of possible genetic models: the recessive (REC), additive (ADD), multiplicative (MUL), dominant (DOM), and overdominant models. The results show that MAX3 and CLRT have greater efficiency robustness than other tests when the REC, ADD/MUL, and DOM models are possible, whereas Pearson's test and MIN2 have greater efficiency robustness if the possible genetic models also include the overdominant model. We conclude that robust tests (MAX3, MIN2, CLRT, and Pearson's test) are preferable to a single trend test for initial GWA studies. The four robust tests are applied to more than 100 SNPs associated with 11 common diseases identified by the two WTCCC GWA studies.     

Distance-based tests for homogeneity of multivariate dispersions

Summary The traditional likelihood‐based test for differences in multivariate dispersions is known to be sensitive to nonnormality. It is also impossible to use when the number of variables exceeds the number of observations. Many biological and ecological data sets have many variables, are highly skewed, and are zero‐inflated. The traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non‐Euclidean dissimilarity would be more appropriate. Distance‐based tests of homogeneity of multivariate dispersions, which can be based on any dissimilarity measure of choice, are proposed here. They rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes. The tests are straightforward multivariate extensions of Levene's test, with P‐values obtained either using the traditional F‐distribution or using permutation of either least‐squares or LAD residuals. Examples illustrate the utility of the approach, including the analysis of stabilizing selection in sparrows, biodiversity of New Zealand fish assemblages, and the response of Indonesian reef corals to an El Niño. Monte Carlo simulations from the real data sets show that the distance‐based tests are robust and powerful for relevant alternative hypotheses of real differences in spread.     

Berkson's paradox and weighted distributions: An application to Alzheimer's disease

One reason for observing in practice a false positive or negative correlation between two random variables, which are either not correlated or correlated with a different direction, is the overrepresentation in the sample of individuals satisfying specific properties. In 1946, Berkson first illustrated the presence of a false correlation due to this last reason, which is known as Berkson's paradox and is one of the most famous paradox in probability and statistics. In this paper, the concept of weighted distributions is utilized to describe Berskon's paradox. Moreover, a proper procedure is suggested to make inference for the population given a biased sample which possesses all the characteristics of Berkson's paradox. A real data application for patients with dementia due to Alzheimer's disease demonstrates that the proposed method reveals characteristics of the population that are masked by the sampling procedure.     

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.     

POWSIM: a computer program for assessing statistical power when testing for genetic differentiation

Knowledge of statistical power is essential for sampling design and data evaluation when testing for genetic differentiation. Yet, such information is typically missing in studies of conservation and evolutionary genetics, most likely because of complex interactions between the many factors that affect power. powsim is a 32‐bit Windows/DOS simulation‐based computer program that estimates power (and α error) for chi‐square and Fisher's exact tests when evaluating the hypothesis of genetic homogeneity. Optional combinations include the number of samples, sample sizes, number of loci and alleles, allele frequencies, and degree of differentiation (quantified as F_ST). powsim is available at http://www.zoologi.su.se/~ryman .     

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.     

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.     

Sample size calculation for the combination test under nonproportional hazards

For sample size calculation in clinical trials with survival endpoints, the logrank test, which is the optimal method under the proportional hazard (PH) assumption, is predominantly used. In reality, the PH assumption may not hold. For example, in immuno-oncology trials, delayed treatment effects are often expected. The sample size without considering the potential violation of the PH assumption may lead to an underpowered study. In recent years, combination tests such as the maximum weighted logrank test have received great attention because of their robust performance in various hazards scenarios. In this paper, we propose a flexible simulation-free procedure to calculate the sample size using combination tests. The procedure extends the Lakatos' Markov model and allows for complex situations encountered in a clinical trial, like staggered entry, dropouts, etc. We evaluate the procedure using two maximum weighted logrank tests, one projection-type test, and three other commonly used tests under various hazards scenarios. The simulation studies show that the proposed method can achieve the target power for all compared tests in most scenarios. The combination tests exhibit robust performance under correct specification and misspecification scenarios and are highly recommended when the hazard-changing patterns are unknown beforehand. Finally, we demonstrate our method using two clinical trial examples and provide suggestions about the sample size calculations under nonproportional hazards.     

Variance and variability,uncovering an underappreciated component of reproductive isolation

Estimating the fitness of line crosses has been a key element in studies of inbreeding depression, hybridization, and speciation. Fitness values are typically compared using differences in the arithmetic mean of a fitness component between types of crosses. One aspect of fitness that is often overlooked is variance in offspring fitness over time. In the majority of studies, ignoring this aspect of fitness is unavoidable because it is impossible to estimate variance in offspring fitness over long time periods. Here, I describe a method of estimating variance in offspring fitness by substituting spatial variation for temporal variation and provide an empirical example. The method is based on Levene's test of homogeneity of variances. It is implemented by quantifying differences in residual variation among cross types. In a previous study, I performed crosses between populations of the annual plant Diodia teres and quantified hybrid fitness. In this study, another component of isolation and heterosis was revealed when considering variance in offspring fitness. When taking into account variance in offspring fitness using geometric mean fitness as the measure of performance, hybrids between populations from different habitats showed less heterosis than when calculating fitness based on arithmetic mean. This study demonstrates that variance in offspring fitness can be an important aspect of fitness that should be measured more frequently.     

A Monte Carlo Test for Variance Homogeneity in Linear Models

A Monte Carlo procedure is proposed for testing homogeneity of variances in linear models. The method is applicable to a variety of common experimental designs. It is valid when errors are independently normally distributed. Under nonnormality the test is expected to behave robust in a similar fashion as Levene's test. Three examples are given to demonstrate the method.     

A Comparison of Robust Estimators in the Presence of Arbitrarily Right Censored Data

Robust estimation of a location parameter is considered when the data from an unknown symmetric population are subject to arbitrary right-censorship. Comparisons are made between various M-estimators, several L-estimators (trimmed means), and the Kaplan-Meier median. Ten sampling distributions, two uniform censoring distributions, and three sample sizes are examined. A Cauchy censoring distribution is also considered when the sample size is equal to twenty for each of the ten sampling distributions. Performance is based on the estimated mean square error.