S-Sample Smooth Goodness of Fit Testing: Rederivation and Monte Carlo Assessment

S-sample smooth goodness of fit tests may be constructed using components from one sample goodness of fit testing. Each sample could be assessed for consistency with a target distribution using these components, although that is not our objective here. Contrasts in the components may be used to assess consistency of the samples with each other. If all the samples are consistent, we could then conveniently perform a one-sample goodness of fit test for the target distribution. If the samples are not consistent, an LSD-type analysis can be performed on the one-sample components to identify where the differences between occur. This approach gives a detailed and informative scrutiny of the data.     

Comparisons of Some Chi-squared Tests for the Test of Independence in Sparse Two-Way Contingency Tables

We consider in this paper, the behaviour of a class of the CRESSIE READ (1984) power divergence test statistics indexed by parameter λ - I (λ), with the modified X₂ test statistics (LU) proposed by LAWAL and UPTON (1984), for sparse contingency tables ranging from the 3×3 to the 10×10. We present a sample of our results here. The results indicate that the LU test out-performs either the Cressie-Read suggested test I(2/3) or the Pearson's test - I(1). Our results further show that the modification to the likelihood ratio test [Y₂ = I'(0)] proposed by WILLIAMS (1976) performs like the parent Y₂ test, very poorly compared with either the I(2/3), X₂ or the LU test statistics. Power results also indicate that the powers of the LU test are in all cases considered in this study slightly higher than those of X₂ and I(2/3) tests. The LU test is therefore strongly recommended for use with sparse two-way contingency tables because in all of the cases considered, none of the other test statistics consistently out-performs the LU test with respect to attained α level or power.     

Use of the Gibbs Sampler to Obtain Conditional Tests, with Applications

A random sample is drawn from a distribution which admits aminimal sufficient statistic for the parameters. The Gibbs sampleris proposed to generate samples, called conditionally sufficientor co-sufficient samples, from the conditional distributionof the sample given its value of the sufficient statistic. Theprocedure is illustrated for the gamma distribution. Co-sufficientsamples may be used to give exact tests of fit; for the gammadistribution these are compared for size and power with approximatetests based on the parametric bootstrap.     

Parametric and semiparametric approaches to testing for seasonal trend in serial count data

We present two tests for seasonal trend in monthly incidence data. The first approach uses a penalized likelihood to choose the number of harmonic terms to include in a parametric harmonic model (which includes time trends and autogression as well as seasonal harmonic terms) and then tests for seasonality using a parametric bootstrap test. The second approach uses a semiparametric regression model to test for seasonal trend. In the semiparametric model, the seasonal pattern is modeled nonparametrically, parametric terms are included for autoregressive effects and a linear time trend, and a parametric bootstrap test is used to test for seasonality. For both procedures, a null distribution is generated under a null Poisson model with time trends and autoregression parameters.We apply the methods to skin melanoma incidence rates collected by the surveillance, epidemiology, and end results (SEER) program of the National Cancer Institute, and perform simulation studies to evaluate the type I error rate and power for the two procedures. These simulations suggest that both procedures are alpha-level procedures. In addition, the harmonic model/bootstrap test had similar or larger power than the semiparametric model/bootstrap test for a wide range of alternatives, and the harmonic model/bootstrap test is much easier to implement. Thus, we recommend the harmonic model/bootstrap test for the analysis of seasonal incidence data.     

Bootstrap Confidence Regions for Canonical Variates: Application to Studies of Evolutionary Differentiation

Theory recently developed to construct confidence regions based on the parametric bootstrap is applied to add inferential information to graphical displays of sample centroids in canonical variate analysis. Problems of morphometric differentiation among subspecies and species are addressed using numerical resampling procedures.     

Size and Power of Test Statistics for Gene Correlation in 2 × 2 Contingency Tables

Monte Carlo simulation of size and power of two proposed tests for linkage disequilibrium between two genes each with two alleles were investigated. Results were compared with two commonly used statistics, the correlation coefficient r and the log-odds ratio tests. Depending on the sign of the linkage disequilibrium, the new tests were found to be more powerful than either of the correlation or log-odds ratio tests. However, on average (positive and negative linkage disequilibrium) the Chi-square test using the correlation coefficient was to a small extent more powerful than the other tests.     

Analysis of ordered categorical data: two score-independent approaches

SUMMARY: A trend test is often employed to analyze ordered categorical data, in which a set of increasing scores is assigned a priori. There is a drawback in this approach, because how to choose a set of scores is not clear. There have been debates on which scores should be used (e.g., Graubard and Korn, 1987, Biometrics 43, 471-476; Ivanova and Berger, 2001, Biometrics 57, 567-570; Senn, 2007, Biometrics 63, 296-298). Conflicting conclusions are often obtained with different sets of scores. Two approaches, which have been applied to genetic case-control studies, are appealing for ordered categorical data, because they take into account the natural order in the data, are score independent, and not contingent on asymptotic theory. These two approaches are applied to a prospective study for detecting association between maternal drinking and congenital malformations.     

Gene expression analysis with the parametric bootstrap

Recent developments in microarray technology make it possible to capture the gene expression profiles for thousands of genes at once. With this data researchers are tackling problems ranging from the identification of 'cancer genes' to the formidable task of adding functional annotations to our rapidly growing gene databases. Specific research questions suggest patterns of gene expression that are interesting and informative: for instance, genes with large variance or groups of genes that are highly correlated. Cluster analysis and related techniques are proving to be very useful. However, such exploratory methods alone do not provide the opportunity to engage in statistical inference. Given the high dimensionality (thousands of genes) and small sample sizes (often <30) encountered in these datasets, an honest assessment of sampling variability is crucial and can prevent the over-interpretation of spurious results. We describe a statistical framework that encompasses many of the analytical goals in gene expression analysis; our framework is completely compatible with many of the current approaches and, in fact, can increase their utility. We propose the use of a deterministic rule, applied to the parameters of the gene expression distribution, to select a target subset of genes that are of biological interest. In addition to subset membership, the target subset can include information about relationships between genes, such as clustering. This target subset presents an interesting parameter that we can estimate by applying the rule to the sample statistics of microarray data. The parametric bootstrap, based on a multivariate normal model, is used to estimate the distribution of these estimated subsets and relevant summary measures of this sampling distribution are proposed. We focus on rules that operate on the mean and covariance. Using Bernstein's Inequality, we obtain consistency of the subset estimates, under the assumption that the sample size converges faster to infinity than the logarithm of the number of genes. We also provide a conservative sample size formula guaranteeing that the sample mean and sample covariance matrix are uniformly within a distance epsilon > 0 of the population mean and covariance. The practical performance of the method using a cluster-based subset rule is illustrated with a simulation study. The method is illustrated with an analysis of a publicly available leukemia data set.     

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.     

A Modified Exact Test for 2 × 2 Contingency Tables

A modified exact test is proposed for 2×2 contingency tables. This test, which is based on a less conservative definition of the concept of significance (STONE, 1969) is compared with a modified form of Pearson's X² test and with Tocher's randomized exact (UMPU) test. The sizes of the new test lie near the nominal 0.05 levels while those of the X² test usually exceed the nominal level, sometimes by a factor of 2 or more. The power of the modified test is usually close to that of the UMPU test.     

On the Asymptotic Relative Efficiency of the Mantel-Haenszel Test

The Mantel-Haenszel test is optimal when the odds ratio is constant. This paper investigates the effects of departures from the assumption of a constant odds ratio on the behavior of the Mantel-Haenzel test. A simple approximation is proposed for the non-null distribution of the test statistic. Based on this approximation, the asymptotic relative efficiency of the Mantel-Haenszel test, compared to the overall χ² test for no partial association, is calculated. For the case of 2 strata, it is shown that the Mantel-Haenszel test is efficient as long as the logarithms of the odds ratios are of the same sign and their absolute values exceed 1.     

Tests of Significance for Combined Measures of Plant Stability and Performance

Of significance in plant breeding is to develop varieties that are stable and have optimum performance, such as high yield, in different target environments. As such, selection should be on a measure that combines stability and performance. In this study, we examine the distributions for three nonparametric and three parametric measures for combined stability and performance and show that the F statistic from an analysis of variance may be used to test for differences among genotypes with regard to these measures. Recommendations are made based on the size and power of the F statistic for each of the measures. Application of the theoretical results is demonstrated using data on 14 varieties of oats in 18 locations.     

Test Statistics for Three-Factor Interaction in 2 × 2 × 2 Contingency Tables

A Monte Carlo simulation was conducted in order to determine the size and power of two proposed tests (the covariance and correlation tests) for three-factor interaction in 2 × 2 × 2 contingency tables. Results were compared to the log-odds ratio test statistic. Simulation showed the correlation test to be more conservative than the covariance test, but less so than the log-odds ratio test. However, the correlation test was the most powerful among the three tests.