A joint logistic regression and covariate‐adjusted continuous‐time Markov chain model

The use of longitudinal measurements to predict a categorical outcome is an increasingly common goal in research studies. Joint models are commonly used to describe two or more models simultaneously by considering the correlated nature of their outcomes and the random error present in the longitudinal measurements. However, there is limited research on joint models with longitudinal predictors and categorical cross‐sectional outcomes. Perhaps the most challenging task is how to model the longitudinal predictor process such that it represents the true biological mechanism that dictates the association with the categorical response. We propose a joint logistic regression and Markov chain model to describe a binary cross‐sectional response, where the unobserved transition rates of a two‐state continuous‐time Markov chain are included as covariates. We use the method of maximum likelihood to estimate the parameters of our model. In a simulation study, coverage probabilities of about 95%, standard deviations close to standard errors, and low biases for the parameter values show that our estimation method is adequate. We apply the proposed joint model to a dataset of patients with traumatic brain injury to describe and predict a 6‐month outcome based on physiological data collected post‐injury and admission characteristics. Our analysis indicates that the information provided by physiological changes over time may help improve prediction of long‐term functional status of these severely ill subjects. Copyright © 2017 John Wiley & Sons, Ltd.     

A tutorial on Bayesian bivariate meta‐analysis of mixed binary‐continuous outcomes with missing treatment effects

Bivariate random‐effects meta‐analysis (BVMA) is a method of data synthesis that accounts for treatment effects measured on two outcomes. BVMA gives more precise estimates of the population mean and predicted values than two univariate random‐effects meta‐analyses (UVMAs). BVMA also addresses bias from incomplete reporting of outcomes. A few tutorials have covered technical details of BVMA of categorical or continuous outcomes. Limited guidance is available on how to analyze datasets that include trials with mixed continuous‐binary outcomes where treatment effects on one outcome or the other are not reported. Given the advantages of Bayesian BVMA for handling missing outcomes, we present a tutorial for Bayesian BVMA of incompletely reported treatment effects on mixed bivariate outcomes. This step‐by‐step approach can serve as a model for our intended audience, the methodologist familiar with Bayesian meta‐analysis, looking for practical advice on fitting bivariate models. To facilitate application of the proposed methods, we include our WinBUGS code. As an example, we use aggregate‐level data from published trials to demonstrate the estimation of the effects of vitamin K and bisphosphonates on two correlated bone outcomes, fracture, and bone mineral density. We present datasets where reporting of the pairs of treatment effects on both outcomes was ‘partially’ complete (i.e., pairs completely reported in some trials), and we outline steps for modeling the incompletely reported data. To assess what is gained from the additional work required by BVMA, we compare the resulting estimates to those from separate UVMAs. We discuss methodological findings and make four recommendations. Copyright © 2015 John Wiley & Sons, Ltd.     

A general score‐independent test for order‐restricted inference

In the analysis of ordered categorical data, the categories are often assigned a set of subjectively chosen order‐restricted scores. To overcome the arbitrariness involved in the assignment of the scores, several score‐independent tests have been proposed. However, these methods are limited to 2 × K contingency tables, where K is the number of ordered categories. We present an efficiency robust score‐independent test that is applicable to more general situations. The test is embedded into a flexible framework for conditional inference and provides a natural generalization of many familiar tests involving ordered categorical data, such as the generalized Cochran‐Mantel‐Haenszel test for singly or doubly ordered contingency tables, the Page test for randomized block designs and the Tarone‐Ware trend test for survival data. The proposed method is illustrated by several numerical examples.     

Improved Evidence for Linkage on 6p and 5p with Retrospective Pooling of Data from Three Asthma Genome Screens

To improve our ability to identify genes for complex diseases, evaluation of new methods that retrospectively pool genotype and phenotype data collected by multiple centers is important. Availability of three whole‐genome screens enabled us to compare two methods, pooling raw data and meta‐analysis. Multipoint linkage analyses were performed on two outcomes, total serum IgE levels and asthma affection status, using an improved Haseman‐Elston algorithm. Two regions showed stronger evidence for linkage using covariate‐adjusted pooled data, compared with any individual sample. Both methods for pooling data identified strong linkage to Z‐transformed log_eIgE levels at a location between D6S1019 and D6S426, and to the asthma trait at D5S268. In conclusion, retrospective analysis of pooled genome scan data is a potentially powerful and useful method to examine both positive and negative evidence for linkage of quantitative and categorical phenotypes across populations. © 2001 Wiley‐Liss, Inc.     

Noninferiority testing for matched-pair ordinal data with misclassification

New treatments that are noninferior or equivalent to—but not necessarily superior to—the reference treatment may still be beneficial to patients because they have fewer side effects, are more convenient, take less time, or cost less. The noninferiority test is widely used in medical research to provide guidance in such situation. In addition, categorical variables are frequently encountered in medical research, such as in studies involving patient-reported outcomes. In this paper, we develop a noninferiority testing procedure for correlated ordinal categorical variables based on a paired design with a latent normal distribution approach. Misclassification is frequently encountered in the collection of ordinal categorical data; therefore, we further extend the procedure to account for misclassification using information in the partially validated data. Simulation studies are conducted to investigate the accuracy of the estimates, the type I error rates, and the power of the proposed procedure. Finally, we analyze one substantive example to demonstrate the utility of the proposed approach.     

Complete imputation of missing repeated categorical data: one-sample applications.

Longitudinal studies with repeated measures are often subject to non-response. Methods currently employed to alleviate the difficulties caused by missing data are typically unsatisfactory, especially when the cause of the missingness is related to the outcomes. We present an approach for incomplete categorical data in the repeated measures setting that allows missing data to depend on other observed outcomes for a study subject. The proposed methodology also allows a broader examination of study findings through interpretation of results in the framework of the set of all possible test statistics that might have been observed had no data been missing. The proposed approach consists of the following general steps. First, we generate all possible sets of missing values and form a set of possible complete data sets. We then weight each data set according to clearly defined assumptions and apply an appropriate statistical test procedure to each data set, combining the results to give an overall indication of significance. We make use of the EM algorithm and a Bayesian prior in this approach. While not restricted to the one-sample case, the proposed methodology is illustrated for one-sample data and compared to the common complete-case and available-case analysis methods.     

Point and interval estimation in two-stage adaptive designs with time to event data and biomarker-driven subpopulation selection

In personalized medicine, it is often desired to determine if all patients or only a subset of them benefit from a treatment. We consider estimation in two-stage adaptive designs that in stage 1 recruit patients from the full population. In stage 2, patient recruitment is restricted to the part of the population, which, based on stage 1 data, benefits from the experimental treatment. Existing estimators, which adjust for using stage 1 data for selecting the part of the population from which stage 2 patients are recruited, as well as for the confirmatory analysis after stage 2, do not consider time to event patient outcomes. In this work, for time to event data, we have derived a new asymptotically unbiased estimator for the log hazard ratio and a new interval estimator with good coverage probabilities and probabilities that the upper bounds are below the true values. The estimators are appropriate for several selection rules that are based on a single or multiple biomarkers, which can be categorical or continuous.     

分类重复测量数据的非线性混合效应模型

[目的]探讨分类重复测量数据的非线性混合效应模型及SAS8.0软件NLMIXED过程实现。[方法]直接拟合分类反应变量的非线性概率模型,结合重复测量资料的特点,采用附加高斯积分来获得最大似然的参数估计。[结果]非线性混合效应模型能很好地拟合分类反应变量的重复测量资料,它允许固定效应和随机效应进入模型的非线性部分,可方便地分析随机缺失等非均衡数据。[结论]分类反应变量重复测量资料的非线性混合效应模型分析结果合理、容易解释,为分类重复测量资料提供一种新的分析思路。     

A high-dimensional joint model for longitudinal outcomes of different nature

In repeated dose-toxicity studies, many outcomes are repeatedly measured on the same animal to study the toxicity of a compound of interest. This is only one example in which one is confronted with the analysis of many outcomes, possibly of a different type. Probably the most common situation is that of an amalgamation of continuous and categorical outcomes. A possible approach towards the joint analysis of two longitudinal outcomes of a different nature is the use of random-effects models (Models for Discrete Longitudinal Data. Springer Series in Statistics. Springer: New York, 2005). Although a random-effects model can easily be extended to jointly model many outcomes of a different nature, computational problems arise as the number of outcomes increases. To avoid maximization of the full likelihood expression, Fieuws and Verbeke (Biometrics 2006; 62:424-431) proposed a pairwise modeling strategy in which all possible pairs are modeled separately, using a mixed model, yielding several different estimates for the same parameters. These latter estimates are then combined into a single set of estimates. Also inference, based on pseudo-likelihood principles, is indirectly derived from the separate analyses. In this paper, we extend the approach of Fieuws and Verbeke (Biometrics 2006; 62:424-431) in two ways: the method is applied to different types of outcomes and the full pseudo-likelihood expression is maximized at once, leading directly to unique estimates as well as direct application of pseudo-likelihood inference. This is very appealing when interested in hypothesis testing. The method is applied to data from a repeated dose-toxicity study designed for the evaluation of the neurofunctional effects of a psychotrophic drug. The relative merits of both methods are discussed. Copyright (c) 2008 John Wiley & Sons, Ltd.     

Analysis of Data that is in the Form of Categories Part II: Beyond the 2*2 Contingency Table—Partitioning Chi-Square Tests

This article extends the analysis of 2*2 tables considered in the preceding paper of this three-part series. The methods described in the previous paper for analysis of frequency counts of categorical variables in 2*2 tables, including overall tests, partitioned tests, odds ratios, and estimation of required sample size, are applied to tables with multiple levels, such as those with more than two response factors and/or more than two levels for each factor.     

A spatial scan statistic for multinomial data

As a geographical cluster detection analysis tool, the spatial scan statistic has been developed for different types of data such as Bernoulli, Poisson, ordinal, exponential and normal. Another interesting data type is multinomial. For example, one may want to find clusters where the disease‐type distribution is statistically significantly different from the rest of the study region when there are different types of disease. In this paper, we propose a spatial scan statistic for such data, which is useful for geographical cluster detection analysis for categorical data without any intrinsic order information. The proposed method is applied to meningitis data consisting of five different disease categories to identify areas with distinct disease‐type patterns in two counties in the U.K. The performance of the method is evaluated through a simulation study. Copyright © 2010 John Wiley & Sons, Ltd.     

A multistate Markov chain model for longitudinal, categorical quality-of-life data subject to non-ignorable missingness

Quality-of-life (QOL) is an important outcome in clinical research, particularly in cancer clinical trials. Typically, data are collected longitudinally from patients during treatment and subsequent follow-up. Missing data are a common problem, and missingness may arise in a non-ignorable fashion. In particular, the probability that a patient misses an assessment may depend on the patient's QOL at the time of the scheduled assessment. We propose a Markov chain model for the analysis of categorical outcomes derived from QOL measures. Our model assumes that transitions between QOL states depend on covariates through generalized logit models or proportional odds models. To account for non-ignorable missingness, we incorporate logistic regression models for the conditional probabilities of observing measurements, given their actual values. The model can accommodate time-dependent covariates. Estimation is by maximum likelihood, summing over all possible values of the missing measurements. We describe options for selecting parsimonious models, and we study the finite-sample properties of the estimators by simulation. We apply the techniques to data from a breast cancer clinical trial in which QOL assessments were made longitudinally, and in which missing data frequently arose.     

重复测量分类数据的分析

重复测量问题常常涉及到分类数据。本文介绍分析重复测量分类数据的一般统计方法，并用临床资料进行实例分析。     

A Bayesian approach to functional mixed‐effects modeling for longitudinal data with binomial outcomes

Longitudinal growth patterns are routinely seen in medical studies where individual growth and population growth are followed up over a period of time. Many current methods for modeling growth presuppose a parametric relationship between the outcome and time (e.g., linear and quadratic); however, these relationships may not accurately capture growth over time. Functional mixed‐effects (FME) models provide flexibility in handling longitudinal data with nonparametric temporal trends. Although FME methods are well developed for continuous, normally distributed outcome measures, nonparametric methods for handling categorical outcomes are limited. We consider the situation with binomially distributed longitudinal outcomes. Although percent correct data can be modeled assuming normality, estimates outside the parameter space are possible, and thus, estimated curves can be unrealistic. We propose a binomial FME model using Bayesian methodology to account for growth curves with binomial (percentage) outcomes. The usefulness of our methods is demonstrated using a longitudinal study of speech perception outcomes from cochlear implant users where we successfully model both the population and individual growth trajectories. Simulation studies also advocate the usefulness of the binomial model particularly when outcomes occur near the boundary of the probability parameter space and in situations with a small number of trials. Copyright © 2014 John Wiley & Sons, Ltd.     

An analysis of clustered categorical data--application in dental health

Quite often in medical studies multiple discrete indicators are used to measure some characters that are defined only conceptually and are difficult to measure directly. Studies of this type exhibit categorical responses of dependent nature. Analysis of such categorical data appears to be extremely difficult (intractable) particularly in the presence of risk (causal) factors. In the present article, our purpose is to develop a latent mixture regression model for analysing such multivariate categorical data. Such a mixture model accommodates correlated and overdispersed data through the incorporation of random effects. Unfortunately, a full likelihood analysis is often hampered by the need for numerical integration. Two different procedures have been considered here. Both involve intensive computations. Numerical investigation has been carried out on the basis of a survey data covering 220 individuals from medical colleges in and around Calcutta (India). The purpose of the study is to compare tooth cleaning efficiency of brushes manufactured by different companies.     

Analysis of mixed categorical data in 2 x K contingency tables

2 x K contingency tables having both ordinal and nominal categories are often encountered in various types of studies. Such data are referred to as 'mixed' categorical data in this article. To apply a method for ordered categorical data one has to discard the nominal categories, and to apply a method for nominal categories one has to discard the ordering information inherent in the ordered categories. Therefore, investigators often either discard observations in nominal categories or discard the ordering of the categories before analysing such data. Some information will be lost in both approaches. A method for analysing data in 2 x K 'mixed' tables is proposed in this paper which can be considered as an extension of well known methods for nominal and ordered categories. The proposed method utilizes observations in the nominal categories as well as the ordering information. If all the categories were ordered then the proposed method reduces to the trend test, and if all the categories were nominal then the proposed method reduced to Pearson's chi-square test.     

Maximum likelihood estimation with missing outcomes: From simplicity to complexity

Many clinical or prevention studies involve missing or censored outcomes. Maximum likelihood (ML) methods provide a conceptually straightforward approach to estimation when the outcome is partially missing. Methods of implementing ML methods range from the simple to the complex, depending on the type of data and the missing-data mechanism. Simple ML methods for ignorable missing-data mechanisms (when data are missing at random) include complete-case analysis, complete-case analysis with covariate adjustment, survival analysis with covariate adjustment, and analysis via propensity-to-be-missing scores. More complex ML methods for ignorable missing-data mechanisms include the analysis of longitudinal dropouts via a marginal model for continuous data or a conditional model for categorical data. A moderately complex ML method for categorical data with a saturated model and either ignorable or nonignorable missing-data mechanisms is a perfect fit analysis, an algebraic method involving closed-form estimates and variances. A complex and flexible ML method with categorical data and either ignorable or nonignorable missing-data mechanisms is the method of composite linear models, a matrix method requiring specialized software. Except for the method of composite linear models, which can involve challenging matrix specifications, the implementation of these ML methods ranges in difficulty from easy to moderate.     

Gaussian-based routines to impute categorical variables in health surveys

The multivariate normal (MVN) distribution is arguably the most popular parametric model used in imputation and is available in most software packages (e.g., SAS PROC MI, R package norm). When it is applied to categorical variables as an approximation, practitioners often either apply simple rounding techniques for ordinal variables or create a distinct 'missing' category and/or disregard the nominal variable from the imputation phase. All of these practices can potentially lead to biased and/or uninterpretable inferences. In this work, we develop a new rounding methodology calibrated to preserve observed distributions to multiply impute missing categorical covariates. The major attractiveness of this method is its flexibility to use any 'working' imputation software, particularly those based on MVN, allowing practitioners to obtain usable imputations with small biases. A simulation study demonstrates the clear advantage of the proposed method in rounding ordinal variables and, in some scenarios, its plausibility in imputing nominal variables. We illustrate our methods on a widely used National Survey of Children with Special Health Care Needs where incomplete values on race posed a valid threat on inferences pertaining to disparities.     

Semi-parametric and non-parametric methods for the analysis of repeated measurements with applications to clinical trials.

Techniques applicable for the analysis of longitudinal data when the response variable is non-normal are not nearly as comprehensive as for normally-distributed outcomes. However, there have been several recent developments. Semi-parametric and non-parametric methodology for the analysis of repeated measurements is reviewed. The commonly encountered design in which, for each subject, one assesses a univariate response variable at multiple fixed time points, is considered. The types of outcomes considered include binary, ordered categorical, and continuous (but extremely non-normal) response variables. All of the methods considered allow for incomplete data due to the occurrence of missing observations. In addition, discrete and/or continuous covariates, which may be time-dependent, are accommodated by some of the approaches. The methods are demonstrated using data from three clinical trials.     

Cross-over trials in Statistics in Medicine: the first '25' years

Papers on cross-over trials that have appeared in the first 25 years of Statistics in Medicine are reviewed. Papers on bioequivalence are also considered. After a brief statistical summary, individual papers are discussed under seven headings: 1. The two-stage analysis of AB/BA trials, 2. Baselines, 3. Binary and categorical data, 4. Survival data, 5. Modelling carry-over, 6. Bioequivalence and 7. Components of variation. Finally, a brief assessment of the importance in this field of Statistics in Medicine is given.