Estimation of covariate effects in generalized linear mixed models with a misspecified distribution of random intercepts and slopes

Generalized linear mixed models with random intercepts and slopes provide useful analyses of clustered and longitudinal data and typically require the specification of the distribution of the random effects. Previous work for models with only random intercepts has shown that misspecifying the shape of this distribution may bias estimates of the intercept, but typically leads to little bias in estimates of covariate effects. Very few papers have examined the effects of misspecifying the joint distribution of random intercepts and slopes. However, simulation results in a recent paper suggest that misspecifying the shape of the random slope distribution can yield severely biased estimates of all model parameters. Using analytic results, simulation studies and fits to example data, this paper examines the bias in parameter estimates due to misspecification of the shape of the joint distribution of random intercepts and slopes. Consistent with results for models with only random intercepts, and contrary to the claims of severe bias in a recent paper, we show that misspecification of the joint distribution typically yields little bias in estimates of covariate effects and is restricted to covariates associated with the misspecified random effects distributions. We also show that misspecification of the distribution of random effects has little effect on confidence interval performance. Coverage rates based on the model‐based standard errors from fitted likelihoods were generally quite close to nominal. Copyright © 2012 John Wiley & Sons, Ltd.     

Misspecifying the covariance structure in a linear mixed model under MAR drop-out

Misspecification of the covariance structure in a linear mixed model (LMM) can lead to biased population parameters' estimates under MAR drop-out. In our motivating example of modeling CD4 cell counts during untreated HIV infection, random intercept and slope LMMs are frequently used. In this article, we evaluate the performance of LMMs with specific covariance structures, in terms of bias in the fixed effects estimates, under specific MAR drop-out mechanisms, and adopt a Bayesian model comparison criterion to discriminate between the examined approaches in real-data applications. We analytically show that using a random intercept and slope structure when the true one is more complex can lead to seriously biased estimates, with the degree of bias depending on the magnitude of the MAR drop-out. Under misspecified covariance structure, we compare in terms of induced bias the approach of adding a fractional Brownian motion (BM) process on top of random intercepts and slopes with the approach of using splines for the random effects. In general, the performance of both approaches was satisfactory, with the BM model leading to smaller bias in most cases. A simulation study is carried out to evaluate the performance of the proposed Bayesian criterion in identifying the model with the correct covariance structure. Overall, the proposed method performs better than the AIC and BIC criteria under our specific simulation setting. The models under consideration are applied to real data from the CASCADE study; the most plausible model is identified by all examined criteria.     

Variance reduction in randomised trials by inverse probability weighting using the propensity score

In individually randomised controlled trials, adjustment for baseline characteristics is often undertaken to increase precision of the treatment effect estimate. This is usually performed using covariate adjustment in outcome regression models. An alternative method of adjustment is to use inverse probability‐of‐treatment weighting (IPTW), on the basis of estimated propensity scores. We calculate the large‐sample marginal variance of IPTW estimators of the mean difference for continuous outcomes, and risk difference, risk ratio or odds ratio for binary outcomes. We show that IPTW adjustment always increases the precision of the treatment effect estimate. For continuous outcomes, we demonstrate that the IPTW estimator has the same large‐sample marginal variance as the standard analysis of covariance estimator. However, ignoring the estimation of the propensity score in the calculation of the variance leads to the erroneous conclusion that the IPTW treatment effect estimator has the same variance as an unadjusted estimator; thus, it is important to use a variance estimator that correctly takes into account the estimation of the propensity score. The IPTW approach has particular advantages when estimating risk differences or risk ratios. In this case, non‐convergence of covariate‐adjusted outcome regression models frequently occurs. Such problems can be circumvented by using the IPTW adjustment approach. © 2013 The authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Variable selection for joint models of multivariate longitudinal measurements and event time data

Joint modeling of longitudinal and survival data has attracted a great deal of attention. Some research has been undertaken to extend the joint model to incorporate multivariate longitudinal measurements recently. However, there is a lack of variable selection methods in the joint modeling of multivariate longitudinal measurements and survival time. In this article, we develop penalized likelihood methods for the selection of longitudinal features in the survival submodel. A multivariate linear mixed effect model is used to model multiple longitudinal processes where random intercepts and slopes serve as essential features of the trajectories. We introduce L1 penalty functions to select both random effects in the survival submodel and off‐diagonal elements in the covariance matrix of random effects. An estimation procedure is developed based on Laplace approximation. Our simulations demonstrate excellent selection properties of the proposed procedure. We apply our methods to explore the relationship between mortality and multiple longitudinal processes for end stage renal disease patients on hemodialysis. We find that lower levels of albumin, higher levels of neutrophil‐to‐lymphocyte ratio, and higher levels of interdialytic weight gain at the beginning of the follow‐up time, as well as decrease in predialysis systolic blood pressure and increase of neutrophil‐to‐lymphocyte ratio over time are associated with higher mortality hazard rates.     

STATISTICAL ANALYSIS OF ZIDOVUDINE (AZT) EFFECT ON CD4 CELL COUNTS IN HIV DISEASE

We fit a class of random effects linear growth curve models for the square root of CD4 count to serial marker data from 164 HIV-positive individuals with known (or accurately estimated) dates of seroconversion and at least 10 CD4 measurements each (median 16). We do so by adopting a Bayesian viewpoint and using the Markov chain Monte Carlo technique Gibbs sampling. In particular, we examine the effect of the antiretroviral treatment zidovudine on the √CD4 series for the 136 patients who took the drug. Treatment effects are modelled by positing recoveries in √CD4 level proportional to current immuno-competence and changes in slope proportional to current rate of √CD4 loss. Both fixed and random treatment effects are considered and models are criticized and compared using Bayesian predictive methodology and checking data which comprise 424 new observations. Results indicate re-elevation of √CD4 level is associated with treatment but the effect, though significant, is mostly of small magnitude and is possibly transient; models neglecting consideration of treatment fit the checking data almost as well. Best overall model estimates mean rate of √CD4 loss per annum to be 2⋅1 (standard error 0⋅12); mean seroconversion value of √CD4 is 28⋅4 (SE 0⋅65). The estimated variance of individual slopes is 1⋅9 (SE 0⋅28), there being considerable individual variation in rate of CD4 loss, and a recovery in level of 0⋅047 (SE 0⋅014) times current √CD4 level is estimated at treatment uptake.     

The bivariate combined model for spatial data analysis

To describe the spatial distribution of diseases, a number of methods have been proposed to model relative risks within areas. Most models use Bayesian hierarchical methods, in which one models both spatially structured and unstructured extra‐Poisson variance present in the data. For modelling a single disease, the conditional autoregressive (CAR) convolution model has been very popular. More recently, a combined model was proposed that ‘combines’ ideas from the CAR convolution model and the well‐known Poisson‐gamma model. The combined model was shown to be a good alternative to the CAR convolution model when there was a large amount of uncorrelated extra‐variance in the data. Less solutions exist for modelling two diseases simultaneously or modelling a disease in two sub‐populations simultaneously. Furthermore, existing models are typically based on the CAR convolution model. In this paper, a bivariate version of the combined model is proposed in which the unstructured heterogeneity term is split up into terms that are shared and terms that are specific to the disease or subpopulation, while spatial dependency is introduced via a univariate or multivariate Markov random field. The proposed method is illustrated by analysis of disease data in Georgia (USA) and Limburg (Belgium) and in a simulation study. We conclude that the bivariate combined model constitutes an interesting model when two diseases are possibly correlated. As the choice of the preferred model differs between data sets, we suggest to use the new and existing modelling approaches together and to choose the best model via goodness‐of‐fit statistics. Copyright © 2016 John Wiley & Sons, Ltd.     

Separable covariance models for health care quality measures across years and topics

Public quality reports for Medicare Advantage health plans include 11 measures of patient experiences reported in the annual Consumer Assessment of Healthcare Providers and Systems surveys. Computing summaries at the health plan level (of multiple measures in multiple years) yields an array‐structured random variable. To summarize associations among measures and years, we model the variance‐covariance matrix governing the plan‐level vectors of yearly quality measures as a Kronecker product of an across‐measure matrix and an across‐year matrix, or a sum of such Kronecker products. This approach extends separable covariance structure to Fay‐Herriot models. In addition, we develop linear combinations of Kronecker products similar to principal components for array random variables. To each Kronecker‐product term, we apply post hoc analyses suited to the corresponding dimension of the cross‐classification: 1‐way factor analysis for the across‐measure factor and time‐series analysis to the across‐year factor. These methods draw out key patterns of variation in the quality measures over time and suggest new strategies for reporting quality information to consumers.     

Generalized survival models for correlated time‐to‐event data

Our aim is to develop a rich and coherent framework for modeling correlated time‐to‐event data, including (1) survival regression models with different links and (2) flexible modeling for time‐dependent and nonlinear effects with rich postestimation. We extend the class of generalized survival models, which expresses a transformed survival in terms of a linear predictor, by incorporating a shared frailty or random effects for correlated survival data. The proposed approach can include parametric or penalized smooth functions for time, time‐dependent effects, nonlinear effects, and their interactions. The maximum (penalized) marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects. The optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood‐based cross‐validation criterion. For models with normal random effects, Gauss‐Hermite quadrature can be used to obtain the cluster‐level marginal likelihoods. The Akaike Information Criterion can be used to compare models and select the link function. We have implemented these methods in the R package rstpm2. Simulating for both small and larger clusters, we find that this approach performs well. Through 2 applications, we demonstrate (1) a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and (2) the estimation of time‐varying effects on the log‐time scale, age‐varying effects for a specific treatment, and two‐dimensional splines for time and age.     

A model selection approach to analysis of variance and covariance

An alternative to analysis of variance is a model selection approach where every partition of the treatment means into clusters with equal value is treated as a separate model. The null hypothesis that all treatments are equal corresponds to the partition with all means in a single cluster. The alternative hypothesis correspond to the set of all other partitions of treatment means. A model selection approach can also be used for a treatment by covariate interaction, where the null hypothesis and each alternative correspond to a partition of treatments into clusters with equal covariate effects. We extend the partition‐as‐model approach to simultaneous inference for both treatment main effect and treatment interaction with a continuous covariate with separate partitions for the intercepts and treatment‐specific slopes. The model space is the Cartesian product of the intercept partition and the slope partition, and we develop five joint priors for this model space. In four of these priors the intercept and slope partition are dependent. We advise on setting priors over models, and we use the model to analyze an orthodontic data set that compares the frictional resistance created by orthodontic fixtures. Copyright © 2009 John Wiley & Sons, Ltd.     

Modelling the random effects covariance matrix in longitudinal data

A common class of models for longitudinal data are random effects (mixed) models. In these models, the random effects covariance matrix is typically assumed constant across subject. However, in many situations this matrix may differ by measured covariates. In this paper, we propose an approach to model the random effects covariance matrix by using a special Cholesky decomposition of the matrix. In particular, we will allow the parameters that result from this decomposition to depend on subject-specific covariates and also explore ways to parsimoniously model these parameters. An advantage of this parameterization is that there is no concern about the positive definiteness of the resulting estimator of the covariance matrix. In addition, the parameters resulting from this decomposition have a sensible interpretation. We propose fully Bayesian modelling for which a simple Gibbs sampler can be implemented to sample from the posterior distribution of the parameters. We illustrate these models on data from depression studies and examine the impact of heterogeneity in the covariance matrix on estimation of both fixed and random effects.     

Exploratory assessment of treatment‐dependent random‐effects distribution using gradient functions

In analyzing repeated measurements from randomized controlled trials with mixed‐effects models, it is important to carefully examine the conventional normality assumption regarding the random‐effects distribution and its dependence on treatment allocation in order to avoid biased estimation and correctly interpret the estimated random‐effects distribution. In this article, we propose the use of a gradient function method in modeling with the different random‐effects distributions depending on the treatment allocation. This method can be effective for considering in advance whether a proper fit requires a model that allows dependence of the random‐effects distribution on covariates, or for finding the subpopulations in the random effects.     

Mapping gender variation in the spatial pattern of alcohol-related mortality: a Bayesian analysis using data from South Yorkshire, United Kingdom

Gender variation in the spatial pattern of alcohol-related deaths in South Yorkshire, UK for the period 1999 and 2003 was explored using two Bayesian modelling approaches. Firstly, separate models were fitted to male and female deaths, each with a fixed effect deprivation covariate and a random effect with unstructured and spatially structured terms. In a modification to the initial models, covariates were assumed estimated with error rather than known with certainty. In the second modelling approach male and female deaths were modelled jointly with a shared component for random effects. A range of different unstructured and spatially structured specifications for the shared and gender-specific random effects were fitted. In the best fitting shared component model a spatially structured prior was assumed for the shared component, while gender-specific components were assumed unstructured. Deprivation coefficients and random effect standard deviations were very similar between the gender-specific and shared component models. In each case the effect of deprivation was observed to be greater in males than in females, and slightly larger in the measurement error models than in the fixed covariate models. Greater variation was observed in the spatially smoothed estimates of risk for males versus females in both gender-specific and shared component models. The shared component explained a greater proportion of the male risk than it did the female risk. The analysis approach reveals the residual (unexplained by deprivation) gender-specific and shared risk surfaces, information which may be useful for guiding public health action.     

Fractional Brownian motion and multivariate‐t models for longitudinal biomedical data,with application to CD4 counts in HIV‐positive patients

Longitudinal data are widely analysed using linear mixed models, with ‘random slopes’ models particularly common. However, when modelling, for example, longitudinal pre‐treatment CD4 cell counts in HIV‐positive patients, the incorporation of non‐stationary stochastic processes such as Brownian motion has been shown to lead to a more biologically plausible model and a substantial improvement in model fit. In this article, we propose two further extensions. Firstly, we propose the addition of a fractional Brownian motion component, and secondly, we generalise the model to follow a multivariate‐t distribution. These extensions are biologically plausible, and each demonstrated substantially improved fit on application to example data from the Concerted Action on SeroConversion to AIDS and Death in Europe study. We also propose novel procedures for residual diagnostic plots that allow such models to be assessed. Cohorts of patients were simulated from the previously reported and newly developed models in order to evaluate differences in predictions made for the timing of treatment initiation under different clinical management strategies. A further simulation study was performed to demonstrate the substantial biases in parameter estimates of the mean slope of CD4 decline with time that can occur when random slopes models are applied in the presence of censoring because of treatment initiation, with the degree of bias found to depend strongly on the treatment initiation rule applied. Our findings indicate that researchers should consider more complex and flexible models for the analysis of longitudinal biomarker data, particularly when there are substantial missing data, and that the parameter estimates from random slopes models must be interpreted with caution. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

A comparison of arm-based and contrast-based models for network meta-analysis

Differences between arm-based (AB) and contrast-based (CB) models for network meta-analysis (NMA) are controversial. We compare the CB model of Lu and Ades (2006), the AB model of Hong et al(2016), and two intermediate models, using hypothetical data and a selected real data set. Differences between models arise primarily from study intercepts being fixed effects in the Lu-Ades model but random effects in the Hong model, and we identify four key difference. (1) If study intercepts are fixed effects then only within-study information is used, but if they are random effects then between-study information is also used and can cause important bias. (2) Models with random study intercepts are suitable for deriving a wider range of estimands, eg, the marginal risk difference, when underlying risk is derived from the NMA data; but underlying risk is usually best derived from external data, and then models with fixed intercepts are equally good. (3) The Hong model allows treatment effects to be related to study intercepts, but the Lu-Ades model does not. (4) The Hong model is valid under a more relaxed missing data assumption, that arms (rather than contrasts) are missing at random, but this does not appear to reduce bias. We also describe an AB model with fixed study intercepts and a CB model with random study intercepts. We conclude that both AB and CB models are suitable for the analysis of NMA data, but using random study intercepts requires a strong rationale such as relating treatment effects to study intercepts.     

Specification of covariance structure in longitudinal data analysis for randomized clinical trials

Misspecification of the covariance structure for repeated measurements in longitudinal analysis may lead to biased estimates of the regression parameters and under or overestimation of the corresponding standard errors in the presence of missing data. The so‐called sandwich estimator can ‘correct’ the variance, but it does not reduce the bias in point estimates. Removing all assumptions from the covariance structure (i.e. using an unstructured (UN) covariance) will remove such biases. However, an excessive amount of missing data may cause convergence problems for iterative algorithms, such as the default Newton–Raphson algorithm in the popular SAS PROC MIXED. This article examines, both through theory and simulations, the existence and the magnitude of these biases. We recommend the use of UN covariance as the default strategy for analyzing longitudinal data from randomized clinical trials with moderate to large number of subjects and small to moderate number of time points. We also present an algorithm to assist in the convergence when the UN covariance is used. Copyright © 2009 John Wiley & Sons, Ltd.     

Effects of covariance model assumptions on hypothesis tests for repeated measurements: analysis of ovarian hormone data and pituitary-pteryomaxillary distance data

In the analysis of repeated measurements, multivariate methods which account for the correlations among the observations from the same experimental unit are widely used. Two commonly-used multivariate methods are the unstructured multivariate approach and the mixed model approach. The unstructured multivariate approach uses MANOVA types of models and does not require assumptions on the covariance structure. The mixed model approach uses multivariate linear models with random effects and requires covariance structure assumptions. In this paper, we describe the characteristics of tests based on these two methods of analysis and investigate the performance of these tests. We focus particularly on tests for group effects and parallelism of response profiles.     

Two‐step estimation in ratio‐of‐mediator‐probability weighted causal mediation analysis

This study investigates appropriate estimation of estimator variability in the context of causal mediation analysis that employs propensity score‐based weighting. Such an analysis decomposes the total effect of a treatment on the outcome into an indirect effect transmitted through a focal mediator and a direct effect bypassing the mediator. Ratio‐of‐mediator‐probability weighting estimates these causal effects by adjusting for the confounding impact of a large number of pretreatment covariates through propensity score‐based weighting. In step 1, a propensity score model is estimated. In step 2, the causal effects of interest are estimated using weights derived from the prior step's regression coefficient estimates. Statistical inferences obtained from this 2‐step estimation procedure are potentially problematic if the estimated standard errors of the causal effect estimates do not reflect the sampling uncertainty in the estimation of the weights. This study extends to ratio‐of‐mediator‐probability weighting analysis a solution to the 2‐step estimation problem by stacking the score functions from both steps. We derive the asymptotic variance‐covariance matrix for the indirect effect and direct effect 2‐step estimators, provide simulation results, and illustrate with an application study. Our simulation results indicate that the sampling uncertainty in the estimated weights should not be ignored. The standard error estimation using the stacking procedure offers a viable alternative to bootstrap standard error estimation. We discuss broad implications of this approach for causal analysis involving propensity score‐based weighting.     

A Bayesian semiparametric model for bivariate sparse longitudinal data

Mixed‐effects models have recently become popular for analyzing sparse longitudinal data that arise naturally in biological, agricultural and biomedical studies. Traditional approaches assume independent residuals over time and explain the longitudinal dependence by random effects. However, when bivariate or multivariate traits are measured longitudinally, this fundamental assumption is likely to be violated because of intertrait dependence over time. We provide a more general framework where the dependence of the observations from the same subject over time is not assumed to be explained completely by the random effects of the model. We propose a novel, mixed model‐based approach and estimate the error–covariance structure nonparametrically under a generalized linear model framework. We use penalized splines to model the general effect of time, and we consider a Dirichlet process mixture of normal prior for the random‐effects distribution. We analyze blood pressure data from the Framingham Heart Study where body mass index, gender and time are treated as covariates. We compare our method with traditional methods including parametric modeling of the random effects and independent residual errors over time. We conduct extensive simulation studies to investigate the practical usefulness of the proposed method. The current approach is very helpful in analyzing bivariate irregular longitudinal traits. Copyright © 2013 John Wiley & Sons, Ltd.     

Reply to discussion of ‘Empirical vs natural weighting in random effects meta‐analysis’

Three discussion papers raise interesting points, but at the end of the day, we continue to recommend that past meta‐analyses which have influenced public policy or clinical paradigms be reanalyzed by unweighted methods. For all future meta‐analyses that employ either (a) fixed effects where concomitant treatment and/or eligibility are diverse or (b) classical random effects weighted methods, unweighted methods should serve as the primary analysis. Other analyses would be reasonable as secondary approaches. Two commentaries suggest that weights are only random to the extent of estimation errors in the between study and within study variance components. We shall demonstrate that even if these components are known, there is still considerable random variability in the weights. In fact, methods that try to weight the estimates inversely proportional to the variance have a number of undesirable properties, including bias, incorrect standard errors, inconsistency (including coverage of confidence intervals), and counter‐intuitive properties that the expectation of the estimator changes both with the number of studies sampled and with constant multiples of sample size across all studies. These adverse properties do not exist for the unweighted approach. From the numerical example of phenylephrine 10 mg, despite the arguments of Waksman, the proper conclusion is that the collection of studies does not constitute evidence‐based support for efficacy in terms of lowering nasal airway resistance. In the final section, we present two compelling examples where questionable inferences were made, with potential major public implications. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimating overall exposure effects for the clustered and censored outcome using random effect Tobit regression models

The random effect Tobit model is a regression model that accommodates both left‐ and/or right‐censoring and within‐cluster dependence of the outcome variable. Regression coefficients of random effect Tobit models have conditional interpretations on a constructed latent dependent variable and do not provide inference of overall exposure effects on the original outcome scale. Marginalized random effects model (MREM) permits likelihood‐based estimation of marginal mean parameters for the clustered data. For random effect Tobit models, we extend the MREM to marginalize over both the random effects and the normal space and boundary components of the censored response to estimate overall exposure effects at population level. We also extend the ‘Average Predicted Value’ method to estimate the model‐predicted marginal means for each person under different exposure status in a designated reference group by integrating over the random effects and then use the calculated difference to assess the overall exposure effect. The maximum likelihood estimation is proposed utilizing a quasi‐Newton optimization algorithm with Gauss–Hermite quadrature to approximate the integration of the random effects. We use these methods to carefully analyze two real datasets. Copyright © 2016 John Wiley & Sons, Ltd.