Use of generalized estimating equations in a trial in influenza to explore treatment effects over time

For trials with repeated measurements of outcome, analyses often focus on univariate outcomes, such as analysis of summary measures or of the last on‐treatment observation. Methods which model the whole data set provide a rich source of approaches to analysis. For continuous data, mixed‐effect modelling is increasingly used. For binary and categorical data, models based on use of generalized estimating equations account for intra‐subject correlation and allow exploration of the time course of response, as well as providing a useful way to account for missing data, when such data can be maintained as missing in the analysis. The utility of this approach is illustrated by an example from a trial in influenza. Copyright © 2004 John Wiley & Sons Ltd.     

A mixed effects model for analyzing area under the curve of longitudinally measured biomarkers with missing data

A simple approach for analyzing longitudinally measured biomarkers is to calculate summary measures such as the area under the curve (AUC) for each individual and then compare the mean AUC between treatment groups using methods such as t test. This two-step approach is difficult to implement when there are missing data since the AUC cannot be directly calculated for individuals with missing measurements. Simple methods for dealing with missing data include the complete case analysis and imputation. A recent study showed that the estimated mean AUC difference between treatment groups based on the linear mixed model (LMM), rather than on individually calculated AUCs by simple imputation, has negligible bias under random missing assumptions and only small bias when missing is not at random. However, this model assumes the outcome to be normally distributed, which is often violated in biomarker data. In this paper, we propose to use a LMM on log-transformed biomarkers, based on which statistical inference for the ratio, rather than difference, of AUC between treatment groups is provided. The proposed method can not only handle the potential baseline imbalance in a randomized trail but also circumvent the estimation of the nuisance variance parameters in the log-normal model. The proposed model is applied to a recently completed large randomized trial studying the effect of nicotine reduction on biomarker exposure of smokers.     

Sample‐size calculations for multi‐group comparison in population pharmacokinetic experiments

This paper describes an approach for calculating sample size for population pharmacokinetic experiments that involve hypothesis testing based on multi‐group comparison detecting the difference in parameters between groups under mixed‐effects modelling. This approach extends what has been described for generalized linear models and nonlinear population pharmacokinetic models that involve only binary covariates to more complex nonlinear population pharmacokinetic models. The structural nonlinear model is linearized around the random effects to obtain the marginal model and the hypothesis testing involving model parameters is based on Wald's test. This approach provides an efficient and fast method for calculating sample size for hypothesis testing in population pharmacokinetic models. The approach can also handle different design problems such as unequal allocation of subjects to groups and unbalanced sampling times between and within groups. The results obtained following application to a one compartment intravenous bolus dose model that involved three different hypotheses under different scenarios showed good agreement between the power obtained from NONMEM simulations and nominal power. Copyright © 2009 John Wiley & Sons, Ltd.     

Sampling from the posterior distribution in generalized linear mixed models

Generalized linear mixed models provide a unified framework for treatment of exponential family regression models, overdispersed data and longitudinal studies. These problems typically involve the presence of random effects and this paper presents a new methodology for making Bayesian inference about them. The approach is simulation-based and involves the use of Markov chain Monte Carlo techniques. The usual iterative weighted least squares algorithm is extended to include a sampling step based on the Metropolis–Hastings algorithm thus providing a unified iterative scheme. Non-normal prior distributions for the regression coefficients and for the random effects distribution are considered. Random effect structures with nesting required by longitudinal studies are also considered. Particular interests concern the significance of regression coefficients and assessment of the form of the random effects. Extensions to unknown scale parameters, unknown link functions, survival and frailty models are outlined.     

Mixture of linear mixed models using multivariate t distribution

Linear mixed models are widely used when multiple correlated measurements are made on each unit of interest. In many applications, the units may form several distinct clusters, and such heterogeneity can be more appropriately modelled by a finite mixture linear mixed model. The classical estimation approach, in which both the random effects and the error parts are assumed to follow normal distribution, is sensitive to outliers, and failure to accommodate outliers may greatly jeopardize the model estimation and inference. We propose a new mixture linear mixed model using multivariate t distribution. For each mixture component, we assume the response and the random effects jointly follow a multivariate t distribution, to conveniently robustify the estimation procedure. An efficient expectation conditional maximization algorithm is developed for conducting maximum likelihood estimation. The degrees of freedom parameters of the t distributions are chosen data adaptively, for achieving flexible trade-off between estimation robustness and efficiency. Simulation studies and an application on analysing lung growth longitudinal data showcase the efficacy of the proposed approach.     

Multilevel modelling of complex survey data

Summary.  Multilevel modelling is sometimes used for data from complex surveys involving multistage sampling, unequal sampling probabilities and stratification. We consider generalized linear mixed models and particularly the case of dichotomous responses. A pseudolikelihood approach for accommodating inverse probability weights in multilevel models with an arbitrary number of levels is implemented by using adaptive quadrature. A sandwich estimator is used to obtain standard errors that account for stratification and clustering. When level 1 weights are used that vary between elementary units in clusters, the scaling of the weights becomes important. We point out that not only variance components but also regression coefficients can be severely biased when the response is dichotomous. The pseudolikelihood methodology is applied to complex survey data on reading proficiency from the American sample of the 'Program for international student assessment' 2000 study, using the Stata program gllamm which can estimate a wide range of multilevel and latent variable models. Performance of pseudo-maximum-likelihood with different methods for handling level 1 weights is investigated in a Monte Carlo experiment. Pseudo-maximum-likelihood estimators of (conditional) regression coefficients perform well for large cluster sizes but are biased for small cluster sizes. In contrast, estimators of marginal effects perform well in both situations. We conclude that caution must be exercised in pseudo-maximum-likelihood estimation for small cluster sizes when level 1 weights are used.     

Bayesian estimation and case influence diagnostics for the zero-inflated negative binomial regression model

In recent years, there has been considerable interest in regression models based on zero-inflated distributions. These models are commonly encountered in many disciplines, such as medicine, public health, and environmental sciences, among others. The zero-inflated Poisson (ZIP) model has been typically considered for these types of problems. However, the ZIP model can fail if the non-zero counts are overdispersed in relation to the Poisson distribution, hence the zero-inflated negative binomial (ZINB) model may be more appropriate. In this paper, we present a Bayesian approach for fitting the ZINB regression model. This model considers that an observed zero may come from a point mass distribution at zero or from the negative binomial model. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. The approach can be easily implemented using standard Bayesian software, such as WinBUGS. The performance of the proposed method is evaluated with a simulation study. Further, a real data set is analyzed, where we show that ZINB regression models seems to fit the data better than the Poisson counterpart.     

The relationship between osteoporotic fracture risk and a surrogate: Apparent discrepancies between analyses based on individual patient data and summary statistics

There is debate within the osteoporosis research community about the relationship between the risk of osteoporotic fracture and the surrogate measures of fracture risk. Meta‐regression analyses based on summary data have shown a linear relationship between fracture risk and surrogate measures, whereas analyses based on individual patient data (IPD) have shown a nonlinear relationship. We investigated the association between changes in a surrogate measure of fracture incidence, in this case a bone turnover marker for resorption assessed in the three risedronate phase III clinical programmes, and incident osteoporosis‐related fracture risk using regression models based on patient‐level and trial‐level information. The relationship between osteoporosis‐related fracture risk and changes in bone resorption was different when analysed on the basis of IPD than when analysed on the basis of a meta‐analytic approach (i.e., meta‐regression) using summary data (e.g., treatment effect based on treatment group estimates). This inconsistency in our findings was consistent with those in the published literature. Meta‐regression based on summary statistics at the trial level is not expected to reflect causal relationships between a clinical outcome and surrogate measures. Analyses based on IPD make possible a more comprehensive analysis since all relevant data on a patient level are available. Copyright © 2004 John Wiley & Sons Ltd.     

Approximate testing in two-stage nonlinear mixed models

We investigate here small sample properties of approximate F-tests about fixed effects parameters in nonlinear mixed models. For estimation of population fixed effects parameters as well as variance components, we apply the two-stage approach. This method is useful and popular when the number of observations per sampling unit is large enough. The approximate F-test is constructed based on large-sample approximation to the distribution of nonlinear least-squares estimates of subject-specific parameters. We recommend a modified test statistic that takes into consideration approximation to the large-sample Fisher information matrix (See [Volaufova J, Burton JH. Note on hypothesis testing in mixed models. Oral presentation at: LINSTAT 2012/21st IWMS; 2012; Bedlewo, Poland]). Our main focus is on comparing finite sample properties of broadly used approximate tests (Wald test and likelihood ratio test) and the modified F-test under the null hypothesis, especially accuracy of p-values (See [Volaufova J, LaMotte L. Comparison of approximate tests of fixed effects in linear repeated measures design models with covariates. Tatra Mountains. 2008;39:17–25]). For that purpose two extensive simulation studies are conducted based on pharmacokinetic models (See [Hartford A, Davidian M. Consequences of misspecifying assumptions in nonlinear mixed effects models. Comput Stat and Data Anal. 2000;34:139–164; Pinheiro J, Bates D. Approximations to the log-likelihood function in the non-linear mixed-effects model. J Comput Graph Stat. 1995;4(1):12–35]).     

Causal Reasoning from Longitudinal Data*

Abstract. This paper reviews some of the key statistical ideas that are encountered when trying to find empirical support to causal interpretations and conclusions, by applying statistical methods on experimental or observational longitudinal data. In such data, typically a collection of individuals are followed over time, then each one has registered a sequence of covariate measurements along with values of control variables that in the analysis are to be interpreted as causes, and finally the individual outcomes or responses are reported. Particular attention is given to the potentially important problem of confounding. We provide conditions under which, at least in principle, unconfounded estimation of the causal effects can be accomplished. Our approach for dealing with causal problems is entirely probabilistic, and we apply Bayesian ideas and techniques to deal with the corresponding statistical inference. In particular, we use the general framework of marked point processes for setting up the probability models, and consider posterior predictive distributions as providing the natural summary measures for assessing the causal effects. We also draw connections to relevant recent work in this area, notably to Judea Pearl's formulations based on graphical models and his calculus of so‐called do‐probabilities. Two examples illustrating different aspects of causal reasoning are discussed in detail.     

Simplified hierarchical linear models for the evaluation of surrogate endpoints

The linear mixed-effects model (Verbeke and Molenberghs, 2000) has become a standard tool for the analysis of continuous hierarchical data such as, for example, repeated measures or data from meta-analyses. However, in certain situations the model does pose insurmountable computational problems. Precisely this has been the experience of Buyse et al. (2000a) who proposed an estimation- and prediction-based approach for evaluating surrogate endpoints. Their approach requires fitting linear mixed models to data from several clinical trials. In doing so, these authors built on the earlier, single-trial based, work by Prentice (1989), Freedman et al. (1992), and Buyse and Molenberghs (1998). While Buyse et al. (2000a) claim their approach has a number of advantages over the classical single-trial methods, a solution needs to be found for the computational complexity of the corresponding linear mixed model. In this paper, we propose and study a number of possible simplifications. This is done by means of a simulation study and by applying the various strategies to data from three clinical studies: Pharmacological Therapy for Macular Degeneration Study Group (1977), Ovarian Cancer Meta-analysis Project (1991) and Corfu-A Study Group (1995).     

Spatial generalized linear mixed models in small area estimation

In survey sampling, policy decisions regarding the allocation of resources to sub‐groups of a population depend on reliable predictors of their underlying parameters. However, in some sub‐groups, called small areas due to small sample sizes relative to the population, the information needed for reliable estimation is typically not available. Consequently, data on a coarser scale are used to predict the characteristics of small areas. Mixed models are the primary tools in small area estimation (SAE) and also borrow information from alternative sources (e.g., previous surveys and administrative and census data sets). In many circumstances, small area predictors are associated with location. For instance, in the case of chronic disease or cancer, it is important for policy makers to understand spatial patterns of disease in order to determine small areas with high risk of disease and establish prevention strategies. The literature considering SAE with spatial random effects is sparse and mostly in the context of spatial linear mixed models. In this article, small area models are proposed for the class of spatial generalized linear mixed models to obtain small area predictors and corresponding second‐order unbiased mean squared prediction errors via Taylor expansion and a parametric bootstrap approach. The performance of the proposed approach is evaluated through simulation studies and application of the models to a real esophageal cancer data set from Minnesota, U.S.A. The Canadian Journal of Statistics 47: 426–437; 2019 © 2019 Statistical Society of Canada     

Semi‐parametric small‐area estimation by combining time‐series and cross‐sectional data methods

In survey sampling, policymaking regarding the allocation of resources to subgroups (called small areas) or the determination of subgroups with specific properties in a population should be based on reliable estimates. Information, however, is often collected at a different scale than that of these subgroups; hence, the estimation can only be obtained on finer scale data. Parametric mixed models are commonly used in small‐area estimation. The relationship between predictors and response, however, may not be linear in some real situations. Recently, small‐area estimation using a generalised linear mixed model (GLMM) with a penalised spline (P‐spline) regression model, for the fixed part of the model, has been proposed to analyse cross‐sectional responses, both normal and non‐normal. However, there are many situations in which the responses in small areas are serially dependent over time. Such a situation is exemplified by a data set on the annual number of visits to physicians by patients seeking treatment for asthma, in different areas of Manitoba, Canada. In cases where covariates that can possibly predict physician visits by asthma patients (e.g. age and genetic and environmental factors) may not have a linear relationship with the response, new models for analysing such data sets are required. In the current work, using both time‐series and cross‐sectional data methods, we propose P‐spline regression models for small‐area estimation under GLMMs. Our proposed model covers both normal and non‐normal responses. In particular, the empirical best predictors of small‐area parameters and their corresponding prediction intervals are studied with the maximum likelihood estimation approach being used to estimate the model parameters. The performance of the proposed approach is evaluated using some simulations and also by analysing two real data sets (precipitation and asthma).     

Bayesian Inference for Skew-normal Linear Mixed Models

Linear mixed models (LMM) are frequently used to analyze repeated measures data, because they are more flexible to modelling the correlation within-subject, often present in this type of data. The most popular LMM for continuous responses assumes that both the random effects and the within-subjects errors are normally distributed, which can be an unrealistic assumption, obscuring important features of the variations present within and among the units (or groups). This work presents skew-normal liner mixed models (SNLMM) that relax the normality assumption by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in mixed models. The MCMC scheme is derived and the results of a simulation study are provided demonstrating that standard information criteria may be used to detect departures from normality. The procedures are illustrated using a real data set from a cholesterol study.     

Generalised partial linear single-index mixed models for repeated measures data

In this paper, we propose generalised partial linear single-index mixed models for analysing repeated measures data. A penalised quasi-likelihood approach using P-spline is used to estimate the nonparametric function, linear parameters, and single-index coefficients. Asymptotic properties of the estimators are developed when the dimension of spline basis grows with increasing sample size. Simulation examples and two applications: the study of health effects of air pollution in North Carolina, and treatment effect of naltrexone on health costs for alcohol-dependent individuals, illustrate the effectiveness of our approach.     

Design optimisation for pharmacokinetic modeling of a cocktail of phenotyping drugs

Our paper proposes a methodological strategy to select optimal sampling designs for phenotyping studies including a cocktail of drugs. A cocktail approach is of high interest to determine the simultaneous activity of enzymes responsible for drug metabolism and pharmacokinetics, therefore useful in anticipating drug–drug interactions and in personalized medicine. Phenotyping indexes, which are area under the concentration‐time curves, can be derived from a few samples using nonlinear mixed effect models and maximum a posteriori estimation. Because of clinical constraints in phenotyping studies, the number of samples that can be collected in individuals is limited and the sampling times must be as flexible as possible. Therefore to optimize joint design for several drugs (i.e., to determine a compromise between informative times that best characterize each drug's kinetics), we proposed to use a compound optimality criterion based on the expected population Fisher information matrix in nonlinear mixed effect models. This criterion allows weighting different models, which might be useful to take into account the importance accorded to each target in a phenotyping test. We also computed windows around the optimal times based on recursive random sampling and Monte‐Carlo simulation while maintaining a reasonable level of efficiency for parameter estimation. We illustrated this strategy for two drugs often included in phenotyping cocktails, midazolam (probe for CYP3A) and digoxin (P‐glycoprotein), based on the data of a previous study, and were able to find a sparse and flexible design. The obtained design was evaluated by clinical trial simulations and shown to be efficient for the estimation of population and individual parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

Unconditional empirical likelihood approach for analytic use of public survey data

Modeling survey data often requires having the knowledge of design and weighting variables. With public-use survey data, some of these variables may not be available for confidentiality reasons. The proposed approach can be used in this situation, as long as calibrated weights and variables specifying the strata and primary sampling units are available. It gives consistent point estimation and a pivotal statistics for testing and confidence intervals. The proposed approach does not rely on with-replacement sampling, single-stage, negligible sampling fractions, or noninformative sampling. Adjustments based on design effects, eigenvalues, joint-inclusion probabilities or bootstrap, are not needed. The inclusion probabilities and auxiliary variables do not have to be known. Multistage designs with unequal selection of primary sampling units are considered. Nonresponse can be easily accommodated if the calibrated weights include reweighting adjustment for nonresponse. We use an unconditional approach, where the variables and sample are random variables. The design can be informative.     

Linear quantile mixed models

Dependent data arise in many studies. Frequently adopted sampling designs, such as cluster, multilevel, spatial, and repeated measures, may induce this dependence, which the analysis of the data needs to take into due account. In a previous publication (Geraci and Bottai in Biostatistics 8:140–154, 2007), we proposed a conditional quantile regression model for continuous responses where subject-specific random intercepts were included to account for within-subject dependence in the context of longitudinal data analysis. The approach hinged upon the link existing between the minimization of weighted absolute deviations, typically used in quantile regression, and the maximization of a Laplace likelihood. Here, we consider an extension of those models to more complex dependence structures in the data, which are modeled by including multiple random effects in the linear conditional quantile functions. We also discuss estimation strategies to reduce the computational burden and inefficiency associated with the Monte Carlo EM algorithm we have proposed previously. In particular, the estimation of the fixed regression coefficients and of the random effects’ covariance matrix is based on a combination of Gaussian quadrature approximations and non-smooth optimization algorithms. Finally, a simulation study and a number of applications of our models are presented.     

Likelihood analysis for a class of beta mixed models

Beta regression is a suitable choice for modelling continuous response variables taking values on the unit interval. Data structures such as hierarchical, repeated measures and longitudinal typically induce extra variability and/or dependence and can be accounted for by the inclusion of random effects. In this sense, Statistical inference typically requires numerical methods, possibly combined with sampling algorithms. A class of Beta mixed models is adopted for the analysis of two real problems with grouped data structures. We focus on likelihood inference and describe the implemented algorithms. The first is a study on the life quality index of industry workers with data collected according to an hierarchical sampling scheme. The second is a study assessing the impact of hydroelectric power plants upon measures of water quality indexes up, downstream and at the reservoirs of the dammed rivers, with a nested and longitudinal data structure. Results from different algorithms are reported for comparison including from data-cloning, an alternative to numerical approximations which also allows assessing identifiability. Confidence intervals based on profiled likelihoods are compared with those obtained by asymptotic quadratic approximations, showing relevant differences for parameters related to the random effects. In both cases, the scientific hypothesis of interest was investigated by comparing alternative models, leading to relevant interpretations of the results within each context.     

An efficient Monte Carlo EM algorithm for Bayesian lasso

The lasso is a popular technique of simultaneous estimation and variable selection in many research areas. The marginal posterior mode of the regression coefficients is equivalent to estimates given by the non-Bayesian lasso when the regression coefficients have independent Laplace priors. Because of its flexibility of statistical inferences, the Bayesian approach is attracting a growing body of research in recent years. Current approaches are primarily to either do a fully Bayesian analysis using Markov chain Monte Carlo (MCMC) algorithm or use Monte Carlo expectation maximization (MCEM) methods with an MCMC algorithm in each E-step. However, MCMC-based Bayesian method has much computational burden and slow convergence. Tan et al. [An efficient MCEM algorithm for fitting generalized linear mixed models for correlated binary data. J Stat Comput Simul. 2007;77:929–943] proposed a non-iterative sampling approach, the inverse Bayes formula (IBF) sampler, for computing posteriors of a hierarchical model in the structure of MCEM. Motivated by their paper, we develop this IBF sampler in the structure of MCEM to give the marginal posterior mode of the regression coefficients for the Bayesian lasso, by adjusting the weights of importance sampling, when the full conditional distribution is not explicit. Simulation experiments show that the computational time is much reduced with our method based on the expectation maximization algorithm and our algorithms and our methods behave comparably with other Bayesian lasso methods not only in prediction accuracy but also in variable selection accuracy and even better especially when the sample size is relatively large.