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.     

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.     

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.     

An intrinsic rank test for k independent samples

Considerations on the sensitivity of the KRUSKAL-WALLIS Analysis of Variance by Ranks suggest an alternative method that is less sensitive to spurious interchange of ranks and tied ranks yet offers sufficient power. The Intrinsic Rank Test for k Independent Samples is proposed in this sense as an alternative method to the KRUSKAL- WALLIS test. The Intrinsic Rank Test is based on rank information, but the criterion ranks (intrinsic ranks) are not directly derïved from the observations, but rather indirectly, by means of a further transformation, from the ordinal ranks assigned in the usual manner. The incidence of the intrinsic ranks upon the k samples is then used to calculate a X²-like test statistic, which is readily evaluated with the Gamma distribution.     

Sample Size Determination for Designing a Strata-Matched Case-Control Study to Detect Multiple Risk Factors

Investigations of sample size for planning case-control studies have usually been limited to detecting a single factor. In this paper, we investigate sample size for multiple risk factors in strata-matched case-control studies. We construct an omnibus statistic for testing M different risk factors based on the jointly sufficient statistics of parameters associated with the risk factors. The statistic is non-iterative, and it reduces to the Cochran statistic when M = 1. The asymptotic power function of the test is a non-central chi-square with M degrees of freedom and the sample size required for a specific power can be obtained by the inverse relationship. We find that the equal sample allocation is optimum. A Monte Carlo experiment demonstrates that an approximate formula for calculating sample size is satisfactory in typical epidemiologic studies. An approximate sample size obtained using Bonferroni's method for multiple comparisons is much larger than that obtained using the omnibus test. Approximate sample size formulas investigated in this paper using the omnibus test, as well as the individual tests, can be useful in designing case-control studies for detecting multiple risk factors.     

COMPARISON OF CORRELATED PROPORTIONS IN REPLICATED PRODUCT TESTS

Product testing with multiple matched samples is often encountered. If the response for each sample is binary, the test involves comparison of multiple proportions of dependent samples. In the paper, a test statistic due to Bennett (1967, 1968) is discussed and extended to the replicated testing situation. An adjusted Bennett's statistic is proposed based on the Dirichlet-multinomial distribution. The conventional and adjusted Bennett's statistics are illustrated by a numerical example in product testing.     

On Exact Unconditional Test for Linear Trend in Dose‐Response Studies

The one‐degree‐of‐freedom Cochran‐Armitage (C‐A) test statistic for linear trend has been widely applied in various dose‐response studies (e.g., anti‐ulcer medications and short‐term antibiotics, animal carcinogenicity bioassays and occupational toxicant studies). This approximate statistic relies, however, on asymptotic theory that is reliable only when the sample sizes are reasonably large and well balanced across dose levels. For small, sparse, or skewed data, the asymptotic theory is suspect and exact conditional method (based on the C‐A statistic) seems to provide a dependable alternative. Unfortunately, the exact conditional method is only practical for the linear logistic model from which the sufficient statistics for the regression coefficients can be obtained explicitly. In this article, a simple and efficient recursive polynomial multiplication algorithm for exact unconditional test (based on the C‐A statistic) for detecting a linear trend in proportions is derived. The method is applicable for all choices of the model with monotone trend including logistic, probit, arcsine, extreme value and one hit. We also show that this algorithm can be easily extended to exact unconditional power calculation for studies with up to a moderately large sample size. A real example is given to illustrate the applicability of the proposed method.     

Testing substitution models within a phylogenetic tree

Phylogenetic tree reconstruction frequently assumes the homogeneity of the substitution process over the whole tree. To test this assumption statistically, we propose a test based on the sample covariance matrix of the set of substitution rate matrices estimated from pairwise sequence comparison. The sample covariance matrix is condensed into a one-dimensional test statistic Delta = sum ln(1 + delta(i)), where delta(i) are the eigenvalues of the sample covariance matrix. The test does not assume a specific mutational model. It analyses the variation in the estimated rate matrices. The distribution of this test statistic is determined by simulations based on the phylogeny estimated from the data. We study the power of the test under various scenarios and apply the test to X chromosome and mtDNA primate sequence data. Finally, we demonstrate how to include rate variation in the test.     

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.     

Nonparametric Location Tests Based on the Empirical Distribution

We present a unified approach to the nonparametric comparison of locations in the one- and two sample case using the empirical distribution functions. This approach reveals how to define a simple test statistic for this comparison in the case of large sample sizes which is equivalent to the known Wilcoxon-Mann-Whitney statistics and is based on the grouping of the data. In the particular case of equal group sizes we find an extremely simple form for this location test.     

Assessing the adequacy of variance function in heteroscedastic regression models

Heteroscedastic data arise in many applications. In heteroscedastic regression analysis, the variance is often modeled as a parametric function of the covariates or the regression mean. We propose a kernel-smoothing type nonparametric test for checking the adequacy of a given parametric variance structure. The test does not need to specify a parametric distribution for the random errors. It is shown that the test statistic has an asymptotical normal distribution under the null hypothesis and is powerful against a large class of alternatives. We suggest a simple bootstrap algorithm to approximate the distribution of the test statistic in finite sample size. Numerical simulations demonstrate the satisfactory performance of the proposed test. We also illustrate the application by the analysis of a radioimmunoassay data set.     

A Comparison of Procedures for Adaptive Choice of Location Tests in Flexible Two‐Stage Designs

Although linear rank statistics for the two‐sample problem are distribution free tests, their power depends on the distribution of the data. In the planning phase of an experiment, researchers are often uncertain about the shape of this distribution and so the choice of test statistic for the analysis and the determination of the required sample size are based on vague information. Adaptive designs with interim analysis can potentially overcome both problems. And in particular, adaptive tests based on a selector statistic are a solution to the first. We investigate whether adaptive tests can be usefully implemented in flexible two‐stage designs to gain power. In a simulation study, we compare several methods for choosing a test statistic for the second stage of an adaptive design based on interim data with the procedure that applies adaptive tests in both stages. We find that the latter is a sensible approach that leads to the best results in most situations considered here. The different methods are illustrated using a clinical trial example.     

A bootstrap assessment of variability in pedigree reconstruction based on genetic markers

The problem of assessing the variability in pedigree reconstruction using DNA markers is considered for the special case of single generation samples with no parents present. Error in pedigree reconstruction is measured through a metric imposed on the space of partitions of the individuals into family groups. A confidence set can therefore be taken to be a neighborhood of a point estimate, analogous to the estimation of a parameter in Euclidean space. The coverage probability is estimated using bootstrap techniques. Although the distributional properties of the sample depend on the population genotype frequencies, these are in practice usually unknown. Confidence sets conditioned on a statistic approximately sufficient for these frequencies are compared with confidence sets obtained by substituting frequency estimates directly into the sampling distribution. In two simulation studies, the difference is found to be of some consequence.     

Small Sample Properties of the Mantel-Haenszel Test Statistic for Density Follow-Up Studies

For the analysis of combinations of 2×2 non-contingency tables as obtained from density follow-up studies (relating a number of events to a number of person-years of follow-up) an analogue of the Mantel-Haenszel test for 2×2 contingency tables is widely used. In this paper the small sample properties of this test, both with and without continuity correction, are evaluated. Also the improvement of the test-statistic by using the first four cumulants via the Edgeworth expansion was studied. Results on continuity correction agree with similar studies on the Mantel-Haenszel statistic for 2×2 contingency tables: Continuity correction gives a p-value which approximates the exact p-value better than the p-value obtained without this correction; both the exact test and its approximations show considerable conservatism in small samples; the uncorrected Mantel-Haenszel test statistic gives a p-value that agrees more with the nominal significance level, but can be anti-conservative. The p-value based on the first four cumulants gives a better approximation of the exact p-value than the continuity corrected test, especially when the distribution has marked skewness.     

On the KOLMOGOROV-SMIRNOV Test for the POISSON Distribution with Unknown Mean

The standard tables for the KOLMOGOROV -SMIRNOV test are valid only in the case of testing whether a set of observations is from a completely specified cumulative distribution, F₀(X), with all parameters known. If the parameters are unknown and must be estimated from the sample, then the tables are not valid. A table is given in this paper for use with the KOLMOGOROV -SMIRNOV statistic in the case of testing whether a set of observations is from the POISSON distribution with an unknown mean that must be estimated from the sample. The table is obtained from a Monte Carlo calculation.     

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.     

Determining the dimension of the central subspace and central mean subspace

The central subspace and central mean subspace are two importanttargets of sufficient dimension reduction. We propose a weightedchi-squared test to determine their dimensions based on matriceswhose column spaces are exactly equal to the central subspaceor the central mean subspace. The asymptotic distribution ofthe test statistic is obtained. Simulation examples are usedto demonstrate the performance of this test.     

A Modified Chi-Square Test for Testing Equality of Two Multinomial Populations Against an Ordering Restricted Alternative

A modified chi-square test for testing the equality of two multinomial populations against an ordering restricted alternative in one sample and two sample cases is constructed. The relation between a concept of dependence called dependence by chi-square and stochastic ordering is established. A tabulation of the asymptotic distribution of the test statistic under the null hypothesis is given. Simulations are used to compare the power of this test with the power of the likelihood ratio test of stochastic ordering of the two multinomial populations.     

Optimal sufficient dimension reduction for the conditional mean in multivariate regression

The aim of this article is to develop optimal sufficient dimensionreduction methodology for the conditional mean in multivariateregression. The context is roughly the same as that of a relatedmethod by Cook & Setodji (2003), but the new method hasseveral advantages. It is asymptotically optimal in the sensedescribed herein and its test statistic for dimension alwayshas a chi-squared distribution asymptotically under the nullhypothesis. Additionally, the optimal method allows tests ofpredictor effects. A comparison of the two methods is provided.     

An Exact Paired Rank Test

An exact rank test for two dependent samples based on overall mid‐ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data.