Numerical testing of evolution theories †

Fisher's method of combining independent tests is used to construct tests of means of multivariate normal populations when the covariance matrix has intraclass correlation structure. Monte Carlo studies are reported which show that the tests are more powerful than Hotelling's T ²-test in both one and two sample situations.     

Testing a block exchangeable covariance matrix

Testing hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS 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. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.     

Multivariate tests comparing binomial probabilities, with application to safety studies for drugs

Summary.  In magazine advertisements for new drugs, it is common to see summary tables that compare the relative frequency of several side-effects for the drug and for a placebo, based on results from placebo-controlled clinical trials. The paper summarizes ways to conduct a global test of equality of the population proportions for the drug and the vector of population proportions for the placebo. For multivariate normal responses, the Hotelling T ²-test is a well-known method for testing equality of a vector of means for two independent samples. The tests in the paper are analogues of this test for vectors of binary responses. The likelihood ratio tests can be computationally intensive or have poor asymptotic performance. Simple quadratic forms comparing the two vectors provide alternative tests. Much better performance results from using a score-type version with a null-estimated covariance matrix than from the sample covariance matrix that applies with an ordinary Wald test. For either type of statistic, asymptotic inference is often inadequate, so we also present alternative, exact permutation tests. Follow-up inferences are also discussed, and our methods are applied to safety data from a phase II clinical trial.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

A continuously adaptive nonparametric two–sample test

A two–sample test statistic for detecting shifts in location is developed for a broad range of underlying distributions using adaptive techniques. The test statistic is a linear rank statistics which uses a simple modification of the Wilcoxon test; the scores are Winsorized ranks where the upper and lower Winsorinzing proportions are estimated in the first stage of the adaptive procedure using sample the first stage of the adaptive procedure using sample measures of the distribution's skewness and tailweight. An empirical relationship between the Winsorizing proportions and the sample skewness and tailweight allows for a ‘continuous’ adaptation of the test statistic to the data. The test has good asymptotic properties, and the small sample results are compared with other populatr parametric, nonparametric, and two–stage tests using Monte Carlo methods. Based on these results, this proposed test procedure is recommended for moderate and larger sample sizes.     

Monitoring Variation in a Multivariate Process When the Dimension is Large Relative to the Sample Size

A control procedure is presented for monitoring changes in variation for a multivariate normal process in a Phase II operation where the subgroup size, m, is less than p, the number of variates. The methodology is based on a form of Wilk' statistic, which can be expressed as a function of the ratio of the determinants of two separate estimates of the covariance matrix. One estimate is based on the historical data set from Phase I and the other is based on an augmented data set including new data obtained in Phase II. The proposed statistic is shown to be distributed as the product of independent beta distributions that can be approximated using either a chi-square or F-distribution. An ARL study of the statistic is presented for a range of conditions for the population covariance matrix. Cases are considered where a p-variate process is being monitored using a sample of m observations per subgroup and m < p. Data from an industrial multivariate process is used to illustrate the proposed technique.     

Parametric and permutation testing for multivariate monotonic alternatives

We are firstly interested in testing the homogeneity of k mean vectors against two-sided restricted alternatives separately in multivariate normal distributions. This problem is a multivariate extension of Bartholomew (in Biometrica 46:328–335, 1959b) and an extension of Sasabuchi et al. (in Biometrica 70:465–472, 1983) and Kulatunga and Sasabuchi (in Mem. Fac. Sci., Kyushu Univ. Ser. A: Mathematica 38:151–161, 1984) to two-sided ordered hypotheses. We examine the problem of testing under two separate cases. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but common. For the general case that covariance matrices are known the test statistic is obtained using the likelihood ratio method. When the known covariance matrices are common and diagonal, the null distribution of test statistic is derived and its critical values are computed at different significance levels. A Monte Carlo study is also presented to estimate the power of the test. A test statistic is proposed for the case when the common covariance matrices are unknown. Since it is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown, we first present a reformulation of the test statistic based on the orthogonal projections on the closed convex cones and then determine the upper bounds for its p-values. Also we provide a general nonparametric solution based on the permutation approach and nonparametric combination of dependent tests.     

A Two Sample Test for Mean Vectors with Unequal Covariance Matrices

In this paper, we consider testing the equality of two mean vectors with unequal covariance matrices. In the case of equal covariance matrices, we can use Hotelling’s T² statistic, which follows the F distribution under the null hypothesis. Meanwhile, in the case of unequal covariance matrices, the T² type test statistic does not follow the F distribution, and it is also difficult to derive the exact distribution. In this paper, we propose an approximate solution to the problem by adjusting the degrees of freedom of the F distribution. Asymptotic expansions up to the term of order N^{? 2} for the first and second moments of the U statistic are given, where N is the total sample size minus two. A new approximate degrees of freedom and its bias correction are obtained. Finally, numerical comparison is presented by a Monte Carlo simulation.     

Percentage points of the statistics for testing hypotheses on mean vectors of multivariate normal distributions with missing observations

A problem of testing of hypotheses on the mean vector of a multivariate normal distribution with unknown and positive definite covariance matrix is considered when a sample with a special, though not unusual, pattern of missing observations from that population is available. The approximate percentage points of the test statistic are obtained and their accuracy has been checked by comparing them with some exact percentage points which are calculated for complete samples and some special incomplete samples. The approximate percentage points are in good agreement with exact percentage points. The above work is extended to the problem of testing the hypothesis of equality of two mean vectors of two multivariate normal distributions with the same, unknown covariance matrix     

Multivariate analysis of variance with additional data from an unknown subset of the populations

In this paper we consider the problem of testing the means of k multivariate normal populations with additional data from an unknown subset of the k populations. The purpose of this research is to offer test procedures utilizing all the available data for the multivariate analysis of variance problem because the additional data may contain valuable information about the parameters of the k populations. The standard procedure uses only the data from identified populations. We provide a test using all available data based upon Hotelling' s generalized T²statistic. The power of this test is computed using Betz's approximation of Hotelling' s generalized T²statistic by an F-distribution. A comparison of the power of the test and the standard test procedure is also given.     

Chi-square tests for multivariate normality with application to common stock prices

The theory of chi-square tests with data-dependent cells is applied to provide tests of fit to the family of p-variate normal distributions. The cells are bounded by hyperellipses (x-[Xbar])'S^-1 (x-[Xbar]) = c_i centered at the sample mean [Xbar] and having shape deter-mined by the sample covariance matrix S. The Pearson statistic with these cells is affine-invariant, has a null distribution not depending on the true mean and covariance, and has asymptotic critical points between those of x² (M-1) and x² (M-2) when M cells are employed. The test is insensitive to lack of symmetry, but peakedness, broad shoulders and heavy tails are easily discerned in the cell counts. Multivariate normality of logarithms of relative prices of common stocks, a common assumption in finan-cial markets theory, is studied using the statistic described here and a large data base.     

ROBUST TESTS BASED ON THE SAMPLE CHARACTERISTIC FUNCTION

A new method is described for robust analysis of variance in the balanced fixed effects case. The method uses the empirical characteristic function of the treatment samples, and has an interpretation in terms of S-estimators. The test statistic, under the null hypothesis, asymptotically follows a central chi-square distribution, and under contiguous alternatives a noncentral chi-square distribution. A Monte Carlo study suggests that, for finite samples, this is reasonably well approximated by the usual F distribution used in analysis of variance. The test statistic has a bounded influence function. The new procedure competes well with Huber's and a Wald-type procedure except in very heavy-tailed cases.     

A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

Distribution of multivariate quadratic forms under certain covariance structures

Necessary and sufficient conditions are given for the covariance structure of all the observations in a multivariate factorial experiment under which certain multivariate quadratic forms are independent and distributed as a constant times a Wishart. It is also shown that exact multivariate test statistics can be formed for certain covariance structures of the observations when the assumption of equal covariance matrices for each normal population is relaxed. A characterization is given for the dependency structure between random vectors in which the sample mean and sample covariance matrix have certain properties.     

A Note on Sufficient Conditions for Valid Unmodified t Testing in Correlation Analysis with Autocorrelated and Heteroscedastic Sample Data

Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. “Observation vectors” may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized.     

Improvement of the quality of the chi-square approximation for the ADF test on a covariance matrix with a linear structure

The asymptotically distribution-free (ADF) test statistic was proposed by Browne (1984). It is known that the null distribution of the ADF test statistic is asymptotically distributed according to the chi-square distribution. This asymptotic property is always satisfied, even under nonnormality, although the null distributions of other famous test statistics, e.g., the maximum likelihood test statistic and the generalized least square test statistic, do not converge to the chi-square distribution under nonnormality. However, many authors have reported numerical results which indicate that the quality of the chi-square approximation for the ADF test is very poor, even when the sample size is large and the population distribution is normal. In this paper, we try to improve the quality of the chi-square approximation to the ADF test for a covariance matrix with a linear structure by using the Bartlett correction applicable under the assumption of normality. By conducting numerical studies, we verify that the obtained Bartlett correction can perform well even when the assumption of normality is violated.     

A test for bivariate two sample location problem

A consistent test for difference in locations between two bivariate populations is proposed, The test is similar as the Mann-Whitney test and depends on the exceedances of slopes of the two samples where slope for each sample observation is computed by taking the ratios of the observed values. In terms of the slopes, it reduces to a univariate problem, The power of the test has been compared with those of various existing tests by simulation. The proposed test statistic is compared with Mardia's(1967) test statistics, Peters-Randies(1991) test statistic, Wilcoxon's rank sum test. statistic and Hotelling' T² test statistic using Monte Carlo technique. It performs better than other statistics compared for small differences in locations between two populations when underlying population is population 7(light tailed population) and sample size 15 and 18 respectively. When underlying population is population 6(heavy tailed population) and sample sizes are 15 and 18 it performas better than other statistic compared except Wilcoxon's rank sum test statistics for small differences in location between two populations. It performs better than Mardia's(1967) test statistic for large differences in location between two population when underlying population is bivariate normal mixture with probability p=0.5, population 6, Pearson type II population and Pearson type VII population for sample size 15 and 18 .Under bivariate normal population it performs as good as Mardia' (1967) test statistic for small differences in locations between two populations and sample sizes 15 and 18. For sample sizes 25 and 28 respectively it performs better than Mardia's (1967) test statistic when underlying population is population 6, Pearson type II population and Pearson type VII population     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

Bayesian shrinkage estimates for regression coefficients in m populations

For a linear regression model over m populations with separate regression coefficients but a common error variance, a Bayesian model is employed to obtain regression coefficient estimates which are shrunk toward an overall value. The formulation uses Normal priors on the coefficients and diffuse priors on the grand mean vectors, the error variance, and the between-to-error variance ratios. The posterior density of the parameters which were given diffuse priors is obtained. From this the posterior means and variances of regression coefficients and the predictive mean and variance of a future observation are obtained directly by numerical integration in the balanced case, and with the aid of series expansions in the approximately balanced case. An example is presented and worked out for the case of one predictor variable. The method is an extension of Box & Tiao's Bayesian estimation of means in the balanced one-way random effects model.     

On Hotelling’s T2 test in a special paired sample case

In a special paired sample case, Hotelling’s T² test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.