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.     

Individualizing drug dosage with longitudinal data

We propose a two‐step procedure to personalize drug dosage over time under the framework of a log‐linear mixed‐effect model. We model patients' heterogeneity using subject‐specific random effects, which are treated as the realizations of an unspecified stochastic process. We extend the conditional quadratic inference function to estimate both fixed‐effect coefficients and individual random effects on a longitudinal training data sample in the first step and propose an adaptive procedure to estimate new patients' random effects and provide dosage recommendations for new patients in the second step. An advantage of our approach is that we do not impose any distribution assumption on estimating random effects. Moreover, the new approach can accommodate more general time‐varying covariates corresponding to random effects. We show in theory and numerical studies that the proposed method is more efficient compared with existing approaches, especially when covariates are time varying. In addition, a real data example of a clozapine study confirms that our two‐step procedure leads to more accurate drug dosage recommendations. Copyright © 2016 John Wiley & Sons, Ltd.     

Penalized variable selection for accelerated failure time models with random effects

Accelerated failure time (AFT) models allowing for random effects are linear mixed models under the log-transformation of survival time with censoring and describe dependence in correlated survival data. It is well known that the AFT models are useful alternatives to frailty models. To the best of our knowledge, however, there is no literature on variable selection methods for such AFT models. In this paper, we propose a simple but unified variable-selection procedure of fixed effects in the AFT random-effect models using penalized h-likelihood (HL). We consider four penalty functions (ie, least absolute shrinkage and selection operator (LASSO), adaptive LASSO, smoothly clipped absolute deviation (SCAD), and HL). We show that the proposed method can be easily implemented via a slight modification to existing h-likelihood estimation procedures. We thus demonstrate that the proposed method can also be easily extended to AFT models with multilevel (or nested) structures. Simulation studies also show that the procedure using the adaptive LASSO, SCAD, or HL penalty performs well. In particular, we find via the simulation results that the variable selection method with HL penalty provides a higher probability of choosing the true model than other three methods. The usefulness of the new method is illustrated using two actual datasets from multicenter clinical trials.     

Mixed models,linear dependency,and identification in age‐period‐cohort models

This paper examines the identification problem in age‐period‐cohort models that use either linear or categorically coded ages, periods, and cohorts or combinations of these parameterizations. These models are not identified using the traditional fixed effect regression model approach because of a linear dependency between the ages, periods, and cohorts. However, these models can be identified if the researcher introduces a single just identifying constraint on the model coefficients. The problem with such constraints is that the results can differ substantially depending on the constraint chosen. Somewhat surprisingly, age‐period‐cohort models that specify one or more of ages and/or periods and/or cohorts as random effects are identified. This is the case without introducing an additional constraint. I label this identification as statistical model identification and show how statistical model identification comes about in mixed models and why which effects are treated as fixed and which are treated as random can substantially change the estimates of the age, period, and cohort effects. Copyright © 2017 John Wiley & Sons, Ltd.     

Comparing linear and nonlinear mixed model approaches to cosinor analysis

The cosinor model, used for variables governed by circadian and other biological rhythms, is a nonlinear model in the amplitude and acrophase parameters that has a linear representation upon transformation. With linear cosinor analysis, amplitude and acrophase for each harmonic can be computed as nonlinear functions of the estimated linear regression coefficients. Here a flexible mixed model approach to cosinor analysis is considered, where the fixed effect parameters may enter nonlinearly as acrophase and amplitude for each harmonic or linearly after transformation to regression coefficients. In addition, the random effects may enter nonlinearly as subject-specific deviations from the acrophases and amplitudes or linearly as subject-specific deviations from the regression coefficients. It is also possible for the fixed effects to enter nonlinearly while the random effects enter linearly. Additionally, we evaluate whether including higher order linear harmonic terms as random effects, that is, Rao-Khatri 'covariance adjustment', improves precision. Applying the delta method to nonlinear functions of the parameters from linear mixed cosinor models to obtain approximate variances produces results that are often identical to results from nonlinear mixed models. Consequently, traditional linear cosinor analysis can often be used to estimate and compare the nonlinear parameters of interest, that is, amplitudes and acrophases, via the delta method. This is advantageous since the nonlinear mixed model may have convergence difficulties for more complex models. However, for some multiple-group analyses, the linear cosinor transformation should not be used and we clarify when the two methods are equivalent and when they differ.     

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.     

Proportional hazards model with random effects

We propose a general proportional hazards model with random effects for handling clustered survival data. This generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk, in a way similar to the modelling of random effects in linear, generalized linear and non-linear mixed models. The distribution of the random effects is generally assumed to be multivariate normal, but other (preferably symmetrical) distributions are also possible. Maximum likelihood estimates of the regression parameters, the variance components and the baseline hazard function are obtained via the EM algorithm. The E-step of the algorithm involves computation of the conditional expectations of functions of the random effects, for which we use Markov chain Monte Carlo (MCMC) methods. Approximate variances of the estimates are computed by Louis' formula, and posterior expectations and variances of the individual random effects can be obtained as a by-product of the estimation. The inference procedure is exemplified on two data sets.     

重复测量资料非线性混合效应模型应用与实现

目的探讨重复测量资料非线性分析技术、SAS软件NLMIXED过程及在群体药动学的应用.方法结合重复测量数据特点,采用最大似然原理进行参数估计,建立非线性混合效应参数模型.结果该模型不仅考虑了个体内和个体间变异,而且也考虑了参数间的非线性,允许固定效应和随机效应进入模型的非线性部分;可方便地分析随机缺失等非均衡数据;有助于引入其他解释变量时最佳模型的选择,更客观地解释其对代谢过程的影响.结论当重复测量资料不满足线性条件时,使用非线性混合效应模型能更客观地反映原数据特征,挖掘资料蕴藏的信息,弥补线性理论分析非线性重复测量资料之不足.     

A Bayesian order-restricted model for hormonal dynamics during menstrual cycles of healthy women

We propose a Bayesian framework for analyzing multivariate linear mixed effect models with linear constraints on the fixed effect parameters. The procedure can incorporate both firm and soft restrictions on the parameters and Bayesian model selection for the random effects. The framework is used to analyze data from the BioCycle study. One of the main objectives of the BioCycle study is to investigate the association between markers of oxidative stress and hormone levels during menstrual cycles of healthy women. Contrary to the popular belief that ovarian hormones are negatively associated with level of F (2) -isoprostanes, a known marker for oxidative stress, our analysis finds a positive association between ovarian hormone levels and isoprostane levels. The positive association corroborates the findings from a previous analysis of the BioCycle data. Copyright ? 2011 John Wiley & Sons, Ltd.     

Real longitudinal data analysis for real people: Building a good enough mixed model

Mixed effects models have become very popular, especially for the analysis of longitudinal data. One challenge is how to build a good enough mixed effects model. In this paper, we suggest a systematic strategy for addressing this challenge and introduce easily implemented practical advice to build mixed effects models. A general discussion of the scientific strategies motivates the recommended five‐step procedure for model fitting. The need to model both the mean structure (the fixed effects) and the covariance structure (the random effects and residual error) creates the fundamental flexibility and complexity. Some very practical recommendations help to conquer the complexity. Centering, scaling, and full‐rank coding of all the predictor variables radically improve the chances of convergence, computing speed, and numerical accuracy. Applying computational and assumption diagnostics from univariate linear models to mixed model data greatly helps to detect and solve the related computational problems. Applying computational and assumption diagnostics from the univariate linear models to the mixed model data can radically improve the chances of convergence, computing speed, and numerical accuracy. The approach helps to fit more general covariance models, a crucial step in selecting a credible covariance model needed for defensible inference. A detailed demonstration of the recommended strategy is based on data from a published study of a randomized trial of a multicomponent intervention to prevent young adolescents' alcohol use. The discussion highlights a need for additional covariance and inference tools for mixed models. The discussion also highlights the need for improving how scientists and statisticians teach and review the process of finding a good enough mixed model. Copyright © 2009 John Wiley & Sons, Ltd.     

A semi-parametric Bayesian approach to generalized linear mixed models

The linear mixed effects model with normal errors is a popular model for the analysis of repeated measures and longitudinal data. The generalized linear model is useful for data that have non-normal errors but where the errors are uncorrelated. A descendant of these two models generates a model for correlated data with non-normal errors, called the generalized linear mixed model (GLMM). Frequentist attempts to fit these models generally rely on approximate results and inference relies on asymptotic assumptions. Recent advances in computing technology have made Bayesian approaches to this class of models computationally feasible. Markov chain Monte Carlo methods can be used to obtain ‘exact’ inference for these models, as demonstrated by Zeger and Karim. In the linear or generalized linear mixed model, the random effects are typically taken to have a fully parametric distribution, such as the normal distribution. In this paper, we extend the GLMM by allowing the random effects to have a non-parametric prior distribution. We do this using a Dirichlet process prior for the general distribution of the random effects. The approach easily extends to more general population models. We perform computations for the models using the Gibbs sampler. © 1998 John Wiley & Sons, Ltd.     

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 binning method for analyzing mixed longitudinal data measured at distinct time points

For longitudinal data where the response and time‐dependent predictors within each individual are measured at distinct time points, traditional longitudinal models such as generalized linear mixed effects models or marginal models cannot be directly applied. Instead, some preprocessing such as smoothing is required to temporally align the response and predictors. We propose a binning method, which results in equally spaced bins of time. After incorporating binning, traditional models can be applied. The proposed binning approach was applied on a longitudinal hemodialysis study to look for possible contemporaneous and lagged effects between occurrences of a health event (i.e. infection) and levels of a protein marker of inflammation (i.e. C‐reactive protein). Both Poisson mixed effects models and zero‐inflated Poisson (ZIP) mixed effects models were applied to the subsequent data, and some important biological findings about contemporaneous and lagged associations were uncovered. In addition, a simulation study was conducted to investigate various properties of the binning approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Joint mixed‐effects models for causal inference with longitudinal data

Causal inference with observational longitudinal data and time‐varying exposures is complicated due to the potential for time‐dependent confounding and unmeasured confounding. Most causal inference methods that handle time‐dependent confounding rely on either the assumption of no unmeasured confounders or the availability of an unconfounded variable that is associated with the exposure (eg, an instrumental variable). Furthermore, when data are incomplete, validity of many methods often depends on the assumption of missing at random. We propose an approach that combines a parametric joint mixed‐effects model for the study outcome and the exposure with g‐computation to identify and estimate causal effects in the presence of time‐dependent confounding and unmeasured confounding. G‐computation can estimate participant‐specific or population‐average causal effects using parameters of the joint model. The joint model is a type of shared parameter model where the outcome and exposure‐selection models share common random effect(s). We also extend the joint model to handle missing data and truncation by death when missingness is possibly not at random. We evaluate the performance of the proposed method using simulation studies and compare the method to both linear mixed‐ and fixed‐effects models combined with g‐computation as well as to targeted maximum likelihood estimation. We apply the method to an epidemiologic study of vitamin D and depressive symptoms in older adults and include code using SAS PROC NLMIXED software to enhance the accessibility of the method to applied researchers.     

Longitudinal data analysis for linear gaussian models with random disturbed-highest-derivative-polynomial subject effects

For linear regression analysis of longitudinal data with Gaussian response, I propose a new model to generalize the traditional class of random effects models in which the random effects are deterministic polynomials with coefficients randomly distributed over subjects with mean zero. The generalization is accomplished by adding zero mean Gaussian ‘disturbances’ to the highest derivative of each random coefficient subject polynomial, independently at each observation time. The resulting random effects, which have mean zero at each observation time, are called disturbed highest derivative polynomials (DHDPs). The disturbances induce serial correlation and also allow the subject-specific DHDP time trends to be non-linear. I do not estimate the subject-specific DHDP time trends. Analysis is based on the marginal model, that is, the fixed effects or population model obtained by integrating the random polynomial coefficients and all disturbances out of the joint distribution of themselves and the response vector. This allows a ‘population averaged’ interpretation. One can select the DHDP order by an information criterion. When the population time trend is not correctly modelled, the optimal DHDP order will be larger than when it is correctly modelled. One can make the covariance matrix of the regression coefficients robust to errors in modelling the within-subject dependence. I describe the relationship of a DHDP to a smoothing polynomial spline, and show how to replace the DHDP model with a smoothing polynomial spline model for the within-subject dependence in the marginal model.     

Pseudo maximum likelihood approach for the analysis of multivariate left‐censored longitudinal data

The linear mixed effects model based on a full likelihood is one of the few methods available to model longitudinal data subject to left censoring. However, a full likelihood approach is complicated algebraically because of the large dimension of the numeric computations, and maximum likelihood estimation can be computationally prohibitive when the data are heavily censored. Moreover, for mixed models, the complexity of the computation increases as the dimension of the random effects in the model increases. We propose a method based on pseudo likelihood that simplifies the computational complexities, allows a wide class of multivariate models, and that can be used for many different data structures including settings where the level of censoring is high. The motivation for this work comes from the need for a joint model to assess the joint effect of pro‐inflammatory and anti‐inflammatory biomarker data on 30‐day mortality status while simultaneously accounting for longitudinal left censoring and correlation between markers in the analysis of Genetic and Inflammatory Markers for Sepsis study conducted at the University of Pittsburgh. Two markers, interleukin‐6 and interleukin‐10, which naturally are correlated because of a shared similar biological pathways and are left‐censored because of the limited sensitivity of the assays, are considered to determine if higher levels of these markers is associated with an increased risk of death after accounting for the left censoring and their assumed correlation. Copyright © 2016 John Wiley & Sons, Ltd.     

A likelihood reformulation method in non-normal random effects models

In this paper, we propose a practical computational method to obtain the maximum likelihood estimates (MLE) for mixed models with non-normal random effects. By simply multiplying and dividing a standard normal density, we reformulate the likelihood conditional on the non-normal random effects to that conditional on the normal random effects. Gaussian quadrature technique, conveniently implemented in SAS Proc NLMIXED, can then be used to carry out the estimation process. Our method substantially reduces computational time, while yielding similar estimates to the probability integral transformation method (J. Comput. Graphical Stat. 2006; 15:39-57). Furthermore, our method can be applied to more general situations, e.g. finite mixture random effects or correlated random effects from Clayton copula. Simulations and applications are presented to illustrate our method.     

Bayesian semiparametric variable selection with applications to periodontal data

A normality assumption is typically adopted for the random effects in a clustered or longitudinal data analysis using a linear mixed model. However, such an assumption is not always realistic, and it may lead to potential biases of the estimates, especially when variable selection is taken into account. Furthermore, flexibility of nonparametric assumptions (e.g., Dirichlet process) on these random effects may potentially cause centering problems, leading to difficulty of interpretation of fixed effects and variable selection. Motivated by these problems, we proposed a Bayesian method for fixed and random effects selection in nonparametric random effects models. We modeled the regression coefficients via centered latent variables which are distributed as probit stick‐breaking scale mixtures. By using the mixture priors for centered latent variables along with covariance decomposition, we could avoid the aforementioned problems and allow efficient selection of fixed and random effects from the model. We demonstrated the advantages of our proposed approach over other competing alternatives through a simulated example and also via an illustrative application to a data set from a periodontal disease study. Copyright © 2017 John Wiley & Sons, Ltd.