Comparison of Within-Subject Covariance Matrices in 2 × 2 Crossover Trials with Multivariate Response

Abstract.  In a multivariate framework, it is shown how the two treatments in a 2 × 2 crossover trial with multivariate response can be compared with respect to mean vectors, i.e. fixed treatment effects, as well as within-subject covariance matrices, marginally and simultaneously. No distributional assumption is made about the between-subject variability, whereas multivariate normality is assumed for the within-subject variability. The proposed exact statistical inferences are valid even with few subjects. Data from a crossover trial with bivariate response are analysed with the proposed multivariate methods as well as with univariate methods.     

Non-penalty shrinkage estimation of random effect models for longitudinal data with AR(1) errors

In this paper, we consider the non-penalty shrinkage estimation method of random effect models with autoregressive errors for longitudinal data when there are many covariates and some of them may not be active for the response variable. In observational studies, subjects are followed over equally or unequally spaced visits to determine the continuous response and whether the response is associated with the risk factors/covariates. Measurements from the same subject are usually more similar to each other and thus are correlated with each other but not with observations of other subjects. To analyse this data, we consider a linear model that contains both random effects across subjects and within-subject errors that follows autoregressive structure of order 1 (AR(1)). Considering the subject-specific random effect as a nuisance parameter, we use two competing models, one includes all the covariates and the other restricts the coefficients based on the auxiliary information. We consider the non-penalty shrinkage estimation strategy that shrinks the unrestricted estimator in the direction of the restricted estimator. We discuss the asymptotic properties of the shrinkage estimators using the notion of asymptotic biases and risks. A Monte Carlo simulation study is conducted to examine the relative performance of the shrinkage estimators with the unrestricted estimator when the shrinkage dimension exceeds two. We also numerically compare the performance of the shrinkage estimators to that of the LASSO estimator. A longitudinal CD4 cell count data set will be used to illustrate the usefulness of shrinkage and LASSO estimators.     

Semiparametric Methods for Clustered Recurrent Event Data

In biomedical studies, the event of interest is often recurrent and within-subject events cannot usually be assumed independent. In addition, individuals within a cluster might not be independent; for example, in multi-center or familial studies, subjects from the same center or family might be correlated. We propose methods of estimating parameters in two semi-parametric proportional rates/means models for clustered recurrent event data. The first model contains a baseline rate function which is common across clusters, while the second model features cluster-specific baseline rates. Dependence structures for patients-within-cluster and events-within-patient are both unspecified. Estimating equations are derived for the regression parameters. For the common baseline model, an estimator of the baseline mean function is proposed. The asymptotic distributions of the model parameters are derived, while finite-sample properties are assessed through a simulation study. Using data from a national organ failure registry, the proposed methods are applied to the analysis of technique failures among Canadian dialysis patients.     

A comparison of methods for simulating correlated binary variables with specified marginal means and correlations

Simulation studies employed to study properties of estimators for parameters in population-average models for clustered or longitudinal data require suitable algorithms for data generation. Methods for generating correlated binary data that allow general specifications of the marginal mean and correlation structures are particularly useful. We compare an algorithm based on dichotomizing multi-normal variates to one based on a conditional linear family (CLF) of distributions [Qaqish BF. A family of multivariate binary distributions for simulating correlated binary variables with specified marginal means and correlations. Biometrika. 2003;90:455–463] with respect to range restrictions induced on correlations. Examples include generating longitudinal binary data and generating correlated binary data compatible with specified marginal means and covariance structures for bivariate, overdispersed binomial outcomes. Results show the CLF method gives a wider range of correlations for longitudinal data having autocorrelated within-subject associations, while the multivariate probit method gives a wider range of correlations for clustered data having exchangeable-type correlations. In the case of a decaying-product correlation structure, it is shown that the CLF method achieves the nonparametric limits on the range of correlations, which cannot be surpassed by any method.     

Crossover designs based on type I orthogonal arrays for a self and simple mixed carryover effects model with correlated errors

We investigate the performance of crossover designs based on type I orthogonal arrays for a self and simple mixed carryover effects model in the presence of correlated errors. Assuming that between-subject errors are independent while within-subject errors behave according to the stationary first-order autoregressive and moving average processes, analytical optimality results for 3-period designs are established and, as an illustration, numerical details for a number of 4-period cases are tabulated.     

An improved alternative estimation procedure for current population mean in presence of positively and negatively correlated auxiliary variables in two-occasion rotation patterns

Abstract

The present study confirms the influential role of a positively and a negatively correlated auxiliary variables in enhancing the precision of estimates of current population mean in two occasion rotation (successive) sampling. Exponential-type estimators of current population mean have been proposed for three different situations: (i) the information on a positively correlated auxiliary variable is readily available on both occasions (ii) the information on a negatively correlated auxiliary variable is readily available on both occasions and (iii) the information on both positively and negatively correlated auxiliary variables are readily available on both the occasions. The characteristics of the proposed estimators have been explored and their efficacious performances are compared with the natural and recent contemporary estimators. Optimum replacement strategies of the proposed estimation procedures have been formulated. Simulation and empirical studies are carried out to justify the proposition of the proposed estimators and appropriate recommendations have been put forward to the survey practitioners.     

THE ANALYSIS OF DISCRETE CHOICE EXPERIMENTS WITH CORRELATED ERROR STRUCTURE

In a stated preference discrete choice experiment each subject is typically presented with several choice sets, and each choice set contains a number of alternatives. The alternatives are defined in terms of their name (brand) and their attributes at specified levels. The task for the subject is to choose from each choice set the alternative with highest utility for them. The multinomial is an appropriate distribution for the responses to each choice set since each subject chooses one alternative, and the multinomial logit is a common model. If the responses to the several choice sets are independent, the likelihood function is simply the product of multinomials. The most common and generally preferred method of estimating the parameters of the model is maximum likelihood (that is, selecting as estimates those values that maximize the likelihood function). If the assumption of within-subject independence to successive choice tasks is violated (it is almost surely violated), the likelihood function is incorrect and maximum likelihood estimation is inappropriate. The most serious errors involve the estimation of the variance-covariance matrix of the model parameter estimates, and the corresponding variances of market shares and changes in market shares.

In this paper we present an alternative method of estimation of the model parameter coefficients that incorporates a first-order within-subject covariance structure. The method involves the familiar log-odds transformation and application of the multivariate delta method. Estimation of the model coefficients after the transformation is a straightforward generalized least squares regression, and the corresponding improved estimate of the variance-covariance matrix is in closed form. Estimates of market share (and change in market share) follow from a second application of the multivariate delta method. The method and comparison with maximum likelihood estimation are illustrated with several simulated and actual data examples.

Advantages of the proposed method are: 1) it incorporates the within-subject covariance structure; 2) it is completely data driven; 3) it requires no additional model assumptions; 4) assuming asymptotic normality, it provides a simple procedure for computing confidence regions on market shares and changes in market shares; and 5) it produces results that are asymptotically equivalent to those produced by maximum likelihood when the data are independent.     

A hierarchical eigenmodel for pooled covariance estimation

Summary.  Although the covariance matrices corresponding to different populations are unlikely to be exactly equal they can still exhibit a high degree of similarity. For example, some pairs of variables may be positively correlated across most groups, whereas the correlation between other pairs may be consistently negative. In such cases much of the similarity across covariance matrices can be described by similarities in their principal axes, which are the axes that are defined by the eigenvectors of the covariance matrices. Estimating the degree of across-population eigenvector heterogeneity can be helpful for a variety of estimation tasks. For example, eigenvector matrices can be pooled to form a central set of principal axes and, to the extent that the axes are similar, covariance estimates for populations having small sample sizes can be stabilized by shrinking their principal axes towards the across-population centre. To this end, the paper develops a hierarchical model and estimation procedure for pooling principal axes across several populations. The model for the across-group heterogeneity is based on a matrix-valued antipodally symmetric Bingham distribution that can flexibly describe notions of 'centre' and 'spread' for a population of orthogonal matrices.     

Marginalized transition random effect models for multivariate longitudinal binary data

Generalized linear models with random effects and/or serial dependence are commonly used to analyze longitudinal data. However, the computation and interpretation of marginal covariate effects can be difficult. This led Heagerty (1999, 2002) to propose models for longitudinal binary data in which a logistic regression is first used to explain the average marginal response. The model is then completed by introducing a conditional regression that allows for the longitudinal, within‐subject, dependence, either via random effects or regressing on previous responses. In this paper, the authors extend the work of Heagerty to handle multivariate longitudinal binary response data using a triple of regression models that directly model the marginal mean response while taking into account dependence across time and across responses. Markov Chain Monte Carlo methods are used for inference. Data from the Iowa Youth and Families Project are used to illustrate the methods.     

Robust estimation of mean and covariance for longitudinal data with dropouts

In this paper, we study estimation of linear models in the framework of longitudinal data with dropouts. Under the assumptions that random errors follow an elliptical distribution and all the subjects share the same within-subject covariance matrix which does not depend on covariates, we develop a robust method for simultaneous estimation of mean and covariance. The proposed method is robust against outliers, and does not require to model the covariance and missing data process. Theoretical properties of the proposed estimator are established and simulation studies show its good performance. In the end, the proposed method is applied to a real data analysis for illustration.     

Modeling longitudinal count data with dropouts

This paper explores the utility of different approaches for modeling longitudinal count data with dropouts arising from a clinical study for the treatment of actinic keratosis lesions on the face and balding scalp. A feature of these data is that as the disease for subjects on the active arm improves their data show larger dispersion compared with those on the vehicle, exhibiting an over‐dispersion relative to the Poisson distribution. After fitting the marginal (or population averaged) model using the generalized estimating equation (GEE), we note that inferences from such a model might be biased as dropouts are treatment related. Then, we consider using a weighted GEE (WGEE) where each subject's contribution to the analysis is weighted inversely by the subject's probability of dropout. Based on the model findings, we argue that the WGEE might not address the concerns about the impact of dropouts on the efficacy findings when dropouts are treatment related. As an alternative, we consider likelihood‐based inference where random effects are added to the model to allow for heterogeneity across subjects. Finally, we consider a transition model where, unlike the previous approaches that model the log‐link function of the mean response, we model the subject's actual lesion counts. This model is an extension of the Poisson autoregressive model of order 1, where the autoregressive parameter is taken to be a function of treatment as well as other covariates to induce different dispersions and correlations for the two treatment arms. We conclude with a discussion about model selection. Published in 2009 by John Wiley & Sons, Ltd.     

Efficient estimation for time-varying coefficient longitudinal models

For estimation of time-varying coefficient longitudinal models, the widely used local least-squares (LS) or covariance-weighted local LS smoothing uses information from the local sample average. Motivated by the fact that a combination of multiple quantiles provides a more complete picture of the distribution, we investigate quantile regression-based methods to improve efficiency by optimally combining information across quantiles. Under the working independence scenario, the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound. In the presence of dependence among within-subject measurements, we adopt a prewhitening technique to transform regression errors into independent innovations and show that the prewhitened optimally weighted quantile average estimator asymptotically achieves the Cramér–Rao bound for the independent innovations. Fully data-driven bandwidth selection and optimal weights estimation are implemented through a two-step procedure. Monte Carlo studies show that the proposed method delivers more robust and superior overall performance than that of the existing methods.     

Estimation of population mean based on dual use of auxiliary information in non response

Whenever there is auxiliary information available in any form, the researchers want to utilize it in the method of estimation to obtain the most efficient estimator. When there exists enough amount of correlation between the study and the auxiliary variables, and parallel to these associations, the ranks of the auxiliary variables are also correlated with the study variable, which can be used a valuable device for enhancing the precision of an estimator accordingly. This article addresses the problem of estimating the finite population mean that utilizes the complementary information in the presence of (i) the auxiliary variable and (ii) the ranks of the auxiliary variable for non response. We suggest an improved estimator for estimating the finite population mean using the auxiliary information in the presence of non response. Expressions for bias and mean squared error of considered estimators are derived up to the first order of approximation. The performance of estimators is compared theoretically and numerically. A numerical study is carried out to evaluate the performances of estimators. It is observed that the proposed estimator is more efficient than the usual sample mean and the regression estimators, and some other families of ratio and exponential type of estimators.     

Modeling Association Between Two or More Categorical Variables that Allow for Multiple Category Choices

Multiple-response (or pick any/c) categorical variables summarize responses to survey questions that ask “pick any” from a set of item responses. Extensions to loglinear model methodology are proposed to model associations between these variables across all their items simultaneously. Because individual item responses to a multiple-response categorical variable are likely to be correlated, the usual chi-square distributional approximations for model-comparison statistics are not appropriate. Adjusted statistics and a new bootstrap procedure are developed to facilitate distributional approximations. Odds ratio and standardized Pearson residual measures are also developed to estimate specific associations and examine deviations from a specified model.     

Combining estimates from multiple early studies to obtain estimates of response: using shrinkage estimates to obtain estimates of response

Development of anti-cancer therapies usually involve small to moderate size studies to provide initial estimates of response rates before initiating larger studies to better quantify response. These early trials often each contain a single tumor type, possibly using other stratification factors. Response rate for a given tumor type is routinely reported as the percentage of patients meeting a clinical criteria (e.g. tumor shrinkage), without any regard to response in the other studies. These estimates (called maximum likelihood estimates or MLEs) on average approximate the true value, but have variances that are usually large, especially for small to moderate size studies. The approach presented here is offered as a way to improve overall estimation of response rates when several small trials are considered by reducing the total uncertainty.The shrinkage estimators considered here (James-Stein/empirical Bayes and hierarchical Bayes) are alternatives that use information from all studies to provide potentially better estimates for each study. While these estimates introduce a small bias, they have a considerably smaller variance, and thus tend to be better in terms of total mean squared error. These procedures provide a better view of drug performance in that group of tumor types as a whole, as opposed to estimating each response rate individually without consideration of the others. In technical terms, the vector of estimated response rates is nearer the vector of true values, on average, than the vector of the usual unbiased MLEs applied to such trials.     

Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

The stratified Cox model is commonly used for stratified clinical trials with time‐to‐event endpoints. The estimated log hazard ratio is approximately a weighted average of corresponding stratum‐specific Cox model estimates using inverse‐variance weights; the latter are optimal only under the (often implausible) assumption of a constant hazard ratio across strata. Focusing on trials with limited sample sizes (50‐200 subjects per treatment), we propose an alternative approach in which stratum‐specific estimates are obtained using a refined generalized logrank (RGLR) approach and then combined using either sample size or minimum risk weights for overall inference. Our proposal extends the work of Mehrotra et al, to incorporate the RGLR statistic, which outperforms the Cox model in the setting of proportional hazards and small samples. This work also entails development of a remarkably accurate plug‐in formula for the variance of RGLR‐based estimated log hazard ratios. We demonstrate using simulations that our proposed two‐step RGLR analysis delivers notably better results through smaller estimation bias and mean squared error and larger power than the stratified Cox model analysis when there is a treatment‐by‐stratum interaction, with similar performance when there is no interaction. Additionally, our method controls the type I error rate while the stratified Cox model does not in small samples. We illustrate our method using data from a clinical trial comparing two treatments for colon cancer.     

A class of multivariate distribution-free tests of independence based on graphs

A class of distribution-free tests is proposed for the independence of two subsets of response coordinates. The tests are based on the pairwise distances across subjects within each subset of the response. A complete graph is induced by each subset of response coordinates, with the sample points as nodes and the pairwise distances as the edge weights. The proposed test statistic depends only on the rank order of edges in these complete graphs. The response vector may be of any dimensions. In particular, the number of samples may be smaller than the dimensions of the response. The test statistic is shown to have a normal limiting distribution with known expectation and variance under the null hypothesis of independence. The exact distribution free null distribution of the test statistic is given for a sample of size 14, and its Monte-Carlo approximation is considered for larger sample sizes. We demonstrate in simulations that this new class of tests has good power properties for very general alternatives.     

Discrimination indexes and models for studying sensory functioning in aging

A popular paradigm in experimental psychophysics has subjects estimate sensation magnitude by assigning numbers to stimuli in some way. While it is typical to analyze the central tendency (e.g. means and slopes) of the subjects' psychophysical. functions, there is often a greater need to analyze the internal consistency of these functions. A subject who gives increasing mean responses across increasing stimulus intensities and also gives highly consistent responses within stimulus intensities is showing superior sensory discrimination. We propose new discrimination indexes, based on measures of association and lack-of-fit, that summarize monotonic regressions of the subject's data, as well as non-metric and metric-sensitive measures related to Kendall's coefficient of concordance. We use these indexes in quadratic spline regression models for cross-sectional age trends in sensory discrimination, with covariates included to adjust for task demands and gender differences.Because such data are potentially affected by increasing variability with age, we describe a method to assess this and adjust for it using reweighted least squares.     

Bayesian analysis of joint mean and covariance models for longitudinal data

Efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure. If misspecification occurs, it may lead to inefficient or biased estimators of parameters in the mean. One of the most commonly used methods for handling the covariance matrix is based on simultaneous modeling of the Cholesky decomposition. Therefore, in this paper, we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a fully Bayesian inference for joint mean and covariance models based on this decomposition. A computational efficient Markov chain Monte Carlo method which combines the Gibbs sampler and Metropolis–Hastings algorithm is implemented to simultaneously obtain the Bayesian estimates of unknown parameters, as well as their standard deviation estimates. Finally, several simulation studies and a real example are presented to illustrate the proposed methodology.     

BLOCK-BASED BAYESIAN EPISTASIS ASSOCIATION MAPPING WITH APPLICATION TO WTCCC TYPE 1 DIABETES DATA

Interactions among multiple genes across the genome may contribute to the risks of many complex human diseases. Whole-genome single nucleotide polymorphisms (SNPs) data collected for many thousands of SNP markers from thousands of individuals under the case-control design promise to shed light on our understanding of such interactions. However, nearby SNPs are highly correlated due to linkage disequilibrium (LD) and the number of possible interactions is too large for exhaustive evaluation. We propose a novel Bayesian method for simultaneously partitioning SNPs into LD-blocks and selecting SNPs within blocks that are associated with the disease, either individually or interactively with other SNPs. When applied to homogeneous population data, the method gives posterior probabilities for LD-block boundaries, which not only result in accurate block partitions of SNPs, but also provide measures of partition uncertainty. When applied to case-control data for association mapping, the method implicitly filters out SNP associations created merely by LD with disease loci within the same blocks. Simulation study showed that this approach is more powerful in detecting multi-locus associations than other methods we tested, including one of ours. When applied to the WTCCC type 1 diabetes data, the method identified many previously known T1D associated genes, including PTPN22, CTLA4, MHC, and IL2RA. The method also revealed some interesting two-way associations that are undetected by single SNP methods. Most of the significant associations are located within the MHC region. Our analysis showed that the MHC SNPs form long-distance joint associations over several known recombination hotspots. By controlling the haplotypes of the MHC class II region, we identified additional associations in both MHC class I (HLA-A, HLA-B) and class III regions (BAT1). We also observed significant interactions between genes PRSS16, ZNF184 in the extended MHC region and the MHC class II genes. The proposed method can be broadly applied to the classification problem with correlated discrete covariates.