Rank Analysis of Block Designs Having Different Cell Frequencies

A distribution–free test is considered for testing the treatment effects in block designs with different cell frequencies. A test statistic which is a function of treatment ranks has been proposed which is distributed as chi-square for large samples. The null distribution of the test statistic has been obtained. The entire procedure has been explained by a numerical example.     

Use of Ranks in Comparisons of Variability in the Non-Normal Populations

A method of analysis for comparing the variability of two samples drawn from two populations has been developed. The method is also suitable for the nonnumeric form of data. A test based on ordered observations for testing the null hypothesis of equality of two variances has been given. The test statistic is a function of the sum of ranks assigned to smaller size sample. Ranking procedure has been modified to depict the variability in the data by the sum of ranks. The null distribution of the test-statistic has been worked out for small samples and it turns out to be chi-square distribution for large samples. The analytical procedure has been explained by a numerical example on the productivity and production of rice and wheat in India from 1950–51 to 1983–84.     

A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures

The problem of testing the separability of a covariance matrix against an unstructured variance‐covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first‐order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ² distribution. The tests are implemented on a real dataset from medical studies.     

On the exact null distribution of the generalised likelihood ratio test for analysing unreplicated factorial designs

Al-Shiha and Yang (1999) proposed a multistage procedure for analysing unreplicated factorial experiments, which is based on the statistic that is derived from the generalised likelihood ratio test statistic under the assumption of normality. It was shown by their simulation study that the method is quite competitive with Lenth's (1989) method. In their paper, because of the difficulty of determining the null distribution analytically, the quantiles of the null distribution were empirically simulated. In this paper, we give the exact null distribution of their test statistic, which makes it possible to calculate the critical values of the test.     

Testing the Equality of Dependent Variances

The likelihood ratio test for testing equality of vgE;2 correlated variables is developed. In general, evaluation of the test statistic involves the iterative optimization of a likelihood function with 1 + v(v – 1)/2 parameters. The explicit form of the test statistic is derived in the bivariate case, and an iterative algorithm for determining the maximum likelihood estimates is suggested. A limited Monte Carlo study determines the behavior of the proposed procedure under the null hypothesis and variety of parameter values.     

Multi-Sample Slippage Test for Ordered Observations

A multi-sample slippage test based on ordered observations has been given. The test statistic is based on the sum of ranks of the sample. The probability distribution of the test statistic has been worked out for small sample and it turns out to be chi-square distribution for large sample. The analytical procedure has been explained by a numerical example.     

Testing for differences in beta diversity from plot-to-plot dissimilarities

The central role of beta diversity in describing biodiversity patterns has been investigated in many fields of ecology and biogeography. While a variety of measures of beta diversity have been proposed over the past five decades, the question of how to test for differences in beta diversity among different sets of sampling plots has been addressed only rarely. Here, we describe a simple analytical procedure to test for differences in beta diversity among distinct sets of plots. The advantage of this approach compared to methods that have been previously proposed lies in its randomization procedure. Such a procedure creates a distribution of null values of the test statistic that is compatible with the null hypothesis of no significant difference in multivariate dispersion between the groups. The proposed test was illustrated using a large dataset of plant and water beetle (Coleoptera) assemblages collected from 45 farmland ponds in Ireland.     

A New Statistical Goodness-of-Fit Test Based on Graphical Representation

A statistical goodness-of-fit test, based on representing the sample observations by linked vectors, is developed. The direction and the length of the linked vectors are defined as functions of the expected values of the order statistics and sample order statistics, respectively. The underlying method can be used to test distributional assumptions for any location-scale family. A test statistic Q_n is introduced and some of its properties are studied. It is shown that the proposed test can be generalized to test if two or more independent samples come from the same distribution. The test procedure provides a graphical method of identifying the true distribution when the null hypothesis is rejected.     

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.     

Comparing Power Behaviours for Some Goodness-of-Fit Test Statistics in Sparse Multinomials

This paper is concerned with the power behaviour of four goodness-of-fit test statistics in sparse multinomials with k cells. Most previous work has been concerned only with both Pearson's X² and the likelihood ratio test statistics. We consider in this study, two additional test statistics, namely, the Cressie-Read test statistic – I(2/3) and the modified Freeman-Tukey test (FT) statistic. Because k ≥ 10 in this study, a Monte Carlo procedure based on 1000 simulated samples is used to estimate the powers for the four test statistics. Alternatives on various line segments are employed. Results suggest that none of the test statistics completely dominate the other and that the choice of which test to use depends on the nature of the alternative hypothesis. These results are consistent with those obtained by West and Kempthorne (1972), although, the Pearson's χ² test statistic may be preferred because of its closer approximation to the χ² distribution in terms of the attained α levels.     

Exact tests for the analysis of case-control studies of genetic markers

OBJECTIVES: The association of a candidate gene with disease can be evaluated by a case-control study in which the genotype distribution is compared for diseased cases and unaffected controls. Usually, the data are analyzed with Armitage's test using the asymptotic null distribution of the test statistic. Since this test does not generally guarantee a type I error rate less than or equal to the significance level alpha, tests based on exact null distributions have been investigated. METHODS: An algorithm to generate the exact null distribution for both Armitage's test statistic and a recently proposed modification of the Baumgartner-Weiss-Schindler statistic is presented. I have compared the tests in a simulation study. RESULTS: The asymptotic Armitage test is slightly anticonservative whereas the exact tests control the type I error rate. The exact Armitage test is very conservative, but the exact test based on the modification of the Baumgartner-Weiss-Schindler statistic has a type I error rate close to alpha. The exact Armitage test is the least powerful test; the difference in power between the other two tests is often small and the comparison does not show a clear winner. CONCLUSION: Simulation results indicate that an exact test based on the modification of the Baumgartner-Weiss-Schindler statistic is preferable for the analysis of case-control studies of genetic markers.     

An entropy-based approach for testing genetic epistasis underlying complex diseases

The genetic basis of complex diseases is expected to be highly heterogeneous, with complex interactions among multiple disease loci and environment factors. Due to the multi-dimensional property of interactions among large number of genetic loci, efficient statistical approach has not been well developed to handle the high-order epistatic complexity. In this article, we introduce a new approach for testing genetic epistasis in multiple loci using an entropy-based statistic for a case-only design. The entropy-based statistic asymptotically follows a χ² distribution. Computer simulations show that the entropy-based approach has better control of type I error and higher power compared to the standard χ² test. Motivated by a schizophrenia data set, we propose a method for measuring and testing the relative entropy of a clinical phenotype, through which one can test the contribution or interaction of multiple disease loci to a clinical phenotype. A sequential forward selection procedure is proposed to construct a genetic interaction network which is illustrated through a tree-based diagram. The network information clearly shows the relative importance of a set of genetic loci on a clinical phenotype. To show the utility of the new entropy-based approach, it is applied to analyze two real data sets, a schizophrenia data set and a published malaria data set. Our approach provides a fast and testable framework for genetic epistasis study in a case-only design.     

Change-plane analysis for subgroup detection with a continuous treatment

Detecting and characterizing subgroups with differential effects of a binary treatment has been widely studied and led to improvements in patient outcomes and population risk management. Under the setting of a continuous treatment, however, such investigations remain scarce. We propose a semiparametric change-plane model and consequently a doubly robust test statistic for assessing the existence of two subgroups with differential treatment effects under a continuous treatment. The proposed testing procedure is valid when either the baseline function for the covariate effects or the generalized propensity score function for the continuous treatment is correctly specified. The asymptotic distributions of the test statistic under the null and local alternative hypotheses are established. When the null hypothesis of no subgroup is rejected, the change-plane parameters that define the subgroups can be estimated. This paper provides a unified framework of the change-plane method to handle various types of outcomes, including the exponential family of distributions and time-to-event outcomes. Additional extensions with nonparametric estimation approaches are also provided. We evaluate the performance of our proposed methods through extensive simulation studies under various scenarios. An application to the Health Effects of Arsenic Longitudinal Study with a continuous environmental exposure of arsenic is presented.     

Evaluation of Noether's Method of Sample Size Determination for the Wilcoxon-Mann-Whitney Test

NOETHER (1987) proposed a method of sample size determination for the Wilcoxon-Mann-Whitney test. To obtain a sample size formula, he restricted himself to alternatives that differ only slightly from the null hypothesis, so that the unknown variance o² of the Mann-Whitney statistic can be approximated by the known variance under the null hypothesis which depends only on n. This fact is frequently forgotten in statistical practice. In this paper, we compare Noether's large sample solution against an alternative approach based on upper bounds of σ² which is valid for any alternatives. This comparison shows that Noether's approximation is sufficiently reliable with small and large deviations from the null hypothesis.     

On the use of permutation in and the performance of a class of nonparametric methods to detect differential gene expression

MOTIVATION: Recently a class of nonparametric statistical methods, including the empirical Bayes (EB) method, the significance analysis of microarray (SAM) method and the mixture model method (MMM), have been proposed to detect differential gene expression for replicated microarray experiments conducted under two conditions. All the methods depend on constructing a test statistic Z and a so-called null statistic z. The null statistic z is used to provide some reference distribution for Z such that statistical inference can be accomplished. A common way of constructing z is to apply Z to randomly permuted data. Here we point our that the distribution of z may not approximate the null distribution of Z well, leading to possibly too conservative inference. This observation may apply to other permutation-based nonparametric methods. We propose a new method of constructing a null statistic that aims to estimate the null distribution of a test statistic directly. RESULTS: Using simulated data and real data, we assess and compare the performance of the existing method and our new method when applied in EB, SAM and MMM. Some interesting findings on operating characteristics of EB, SAM and MMM are also reported. Finally, by combining the idea of SAM and MMM, we outline a simple nonparametric method based on the direct use of a test statistic and a null statistic.     

The Reliability of Tests of Hypotheses when Overdispersed Logistic-Linear Models are Fitted by Maximum Quasi-Likelihood

When overdispersed logistic-linear models are fitted by maximum quasi-likelihood hypotheses can be tested by comparing either the Wald statistic, or the quasi-likelihood score statistic, or the quasi likelihood-ratio statistic, with the approximating null X² distribution. This paper reports a simulation study of the reliability of these tests. Some factors affecting their relative reliabilities are identified. An extended quasi-likelihood ratio test is also considered.     

Some Tests for k£2 Upper Outliers in an Exponential Sample

The sequential procedure for testing up to k upper outliers proposed by Kimber (1982) for one-parameter exponential distribution is modified to a two-parameter exponential distribution. Further null distributions of some test statistics for an upper outlier-pair in a complete or censored sample from a two-parameter exponential distribution are given. Percentage points of the statistic T₁ are tabulated.     

The Wald Statistic in Proportional Hazards Hypothesis Testing

In survivorship modelling using the proportional hazards model of Cox (1972, Journal of the Royal Statistical Society, Series B, 34, 187–220), it is often desired to test a subset of the vector of unknown regression parameters β in the expression for the hazard rate at time t. The likelihood ratio test statistic is well behaved in most situations but may be expensive to calculate. The Wald (1943, Transactions of the American Mathematical Society 54, 426–482) test statistic is easier to calculate, but has some drawbacks. In testing a single parameter in a binomial logit model, Hauck and Donner (1977, Journal of the American Statistical Association 72, 851–853) show that the Wald statistic decreases to zero the further the parameter estimate is from the null and that the asymptotic power of the test decreases to the significance level. The Wald statistic is extensively used in statistical software packages for survivorship modelling and it is therefore important to understand its behavior. The present work examines empirically the behavior of the Wald statistic under various departures from the null hypothesis and under the presence of Type I censoring and covariates in the model. It is shown via examples that the Wald statistic's behavior is not as aberrant as found for the logistic model. For the single parameter case, the asymptotic non-null distribution of the Wald statistic is examined.     

Detection of differentially expressed gene sets in a partially paired microarray data set

Partially paired data sets often occur in microarray experiments (Kim et al., 2005; Liu, Liang and Jang, 2006). Discussions of testing with partially paired data are found in the literature (Lin and Stivers 1974; Ekbohm, 1976; Bhoj, 1978). Bhoj (1978) initially proposed a test statistic that uses a convex combination of paired and unpaired t statistics. Kim et al. (2005) later proposed the t3 statistic, which is a linear combination of paired and unpaired t statistics, and then used it to detect differentially expressed (DE) genes in colorectal cancer (CRC) cDNA microarray data. In this paper, we extend Kim et al.'s t3 statistic to the Hotelling's T2 type statistic Tp for detecting DE gene sets of size p. We employ Efron's empirical null principle to incorporate inter-gene correlation in the estimation of the false discovery rate. Then, the proposed Tp statistic is applied to Kim et al's CRC data to detect the DE gene sets of sizes p=2 and p=3. Our results show that for small p, particularly for p=2 and marginally for p=3, the proposed Tp statistic compliments the univariate procedure by detecting additional DE genes that were undetected in the univariate test procedure. We also conduct a simulation study to demonstrate that Efron's empirical null principle is robust to the departure from the normal assumption.     

Testing for Homogeneity in Meta‐Analysis I. The One‐Parameter Case: Standardized Mean Difference

Summary Meta‐analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran’s Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi‐square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi‐square moment in the one‐parameter case. We apply the result to the homogeneity test for meta‐analyses in which the effects are measured by the standardized mean difference (Cohen’s d‐statistic). In this situation, we recommend approximating the null distribution of Q by a chi‐square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.