Analyzing multiple endpoints in a confirmatory randomized clinical trial—an approach that addresses stratification,missing values,baseline imbalance and multiplicity for strictly ordinal outcomes

Confirmatory randomized clinical trials with a stratified design may have ordinal response outcomes, ie, either ordered categories or continuous determinations that are not compatible with an interval scale. Also, multiple endpoints are often collected when 1 single endpoint does not represent the overall efficacy of the treatment. In addition, random baseline imbalances and missing values can add another layer of difficulty in the analysis plan. Therefore, the development of an approach that provides a consolidated strategy to all issues collectively is essential. For a real case example that is from a clinical trial comparing a test treatment and a control for the pain management for patients with osteoarthritis, which has all aforementioned issues, multivariate Mann‐Whitney estimators with stratification adjustment are applicable to the strictly ordinal responses with stratified design. Randomization based nonparametric analysis of covariance is applied to account for the possible baseline imbalances. Several approaches that handle missing values are provided. A global test followed by a closed testing procedure controls the family wise error rate in the strong sense for the analysis of multiple endpoints. Four outcomes indicating joint pain, stiffness, and functional status were analyzed collectively and also individually through the procedures. Treatment efficacy was observed in the combined endpoint as well as in the individual endpoints. The proposed approach is effective in addressing the aforementioned problems simultaneously and straightforward to implement.     

Applying multivariate techniques to high-dimensional temporally correlated fMRI data

In first-level analyses of functional magnetic resonance imaging data, adjustments for temporal correlation as a Satterthwaite approximation or a prewhitening method are usually implemented in the univariate model to keep the nominal test level. In doing so, the temporal correlation structure of the data is estimated, assuming an autoregressive process of order one.We show that this is applicable in multivariate approaches too - more precisely in the so-called stabilized multivariate test statistics. Furthermore, we propose a block-wise permutation method including a random shift that renders an approximation of the temporal correlation structure unnecessary but also approximately keeps the nominal test level in spite of the dependence of sample elements.Although the intentions are different, a comparison of the multivariate methods with the multiple ones shows that the global approach may achieve advantages if applied to suitable regions of interest. This is illustrated using an example from fMRI studies.     

A flexible fixed-sequence testing method for hierarchically ordered correlated multiple endpoints in clinical trials

Statistical approaches for addressing multiplicity in clinical trials range from the very conservative (the Bonferroni method) to the least conservative the fixed sequence approach. Recently, several authors proposed methods that combine merits of the two extreme approaches. Wiens [2003. A fixed sequence Bonferroni procedure for testing multiple endpoints. Pharmaceutical Statist. 2003, 2, 211–215], for example, considered an extension of the Bonferroni approach where the type I error rate (α)$(\alpha )$ is allocated among the endpoints, however, testing proceeds in a pre-determined order allowing the type I error rate to be saved for later use as long as the null hypotheses are rejected. This leads to a higher power of the test in testing later null hypotheses. In this paper, we consider an extension of Wiens’ approach by taking into account correlations among endpoints for achieving higher flexibility in testing. We show strong control of the family-wise type I error rate for this extension and provide critical values and significance levels for testing up to three endpoints with equal correlations and show how to calculate them for other correlation structures. We also present results of a simulation experiment for comparing the power of the proposed method with those of Wiens’ and others. The results of this experiment show that the magnitude of the gain in power of the proposed method depends on the prospective ordering of testing of the endpoints, the magnitude of the treatment effects of the endpoints and the magnitude of correlation between endpoints. Finally, we consider applications of the proposed method for clinical trials with multiple time points and multiple doses, where correlations among endpoints frequently arise.     

On testing for causality in variance between two multivariate time series

Verifying the existence of a relationship between two multivariate time series represents an important consideration. In this article, the procedure developed by Cheung and Ng [A causality-in-variance test and its application to financial market prices, J. Econom. 72 (1996), pp. 33–48] designed to test causality in variance for univariate time series is generalized in several directions. A first approach proposes test statistics based on residual cross-covariance matrices of squared (standardized) residuals and cross products of (standardized) residuals. In a second approach, transformed residuals are defined for each residual vector time series, and test statistics are constructed based on the cross-correlations of these transformed residuals. Test statistics at individual lags and portmanteau-type test statistics are developed. Conditions are given under which the new test statistics converge in distribution towards chi-square distributions. The proposed methodology can be used to determine the directions of causality in variance, and appropriate test statistics are presented. Monte Carlo simulation results show that the new test statistics offer satisfactory empirical properties. An application with two bivariate financial time series illustrates the methods.     

On the Moments of Multivariate Discrete Distributions Using Finite Difference Operators

This article discusses a general approach to finding the moments of two classes of multivariate discrete distributions, which include those widely used in applied and theoretical statistics. The two classes of multivariate discrete distributions are the multivariate generalized power series distributions (GPSD) and the unified multivariate hypergeometric (UMH) Distributions. The results of Link (1981) follow as special cases.     

A Multivariate Generalized Poisson Regression Model

A multivariate generalized Poisson regression model based on the multivariate generalized Poisson distribution is defined and studied. The regression model can be used to describe a count data with any type of dispersion. The model allows for both positive and negative correlation between any pair of the response variables. The parameters of the regression model are estimated by using the maximum likelihood method. Some test statistics are discussed, and two numerical data sets are used to illustrate the applications of the multivariate count data regression model.     

An Empirical Bayes Method for Multivariate Meta-analysis with an Application in Clinical Trials

We propose an empirical Bayes method for evaluating overall and study-specific treatment effects in multivariate meta-analysis with binary outcome. Instead of modeling transformed proportions or risks via commonly used multivariate general or generalized linear models, we directly model the risks without any transformation. The exact posterior distribution of the study-specific relative risk is derived. The hyperparameters in the posterior distribution can be inferred through an empirical Bayes procedure. As our method does not rely on the choice of transformation, it provides a flexible alternative to the existing methods and in addition, the correlation parameter can be intuitively interpreted as the correlation coefficient between risks.     

Direct estimation of the percentile 'p-value' for the one-sample median test

In this paper we outline and illustrate an easy-to-use inference procedure for directly calculating the approximate bootstrap percentile-type p-value for the one-sample median test, i.e. we calculate the bootstrap p -value without resampling, by using a fractional order statistics based approach. The method parallels earlier work on fractionalorder-statistics-based non-parametric bootstrap percentile-type confidence intervals for quantiles. Monte Carlo simulation studies are performed, which illustrate that the fractional-order-statistics-based approach to the one-sample median test has accurate type I error control for small samples over a wide range of distributions; is easy to calculate; and is preferable to the sign test in terms of type I error control and power. Furthermore, the fractional-order-statistics-based median test is easily generalized to testing that any quantile has some hypothesized value; for example, tests for the upper or lower quartile may be performed using the same framework.     

A General formula for the upper tail significance levels of empirical distribution function test statistics

In this paper ve obtain an asymptotic expression for the upper tail area of the distribution of an infinite weighted sum of chi-square random variables and show how this can be applied to distributions of various goodness of fit test statistics. Results obtained by this general approach are comparable with those reported previously in the literature. In the case of the Cramer-von Mises statistic an empirical adjustment is given vhich significantly improves on previous approximations. For the Kuiper statistic the corresponding empirical adjustment leads to an existing highly accurate approximation.     

Testing noninferiority in three‐armed clinical trials based on likelihood ratio statistics

Clinical noninferiority trials with at least three groups have received much attention recently, perhaps due to the fact that regulatory agencies often require that a placebo group be evaluated along with a new experimental drug and an active control. The authors discuss likelihood ratio tests for binary endpoints and various noninferiority hypotheses. They find that, depending on the particular hypothesis, the test reduces asymptotically either to the intersection‐union test or to a test which follows asymptotically a mixture of generalized chi‐squared distributions. They investigate the performance of this asymptotic test and provide an exact modification. They show that this test considerably outperforms multiple testing methods such as the Bonferroni adjustment with respect to power. They illustrate their methods with a cancer study to compare antiemetic agents. Finally, they discuss the extension of the results to other settings, such as Gaussian endpoints.     

An evaluation of methods for testing hypotheses relating to two endpoints in a single clinical trial

The issues and dangers involved in testing multiple hypotheses are well recognised within the pharmaceutical industry. In reporting clinical trials, strenuous efforts are taken to avoid the inflation of type I error, with procedures such as the Bonferroni adjustment and its many elaborations and refinements being widely employed. Typically, such methods are conservative. They tend to be accurate if the multiple test statistics involved are mutually independent and achieve less than the type I error rate specified if these statistics are positively correlated. An alternative approach is to estimate the correlations between the test statistics and to perform a test that is conditional on those estimates being the true correlations. In this paper, we begin by assuming that test statistics are normally distributed and that their correlations are known. Under these circumstances, we explore several approaches to multiple testing, adapt them so that type I error is preserved exactly and then compare their powers over a range of true parameter values. For simplicity, the explorations are confined to the bivariate case. Having described the relative strengths and weaknesses of the approaches under study, we use simulation to assess the accuracy of the approximate theory developed when the correlations are estimated from the study data rather than being known in advance and when data are binary so that test statistics are only approximately normally distributed.     

Inferences on correlation coefficients: One-sample,independent and correlated cases

This article concerns inference on the correlation coefficients of a multivariate normal distribution. Inferential procedures based on the concepts of generalized variables (GVs) and generalized p$p$-values are proposed for elements of a correlation matrix. For the simple correlation coefficient, the merits of the generalized confidence limits and other approximate methods are evaluated using a numerical study. The study indicates that the proposed generalized confidence limits are uniformly most accurate even for samples as small as three. The results are extended for comparing two independent correlations, overlapping and non-overlapping dependent correlations. For each problem, the properties of the GV approach and other asymptotic methods are evaluated using Monte Carlo simulation. The GV approach produces satisfactory results for all the problems considered. The methods are illustrated using a few practical examples.     

Hypothesis Testing of Hazard Ratio Parameters in Marginal Models for Multivariate Failure Time Data

Marginal hazard models for multivariate failure time data have been studied extensively in recent literature. However, standard hypothesis test statistics based on the likelihood method are not exactly appropriate for this kind of model. In this paper, extensions of the three commonly used likelihood hypothesis test statistics are discussed. Generalized Wald, generalized score and generalized likelihood ratio tests for hazard ratio parameters in a marginal hazard model for multivariate failure time data are proposed and their asymptotic distributions examined. The finite sample properties of these statistics are studied through simulations. The proposed method is applied to data from Busselton Population Health Surveys.     

Inferences on the Coefficients of Variation in a Multivariate Normal Population

In this article, the problem of testing the equality of coefficients of variation in a multivariate normal population is considered, and an asymptotic approach and a generalized p-value approach based on the concepts of generalized test variable are proposed. Monte Carlo simulation studies show that the proposed generalized p-value test has good empirical sizes, and it is better than the asymptotic approach. In addition, the problem of hypothesis testing and confidence interval for the common coefficient variation of a multivariate normal population are considered, and a generalized p-value and a generalized confidence interval are proposed. Using Monte Carlo simulation, we find that the coverage probabilities and expected lengths of this generalized confidence interval are satisfactory, and the empirical sizes of the generalized p-value are close to nominal level. We illustrate our approaches using a real data.     

Assessing inter- and intra-agreement for dependent binary data: a Bayesian hierarchical correlation approach

Agreement measures are designed to assess consistency between different instruments rating measurements of interest. When the individual responses are correlated with multilevel structure of nestings and clusters, traditional approaches are not readily available to estimate the inter- and intra-agreement for such complex multilevel settings. Our research stems from conformity evaluation between optometric devices with measurements on both eyes, equality tests of agreement in high myopic status between monozygous twins and dizygous twins, and assessment of reliability for different pathologists in dysplasia. In this paper, we focus on applying a Bayesian hierarchical correlation model incorporating adjustment for explanatory variables and nesting correlation structures to assess the inter- and intra-agreement through correlations of random effects for various sources. This Bayesian generalized linear mixed-effects model (GLMM) is further compared with the approximate intra-class correlation coefficients and kappa measures by the traditional Cohen’s kappa statistic and the generalized estimating equations (GEE) approach. The results of comparison studies reveal that the Bayesian GLMM provides a reliable and stable procedure in estimating inter- and intra-agreement simultaneously after adjusting for covariates and correlation structures, in marked contrast to Cohen’s kappa and the GEE approach.     

Predicting treatment efficacy via quantitative magnetic resonance imaging: a Bayesian joint model

The prognosis for patients with high grade gliomas is poor, with a median survival of 1 year. Treatment efficacy assessment is typically unavailable until 5-6 months post diagnosis. Investigators hypothesize that quantitative magnetic resonance imaging can assess treatment efficacy 3 weeks after therapy starts, thereby allowing salvage treatments to begin earlier. The purpose of this work is to build a predictive model of treatment efficacy by using quantitative magnetic resonance imaging data and to assess its performance. The outcome is 1-year survival status. We propose a joint, two-stage Bayesian model. In stage I, we smooth the image data with a multivariate spatiotemporal pairwise difference prior. We propose four summary statistics that are functionals of posterior parameters from the first-stage model. In stage II, these statistics enter a generalized non-linear model as predictors of survival status. We use the probit link and a multivariate adaptive regression spline basis. Gibbs sampling and reversible jump Markov chain Monte Carlo methods are applied iteratively between the two stages to estimate the posterior distribution. Through both simulation studies and model performance comparisons we find that we can achieve higher overall correct classification rates by accounting for the spatiotemporal correlation in the images and by allowing for a more complex and flexible decision boundary provided by the generalized non-linear model.     

Goodness-of-link tests for multivariate regression models

ABSTRACT

This note presents an approximation to multivariate regression models which is obtained from a first-order series expansion of the multivariate link function. The proposed approach yields a variable-addition approximation of regression models that enables a multivariate generalization of the well-known goodness-of-link specification test, available for univariate generalized linear models. Application of this general methodology is illustrated with models of multinomial discrete choice and multivariate fractional data, in which context it is shown to lead to well-established approximation and testing procedures.     

Fast and Accurate Approximation to Significance Tests in Genome-Wide Association Studies

Genome-wide association studies commonly involve simultaneous tests of millions of single nucleotide polymorphisms (SNP) for disease association. The SNPs in nearby genomic regions, however, are often highly correlated due to linkage disequilibrium (LD, a genetic term for correlation). Simple Bonferonni correction for multiple comparisons is therefore too conservative. Permutation tests, which are often employed in practice, are both computationally expensive for genome-wide studies and limited in their scopes. We present an accurate and computationally efficient method, based on Poisson de-clumping heuristics, for approximating genome-wide significance of SNP associations. Compared with permutation tests and other multiple comparison adjustment approaches, our method computes the most accurate and robust p-value adjustments for millions of correlated comparisons within seconds. We demonstrate analytically that the accuracy and the efficiency of our method are nearly independent of the sample size, the number of SNPs, and the scale of p-values to be adjusted. In addition, our method can be easily adopted to estimate false discovery rate. When applied to genome-wide SNP datasets, we observed highly variable p-value adjustment results evaluated from different genomic regions. The variation in adjustments along the genome, however, are well conserved between the European and the African populations. The p-value adjustments are significantly correlated with LD among SNPs, recombination rates, and SNP densities. Given the large variability of sequence features in the genome, we further discuss a novel approach of using SNP-specific (local) thresholds to detect genome-wide significant associations. This article has supplementary material online.     

Permutation tests for between-unit fixed effects in multivariate generalized linear mixed models

A permutation testing approach in multivariate mixed models is presented. The solutions proposed allow for testing between-unit effect; they are exact under some assumptions, while approximated in the more general case. The classes of models comprised by this approach include generalized linear models, vector generalized additive models and other nonparametric models based on smoothing. Moreover it does not assume observations of different units to have the same distribution. The extensions to a multivariate framework are presented and discussed. The proposed multivariate tests exploit the dependence among variables, hence increasing the power with respect to other standard solutions (e.g. Bonferroni correction) which combine many univariate tests in an overall one. Examples are given of two applications to real data from psychological and ecological studies; a simulation study provides some insight into the unbiasedness of the tests and their power. The methods were implemented in the R package flip, freely available on CRAN.     

Sequential tests for non-proportional hazards data

In clinical trials survival endpoints are usually compared using the log-rank test. Sequential methods for the log-rank test and the Cox proportional hazards model are largely reported in the statistical literature. When the proportional hazards assumption is violated the hazard ratio is ill-defined and the power of the log-rank test depends on the distribution of the censoring times. The average hazard ratio was proposed as an alternative effect measure, which has a meaningful interpretation in the case of non-proportional hazards, and is equal to the hazard ratio, if the hazards are indeed proportional. In the present work we prove that the average hazard ratio based sequential test statistics are asymptotically multivariate normal with the independent increments property. This allows for the calculation of group-sequential boundaries using standard methods and existing software. The finite sample characteristics of the new method are examined in a simulation study in a proportional and a non-proportional hazards setting.