Robustness of a parametric model for informatively censored bivariate longitudinal data under misspecification of its distributional assumptions: A simulation study

Repeated measurements of surrogate markers are frequently used to track disease progression, but these series are often prematurely terminated due to disease progression or death. Analysing such data through standard likelihood-based approaches can yield severely biased estimates if the censoring mechanism is non-ignorable. Motivated by this problem, we have proposed the bivariate joint multivariate random effects (JMRE) model, which has shown that when correctly specified it performs well in terms of bias reduction and precision.The bivariate JMRE model is fully parametric and belongs to the class of shared parameters joint models where a survival model for the dropouts and a mixed model for the markers' evolution are linked through a multivariate normal distribution of random effects. As in every parametric model, robustness under violations of its distributional assumptions is of great importance. In this study we generated 500 simulated data sets assuming that random effects jointly follow a heavy-tailed distribution, two skewed distributions or a mixture of two normal distributions. Moreover, we generated data where level-1 errors or residuals in the survival part of the model follow a skewed distribution. Further sensitivity analysis on the effects of reduced sample size, increased level-1 variances and altered fixed effects values was also performed.We found that fixed effects estimates are almost unaffected, but their standard errors (SEs) may be underestimated especially under heavily skewed distributions. The proposed model seems robust enough, but its performance on smaller data sets or under more extreme departures of its assumptions needs further investigation.     

Estimation and prediction in linear mixed models with skew-normal random effects for longitudinal data

This paper extends the classical linear mixed model by considering a multivariate skew-normal assumption for the distribution of random effects. We present an efficient hybrid ECME-NR algorithm for the computation of maximum-likelihood estimates of parameters. A score test statistic for testing the existence of skewness preference among random effects is developed. The technique for the prediction of future responses under this model is also investigated. The methodology is illustrated through an application to Framingham cholesterol data and a simulation study.     

Analysis of incomplete multivariate data using linear models with structured covariance matrices

Incomplete and unbalanced multivariate data often arise in longitudinal studies due to missing or unequally-timed repeated measurements and/or the presence of time-varying covariates. A general approach to analysing such data is through maximum likelihood analysis using a linear model for the expected responses, and structural models for the within-subject covariances. Two important advantages of this approach are: (1) the generality of the model allows the analyst to consider a wider range of models than were previously possible using classical methods developed for balanced and complete data, and (2) maximum likelihood estimates obtained from incomplete data are often preferable to other estimates such as those obtained from complete cases from the standpoint of bias and efficiency. A variety of applications of the model are discussed, including univariate and multivariate analysis of incomplete repeated measures data, analysis of growth curves with missing data using random effects and time-series models, and applications to unbalanced longitudinal data.     

Sample size and power calculations for correlations between bivariate longitudinal data

The analysis of a baseline predictor with a longitudinally measured outcome is well established and sample size calculations are reasonably well understood. Analysis of bivariate longitudinally measured outcomes is gaining in popularity and methods to address design issues are required. The focus in a random effects model for bivariate longitudinal outcomes is on the correlations that arise between the random effects and between the bivariate residuals. In the bivariate random effects model, we estimate the asymptotic variances of the correlations and we propose power calculations for testing and estimating the correlations. We compare asymptotic variance estimates to variance estimates obtained from simulation studies and compare our proposed power calculations for correlations on bivariate longitudinal data to power calculations for correlations on cross‐sectional data. Copyright © 2010 John Wiley & Sons, Ltd.     

Semiparametric Bayesian inference on skew–normal joint modeling of multivariate longitudinal and survival data

We propose a semiparametric multivariate skew–normal joint model for multivariate longitudinal and multivariate survival data. One main feature of the posited model is that we relax the commonly used normality assumption for random effects and within‐subject error by using a centered Dirichlet process prior to specify the random effects distribution and using a multivariate skew–normal distribution to specify the within‐subject error distribution and model trajectory functions of longitudinal responses semiparametrically. A Bayesian approach is proposed to simultaneously obtain Bayesian estimates of unknown parameters, random effects and nonparametric functions by combining the Gibbs sampler and the Metropolis–Hastings algorithm. Particularly, a Bayesian local influence approach is developed to assess the effect of minor perturbations to within‐subject measurement error and random effects. Several simulation studies and an example are presented to illustrate the proposed methodologies. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayesian hierarchical joint modeling of repeatedly measured continuous and ordinal markers of disease severity: Application to Ugandan diabetes data

Modeling of correlated biomarkers jointly has been shown to improve the efficiency of parameter estimates, leading to better clinical decisions. In this paper, we employ a joint modeling approach to a unique diabetes dataset, where blood glucose (continuous) and urine glucose (ordinal) measures of disease severity for diabetes are known to be correlated. The postulated joint model assumes that the outcomes are from distributions that are in the exponential family and hence modeled as multivariate generalized linear mixed effects model associated through correlated and/or shared random effects. The Markov chain Monte Carlo Bayesian approach is used to approximate posterior distribution and draw inference on the parameters. This proposed methodology provides a flexible framework to account for the hierarchical structure of the highly unbalanced data as well as the association between the 2 outcomes. The results indicate improved efficiency of parameter estimates when blood glucose and urine glucose are modeled jointly. Moreover, the simulation studies show that estimates obtained from the joint model are consistently less biased and more efficient than those in the separate models.     

Improving uncertainty analysis in kinetic evaluations using iteratively reweighted least squares

Kinetic parameters of environmental fate processes are usually inferred by fitting appropriate kinetic models to the data using standard nonlinear least squares (NLS) approaches. Although NLS is appropriate to estimate the optimum parameter values, it implies restrictive assumptions on data variances when the confidence limits of the parameters must also be determined. Particularly in the case of degradation and metabolite formation, the assumption of equal error variance is often not realistic because the parent data usually show higher variances than those of the metabolites. Conventionally, such problems would be tackled by weighted NLS regression, which requires prior knowledge about the data errors. Instead of implicitly assuming equal error variances or giving arbitrary weights decided by the researcher, we use an iteratively reweighted least squares (IRLS) algorithm to obtain the maximum likelihood estimates of the model parameters and the error variances specific for the different species in a model. A study with simulated data shows that IRLS gives reliable results in the case of both unequal and equal error variances. We also compared results obtained by NLS and IRLS, with probability distributions of the parameters inferred with a Markov-Chain Monte-Carlo (MCMC) approach for data from aerobic transformation of different chemicals in soil. Confidence intervals obtained by IRLS and MCMC are consistent, whereas NLS leads to very different results when the error variances are distinctly different between different species. Because the MCMC results can be assumed to reflect the real parameter distribution imposed by the observed data, we conclude that IRLS generally yields more realistic estimates of confidence intervals for model parameters than NLS.     

Design effects for binary regression models fitted to dependent data

Dependent data, such as arise with cluster sampling, typically yield variances of parameter estimates which are larger than would be provided by a simple random sample of the same size. This variance inflation factor is called the design effect of the estimator. Design effects have been derived for cluster sampling designs using simple estimators such as means and proportions, and also for linear regression coefficient estimators. In this paper, we show that a method to derive design effects for linear regression estimators extends to generalized linear models for binary responses. In particular, some simple expressions for design effects in the linear regression model provide accurate approximations for binary regression models such as those based on the logistic, probit and complementary log—log links. We corroborate our findings with two examples and some simulation studies.     

Long-term survivor mixture model with random effects: application to a multi-centre clinical trial of carcinoma

A mixture model incorporating long-term survivors has been adopted in the field of biostatistics where some individuals may never experience the failure event under study. The surviving fractions may be considered as cured. In most applications, the survival times are assumed to be independent. However, when the survival data are obtained from a multi-centre clinical trial, it is conceived that the environmental conditions and facilities shared within clinic affects the proportion cured as well as the failure risk for the uncured individuals. It necessitates a long-term survivor mixture model with random effects. In this paper, the long-term survivor mixture model is extended for the analysis of multivariate failure time data using the generalized linear mixed model (GLMM) approach. The proposed model is applied to analyse a numerical data set from a multi-centre clinical trial of carcinoma as an illustration. Some simulation experiments are performed to assess the applicability of the model based on the average biases of the estimates formed.     

Predicting longitudinal trajectories of health probabilities with random‐effects multinomial logit regression

Researchers often encounter longitudinal health data characterized with three or more ordinal or nominal categories. Random‐effects multinomial logit models are generally applied to account for potential lack of independence inherent in such clustered data. When parameter estimates are used to describe longitudinal processes, however, random effects, both between and within individuals, need to be retransformed for correctly predicting outcome probabilities. This study attempts to go beyond existing work by developing a retransformation method that derives longitudinal growth trajectories of unbiased health probabilities. We estimated variances of the predicted probabilities by using the delta method. Additionally, we transformed the covariates’ regression coefficients on the multinomial logit function, not substantively meaningful, to the conditional effects on the predicted probabilities. The empirical illustration uses the longitudinal data from the Asset and Health Dynamics among the Oldest Old. Our analysis compared three sets of the predicted probabilities of three health states at six time points, obtained from, respectively, the retransformation method, the best linear unbiased prediction, and the fixed‐effects approach. The results demonstrate that neglect of retransforming random errors in the random‐effects multinomial logit model results in severely biased longitudinal trajectories of health probabilities as well as overestimated effects of covariates on the probabilities. Copyright © 2012 John Wiley & Sons, Ltd.     

Impact of missing data due to drop-outs on estimators for rates of change in longitudinal studies: a simulation study.

Many cohort studies and clinical trials are designed to compare rates of change over time in one or more disease markers in several groups. One major problem in such longitudinal studies is missing data due to patient drop-out. The bias and efficiency of six different methods to estimate rates of changes in longitudinal studies with incomplete observations were compared: generalized estimating equation estimates (GEE) proposed by Liang and Zeger (1986); unweighted average of ordinary least squares (OLSE) of individual rates of change (UWLS); weighted average of OLSE (WLS); conditional linear model estimates (CLE), a covariate type estimates proposed by Wu and Bailey (1989); random effect (RE), and joint multivariate RE (JMRE) estimates. The latter method combines a linear RE model for the underlying pattern of the marker with a log-normal survival model for informative drop-out process. The performance of these methods in the presence of missing data completely at random (MCAR), at random (MAR) and non-ignorable (NIM) were compared in simulation studies. Data for the disease marker were generated under the linear random effects model with parameter values derived from realistic examples in HIV infection. Rates of drop-out, assumed to increase over time, were allowed to be independent of marker values or to depend either only on previous marker values or on both previous and current marker values. Under MACR all six methods yielded unbiased estimates of both group mean rates and between-group difference. However, the cross-sectional view of the data in the GEE method resulted in seriously biased estimates under MAR and NIM drop-out process. The bias in the estimates ranged from 30 per cent to 50 per cent. The degree of bias in the GEE estimates increases with the severity of non-randomness and with the proportion of MAR data. Under MCAR and MAR all the other five methods performed relatively well. RE and JMRE estimates were more efficient(that is, had smaller variance) than UWLS, WLS and CL estimates. Under NIM, WLS and particularly RE estimates tended to underestimate the average rate of marker change (bias approximately 10 per cent). Under NIM, UWLS, CL and JMRE performed better in terms of bias (3-5 per cent) with the JMRE giving the most efficient estimates. Given that markers are key variables related to disease progression, missing marker data are likely to be at least MAR. Thus, the GEE method may not be appropriate for analysing such longitudinal marker data. The potential biases due to incomplete data require greater recognition in reports of longitudinal studies. Sensitivity analyses to assess the effect of drop-outs on inferences about the target parameters are important.     

Mixture cure model with random effects for the analysis of a multi-center tonsil cancer study

Cure models for clustered survival data have the potential for broad applicability. In this paper, we consider the mixture cure model with random effects and propose several estimation methods based on Gaussian quadrature, rejection sampling, and importance sampling to obtain the maximum likelihood estimates of the model for clustered survival data with a cure fraction. The methods are flexible to accommodate various correlation structures. A simulation study demonstrates that the maximum likelihood estimates of parameters in the model tend to have smaller biases and variances than the estimates obtained from the existing methods. We apply the model to a study of tonsil cancer patients clustered by treatment centers to investigate the effect of covariates on the cure rate and on the failure time distribution of the uncured patients. The maximum likelihood estimates of the parameters demonstrate strong correlation among the failure times of the uncured patients and weak correlation among cure statuses in the same center.     

Robust inference for multivariate survival data

Multivariate survival data arise when an individual records multiple survival events or when individuals recording single survival events are grouped into clusters. In this paper we propose a new method for the analysis of multivariate survival data. The technique is a synthesis of the Poisson regression formulation for univariate censored survival analysis and the generalized estimating equation approach for obtaining valid variance estimates for generalized linear models in the presence of clustering. When the survival data are clustered, combining the methods provides not only valid estimates for the variances of regression parameters but also estimates of the dependence between survival times. The approach entails specifying parametric models for the marginal hazards and a dependence structure, but does not require specification of the joint multivariate survival distribution. Properties of the methodology are investigated by simulation and through an illustrative example.     

Estimation of an errors-in-variables regression model when the variances of the measurement errors vary between the observations

It is common in the analysis of aggregate data in epidemiology that the variances of the aggregate observations are available. The analysis of such data leads to a measurement error situation, where the known variances of the measurement errors vary between the observations. Assuming multivariate normal distribution for the 'true' observations and normal distributions for the measurement errors, we derive a simple EM algorithm for obtaining maximum likelihood estimates of the parameters of the multivariate normal distributions. The results also facilitate the estimation of regression parameters between the variables as well as the 'true' values of the observations. The approach is applied to re-estimate recent results of the WHO MONICA Project on cardiovascular disease and its risk factors, where the original estimation of the regression coefficients did not adjust for the regression attenuation caused by the measurement errors.     

Meta‐analysis of absolute mean differences from randomised trials with treatment‐related clustering associated with care providers

Nesting of patients within care providers in trials of physical and talking therapies creates an additional level within the design. The statistical implications of this are analogous to those of cluster randomised trials, except that the clustering effect may interact with treatment and can be restricted to one or more of the arms. The statistical model that is recommended at the trial level includes a random effect for the care provider but allows the provider and patient level variances to differ across arms. Evidence suggests that, while potentially important, such within‐trial clustering effects have rarely been taken into account in trials and do not appear to have been considered in meta‐analyses of these trials. This paper describes summary measures and individual‐patient‐data methods for meta‐analysing absolute mean differences from randomised trials with two‐level nested clustering effects, contrasting fixed and random effects meta‐analysis models. It extends methods for incorporating trials with unequal variances and homogeneous clustering to allow for between‐arm and between‐trial heterogeneity in intra‐class correlation coefficient estimates. The work is motivated by a meta‐analysis of trials of counselling in primary care, where the control is no counselling and the outcome is the Beck Depression Inventory. Assuming equal counsellor intra‐class correlation coefficients across trials, the recommended random‐effects heteroscedastic model gave a pooled absolute mean difference of ?2.53 (95% CI ?5.33 to 0.27) using summary measures and ?2.51 (95% CI ?5.35 to 0.33) with the individual‐patient‐data. Pooled estimates were consistently below a minimally important clinical difference of four to five points on the Beck Depression Inventory. Copyright © 2014 John Wiley & Sons, Ltd.     

Modelling spatial disease rates using maximum likelihood

This paper concerns maximum likelihood estimation for a generalized linear mixed model (GLMM) useful for modelling spatial disease rates. The model allows for log-linear covariate adjustment and local smoothing of rates through estimation of spatially correlated random effects. The covariance structure of the random effects is based on a recently proposed model which parameterizes spatial dependence through the inverse covariance matrix. A Markov chain Monte Carlo algorithm for performing maximum likelihood estimation for this model is described. Results of a computer simulation study that compared maximum likelihood (ML) and penalized quasi-likelihood (PQL) estimators are presented. Compared with PQL, ML produced less biased estimates of the intercept but the ML estimates were slightly more variable. Estimates of the other regression coefficients were unbiased and nearly identical for the two methods. ML estimators of the random effects standard deviation and spatial correlation were more biased than the corresponding PQL estimators. The conclusion is that ML estimators for GLMMs cannot be expected to perform better than PQL for small samples.     

A joint modeling and estimation method for multivariate longitudinal data with mixed types of responses to analyze physical activity data generated by accelerometers

A mixed effect model is proposed to jointly analyze multivariate longitudinal data with continuous, proportion, count, and binary responses. The association of the variables is modeled through the correlation of random effects. We use a quasi‐likelihood type approximation for nonlinear variables and transform the proposed model into a multivariate linear mixed model framework for estimation and inference. Via an extension to the EM approach, an efficient algorithm is developed to fit the model. The method is applied to physical activity data, which uses a wearable accelerometer device to measure daily movement and energy expenditure information. Our approach is also evaluated by a simulation study.     

Robust non‐linear mixed modelling of longitudinal PSA levels after prostate cancer treatment

The objective of this study was to develop a robust non‐linear mixed model for prostate‐specific antigen (PSA) measurements after a high‐intensity focused ultrasound (HIFU) treatment for prostate cancer. The characteristics of these data are the presence of outlying values and non‐normal random effects. A numerical study proved that parameter estimates can be biased if these characteristics are not taken into account. The intra‐patient variability was described by a Student‐t distribution and Dirichlet process priors were assumed for non‐normal random effects; a process that limited the bias and provided more efficient parameter estimates than a classical mixed model with normal residuals and random effects. It was applied to the determination of the best dynamic PSA criterion for the diagnosis of prostate cancer recurrence, but could be used in studies that rely on PSA data to improve prognosis or compare treatment efficiencies and also with other longitudinal biomarkers that, such as PSA, present outlying values and non‐normal random effects. Copyright © 2010 John Wiley & Sons, Ltd.     

A multivariate random-effects model with restricted parameters: application to assessing radiation therapy for brain tumours

In clinical studies, multiple endpoints are often measured for each patient longitudinally. The multivariate random-effects or random coefficient model has been a useful method for analysis. However, medical research problems may impose restrictions on the model parameters of interests. For example, in a paediatric brain tumour study on radiation therapy, there is a natural ordering in the white matter relaxation time of brain tissues among different regions surrounding the primary tumour, i.e. the closer a specific region of brain tissues is to the centre of primary tumour, the shorter is the relaxation time. Such parameter constraints should be accounted for in the analysis. This article proposes a class of multivariate random coefficient models with restricted parameters and derives its maximum likelihood estimates (MLE). We propose a modified EM algorithm for the quadratic optimalization with linear inequality constraints necessary in deriving the MLE. The method is applied to analysing the paediatric brain tumour study.     

Multivariate t nonlinear mixed‐effects models for multi‐outcome longitudinal data with missing values

The multivariate nonlinear mixed‐effects model (MNLMM) has emerged as an effective tool for modeling multi‐outcome longitudinal data following nonlinear growth patterns. In the framework of MNLMM, the random effects and within‐subject errors are assumed to be normally distributed for mathematical tractability and computational simplicity. However, a serious departure from normality may cause lack of robustness and subsequently make invalid inference. This paper presents a robust extension of the MNLMM by considering a joint multivariate t distribution for the random effects and within‐subject errors, called the multivariate t nonlinear mixed‐effects model. Moreover, a damped exponential correlation structure is employed to capture the extra serial correlation among irregularly observed multiple repeated measures. An efficient expectation conditional maximization algorithm coupled with the first‐order Taylor approximation is developed for maximizing the complete pseudo‐data likelihood function. The techniques for the estimation of random effects, imputation of missing responses and identification of potential outliers are also investigated. The methodology is motivated by a real data example on 161 pregnant women coming from a study in a private fertilization obstetrics clinic in Santiago, Chile and used to analyze these data. Copyright © 2014 John Wiley & Sons, Ltd.