A class of goodness of fit tests for a copula based on bivariate right-censored data

The copula of a bivariate distribution, constructed by making marginal transformations of each component, captures all the information in the bivariate distribution about the dependence between two variables. For frailty models for bivariate data the choice of a family of distributions for the random frailty corresponds to the choice of a parametric family for the copula. A class of tests of the hypothesis that the copula is in a given parametric family, with unspecified association parameter, based on bivariate right censored data is proposed. These tests are based on first making marginal Kaplan-Meier transformations of the data and then comparing a non-parametric estimate of the copula to an estimate based on the assumed family of models. A number of options are available for choosing the scale and the distance measure for this comparison. Significance levels of the test are found by a modified bootstrap procedure. The procedure is used to check the appropriateness of a gamma or a positive stable frailty model in a set of survival data on Danish twins.     

A parametric model for studying organism fitness using step-stress experiments

We propose a method based on parametric survival analysis to analyze step-stress data. Step-stress studies are failure time studies in which the experimental stressor is increased at specified time intervals. While this protocol has been frequently employed in industrial reliability studies, it is less common in the life sciences. Possible biological applications include experiments on swimming performance of fish using a step function defining increasing water velocity over time, and treadmill tests on humans. A likelihood-ratio test is developed for comparing the failure times in two groups based on a piecewise constant hazard assumption. The test can be extended to other piecewise distributions and to include covariates. An example data set is used to illustrate the method and highlight experimental design issues. A small simulation study compares this analysis procedure to currently used methods with regard to type I error rate and power.     

A Global View on Parametric and Nonparametric Approaches to the Analysis of Ordered Categorical Data

Rank approaches are very common in the analysis of ordered categorical data but can only be interpreted on an experiment‐wise level. Therefore, parametric tests from linear models, although based on metric structures, are used frequently to analyze this type of data. So the questions arise 1. what parametric tests measure in this context and 2. whether the rank approach could be modified to achieve a global level of interpretation. A possible solution to question 2. offers the so called ridit approach, which is based on known reference distributions. In this paper we discuss a global view that shows how rank analysis and ridit analyses are related and how parametric procedures fit into the same framework. The use of the uniform distribution as a reference in the ridit approach gives an explanation to question 1. The asymptotic multivariate normality of the effect estimators is shown and robust test statistics are discussed. Type I and type II error rates are examined in simulation studies and the approach is applied to a toxicological example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Powerful tests for detecting a gene effect in the presence of possible gene-gene interactions using garrote kernel machines

We propose in this article a powerful testing procedure for detecting a gene effect on a continuous outcome in the presence of possible gene-gene interactions (epistasis) in a gene set, e.g., a genetic pathway or network. Traditional tests for this purpose require a large number of degrees of freedom by testing the main effect and all the corresponding interactions under a parametric assumption, and hence suffer from low power. In this article, we propose a powerful kernel machine based test. Specifically, our test is based on a garrote kernel method and is constructed as a score test. Here, the term garrote refers to an extra nonnegative parameter that is multiplied to the covariate of interest so that our score test can be formulated in terms of this nonnegative parameter. A key feature of the proposed test is that it is flexible and developed for both parametric and nonparametric models within a unified framework, and is more powerful than the standard test by accounting for the correlation among genes and hence often uses a much smaller degrees of freedom. We investigate the theoretical properties of the proposed test. We evaluate its finite sample performance using simulation studies, and apply the method to the Michigan prostate cancer gene expression data.     

Testing equality of survival functions based on both paired and unpaired censored data

We introduce two test procedures for comparing two survival distributions on the basis of randomly right-censored data consisting of both paired and unpaired observations. Our procedures are based on generalizations of a pooled rank test statistic previously proposed for uncensored data. One generalization adapts the Prentice-Wilcoxon score, while the other adapts the Akritas score. The use of these particular scoring systems in pooled rank tests with randomly right-censored paired data has been advocated by several researchers. Our test procedures utilize the permutation distributions of the test statistics based on a novel manner of permuting the scores. Permutation versions of tests for right-censored paired data and for two independent right-censored samples that use the proposed scoring systems are obtained as special cases of our test procedures. Simulation results show that our test procedures have high power for detecting scale and location shifts in exponential and log-logistic distributions for the survival times. We also demonstrate the advantages of our test procedures in terms of utilizing randomly occurring unpaired observations that are discarded in test procedures for paired data. The tests are applied to skin graft data previously reported elsewhere.     

Estimation of rates-across-sites distributions in phylogenetic substitution models

Previous work has shown that it is often essential to account for the variation in rates at different sites in phylogenetic models in order to avoid phylogenetic artifacts such as long branch attraction. In most current models, the gamma distribution is used for the rates-across-sites distributions and is implemented as an equal-probability discrete gamma. In this article, we introduce discrete distribution estimates with large numbers of equally spaced rate categories allowing us to investigate the appropriateness of the gamma model. With large numbers of rate categories, these discrete estimates are flexible enough to approximate the shape of almost any distribution. Likelihood ratio statistical tests and a nonparametric bootstrap confidence-bound estimation procedure based on the discrete estimates are presented that can be used to test the fit of a parametric family. We applied the methodology to several different protein data sets, and found that although the gamma model often provides a good parametric model for this type of data, rate estimates from an equal-probability discrete gamma model with a small number of categories will tend to underestimate the largest rates. In cases when the gamma model assumption is in doubt, rate estimates coming from the discrete rate distribution estimate with a large number of rate categories provide a robust alternative to gamma estimates. An alternative implementation of the gamma distribution is proposed that, for equal numbers of rate categories, is computationally more efficient during optimization than the standard gamma implementation and can provide more accurate estimates of site rates.     

Efficient computation of significance levels for multiple associations in large studies of correlated data, including genomewide association studies

Large exploratory studies, including candidate-gene-association testing, genomewide linkage-disequilibrium scans, and array-expression experiments, are becoming increasingly common. A serious problem for such studies is that statistical power is compromised by the need to control the false-positive rate for a large family of tests. Because multiple true associations are anticipated, methods have been proposed that combine evidence from the most significant tests, as a more powerful alternative to individually adjusted tests. The practical application of these methods is currently limited by a reliance on permutation testing to account for the correlated nature of single-nucleotide polymorphism (SNP)-association data. On a genomewide scale, this is both very time-consuming and impractical for repeated explorations with standard marker panels. Here, we alleviate these problems by fitting analytic distributions to the empirical distribution of combined evidence. We fit extreme-value distributions for fixed lengths of combined evidence and a beta distribution for the most significant length. An initial phase of permutation sampling is required to fit these distributions, but it can be completed more quickly than a simple permutation test and need be done only once for each panel of tests, after which the fitted parameters give a reusable calibration of the panel. Our approach is also a more efficient alternative to a standard permutation test. We demonstrate the accuracy of our approach and compare its efficiency with that of permutation tests on genomewide SNP data released by the International HapMap Consortium. The estimation of analytic distributions for combined evidence will allow these powerful methods to be applied more widely in large exploratory studies.     

A test of homogeneity of distributions when observations are subject to measurement errors

When the observed data are contaminated with errors, the standard two-sample testing approaches that ignore measurement errors may produce misleading results, including a higher type-I error rate than the nominal level. To tackle this inconsistency, a nonparametric test is proposed for testing equality of two distributions when the observed contaminated data follow the classical additive measurement error model. The proposed test takes into account the presence of errors in the observed data, and the test statistic is defined in terms of the (deconvoluted) characteristic functions of the latent variables. Proposed method is applicable to a wide range of scenarios as no parametric restrictions are imposed either on the distribution of the underlying latent variables or on the distribution of the measurement errors. Asymptotic null distribution of the test statistic is derived, which is given by an integral of a squared Gaussian process with a complicated covariance structure. For data-based calibration of the test, a new nonparametric Bootstrap method is developed under the two-sample measurement error framework and its validity is established. Finite sample performance of the proposed test is investigated through simulation studies, and the results show superior performance of the proposed method than the standard tests that exhibit inconsistent behavior. Finally, the proposed method was applied to real data sets from the National Health and Nutrition Examination Survey. An R package MEtest is available through CRAN.     

Hypothesis testing in semiparametric additive mixed models

We consider testing whether the nonparametric function in a semiparametric additive mixed model is a simple fixed degree polynomial, for example, a simple linear function. This test provides a goodness-of-fit test for checking parametric models against nonparametric models. It is based on the mixed-model representation of the smoothing spline estimator of the nonparametric function and the variance component score test by treating the inverse of the smoothing parameter as an extra variance component. We also consider testing the equivalence of two nonparametric functions in semiparametric additive mixed models for two groups, such as treatment and placebo groups. The proposed tests are applied to data from an epidemiological study and a clinical trial and their performance is evaluated through simulations.     

Rank-based statistical methodologies for quantitative trait locus mapping

This article addresses the identification of genetic loci (QTL and elsewhere) that influence nonnormal quantitative traits with focus on experimental crosses. QTL mapping is typically based on the assumption that the traits follow normal distributions, which may not be true in practice. Model-free tests have been proposed. However, nonparametric estimation of genetic effects has not been studied. We propose an estimation procedure based on the linear rank test statistics. The properties of the new procedure are compared with those of traditional likelihood-based interval mapping and regression interval mapping via simulations and a real data example. The results indicate that the nonparametric method is a competitive alternative to the existing parametric methodologies.     

The Baumgartner-Weiss-Schindler test for the detection of differentially expressed genes in replicated microarray experiments

MOTIVATION: An important application of microarray experiments is to identify differentially expressed genes. Because microarray data are often not distributed according to a normal distribution nonparametric methods were suggested for their statistical analysis. Here, the Baumgartner-Weiss-Schindler test, a novel and powerful test based on ranks, is investigated and compared with the parametric t-test as well as with two other nonparametric tests (Wilcoxon rank sum test, Fisher-Pitman permutation test) recently recommended for the analysis of gene expression data. RESULTS: Simulation studies show that an exact permutation test based on the Baumgartner-Weiss-Schindler statistic B is preferable to the other three tests. It is less conservative than the Wilcoxon test and more powerful, in particular in case of asymmetric or heavily tailed distributions. When the underlying distribution is symmetric the differences in power between the tests are relatively small. Thus, the Baumgartner-Weiss-Schindler is recommended for the usual situation that the underlying distribution is a priori unknown. AVAILABILITY: SAS code available on request from the authors.     

Comparison of independent samples of high-dimensional data by pairwise distance measures

Pairwise distance or association measures of sample elements are often used as a basis for hierarchical cluster analyses. They can also be used in tests for the comparison of pre-defined subgroups of the total sample. Usually this is done with permutation tests In this paper, we compare such a procedure with alternative tests for high-dimensional data based on spherically distributed scores in simulation experiments and with real data. The tests based on the pairwise distance or similarity measures perform quite well in this comparison. As the number of possible permutations is small in very small samples, this might restrict the use of the test. Therefore, we propose an exact parametric small sample version of the test using randomly rotated samples.     

Testing the power law model for discrete size data

It is sometimes claimed that different types of size data in biology follow a power law. Here, a formal statistical test of the power law for discrete size data is described. The test is based on embedding the power law in the nonparametric family of distributions for which frequency is nonincreasing with size. A parametric bootstrap is used to assess significance. The test is applied to four data sets concerning the frequency of genera of different sizes. The power law is rejected in three out of four cases.     

Adjusting O'Brien's test to control type I error for the generalized nonparametric Behrens-Fisher problem

O'Brien (1984, Biometrics 40, 1079-1087) introduced a simple nonparametric test procedure for testing whether multiple outcomes in one treatment group have consistently larger values than outcomes in the other treatment group. We first explore the theoretical properties of O'Brien's test. We then extend it to the general nonparametric Behrens-Fisher hypothesis problem when no assumption is made regarding the shape of the distributions. We provide conditions when O'Brien's test controls its error probability asymptotically and when it fails. We also provide adjusted tests when the conditions do not hold. Throughout this article, we do not assume that all outcomes are continuous. Simulations are performed to compare the adjusted tests to O'Brien's test. The difference is also illustrated using data from a Parkinson's disease clinical trial.     

Is the Exact Test Better than the Asymptotic Test for Testing Marginal Homogeneity in 2 × 2 Tables?

McNemar test is commonly used to test for the marginal homogeneity in 2 × 2 contingency tables. McNemar test is an asymptotic test based either on standard normal distribution or on the chi‐square distribution. When the total sample size is small, an exact version of McNemar test is available based on the binomial probabilities. The example in the paper came from a clinical study to investigate the effect of epidermal growth factor for children who had microvillus inclusion diseases. There were only six observations available. The test results differ between the exact test and the asymptotic test. It is a common belief that with this small sample size the exact test be used. However, we claim that McNemar test performs better than the exact test even when the sample size is small. In order to investigate the performances of McNemar test and the exact test, we identify the parameters that affect the test results and then perform sensitivity analysis. In addition, through Monte Carlo simulation studies we compare the empirical sizes and powers of these tests as well as other asymptotic tests such as Wald test and the likelihood ratio test.     

A comparison of tests of the difference in the proportion of patients who are cured.

We compared by simulation the likelihood ratio, Wald, and score tests based on a mixture model similar to that proposed by Farewell (1982, Biometrics 38, 1041-1046), and a simple nonparametric test based on the plateau value of the product-limit estimate, for testing the difference in cured proportions between two groups. The parametric tests obtained their asymptotic properties even in small samples provided that one could assume equal failure rates in the two groups. Otherwise, good agreement with predictions required that essentially all potential failures had been observed. The comparative properties of the parametric tests depended on whether the population survival functions crossed, with the power of the Wald test as good as or better than the others in the common situation when the survival functions do not cross. However, its size was sometimes less than nominal. The score test was often not defined and is therefore of limited value. The product-limit test often performed as well as the parametric tests, and despite being biased in some circumstances, may be a useful alternative to these, especially in small samples when some potential failures have not been observed.     

Distribution-free ROC analysis using binary regression techniques

Receiver operating characteristic (ROC) regression methodology is used to identify factors that affect the accuracy of medical diagnostic tests. In this paper, we consider a ROC model for which the ROC curve is a parametric function of covariates but distributions of the diagnostic test results are not specified. Covariates can be either common to all subjects or specific to those with disease. We propose a new estimation procedure based on binary indicators defined by the test result for a diseased subject exceeding various specified quantiles of the distribution of test results from non-diseased subjects with the same covariate values. This procedure is conceptually and computationally simplified relative to existing procedures. Simulation study results indicate that the approach has fairly high statistical efficiency. The new ROC regression methodology is used to evaluate childhood measurements of body mass index as a predictive marker of adult obesity.     

Marginal tests with sliced average variance estimation

We present a new computationally feasible test for the dimensionof the central subspace in a regression problem based on slicedaverage variance estimation. We also provide a marginal coordinatetest. Under the null hypothesis, both the test of dimensionand the marginal coordinate test involve test statistics thatasymptotically have chi-squared distributions given normallydistributed predictors, and have a distribution that is a linearcombination of chi-squared distributions in general.     

A note on permutation tests for variance components in multilevel generalized linear mixed models

In many applications of generalized linear mixed models to multilevel data, it is of interest to test whether a random effects variance component is zero. It is well known that the usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In this note we propose a permutation test, based on randomly permuting the indices associated with a given level of the model, that has the correct Type I error rate under the null. Results from a simulation study suggest that it is more powerful than tests based on mixtures of chi-square distributions. The proposed test is illustrated using data on the familial aggregation of sleep disturbance.