Fractionation in Triad Comparisons

A statistical procedure for comparing the performance of a new product with the existing products on the basis of sensory characters has been developed. A test statistic was evolved for testing the null hypothesis of equality of treatment effects in the case of fractional triad comparisons. The null distribution of the test statistic has been obtained and it is found that it has a χ²-distribution for large number of observations. The procedure is quite simple and is based on a distribution-free test requiring only ordinal scale measurement. The method of analysis has been explained by a numerical example.     

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.     

On using Lehmann alternatives with nonresponders.

The problem of testing for treatment effect when some subjects in the treatment group may be unaffected by the treatment is considered. A form of the Lehmann alternative suggested by Conover and Salsburg is used that assumes that each control score has the same distribution as the minimum of the known number of responses in the treatment group. It is shown that the locally most powerful test leads to a test statistic that, under the hypothesis of no treatment effect, is the sum of independent pareto random variables whereas under the alternative hypothesis it is the sum of independent random variables from a mixture of two pareto distributions. The limiting distribution of the test statistic under both hypotheses is in the domain of attraction of a stable distribution whose indices are derived. The power of the test is given, and its properties are discussed. A set of data from clinical research involving development of a new drug is used to show application of the procedure and demonstrate its usefulness.     

A global goodness-of-fit statistic for Cox regression models

In this paper, a global goodness-of-fit test statistic for a Cox regression model, which has an approximate chi-squared distribution when the model has been correctly specified, is proposed. Our goodness-of-fit statistic is global and has power to detect if interactions or higher order powers of covariates in the model are needed. The proposed statistic is similar to the Hosmer and Lemeshow (1980, Communications in Statistics A10, 1043-1069) goodness-of-fit statistic for binary data as well as Schoenfeld's (1980, Biometrika 67, 145-153) statistic for the Cox model. The methods are illustrated using data from a Mayo Clinic trial in primary billiary cirrhosis of the liver (Fleming and Harrington, 1991, Counting Processes and Survival Analysis), in which the outcome is the time until liver transplantation or death. The are 17 possible covariates. Two Cox proportional hazards models are fit to the data, and the proposed goodness-of-fit statistic is applied to the fitted models.     

Selection of Judges for Sensory Evaluation

A test based on ordered observations for selection of consistent judges for sensory evaluation has been given. The test–statistic depends on the sum of ranks of different products under consideration. 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 of a taste-testing experiment.     

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.     

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.     

To permute or not to permute

Permutation test is a popular technique for testing a hypothesis of no effect, when the distribution of the test statistic is unknown. To test the equality of two means, a permutation test might use a test statistic which is the difference of the two sample means in the univariate case. In the multivariate case, it might use a test statistic which is the maximum of the univariate test statistics. A permutation test then estimates the null distribution of the test statistic by permuting the observations between the two samples. We will show that, for such tests, if the two distributions are not identical (as for example when they have unequal variances, correlations or skewness), then a permutation test for equality of means based on difference of sample means can have an inflated Type I error rate even when the means are equal. Our results illustrate permutation testing should be confined to testing for non-identical distributions. CONTACT: calian@raunvis.hi.is.     

Correlation-based inference for linkage disequilibrium with multiple alleles

The correlation between alleles at a pair of genetic loci is a measure of linkage disequilibrium. The square of the sample correlation multiplied by sample size provides the usual test statistic for the hypothesis of no disequilibrium for loci with two alleles and this relation has proved useful for study design and marker selection. Nevertheless, this relation holds only in a diallelic case, and an extension to multiple alleles has not been made. Here we introduce a similar statistic, R(2), which leads to a correlation-based test for loci with multiple alleles: for a pair of loci with k and m alleles, and a sample of n individuals, the approximate distribution of n(k - 1)(m - 1)/(km)R(2) under independence between loci is chi((k-1)(m-1))(2). One advantage of this statistic is that it can be interpreted as the total correlation between a pair of loci. When the phase of two-locus genotypes is known, the approach is equivalent to a test for the overall correlation between rows and columns in a contingency table. In the phase-known case, R(2) is the sum of the squared sample correlations for all km 2 x 2 subtables formed by collapsing to one allele vs. the rest at each locus. We examine the approximate distribution under the null of independence for R(2) and report its close agreement with the exact distribution obtained by permutation. The test for independence using R(2) is a strong competitor to approaches such as Pearson's chi square, Fisher's exact test, and a test based on Cressie and Read's power divergence statistic. We combine this approach with our previous composite-disequilibrium measures to address the case when the genotypic phase is unknown. Calculation of the new multiallele test statistic and its P-value is very simple and utilizes the approximate distribution of R(2). We provide a computer program that evaluates approximate as well as "exact" permutational P-values.     

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 Exact Evaluation of 2-way Cross-classifications Sequel: A Fugal Algorithm

A generalized algorithm is presented for generating the ensemble of unique r × c contingency tables with arbitrary, fixed row and column sums, and of arbitrary, even large, sizes. This further contributes to the operational solution of the exact ensemble distribution of 2-way contingency tables and cross classifications, along with the sampling distribution of the conventional χ²-like test statistic, or of any other test statistic calculated from the tables. The associated enumeration problem of Integer arrays with fixed row and column sums has been operationally solved.     

Valid inference in random effects meta-analysis

The standard approach to inference for random effects meta-analysis relies on approximating the null distribution of a test statistic by a standard normal distribution. This approximation is asymptotic on k, the number of studies, and can be substantially in error in medical meta-analyses, which often have only a few studies. This paper proposes permutation and ad hoc methods for testing with the random effects model. Under the group permutation method, we randomly switch the treatment and control group labels in each trial. This idea is similar to using a permutation distribution for a community intervention trial where communities are randomized in pairs. The permutation method theoretically controls the type I error rate for typical meta-analyses scenarios. We also suggest two ad hoc procedures. Our first suggestion is to use a t-reference distribution with k-1 degrees of freedom rather than a standard normal distribution for the usual random effects test statistic. We also investigate the use of a simple t-statistic on the reported treatment effects.     

A class of multiplicity adjusted tests for spatial clustering based on case-control point data

A class of tests with quadratic forms for detecting spatial clustering of health events based on case-control point data is proposed. It includes Cuzick and Edwards's test statistic (1990, Journal of the Royal Statistical Society, Series B 52, 73-104). Although they used the property of asymptotic normality of the test statistic, we show that such an approximation is generally poor for moderately large sample sizes. Instead, we suggest a central chi-square distribution as a better approximation to the asymptotic distribution of the test statistic. Furthermore, not only to estimate the optimal value of the unknown parameter on the scale of cluster but also to adjust for multiple testing due to repeating the procedure by changing the parameter value, we propose the minimum of the profile p-value of the test statistic for the parameter as an integrated test statistic. We also provide a statistic to estimate the areas or cases which make large contributions to significant clustering. The proposed methods are illustrated with a data set concerning the locations of cases of childhood leukemia and lymphoma and another on early medieval grave site locations consisting of affected and nonaffected grave sites.     

A multivariate two-sample mean test for small sample size and missing data

We develop a new statistic for testing the equality of two multivariate mean vectors. A scaled chi-squared distribution is proposed as an approximating null distribution. Because the test statistic is based on componentwise statistics, it has the advantage over Hotelling's T2 test of being applicable to the case where the dimension of an observation exceeds the number of observations. An appealing feature of the new test is its ability to handle missing data by relying on only componentwise sample moments. Monte Carlo studies indicate good power compared to Hotelling's T2 and a recently proposed test by Srivastava (2004, Technical Report, University of Toronto). The test is applied to drug discovery data.     

Construction of null statistics in permutation-based multiple testing for multi-factorial microarray experiments

MOTIVATION: The parametric F-test has been widely used in the analysis of factorial microarray experiments to assess treatment effects. However, the normality assumption is often untenable for microarray experiments with small replications. Therefore, permutation-based methods are called for help to assess the statistical significance. The distribution of the F-statistics across all the genes on the array can be regarded as a mixture distribution with a proportion of statistics generated from the null distribution of no differential gene expression whereas the other proportion of statistics generated from the alternative distribution of genes differentially expressed. This results in the fact that the permutation distribution of the F-statistics may not approximate well to the true null distribution of the F-statistics. Therefore, the construction of a proper null statistic to better approximate the null distribution of F-statistic is of great importance to the permutation-based multiple testing in microarray data analysis. RESULTS: In this paper, we extend the ideas of constructing null statistics based on pairwise differences to neglect the treatment effects from the two-sample comparison problem to the multifactorial balanced or unbalanced microarray experiments. A null statistic based on a subpartition method is proposed and its distribution is employed to approximate the null distribution of the F-statistic. The proposed null statistic is able to accommodate unbalance in the design and is also corrected for the undue correlation between its numerator and denominator. In the simulation studies and real biological data analysis, the number of true positives and the false discovery rate (FDR) of the proposed null statistic are compared with those of the permutated version of the F-statistic. It has been shown that our proposed method has a better control of the FDRs and a higher power than the standard permutation method to detect differentially expressed genes because of the better approximated tail probabilities.     

A Maximum Rank-Sum Test for One-Pulse Variation in Monthly Data

A nonparametric test to detect a pulse in monthly data is presented. This test is a maximum rank-sum test. The test statistic can be computed from frequencies or rates. The exact null distribution of the test statistic is tabulated for pulses that last 3, 4, 5, or 6 months. Estimates from a simulation study of the test's type I error rate and power are presented. The statistical modeling of the data is discussed. Several examples are given to illustrate the application of the test and the modeling procedures. Practical matters such as the treatment of tied observations, the effect of unequal lengths in the months, sample-size calculation, and post-test power analysis are discussed and illustrated with examples.     

Efficient p-value evaluation for resampling-based tests

The resampling-based test, which often relies on permutation or bootstrap procedures, has been widely used for statistical hypothesis testing when the asymptotic distribution of the test statistic is unavailable or unreliable. It requires repeated calculations of the test statistic on a large number of simulated data sets for its significance level assessment, and thus it could become very computationally intensive. Here, we propose an efficient p-value evaluation procedure by adapting the stochastic approximation Markov chain Monte Carlo algorithm. The new procedure can be used easily for estimating the p-value for any resampling-based test. We show through numeric simulations that the proposed procedure can be 100-500 000 times as efficient (in term of computing time) as the standard resampling-based procedure when evaluating a test statistic with a small p-value (e.g. less than 10( - 6)). With its computational burden reduced by this proposed procedure, the versatile resampling-based test would become computationally feasible for a much wider range of applications. We demonstrate the application of the new method by applying it to a large-scale genetic association study of prostate cancer.     

An entropy-based statistic for genomewide association studies

Efficient genotyping methods and the availability of a large collection of single-nucleotide polymorphisms provide valuable tools for genetic studies of human disease. The standard chi2 statistic for case-control studies, which uses a linear function of allele frequencies, has limited power when the number of marker loci is large. We introduce a novel test statistic for genetic association studies that uses Shannon entropy and a nonlinear function of allele frequencies to amplify the differences in allele and haplotype frequencies to maintain statistical power with large numbers of marker loci. We investigate the relationship between the entropy-based test statistic and the standard chi2 statistic and show that, in most cases, the power of the entropy-based statistic is greater than that of the standard chi2 statistic. The distribution of the entropy-based statistic and the type I error rates are validated using simulation studies. Finally, we apply the new entropy-based test statistic to two real data sets, one for the COMT gene and schizophrenia and one for the MMP-2 gene and esophageal carcinoma, to evaluate the performance of the new method for genetic association studies. The results show that the entropy-based statistic obtained smaller P values than did the standard chi2 statistic.     

On Testing Equality of Means of Correlated Variates with Missing Data on One Response

A statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. The distribution of the statistic is approximated by a normal distribution under the null hypothesis. The empirical powers of the statistic are computed and compared with those of the conventional paired t and the other known statistics. The power comparisons support the use of the proposed test.     

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.