A comparison of bootstrap approaches for estimating uncertainty of parameters in linear mixed‐effects models

A version of the nonparametric bootstrap, which resamples the entire subjects from original data, called the case bootstrap, has been increasingly used for estimating uncertainty of parameters in mixed‐effects models. It is usually applied to obtain more robust estimates of the parameters and more realistic confidence intervals (CIs). Alternative bootstrap methods, such as residual bootstrap and parametric bootstrap that resample both random effects and residuals, have been proposed to better take into account the hierarchical structure of multi‐level and longitudinal data. However, few studies have been performed to compare these different approaches. In this study, we used simulation to evaluate bootstrap methods proposed for linear mixed‐effect models. We also compared the results obtained by maximum likelihood (ML) and restricted maximum likelihood (REML). Our simulation studies evidenced the good performance of the case bootstrap as well as the bootstraps of both random effects and residuals. On the other hand, the bootstrap methods that resample only the residuals and the bootstraps combining case and residuals performed poorly. REML and ML provided similar bootstrap estimates of uncertainty, but there was slightly more bias and poorer coverage rate for variance parameters with ML in the sparse design. We applied the proposed methods to a real dataset from a study investigating the natural evolution of Parkinson's disease and were able to confirm that the methods provide plausible estimates of uncertainty. Given that most real‐life datasets tend to exhibit heterogeneity in sampling schedules, the residual bootstraps would be expected to perform better than the case bootstrap. Copyright © 2013 John Wiley & Sons, Ltd.     

A NONPARAMETRIC MIXED‐EFFECTS MODEL FOR CANCER MORTALITY

There are several ways to handle within‐subject correlations with a longitudinal discrete outcome, such as mortality. The most frequently used models are either marginal or random‐effects types. This paper deals with a random‐effects‐based approach. We propose a nonparametric regression model having time‐varying mixed effects for longitudinal cancer mortality data. The time‐varying mixed effects in the proposed model are estimated by combining kernel‐smoothing techniques and a growth‐curve model. As an illustration based on real data, we apply the proposed method to a set of prefecture‐specific data on mortality from large‐bowel cancer in Japan.     

Robust transformation mixed‐effects models for longitudinal continuous proportional data

The authors propose a robust transformation linear mixed‐effects model for longitudinal continuous proportional data when some of the subjects exhibit outlying trajectories over time. It becomes troublesome when including or excluding such subjects in the data analysis results in different statistical conclusions. To robustify the longitudinal analysis using the mixed‐effects model, they utilize the multivariate t distribution for random effects or/and error terms. Estimation and inference in the proposed model are established and illustrated by a real data example from an ophthalmology study. Simulation studies show a substantial robustness gain by the proposed model in comparison to the mixed‐effects model based on Aitchison's logit‐normal approach. As a result, the data analysis benefits from the robustness of making consistent conclusions in the presence of influential outliers. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Kernel‐based Generalized Cross‐validation in Non‐parametric Mixed‐effect Models

Abstract. Although generalized cross‐validation (GCV) has been frequently applied to select bandwidth when kernel methods are used to estimate non‐parametric mixed‐effect models in which non‐parametric mean functions are used to model covariate effects, and additive random effects are applied to account for overdispersion and correlation, the optimality of the GCV has not yet been explored. In this article, we construct a kernel estimator of the non‐parametric mean function. An equivalence between the kernel estimator and a weighted least square type estimator is provided, and the optimality of the GCV‐based bandwidth is investigated. The theoretical derivations also show that kernel‐based and spline‐based GCV give very similar asymptotic results. This provides us with a solid base to use kernel estimation for mixed‐effect models. Simulation studies are undertaken to investigate the empirical performance of the GCV. A real data example is analysed for illustration.     

Statistical Inference for Single‐index Panel Data Models

We study estimation and hypothesis testing in single‐index panel data models with individual effects. Through regressing the individual effects on the covariates linearly, we convert the estimation problem in single‐index panel data models to that in partially linear single‐index models. The conversion is valid regardless of the individual effects being random or fixed. We propose an estimating equation approach, which has a desirable double robustness property. We show that our method is applicable in single‐index panel data models with heterogeneous link functions. We further design a chi‐squared test to evaluate whether the individual effects are random or fixed. We conduct simulations to demonstrate the finite sample performance of the method and conduct a data analysis to illustrate its usefulness.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust Bayesian nonlinear mixed‐effects modeling of time to positivity in tuberculosis trials

Early phase 2 tuberculosis (TB) trials are conducted to characterize the early bactericidal activity (EBA) of anti‐TB drugs. The EBA of anti‐TB drugs has conventionally been calculated as the rate of decline in colony forming unit (CFU) count during the first 14 days of treatment. The measurement of CFU count, however, is expensive and prone to contamination. Alternatively to CFU count, time to positivity (TTP), which is a potential biomarker for long‐term efficacy of anti‐TB drugs, can be used to characterize EBA. The current Bayesian nonlinear mixed‐effects (NLME) regression model for TTP data, however, lacks robustness to gross outliers that often are present in the data. The conventional way of handling such outliers involves their identification by visual inspection and subsequent exclusion from the analysis. However, this process can be questioned because of its subjective nature. For this reason, we fitted robust versions of the Bayesian nonlinear mixed‐effects regression model to a wide range of TTP datasets. The performance of the explored models was assessed through model comparison statistics and a simulation study. We conclude that fitting a robust model to TTP data obviates the need for explicit identification and subsequent “deletion” of outliers but ensures that gross outliers exert no undue influence on model fits. We recommend that the current practice of fitting conventional normal theory models be abandoned in favor of fitting robust models to TTP data.     

A generalized Bayesian nonlinear mixed‐effects regression model for zero‐inflated longitudinal count data in tuberculosis trials

In this paper, we investigate Bayesian generalized nonlinear mixed‐effects (NLME) regression models for zero‐inflated longitudinal count data. The methodology is motivated by and applied to colony forming unit (CFU) counts in extended bactericidal activity tuberculosis (TB) trials. Furthermore, for model comparisons, we present a generalized method for calculating the marginal likelihoods required to determine Bayes factors. A simulation study shows that the proposed zero‐inflated negative binomial regression model has good accuracy, precision, and credibility interval coverage. In contrast, conventional normal NLME regression models applied to log‐transformed count data, which handle zero counts as left censored values, may yield credibility intervals that undercover the true bactericidal activity of anti‐TB drugs. We therefore recommend that zero‐inflated NLME regression models should be fitted to CFU count on the original scale, as an alternative to conventional normal NLME regression models on the logarithmic scale.     

Testing for autocorrelation and random-effects in nonlinear mixed effects models based on M-estimation

M-estimation (robust estimation) for the parameters in nonlinear mixed effects models using Fisher scoring method is investigated in the article, which shares some of the features of the existing maximum likelihood estimation: consistency and asymptotic normality. Score tests for autocorrelation and random effects based on M-estimation, together with their asymptotic distribution are also studied. The performance of the test statistics are evaluated via simulations and a real data analysis of plasma concentrations data.     

Estimation for High‐Dimensional Linear Mixed‐Effects Models Using ℓ1‐Penalization

Abstract. We propose an ?₁‐penalized estimation procedure for high‐dimensional linear mixed‐effects models. The models are useful whenever there is a grouping structure among high‐dimensional observations, that is, for clustered data. We prove a consistency and an oracle optimality result and we develop an algorithm with provable numerical convergence. Furthermore, we demonstrate the performance of the method on simulated and a real high‐dimensional data set.     

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.     

Estimating multiple-membership logit models with mixed effects: indirect inference versus data cloning

Multiple-membership logit models with random effects are models for clustered binary data, where each statistical unit can belong to more than one group. The likelihood function of these models is analytically intractable. We propose two different approaches for parameter estimation: indirect inference and data cloning (DC). The former is a non-likelihood-based method which uses an auxiliary model to select reasonable estimates. We propose an auxiliary model with the same dimension of parameter space as the target model, which is particularly convenient to reach good estimates very fast. The latter method computes maximum likelihood estimates through the posterior distribution of an adequate Bayesian model, fitted to cloned data. We implement a DC algorithm specifically for multiple-membership models. A Monte Carlo experiment compares the two methods on simulated data. For further comparison, we also report Bayesian posterior mean and Integrated Nested Laplace Approximation hybrid DC estimates. Simulations show a negligible loss of efficiency for the indirect inference estimator, compensated by a relevant computational gain. The approaches are then illustrated with two real examples on matched paired data.     

Estimating the variance for heterogeneity in arm‐based network meta‐analysis

Network meta‐analysis can be implemented by using arm‐based or contrast‐based models. Here we focus on arm‐based models and fit them using generalized linear mixed model procedures. Full maximum likelihood (ML) estimation leads to biased trial‐by‐treatment interaction variance estimates for heterogeneity. Thus, our objective is to investigate alternative approaches to variance estimation that reduce bias compared with full ML. Specifically, we use penalized quasi‐likelihood/pseudo‐likelihood and hierarchical (h) likelihood approaches. In addition, we consider a novel model modification that yields estimators akin to the residual maximum likelihood estimator for linear mixed models. The proposed methods are compared by simulation, and 2 real datasets are used for illustration. Simulations show that penalized quasi‐likelihood/pseudo‐likelihood and h‐likelihood reduce bias and yield satisfactory coverage rates. Sum‐to‐zero restriction and baseline contrasts for random trial‐by‐treatment interaction effects, as well as a residual ML‐like adjustment, also reduce bias compared with an unconstrained model when ML is used, but coverage rates are not quite as good. Penalized quasi‐likelihood/pseudo‐likelihood and h‐likelihood are therefore recommended.     

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).     

Local influence of nonlinear mixed effects model based on M-estimation

Abstract

In this work we mainly study the local influence in nonlinear mixed effects model with M-estimation. A robust method to obtain maximum likelihood estimates for parameters is presented, and the local influence of nonlinear mixed models based on robust estimation (M-estimation) by use of the curvature method is systematically discussed. The counting formulas of curvature for case weights perturbation, response variable perturbation and random error covariance perturbation are derived. Simulation studies are carried to access performance of the methods we proposed. We illustrate the diagnostics by an example presented in Davidian and Giltinan, which was analyzed under the non-robust situation.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

The impact of period effects on dose level contrasts in alternating cross‐over designs for first‐time‐in‐human studies

For first‐time‐in‐human studies with small molecules alternating cross‐over designs are often employed and at study end are analyzed using linear models. We discuss the impact of including a period effect in the model on the precision with which dose level contrasts can be estimated and quantify the bias of least squares estimators if a period effect is inherent in the data that is not accounted for in the model. We also propose two alternative designs that allow a more precise estimation of dose level contrasts compared with the standard design when period effects are included in the model. Copyright © 2010 John Wiley & Sons, Ltd.     

Covariate Decomposition Methods for Longitudinal Missing‐at‐Random Data and Predictors Associated with Subject‐Specific Effects

Investigators often gather longitudinal data to assess changes in responses over time within subjects and to relate these changes to within‐subject changes in predictors. Missing data are common in such studies and predictors can be correlated with subject‐specific effects. Maximum likelihood methods for generalized linear mixed models provide consistent estimates when the data are ‘missing at random’ (MAR) but can produce inconsistent estimates in settings where the random effects are correlated with one of the predictors. On the other hand, conditional maximum likelihood methods (and closely related maximum likelihood methods that partition covariates into between‐ and within‐cluster components) provide consistent estimation when random effects are correlated with predictors but can produce inconsistent covariate effect estimates when data are MAR. Using theory, simulation studies, and fits to example data this paper shows that decomposition methods using complete covariate information produce consistent estimates. In some practical cases these methods, that ostensibly require complete covariate information, actually only involve the observed covariates. These results offer an easy‐to‐use approach to simultaneously protect against bias from both cluster‐level confounding and MAR missingness in assessments of change.     

Mixed models for data from thorough QT studies: part 2. One-step assessment of conditional QT prolongation

We investigate mixed analysis of covariance models for the 'one-step' assessment of conditional QT prolongation. Initially, we consider three different covariance structures for the data, where between-treatment covariance of repeated measures is modelled respectively through random effects, random coefficients, and through a combination of random effects and random coefficients. In all three of those models, an unstructured covariance pattern is used to model within-treatment covariance. In a fourth model, proposed earlier in the literature, between-treatment covariance is modelled through random coefficients but the residuals are assumed to be independent identically distributed (i.i.d.). Finally, we consider a mixed model with saturated covariance structure. We investigate the precision and robustness of those models by fitting them to a large group of real data sets from thorough QT studies. Our findings suggest: (i) Point estimates of treatment contrasts from all five models are similar. (ii) The random coefficients model with i.i.d. residuals is not robust; the model potentially leads to both under- and overestimation of standard errors of treatment contrasts and therefore cannot be recommended for the analysis of conditional QT prolongation. (iii) The combined random effects/random coefficients model does not always converge; in the cases where it converges, its precision is generally inferior to the other models considered. (iv) Both the random effects and the random coefficients model are robust. (v) The random effects, the random coefficients, and the saturated model have similar precision and all three models are suitable for the one-step assessment of conditional QT prolongation.     

A novel quantification of information for longitudinal data analyzed by mixed‐effects modeling

Nonlinear mixed‐effects (NLME) modeling is one of the most powerful tools for analyzing longitudinal data especially under the sparse sampling design. The determinant of the Fisher information matrix is a commonly used global metric of the information that can be provided by the data under a given model. However, in clinical studies, it is also important to measure how much information the data provide for a certain parameter of interest under the assumed model, for example, the clearance in population pharmacokinetic models. This paper proposes a new, easy‐to‐interpret information metric, the “relative information” (RI), which is designed for specific parameters of a model and takes a value between 0% and 100%. We establish the relationship between interindividual variability for a specific parameter and the variance of the associated parameter estimator, demonstrating that, under a “perfect” experiment (eg, infinite samples or/and minimum experimental error), the RI and the variance of the model parameter estimator converge, respectively, to 100% and the ratio of the interindividual variability for that parameter and the number of subjects. Extensive simulation experiments and analyses of three real datasets show that our proposed RI metric can accurately characterize the information for parameters of interest for NLME models. The new information metric can be readily used to facilitate study designs and model diagnosis.