Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Estimation of group means in generalized linear mixed models

In this study, we investigate the concept of the mean response for a treatment group mean as well as its estimation and prediction for generalized linear models with a subject‐wise random effect. Generalized linear models are commonly used to analyze categorical data. The model‐based mean for a treatment group usually estimates the response at the mean covariate. However, the mean response for the treatment group for studied population is at least equally important in the context of clinical trials. New methods were proposed to estimate such a mean response in generalized linear models; however, this has only been done when there are no random effects in the model. We suggest that, in a generalized linear mixed model (GLMM), there are at least two possible definitions of a treatment group mean response that can serve as estimation/prediction targets. The estimation of these treatment group means is important for healthcare professionals to be able to understand the absolute benefit vs risk. For both of these treatment group means, we propose a new set of methods that suggests how to estimate/predict both of them in a GLMMs with a univariate subject‐wise random effect. Our methods also suggest an easy way of constructing corresponding confidence and prediction intervals for both possible treatment group means. Simulations show that proposed confidence and prediction intervals provide correct empirical coverage probability under most circumstances. Proposed methods have also been applied to analyze hypoglycemia data from diabetes clinical trials.     

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.     

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.     

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.     

Functional Mixed Effects Model for Small Area Estimation

Functional data analysis has become an important area of research because of its ability of handling high‐dimensional and complex data structures. However, the development is limited in the context of linear mixed effect models and, in particular, for small area estimation. The linear mixed effect models are the backbone of small area estimation. In this article, we consider area‐level data and fit a varying coefficient linear mixed effect model where the varying coefficients are semiparametrically modelled via B‐splines. We propose a method of estimating the fixed effect parameters and consider prediction of random effects that can be implemented using a standard software. For measuring prediction uncertainties, we derive an analytical expression for the mean squared errors and propose a method of estimating the mean squared errors. The procedure is illustrated via a real data example, and operating characteristics of the method are judged using finite sample simulation studies.     

Nonlinear mixed‐effects models with misspecified random‐effects distribution

Nonlinear mixed‐effects models are being widely used for the analysis of longitudinal data, especially from pharmaceutical research. They use random effects which are latent and unobservable variables so the random‐effects distribution is subject to misspecification in practice. In this paper, we first study the consequences of misspecifying the random‐effects distribution in nonlinear mixed‐effects models. Our study is focused on Gauss‐Hermite quadrature, which is now the routine method for calculation of the marginal likelihood in mixed models. We then present a formal diagnostic test to check the appropriateness of the assumed random‐effects distribution in nonlinear mixed‐effects models, which is very useful for real data analysis. Our findings show that the estimates of fixed‐effects parameters in nonlinear mixed‐effects models are generally robust to deviations from normality of the random‐effects distribution, but the estimates of variance components are very sensitive to the distributional assumption of random effects. Furthermore, a misspecified random‐effects distribution will either overestimate or underestimate the predictions of random effects. We illustrate the results using a real data application from an intensive pharmacokinetic study.     

Diagnostic tests for the distribution of random effects in multivariate mixed effects models

Abstract

Fourier methods are proposed for testing the distribution of random effects in classical and robust multivariate mixed effects models. The test statistics involve estimation of the characteristic function of random effects. Theoretical and computational issues are addressed while Monte Carlo results show that the new procedures compare favorably with other methods.     

Variance Components Testing in ANOVA‐Type Mixed Models

The purpose of this article is threefold. First, variance components testing for ANOVA ‐type mixed models is considered, in which response may not be divided into independent sub‐vectors, whereas most of existing methods are for models where response can be divided into independent sub‐vectors. Second, testing that a certain subset of variance components is zero. Third, as normality is often violated in practice, it is desirable to construct tests under very mild assumptions. To achieve these goals, an adaptive difference‐based test and an adaptive trace‐based test are constructed. The test statistics are asymptotically normal under the null hypothesis, are consistent against all global alternatives and can detect local alternatives distinct from the null at a rate as close to n ^{? 1 ∕ 2} as possible with n being the sample size. Moreover, when the dimensions of variance components in different sets are bounded, we develop a test with chi‐square as its limiting null distribution. The finite sample performance of the tests is examined via simulations, and a real data set is analysed for illustration.     

The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.     

Analyzing Binomial Data in a Split-Plot Design: Classical Approach or Modern Techniques?

Modeling data that are non-normally distributed with random effects is the major challenge in analyzing binomial data in split-plot designs. Seven methods for analyzing such data using mixed, generalized linear, or generalized linear mixed models are compared for the size and power of the tests. This study shows that analyzing random effects properly is more important than adjusting the analysis for non-normality. Methods based on mixed and generalized linear mixed models hold Type I error rates better than generalized linear models. Mixed model methods tend to have higher power than generalized linear mixed models when the sample size is small.     

On Variance Components in Semiparametric Mixed Models for Longitudinal Data

Abstract. First, to test the existence of random effects in semiparametric mixed models (SMMs) under only moment conditions on random effects and errors, we propose a very simple and easily implemented non‐parametric test based on a difference between two estimators of the error variance. One test is consistent only under the null and the other can be so under both the null and alternatives. Instead of erroneously solving the non‐standard two‐sided testing problem, as in most papers in the literature, we solve it correctly and prove that the asymptotic distribution of our test statistic is standard normal. This avoids Monte Carlo approximations to obtain p ‐values, as is needed for many existing methods, and the test can detect local alternatives approaching the null at rates up to root n. Second, as the higher moments of the error are necessarily estimated because the standardizing constant involves these quantities, we propose a general method to conveniently estimate any moments of the error. Finally, a simulation study and a real data analysis are conducted to investigate the properties of our procedures.     

A test of the missing data mechanism for repeated measures data

The occurrence of missing data is an often unavoidable consequence of repeated measures studies. Fortunately, multivariate general linear models such as growth curve models and linear mixed models with random effects have been well developed to analyze incomplete normally-distributed repeated measures data. Most statistical methods have assumed that the missing data occur at random. This assumption may include two types of missing data mechanism: missing completely at random (MCAR) and missing at random (MAR) in the sense of Rubin (1976). In this paper, we develop a test procedure for distinguishing these two types of missing data mechanism for incomplete normally-distributed repeated measures data. The proposed test is similar in spiril to the test of Park and Davis (1992). We derive the test for incomplete normally-distribrlted repeated measures data using linear mixed models. while Park and Davis (1992) cleirved thr test for incomplete repeatctl categorical data in the framework of Grizzle Starmer. and Koch (1969). Thr proposed procedure can be applied easily to any other multivariate general linear model which allow for missing data. The test is illustrated using the hip-replacernent patient.data from Crowder and Hand (1990).     

Robust Variable Selection in Linear Mixed Models

In this article, we develop a robust variable selection procedure jointly for fixed and random effects in linear mixed models for longitudinal data. We propose a penalized robust estimator for both the regression coefficients and the variance of random effects based on a re-parametrization of the linear mixed models. Under some regularity conditions, we show the oracle properties of the proposed robust variable selection method. Simulation study shows the robustness of the proposed method against outliers. In the end, the proposed methods is illustrated in the analysis of a real data set.     

Assessing additivity in nonparametric models —A kernel‐based method

In this article, we address the testing problem for additivity in nonparametric regression models. We develop a kernel‐based consistent test of a hypothesis of additivity in nonparametric regression, and establish its asymptotic distribution under a sequence of local alternatives. Compared to other existing kernel‐based tests, the proposed test is shown to effectively ameliorate the influence from estimation bias of the additive component of the nonparametric regression, and hence increase its efficiency. Most importantly, it avoids the tuning difficulties by using estimation‐based optimal criteria, while there is no direct tuning strategy for other existing kernel‐based testing methods. We discuss the usage of the new test and give numerical examples to demonstrate the practical performance of the test. The Canadian Journal of Statistics 39: 632–655; 2011. © 2011 Statistical Society of Canada     

A test of separate hypotheses for comparing linear mixed models with non nested fixed effects

As researchers increasingly rely on linear mixed models to characterize longitudinal data, there is a need for improved techniques for selecting among this class of models which requires specification of both fixed and random effects via a mean model and variance-covariance structure. The process is further complicated when fixed and/or random effects are non nested between models. This paper explores the development of a hypothesis test to compare non nested linear mixed models based on extensions of the work begun by Sir David Cox. We assess the robustness of this approach for comparing models containing correlated measures of body fat for predicting longitudinal cardiometabolic risk.     

A comparative study on estimation methods to deal with the endogeneity in linear random-intercept models with an extension

In this paper, we investigate estimation methods to deal with situations where random intercepts are associated to time-varying covariates in the context of linear mixed models. First, a review of previous ways to deal with this so-called endogeneity issue is presented, then a new method based on shared random effects is proposed. Simulation studies and an empirical example are utilized to assess the performance of our proposed method. It is shown that our new approach is more efficient than most competitors and is robust to the misspecification of the random-effects distributions.     

Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.     

An Alternative Estimation Approach for the Heterogeneity Linear Mixed Model

In this article, an alternative estimation approach is proposed to fit linear mixed effects models where the random effects follow a finite mixture of normal distributions. This heterogeneity linear mixed model is an interesting tool since it relaxes the classical normality assumption and is also perfectly suitable for classification purposes, based on longitudinal profiles. Instead of fitting directly the heterogeneity linear mixed model, we propose to fit an equivalent mixture of linear mixed models under some restrictions which is computationally simpler. Unlike the former model, the latter can be maximized analytically using an EM-algorithm and the obtained parameter estimates can be easily used to compute the parameter estimates of interest.     

Analysis of Contingency Tables with Clustered Observations

A new approach is presented for testing independence in contingency tables with clustered observations. The approach is based on the framework of generalized linear mixed models. Under the multinomial logistic link function, the category counts are modelled with random cluster effects and a modified likelihood ratio statistic is used for testing independence. The method is applicable to multi-way tables, and can accommodate multiple levels of clustering. It is illustrated using a benchmark dataset.