A mixed-effects regression model for three-level ordinal response data

Three-level data occur frequently in behaviour and medical sciences. For example, in a multi-centre trial, subjects within a given site are randomly assigned to treatments and then studied over time. In this example, the repeated observations (level-1) are nested within subjects (level-2) who are nested within sites (level-3). Similarly, in twin studies, repeated measurements (level-1) are taken on each twin (level-2) within each twin pair (level-3). A three-level mixed-effects regression model is described here. Random effects at the second and third level are included in the model. Additionally, both proportional odds and non-proportional odds models are developed. The latter allows the effects of explanatory variables to vary across the cumulative logits of the model. A maximum marginal likelihood (MML) solution is described and Gauss-Hermite numerical quadrature is used to integrate over the distribution of random effects. The random effects are normally distributed in this instance. Features of this model are illustrated using data from a school-based smoking prevention trial and an Alzheimer's disease clinical trial.     

Sample size and power determination for clustered repeated measurements

It is common in epidemiological and clinical studies that each subject has repeated measurements on a single common variable, while the subjects are also 'clustered'. To compute sample size or power of a test, we have to consider two types of correlation: correlation among repeated measurements within the same subject, and correlation among subjects in the same cluster. We develop, based on generalized estimating equations, procedures for computing sample size and power with clustered repeated measurements. Explicit formulae are derived for comparing two means, two slopes and two proportions, under several simple correlation structures.     

Mixed treatment comparison of repeated measurements of a continuous endpoint: an example using topical treatments for primary open‐angle glaucoma and ocular hypertension

Mixed treatment comparison (MTC) meta‐analyses estimate relative treatment effects from networks of evidence while preserving randomisation. We extend the MTC framework to allow for repeated measurements of a continuous endpoint that varies over time. We used, as a case study, a systematic review and meta‐analysis of intraocular pressure (IOP) measurements from randomised controlled trials evaluating topical ocular hypotensives in primary open‐angle glaucoma or ocular hypertension because IOP varies over the day and over the treatment course, and repeated measurements are frequently reported. We adopted models for conducting MTC in W inBUGS (The BUGS Project, Cambridge, UK) to allow for repeated IOP measurements and to impute missing standard deviations of the raw data using the predictive distribution from observations with standard deviations. A flexible model with an unconstrained baseline for IOP variations over time and time‐invariant random treatment effects fitted the data well. We also adopted repeated measures models to allow for class effects; assuming treatment effects to be exchangeable within classes slightly improved model fit but could bias estimated treatment effects if exchangeability assumptions were not valid. We enabled all timepoints to be included in the analysis, allowing for repeated measures to increase precision around treatment effects and avoid bias associated with selecting timepoints for meta‐analysis.The methods we developed for modelling repeated measures and allowing for missing data may be adapted for use in other MTC meta‐analyses. Copyright © 2011 John Wiley & Sons, Ltd.     

Relationships between undernutrition prevalence among children and adult women at national and subnational level

OBJECTIVES: The objective of this study is to analyze the relationships between undernutrition prevalence rates among children and adults, both at the level of countries and at the level of smaller geographical subunits within countries (districts, provinces). Results are considered of relevance for evaluation and proper usage of anthropometric information in poverty and food security assessment. DESIGN: Anthropometric information on both children and adults, as reported in the Demographic and Health Surveys, has been the primary source of data. In addition, data published by WHO, FAO, and data from some country specific reports have been used. The final analysis is based on data from 289 subnational geographical units divided over 56 countries in Africa, Asia and Latin America. Ordinary least squares has been used for regression analysis and F-tests for testing differences of variances. RESULTS: At the level of countries, results reveal a strong positive relationship between undernutrition prevalence rates among children and adults. At the level of smaller geographical units, high levels of undernutrition in adult women are almost invariably associated with high levels of undernutrition in children. At the same time, however, low or intermediate levels of undernutrition among adult women are no guarantee that undernutrition levels among children are also low or moderate. CONCLUSION: At the level of countries, information on undernutrition prevalence in children can be considered a proximate of the overall nutritional and food security conditions in a country. At the level of smaller geographical units, relationships are less straightforward, and are hypothesized to depend, at least partially, on the relative importance of food and nonfood factors in the causation of undernutrition.     

A random-effects Markov transition model for Poisson-distributed repeated measures with non-ignorable missing values

In biomedical research with longitudinal designs, missing values due to intermittent non-response or premature withdrawal are usually 'non-ignorable' in the sense that unobserved values are related to the patterns of missingness. By drawing the framework of a shared-parameter mechanism, the process yielding the repeated count measures and the process yielding missing values can be modelled separately, conditionally on a group of shared parameters. For chronic diseases, Markov transition models can be used to study the transitional features of the pathologic processes. In this paper, Markov Chain Monte Carlo algorithms are developed to fit a random-effects Markov transition model for incomplete count repeated measures, within which random effects are shared by the counting process and the missing-data mechanism. Assuming a Poisson distribution for the count measures, the transition probabilities are estimated using a Poisson regression model. The missingness mechanism is modelled with a multinomial-logit regression to calculate the transition probabilities of the missingness indicators. The method is demonstrated using both simulated data sets and a practical data set from a smoking cessation clinical trial.     

Testing moderation in network meta‐analysis with individual participant data

Meta‐analytic methods for combining data from multiple intervention trials are commonly used to estimate the effectiveness of an intervention. They can also be extended to study comparative effectiveness, testing which of several alternative interventions is expected to have the strongest effect. This often requires network meta‐analysis (NMA), which combines trials involving direct comparison of two interventions within the same trial and indirect comparisons across trials. In this paper, we extend existing network methods for main effects to examining moderator effects, allowing for tests of whether intervention effects vary for different populations or when employed in different contexts. In addition, we study how the use of individual participant data may increase the sensitivity of NMA for detecting moderator effects, as compared with aggregate data NMA that employs study‐level effect sizes in a meta‐regression framework. A new NMA diagram is proposed. We also develop a generalized multilevel model for NMA that takes into account within‐trial and between‐trial heterogeneity and can include participant‐level covariates. Within this framework, we present definitions of homogeneity and consistency across trials. A simulation study based on this model is used to assess effects on power to detect both main and moderator effects. Results show that power to detect moderation is substantially greater when applied to individual participant data as compared with study‐level effects. We illustrate the use of this method by applying it to data from a classroom‐based randomized study that involved two sub‐trials, each comparing interventions that were contrasted with separate control groups. Copyright © 2016 John Wiley & Sons, Ltd.     

Sample size requirements to detect an intervention by time interaction in longitudinal cluster randomized clinical trials

In designing a longitudinal cluster randomized clinical trial (cluster‐RCT), the interventions are randomly assigned to clusters such as clinics. Subjects within the same clinic will receive the identical intervention. Each will be assessed repeatedly over the course of the study. A mixed‐effects linear regression model can be applied in a cluster‐RCT with three‐level data to test the hypothesis that the intervention groups differ in the course of outcome over time. Using a test statistic based on maximum likelihood estimates, we derived closed‐form formulae for statistical power to detect the intervention by time interaction and the sample size requirements for each level. Importantly, the sample size does not depend on correlations among second‐level data units and the statistical power function depends on the number of second‐ and third‐level data units through their product. A simulation study confirmed that theoretical power estimates based on the derived formulae are nearly identical to empirical estimates. Copyright © 2009 John Wiley & Sons, Ltd.     

Joint analysis of multi-level repeated measures data and survival: an application to the end stage renal disease (ESRD) data

Shared random effects models have been increasingly common in the joint analyses of repeated measures (e.g. CD4 counts, hemoglobin levels) and a correlated failure time such as death. In this paper we study several shared random effects models in the multi-level repeated measures data setting with dependent failure times. Distinct random effects are used to characterize heterogeneity in repeated measures at different levels. The hazard of death may be dependent on random effects from various levels. To simplify the estimation procedure, we adopt the Gaussian quadrature technique with a piecewise log-linear baseline hazard for the death process, which can be conveniently implemented in the freely available software aML. As an example, we analyze repeated measures of hematocrit level and survival for end stage renal disease patients clustered within a randomly selected 126 dialysis centers in the U.S. renal data system data set. Our model is very comprehensive yet easy to implement, making it appealing to general statistical practitioners.     

Regression models for the analysis of longitudinal Gaussian data from multiple sources

We present a regression model for the joint analysis of longitudinal multiple source Gaussian data. Longitudinal multiple source data arise when repeated measurements are taken from two or more sources, and each source provides a measure of the same underlying variable and on the same scale. This type of data generally produces a relatively large number of observations per subject; thus estimation of an unstructured covariance matrix often may not be possible. We consider two methods by which parsimonious models for the covariance can be obtained for longitudinal multiple source data. The methods are illustrated with an example of multiple informant data arising from a longitudinal interventional trial in psychiatry.     

A mixed‐effect model for positive responses augmented by zeros

In this research article, we propose a class of models for positive and zero responses by means of a zero‐augmented mixed regression model. Under this class, we are particularly interested in studying positive responses whose distribution accommodates skewness. At the same time, responses can be zero, and therefore, we justify the use of a zero‐augmented mixture model. We model the mean of the positive response in a logarithmic scale and the mixture probability in a logit scale, both as a function of fixed and random effects. Moreover, the random effects link the two random components through their joint distribution and incorporate within‐subject correlation because of the repeated measurements and between‐subject heterogeneity. A Markov chain Monte Carlo algorithm is tailored to obtain Bayesian posterior distributions of the unknown quantities of interest, and Bayesian case‐deletion influence diagnostics based on the q‐divergence measure is performed. We apply the proposed method to a dataset from a 24hour dietary recall study conducted in the city of São Paulo and present a simulation study to evaluate the performance of the proposed methods. Copyright © 2015 John Wiley & Sons, Ltd.     

双反应变量重复测量资料分析及MIXED过程实现

目的探讨双反应变量重复测量资料的分析原理与方法及SAS软件PROCMIXED过程的应用。方法结合双反应变量重复测量数据的特点,采用SAS软件的MIXED过程对其进行分析,建立线性混合效应模型。结果该模型不仅考虑了每个变量多次重复测量结果之间的相关性,也考虑了两个变量之间的相关性,同时还引入固定效应和随机效应,结合数据特征分析,结果更为可信。结论对双反应变量非独立重复测量资料,可以把数据之间的相关性分解为重复测量间相关性和变量间相关性两部分,采用MIXED过程不仅可对其相关性做出明晰深入的分析,且可保证数据分析结果解释更符合实际。     

Bayesian joint modeling of longitudinal and spatial survival AIDS data

Joint analysis of longitudinal and survival data has received increasing attention in the recent years, especially for analyzing cancer and AIDS data. As both repeated measurements (longitudinal) and time‐to‐event (survival) outcomes are observed in an individual, a joint modeling is more appropriate because it takes into account the dependence between the two types of responses, which are often analyzed separately. We propose a Bayesian hierarchical model for jointly modeling longitudinal and survival data considering functional time and spatial frailty effects, respectively. That is, the proposed model deals with non‐linear longitudinal effects and spatial survival effects accounting for the unobserved heterogeneity among individuals living in the same region. This joint approach is applied to a cohort study of patients with HIV/AIDS in Brazil during the years 2002–2006. Our Bayesian joint model presents considerable improvements in the estimation of survival times of the Brazilian HIV/AIDS patients when compared with those obtained through a separate survival model and shows that the spatial risk of death is the same across the different Brazilian states. Copyright © 2016 John Wiley & Sons, Ltd.     

Interval estimation of generalized odds ratio in data with repeated measurements

When the underlying responses are on an ordinal scale, the generalized odds ratio (GOR), defined as the ratio of the proportions of concordant and discordant pairs, is a useful index to summarize the difference between two stochastically ordered distributions of an ordinal categorical variable. We discuss interval estimation of the GOR for ordinal data with repeated measurements. On the basis of the Dirichlet-multinomial model, we develop three asymptotic interval estimators of the GOR using Wald's test statistic, a logarithmic transformation, and a method analogous to Fieller's theorem, respectively. To evaluate and compare the finite-sample performance of these estimators, we apply Monte Carlo simulation. We find that when the number of subjects per group is not large, the coverage probability of interval estimator using Wald's test statistic is likely to be less than the desired confidence level. By contrast, the coverage probability of the other two estimators are approximately equal to or larger than the desired confidence level. When the number of subjects per group is small and the intraclass correlation between repeated measurements within subjects is large, we note that applying the interval estimator derived from a method analogous to Fieller's theorem can lose efficiency. We also note that the interval estimator using the logarithmic transformation is generally preferable to the other two estimators with respect to both the coverage probability and the average length. Finally, on the basis of a few preliminary simulations, we do find some robustness for all the estimators developed here. We include an example comparing the inflammation grade after lung transplant between surgeries to illustrate the use of the proposed interval estimators.     

Nutrition survey of Finnish rural children. VI. Methodological study comparing the 24-hour recall and the dietary history interview

The aim of the study was to evaluate and to compare the 24-hr recall method with the dietary history method as used in a food consumption survey of children. Information on the dietary intkake was obtained by 24-hour recall from 158 children and by the history method from 134. The interviews are repeated 7 months later. In addition, 741 children were interviewed by both methods on the same occasion. The repeatability of the results was analyzed both at the individual and at the group level. The correlation coefficients between the first and second interview in terms of the individual intakes of energy and nutrients were fairly low for both methods. At the group level the results of repeated 24-hr recalls were in good agreement. The dietary history method, however, gave significantly different mean intakes when repeated. The correlation coefficients between the values obtained by the 24-hr recall and the history method varied from 0.20 (vitamin A) to 0.50 (energy). The history method gave consistently higher mean values than the 24-hr recall. Neither of the methods can be considered suitable for the measurement of an individual child's dietary intake. The 24-hr recall is preferable for food consumption surveys of groups of children.     

Models for assisted conception data with embryo-specific covariates

Assisted conception routinely involves multiple embryo implantation within each recipient mother, with the outcome of interest being the number and multiplicity of live births. Here we consider the situation in which covariate information, potentially predictive of outcome, is available at the embryo level for each individual implanted embryo. This presents two challenges: firstly the outcome is measured at a higher, recipient, level than the covariates of interest; and secondly it is generally not known which of the implanted embryos developed to give a successful pregnancy. In practice such data have usually been analysed by aggregation of the embryo-level covariates to the recipient-level. Here we consider and compare two alternative approaches which respect the structure of the data alongside the aggregated approach. The first is a deterministic model with separate embryo and recipient success probabilities, each determined by a set of covariates, as first proposed by Spiers and extended by Zhou and Weinberg. The second is based on a multilevel model with the correlations between embryos in the same recipient modelled by a recipient level random effect. These models are compared using two real data sets, and the model properties further explored in a simulation study.     

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.     

Multilevel time series models with applications to repeated measures data

The analysis of repeated measures data can be conducted efficiently using a two-level random coefficients model. A standard assumption is that the within-individual (level 1) residuals are uncorrelated. In some cases, especially where measurements are made close together in time, this may not be reasonable and this additional correlation structure should also be modelled. A time series model for such data is proposed which consists of a standard multilevel model for repeated measures data augmented by an autocorrelation model for the level 1 residuals. First- and second-order autoregressive models are considered in detail, together with a seasonal component. Both discrete and continuous time are considered and it is shown how the autocorrelation parameters can themselves be structured in terms of further explanatory variables. The models are fitted to a data set consisting of repeated height measurements on children.     

Rare‐variant association tests in longitudinal studies,with an application to the Multi‐Ethnic Study of Atherosclerosis (MESA)

Over the past few years, an increasing number of studies have identified rare variants that contribute to trait heritability. Due to the extreme rarity of some individual variants, gene‐based association tests have been proposed to aggregate the genetic variants within a gene, pathway, or specific genomic region as opposed to a one‐at‐a‐time single variant analysis. In addition, in longitudinal studies, statistical power to detect disease susceptibility rare variants can be improved through jointly testing repeatedly measured outcomes, which better describes the temporal development of the trait of interest. However, usual sandwich/model‐based inference for sequencing studies with longitudinal outcomes and rare variants can produce deflated/inflated type I error rate without further corrections. In this paper, we develop a group of tests for rare‐variant association based on outcomes with repeated measures. We propose new perturbation methods such that the type I error rate of the new tests is not only robust to misspecification of within‐subject correlation, but also significantly improved for variants with extreme rarity in a study with small or moderate sample size. Through extensive simulation studies, we illustrate that substantially higher power can be achieved by utilizing longitudinal outcomes and our proposed finite sample adjustment. We illustrate our methods using data from the Multi‐Ethnic Study of Atherosclerosis for exploring association of repeated measures of blood pressure with rare and common variants based on exome sequencing data on 6,361 individuals.     

Modeling repeated labor curves in consecutive pregnancies: Individualized prediction of labor progression from previous pregnancy data

The measurement of cervical dilation of a pregnant woman is used to monitor the progression of labor until 10 cm when pushing begins. There is anecdotal evidence that labor tracks across repeated pregnancies; moreover, no statistical methodology has been developed to address this important issue, which can help obstetricians make more informed clinical decisions about an individual woman's progression. Motivated by the NICHD Consecutive Pregnancies Study (CPS), we propose new methodology for analyzing labor curves across consecutive pregnancies. Our focus is both on studying the correlation between repeated labor curves on the same woman and on using the cervical dilation data from prior pregnancies to predict subsequent labor curves. We propose a hierarchical random effects model with a random change point that characterizes repeated labor curves within and between women to address these issues. We employ Bayesian methodology for parameter estimation and prediction. Model diagnostics to examine the appropriateness of the hierarchical random effects structure for characterizing the dependence structure across consecutive pregnancies are also proposed. The methodology was used in analyzing the CPS data and in developing a predictor for labor progression that can be used in clinical practice.