An appraisal of methods for the analysis of longitudinal categorical data with MAR drop-outs

A number of methods for analysing longitudinal ordinal categorical data with missing-at-random drop-outs are considered. Two are maximum-likelihood methods (MAXLIK) which employ marginal global odds ratios to model associations. The remainder use weighted or unweighted generalized estimating equations (GEE). Two of the GEE use Cholesky-decomposed standardized residuals to model the association structure, while another three extend methods developed for longitudinal binary data in which the association structures are modelled using either Gaussian estimation, multivariate normal estimating equations or conditional residuals. Simulated data sets were used to discover differences among the methods in terms of biases, variances and convergence rates when the association structure is misspecified. The methods were also applied to a real medical data set. Two of the GEE methods, referred to as Cond and ML-norm in this paper and by their originators, were found to have relatively good convergence rates and mean squared errors for all sample sizes (80, 120, 300) considered, and one more, referred to as MGEE in this paper and by its originators, worked fairly well for all but the smallest sample size, 80.     

Sensitivity analysis of longitudinal normal data with drop-outs

We propose to perform a sensitivity analysis to evaluate the extent to which results from a longitudinal study can be affected by informative drop-outs. The method is based on a selection model, where the parameter relating the dropout probability to the current observation is not estimated, but fixed to a set of values. This allows to evaluate several hypotheses for the degree of informativeness of the drop-out process. Expectation and variance of missing data, conditional on the drop-out time are computed, and a stochastic EM algorithm is used to obtain maximum likelihood estimates. Simulations show that when the drop-out parameter is correctly specified, unbiased estimates of the other parameters are obtained, and coverage percentages of their confidence intervals are close to their theoretical value. More interestingly, misspecification of the drop-out parameter does not considerably alter these results. This method was applied to a randomized clinical trial, designed to demonstrate non-inferiority of an inhaled corticosteroid in terms of bone density, compared with a reference treatment. Sensitivity analysis showed that the conclusion of non-inferiority was robust against different hypotheses for the drop-out process.     

A shared random effect parameter approach for longitudinal dementia data with non-ignorable missing data

A significant source of missing data in longitudinal epidemiologic studies on elderly individuals is death. It is generally believed that these missing data by death are non-ignorable to likelihood based inference. Inference based on data only from surviving participants in the study may lead to biased results. In this paper we model both the probability of disease and the probability of death using shared random effect parameters. We also propose to use the Laplace approximation for obtaining an approximate likelihood function so that high dimensional integration over the distributions of the random effect parameters is not necessary. Parameter estimates can be obtained by maximizing the approximate log-likelihood function. Data from a longitudinal dementia study will be used to illustrate the approach. A small simulation is conducted to compare parameter estimates from the proposed method to the 'naive' method where missing data is considered at random.     

An autoregressive linear mixed effects model for the analysis of unequally spaced longitudinal data with dose-modification

The assessment of the dose-response relationship is important but not straightforward when the therapeutic agent is administered repeatedly with dose-modification in each patient and a continuous response is measured repeatedly. We recently proposed an autoregressive linear mixed effects model for such data in which the current response is regressed on the previous response, fixed effects, and random effects. The model represents profiles approaching each patient's asymptote, takes into account the past dose history, and provides a dose-response relationship of the asymptote as a summary measure. In an autoregressive model, intermittent missing data mean the missing values in previous responses as covariates. We previously provided the marginal (unconditional on the previous response) form of the proposed model to deal with intermittent missing data. Irregular timings of dose-modification or measurement can also be treated as equally spaced data with intermittent missing values by selecting an adequately small unit of time. The likelihood is, however, expressed by matrices whose sizes depend on the number of observations for a patient, and the computational burden is large. In this study, we propose a state space form of the autoregressive linear mixed effects model to calculate the marginal likelihood without using large matrices. The regression coefficients of the fixed effects can be concentrated out of the likelihood in this model by the same way of a linear mixed effects model. As an illustration of the approach, we analyzed immunologic data from a clinical trial for multiple sclerosis patients and estimated the dose-response curves for each patient and the population mean.     

Pseudo-likelihood methods for longitudinal binary data with non-ignorable missing responses and covariates

In this paper we consider longitudinal studies in which the outcome to be measured over time is binary, and the covariates of interest are categorical. In longitudinal studies it is common for the outcomes and any time-varying covariates to be missing due to missed study visits, resulting in non-monotone patterns of missingness. Moreover, the reasons for missed visits may be related to the specific values of the response and/or covariates that should have been obtained, i.e. missingness is non-ignorable. With non-monotone non-ignorable missing response and covariate data, a full likelihood approach is quite complicated, and maximum likelihood estimation can be computationally prohibitive when there are many occasions of follow-up. Furthermore, the full likelihood must be correctly specified to obtain consistent parameter estimates. We propose a pseudo-likelihood method for jointly estimating the covariate effects on the marginal probabilities of the outcomes and the parameters of the missing data mechanism. The pseudo-likelihood requires specification of the marginal distributions of the missingness indicator, outcome, and possibly missing covariates at each occasions, but avoids making assumptions about the joint distribution of the data at two or more occasions. Thus, the proposed method can be considered semi-parametric. The proposed method is an extension of the pseudo-likelihood approach in Troxel et al. to handle binary responses and possibly missing time-varying covariates. The method is illustrated using data from the Six Cities study, a longitudinal study of the health effects of air pollution.     

Marginalized transition models for longitudinal binary data with ignorable and non-ignorable drop-out

We extend the marginalized transition model of Heagerty to accommodate non-ignorable monotone drop-out. Using a selection model, weakly identified drop-out parameters are held constant and their effects evaluated through sensitivity analysis. For data missing at random (MAR), efficiency of inverse probability of censoring weighted generalized estimating equations (IPCW-GEE) is as low as 40 per cent compared to a likelihood-based marginalized transition model (MTM) with comparable modelling burden. MTM and IPCW-GEE regression parameters both display misspecification bias for MAR and non-ignorable missing data, and both reduce bias noticeably by improving model fit.     

A new mixed-effects mixture model for constrained longitudinal data

Biomedical research often features continuous responses bounded by the interval [0, 1]. The well-known beta regression model addresses the constrained nature of these data, while its augmented and mixed-effects variants can address the presence of zeros and/or ones and longitudinal or clustered response values, respectively. However, these models are not robust to the presence of outliers and/or excessive number of observations near the tails. We propose a new augmented mixed-effects regression model based on a special beta mixture distribution that is capable of handling these issues. Extensive simulation studies show the superiority of the proposed model to the models most often used in the literature. The proposed model is applied to two real datasets: one taken from a long-term study of Parkinson's disease and the other taken from a study on reading accuracy.     

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.     

Using the general linear mixed model to analyse unbalanced repeated measures and longitudinal data

The general linear mixed model provides a useful approach for analysing a wide variety of data structures which practising statisticians often encounter. Two such data structures which can be problematic to analyse are unbalanced repeated measures data and longitudinal data. Owing to recent advances in methods and software, the mixed model analysis is now readily available to data analysts. The model is similar in many respects to ordinary multiple regression, but because it allows correlation between the observations, it requires additional work to specify models and to assess goodness-of-fit. The extra complexity involved is compensated for by the additional flexibility it provides in model fitting. The purpose of this tutorial is to provide readers with a sufficient introduction to the theory to understand the method and a more extensive discussion of model fitting and checking in order to provide guidelines for its use. We provide two detailed case studies, one a clinical trial with repeated measures and dropouts, and one an epidemiological survey with longitudinal follow-up. © 1997 John Wiley & Sons, Ltd.     

Performance of weighted estimating equations for longitudinal binary data with drop-outs missing at random

The generalized estimating equations (GEE) approach is commonly used to model incomplete longitudinal binary data. When drop-outs are missing at random through dependence on observed responses (MAR), GEE may give biased parameter estimates in the model for the marginal means. A weighted estimating equations approach gives consistent estimation under MAR when the drop-out mechanism is correctly specified. In this approach, observations or person-visits are weighted inversely proportional to their probability of being observed. Using a simulation study, we compare the performance of unweighted and weighted GEE in models for time-specific means of a repeated binary response with MAR drop-outs. Weighted GEE resulted in smaller finite sample bias than GEE. However, when the drop-out model was misspecified, weighted GEE sometimes performed worse than GEE. Weighted GEE with observation-level weights gave more efficient estimates than a weighted GEE procedure with cluster-level weights.     

A robust mixed linear model analysis for longitudinal data

This paper describes robust procedures for estimating parameters of a mixed effects linear model as applied to longitudinal data. In addition to fixed regression parameters, the model incorporates random subject effects to accommodate between-subjects variability and autocorrelation for within-subject variability. Robust empirical Bayesian estimation of subject effects is briefly discussed. As an illustration, the procedures are applied to data from a multiple sclerosis clinical trial.     

A local sensitivity analysis approach to longitudinal non-Gaussian data with non-ignorable dropout

Longitudinal non-Gaussian data subject to potentially non-ignorable dropout is a challenging problem. Frequently an analysis has to rely on some strong but unverifiable assumptions, among which ignorability is a key one. Sensitivity analysis has been advocated to assess the likely effect of alternative assumptions about dropout mechanism on such an analysis. Previously, Ma et al. applied a general index of local sensitivity to non-ignorability (ISNI) to measure the sensitivity of missing at random (MAR) estimates to small departures from ignorability for multivariate normal outcomes. In this paper, we extend the ISNI methodology to handle longitudinal non-Gaussian data subject to non-ignorable dropout. Specifically, we propose to quantify the sensitivity of inferences in the neighborhood of an MAR generalized linear mixed model for longitudinal data. Through a simulation study, we evaluate the performance of the proposed methodology. We then illustrate the methodology in one real example: smoking-cessation data.     

Analysis of non-ignorable missing and left-censored longitudinal data using a weighted random effects tobit model

In a longitudinal study with response data collected during a hospital stay, observations may be missing because of the subject's discharge from the hospital prior to completion of the study or the death of the subject, resulting in non-ignorable missing data. In addition to non-ignorable missingness, there is left-censoring in the response measurements because of the inherent limit of detection. For analyzing non-ignorable missing and left-censored longitudinal data, we have proposed to extend the theory of random effects tobit regression model to weighted random effects tobit regression model. The weights are computed on the basis of inverse probability weighted augmented methodology. An extensive simulation study was performed to compare the performance of the proposed model with a number of competitive models. The simulation study shows that the estimates are consistent and that the root mean square errors of the estimates are minimal for the use of augmented inverse probability weights in the random effects tobit model. The proposed method is also applied to the non-ignorable missing and left-censored interleukin-6 biomarker data obtained from the Genetic and Inflammatory Markers of Sepsis study.     

Modelling RNA-Seq data with a zero-inflated mixture Poisson linear model

RNA sequencing (RNA-Seq) has been frequently used in genomic studies and has generated a vast amount of data. The RNA-Seq data are composed of two parts: (a) a sequence of nucleotides of the genome; and (b) a corresponding sequence of counts, standing for the number of short reads whose mapped positions start at each position of the genome. One common feature of these count data is that they are typically nonuniform; recent studies have revealed that the nonuniformity is partially owing to a systematic bias resulted from the sequencing preference. Existing works in the literature model the nonuniformity with a single component Poisson linear model that incorporates the effects of the sequencing preference. However, we observe consistently that the short reads mapped to a gene may have a mixture structure and can be zero-inflated. A single component model may not suffice to model the complexity of such data. In this paper, we propose a zero-inflated mixture Poisson linear model for the RNA-Seq count data and derive a fast expectation–maximisation-based algorithm for estimating the unknown parameters. Numerical studies are conducted to illustrate the effectiveness of our method.     

A Dirichlet process mixture model for clustering longitudinal gene expression data

Subgroup identification (clustering) is an important problem in biomedical research. Gene expression profiles are commonly utilized to define subgroups. Longitudinal gene expression profiles might provide additional information on disease progression than what is captured by baseline profiles alone. Therefore, subgroup identification could be more accurate and effective with the aid of longitudinal gene expression data. However, existing statistical methods are unable to fully utilize these data for patient clustering. In this article, we introduce a novel clustering method in the Bayesian setting based on longitudinal gene expression profiles. This method, called BClustLonG, adopts a linear mixed‐effects framework to model the trajectory of genes over time, while clustering is jointly conducted based on the regression coefficients obtained from all genes. In order to account for the correlations among genes and alleviate the high dimensionality challenges, we adopt a factor analysis model for the regression coefficients. The Dirichlet process prior distribution is utilized for the means of the regression coefficients to induce clustering. Through extensive simulation studies, we show that BClustLonG has improved performance over other clustering methods. When applied to a dataset of severely injured (burn or trauma) patients, our model is able to identify interesting subgroups. Copyright © 2017 John Wiley & Sons, Ltd.     

A bidimensional finite mixture model for longitudinal data subject to dropout

In longitudinal studies, subjects may be lost to follow up and, thus, present incomplete response sequences. When the mechanism underlying the dropout is nonignorable, we need to account for dependence between the longitudinal and the dropout process. We propose to model such a dependence through discrete latent effects, which are outcome‐specific and account for heterogeneity in the univariate profiles. Dependence between profiles is introduced by using a probability matrix to describe the corresponding joint distribution. In this way, we separately model dependence within each outcome and dependence between outcomes. The major feature of this proposal, when compared with standard finite mixture models, is that it allows the nonignorable dropout model to properly nest its ignorable counterpart. We also discuss the use of an index of (local) sensitivity to nonignorability to investigate the effects that assumptions about the dropout process may have on model parameter estimates. The proposal is illustrated via the analysis of data from a longitudinal study on the dynamics of cognitive functioning in the elderly.     

An autoregressive linear mixed effects model for the analysis of longitudinal data which show profiles approaching asymptotes

In longitudinal data, a continuous response sometimes shows a profile approaching an asymptote. For such data, we propose a new class of models, autoregressive linear mixed effects models in which the current response is regressed on the previous response, fixed effects, and random effects. Asymptotes can shift depending on treatment groups, individuals, and so on, and can be modelled by fixed and random effects. We also propose error structures that are useful in practice. The estimation methods of linear mixed effects models can be used as long as there is no intermittent missing.     

A bivariate autoregressive linear mixed effects model for the analysis of longitudinal data

In clinical studies, dependent bivariate continuous responses may approach equilibrium over time. We propose an autoregressive linear mixed effects model for bivariate longitudinal data in which the current responses are regressed on the previous responses of both variables, fixed effects, and random effects. The equilibria are modeled using fixed and random effects. This model is a bivariate extension of the model for univariate longitudinal data given by Funatogawa et al. (Statist. Med. 2007; 26:2113-2130). As an illustration of the approach we analyze parathyroid hormone and serum calcium measurements in the treatment of secondary hyperparathyroidism in chronic hemodialysis patients.     

A mixture of transition models for heterogeneous longitudinal ordinal data: with applications to longitudinal bacterial vaginosis data

Markov models used to analyze transition patterns in discrete longitudinal data are based on the limiting assumption that individuals follow the common underlying transition process. However, when one is interested in diseases with different disease or severity subtypes, explicitly modeling subpopulation‐specific transition patterns may be appropriate. We propose a model which captures heterogeneity in the transition process through a finite mixture model formulation and provides a framework for identifying subpopulations at different risks. We apply the procedure to longitudinal bacterial vaginosis study data and demonstrate that the model fits the data well. Further, we show that under the mixture model formulation, we can make the important distinction between how covariates affect transition patterns unique to each of the subpopulations and how they affect which subgroup a participant will belong to. Practically, covariate effects on subpopulation‐specific transition behavior and those on subpopulation membership can be interpreted as effects on short‐term and long‐term transition behavior. We further investigate models with higher‐order subpopulation‐specific transition dependence. Copyright © 2014 John Wiley & Sons, Ltd.     

Fast linear mixed model computations for genome‐wide association studies with longitudinal data

Genome‐wide association studies are characterized by a huge number of statistical tests performed to discover new disease‐related genetic variants [in the form of single‐nucleotide polymorphisms (SNPs)] in human DNA. Many SNPs have been identified for cross‐sectionally measured phenotypes. However, there is a growing interest in genetic determinants of the evolution of traits over time. Dealing with correlated observations from the same individual, we need to apply advanced statistical techniques. The linear mixed model is popular but also much more computationally demanding than fitting a linear regression model to independent observations. We propose a conditional two‐step approach as an approximate method to explore the longitudinal relationship between the trait and the SNP. In a simulation study, we compare several fast methods with respect to their accuracy and speed. The conditional two‐step approach is applied to relate SNPs to longitudinal bone mineral density responses collected in the Rotterdam Study. Copyright © 2012 John Wiley & Sons, Ltd.