Geert Verbeke

A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population

Abstract This article investigates the impact of the normality assumption for random effects on their estimates in the linear mixed-effects model. It shows that if the distribution of random effects is a finite mixture of normal distributions, then the random effects may be badly estimated if normality is assumed, and the current methods for inspecting the appropriateness of the model assumptions are not sound. Further, it is argued that a better way to detect the components of the mixture is to build this assumption in the model and then “compare” the fitted model with the Gaussian model. All of this is illustrated on two practical examples.

The Use of Score Tests for Inference on Variance Components

Whenever inference for variance components is required, the choice between one-sided and two-sided tests is crucial. This choice is usually driven by whether or not negative variance components are permitted. For two-sided tests, classical inferential procedures can be followed, based on likelihood ratios, score statistics, or Wald statistics. For one-sided tests, however, one-sided test statistics need to be developed, and their null distribution derived. While this has received considerable attention in the context of the likelihood ratio test, there appears to be much confusion about the related problem for the score test. The aim of this paper is to illustrate that classical (two-sided) score test statistics, frequently advocated in practice, cannot be used in this context, but that well-chosen one-sided counterparts could be used instead. The relation with likelihood ratio tests will be established, and all results are illustrated in an analysis of continuous longitudinal data using linear mixed models.

Longitudinal Data Analysis

Preface. Acknowledgments. Acronyms. 1. Introduction. 1.1 Advantages of Longitudinal Studies. 1.2 Challenges of Longitudinal Data Analysis. 1.3 Some General Notation. 1.4 Data Layout. 1.5 Analysis Considerations. 1.6 General Approaches. 1.7 The Simplest Longitudinal Analysis. 1.8 Summary. 2. ANOVA Approaches to Longitudinal Data. 2.1Single-Sample Repeated Measures ANOVA. 2.2 Multiple-Sample Repeated Measures ANOVA. 2.3 Illustration. 2.4 Summary. 3. MANOVA Approaches to Longitudinal Data. 3.1 Data Layout for ANOVA versus MANOVA. 3.2 MANOVA for Repeated Measurements. 3.3 MANOVA of Repeated Measures-s Sample Case. 3.4 Illustration. 3.5 Summary. 4. Mixed-Effects Regression Models for Continuous Outcomes. 4.1 Introduction. 4.2 A Simple Linear Regression Model. 4.3 Random Intercept MRM. 4.4 Random Intercept and Trend MRM. 4.5 Matrix Formulation. 4.6 Estimation . 4.7 Summary. 5. Mixed-Effects Polynomial Regression Models. 5.1 Introduction. 5.2 Curvilinear Trend Model. 5.3 Orthogonal Polynomials. 5.4 Summary. 6. Covariance Pattern Models. 6.1 Introduction. 6.2 Covariance Pattern Models. 6.3 Model Selection. 6.4 Example. 6.5 Summary. 7. Mixed Regression Models with Autocorrelated Errors. 7.1 Introduction. 7.2 MRMs with AC Errors. 7.3 Model Selection. 7.4 Example. 7.5 Summary. 8. Generalized Estimating Equations (GEE) Models. 8.1 Introduction. 8.2 Generalized Linear Models (GLMs). 8.3 Generalized Estimating Equations (GEE) Models. 8.4 GEE Estimation. 8.5 Example. 8.6 Summary. 9. Mixed-Effects Regression Models for Binary Outcomes. 9.1 Introduction. 9.2 Logistic Regression Model. 9.3 Probit Regression Models. 9.4 Threshold Concept. 9.5 Mixed-Effects Logistic Regression Model. 9.6 Estimation. 9.7 Illustration. 9.8 Summary. 10. Mixed-Effects Regression Models for Ordinal Outcomes. 10.1 Introduction. 10.2 Mixed-Effects Proportional Odds Model. 10.3 Psychiatric Example. 10.4 Health Services Research Example. 10.5 Summary. 11. Mixed-Effects Regression Models for Nominal Data. 11.1 Mixed-Effects Multinomial Regression Model. 11.2 Health Services Research Example. 1 1.3 Competing Risk Survival Models. 11.4 Summary. 12. Mixed-effects Regression Models for Counts. 12.1 Poisson Regression Model. 12.2 Modified Poisson Models. 12.3 The ZIP Model. 12.4 Mixed-Effects Models for Counts. 12.5 Illustration. 12.6 Summary. 13. Mixed-Effects Regression Models for Three-Level Data. 13.1 Three-Level Mixed-Effects Linear Regression Model. 13.1.1 Illustration. 13.2 Three-Level Mixed-Effects Nonlinear Regression Models. 13.3 Summary. 14. Missing Data in Longitudinal Studies. 14.1 Introduction. 14.2 Missing Data Mechanisms. 14.3 Models and Missing Data Mechanisms. 14.4 Testing MCAR. 14.5 Models for Nonignorable Missingness. 14.6 Summary. Bibliography. Topic Index.

A Semi-Parametric Shared Parameter Model to Handle Nonmonotone Nonignorable Missingness

Longitudinal studies often generate incomplete response patterns according to a missing not at random mechanism. Shared parameter models provide an appealing framework for the joint modelling of the measurement and missingness processes, especially in the nonmonotone missingness case, and assume a set of random effects to induce the interdependence. Parametric assumptions are typically made for the random effects distribution, violation of which leads to model misspecification with a potential effect on the parameter estimates and standard errors. In this article we avoid any parametric assumption for the random effects distribution and leave it completely unspecified. The estimation of the model is then made using a semi-parametric maximum likelihood method. Our proposal is illustrated on a randomized longitudinal study on patients with rheumatoid arthritis exhibiting nonmonotone missingness.

The practical use of different strategies to handle dropout in longitudinal studies

In the presence of dropout, valid statistical inferences based on longitudinal data can, in general, only be obtained from modeling the measurement process and the dropout process simultaneously. Many models have been proposed in the statistical literature, most of which have been formulated within the framework of selection models or pattern-mixture models. In this paper, we will use continuous data from a longitudinal clinical trial with a 24% dropout rate to illustrate some of the models frequently used in practice. We emphasize the underlying implicit assumptions made by the different approaches, and the sensitivity of the results with respect to these assumptions. The merits and drawbacks of the procedures are extensively discussed and compared from a practical point of view.

The detection of residual serial correlation in linear mixed models

Diggle (1988) described how the empirical semi-variogram of ordinary least squares residuals can be used to suggest an appropriate serial correlation structure in stationary linear mixed models. In this paper, this approach is extended to non-stationary models which include random effects other than intercepts, and will be applied to prostate cancer data, taken from the Baltimore Longitudinal Study of Aging. A simulation study demonstrates the effectiveness of this extended variogram for improving the covariance structure of the linear mixed model used to describe the prostate data.

An overview of group sequential methods in longitudinal clinical trials

During the last decade, several papers have been published on group sequential methods in general and on sequential longitudinal clinical trials in particular. This paper gives an overview of the proposed methods, emphasizing longitudinal clinical trials. Furthermore, it tries to answer some practical questions that may arise during the conduct of interim analyses in longitudinal trials. Simulations have been carried out to obtain insight in these practical considerations.

The Effect of Drop‐Out on the Efficiency of Longitudinal Experiments

It is shown that drop-out often reduces the efficiency of longitudinal experiments considerably. In the framework of linear mixed models, a general, computationally simple method is provided, for designing longitudinal studies when drop-out is to be expected, such that there is little risk of large losses of efficiency due to the missing data. All the results are extensively illustrated using data from a randomized experiment with rats.

Characterization of the peripheral blood transcriptome in a repeated measures design using a panel of healthy individuals

A repeated measures microarray design with 22 healthy, non-smoking volunteers (aging 32±5years) was set up to study transcriptome profiles in whole blood samples. The results indicate that repeatable data can be obtained with high within-subject correlation. Probes that could discriminate between individuals are associated with immune and inflammatory functions. When investigating possible time trends in the microarray data, we have found no differential expression within a sampling period (within-season effect). Differential expression was observed between sampling seasons and the data suggest a weak response of genes related to immune system functioning. Finally, a high number of probes showed significant season-specific expression variability within subjects. Expression variability increased in springtime and there was an association of the probe list with immune system functioning. Our study suggests that the blood transcriptome of healthy individuals is reproducible over a time period of several months.

Pattern‐Mixture Models

Whereas most models for incomplète longitudinal data are formulated within the sélection model framework, pattern-mixture models hâve gained considérable interest in récent years. We outline several stratégies to fit pattern-mixture models, including the so-called identifying-restrictions stratégies. Multiple imputation is used to apply thèse stratégies to real sets of data. Our ideas are exemplified using qualityof-life data from a longitudinal study on metastatic breast cancer patients and using a longitudinal clinical trial in Alzheimer patients.