A review on linear mixed models for longitudinal data, possibly subject to dropout:
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paper_title: Measuring the quality of life of cancer patients: the Functional Living Index-Cancer: development and validation.

paper_content:

The classical criteria for the evaluation of clinical trials in cancer reflect alterations in physical well-being, but are insensitive to other important factors, such as psychosocial state, sociability, and somatic sensation that may play a critical role in determining the patients' functional response to their illness and its treatment. The Functional Living Index-Cancer is designed for easy, repeated patient self-administration. It is a 22-item questionnaire that has been validated on 837 patients in two cities over a three-year period. Criteria for validity include stability of factor analysis, concurrent validation studies against the Karnofsky, Beck Depression, Spielberger State and Trait Anxiety, and Katz Activities of Daily Living scales, as well as the scaled version of The General Health Questionnaire and The McGill/ Melzack Pain Index. The index is uncontaminated by social desirability issues. The validation studies demonstrate the lack of correlation between traditional measures of patient response and other significant functional factors such as depression and anxiety (r = 0.33), sociability and family interaction, and nausea. These findings elucidate the frequently observed discrepancies between traditional assessments of clinical response and overall functional patient outcome. The index is proposed as an adjunct to clinical trials assessment and may provide additional patient functional information on which to analyse the outcome of clinical trials or offer specific advice to individual patients.
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paper_title: Mixed-effects regression models for studying the natural history of prostate disease

paper_content:

Although prostate cancer and benign prostatic hyperplasia are major health problems in U.S. men, little is known about the early stages of the natural history of prostate disease. A molecular biomarker called prostate specific antigen (PSA), together with a unique longitudinal bank of frozen serum, now allows a historic prospective study of changes in PSA levels for decades prior to the diagnosis of prostate disease. Linear mixed-effects regression models were used to test whether rates of change in PSA were different in men with and without prostate disease. In addition, since the prostate cancer cases developed their tumours at different (and unknown) times during their periods of follow-up, a piece-wise non-linear mixed-effects regression model was used to estimate the time when rapid increases in PSA were first observable beyond the background level of PSA change. These methods have a wide range of applications in biomedical research utilizing repeated measures data such as pharmacokinetic studies, crossover trials, growth and development studies, aging studies, and disease detection.
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paper_title: Normal Human Aging: The Baltimore Longitudinal Study of Aging

paper_content:

The two outstanding longitudinal studies in the United States are the Framingham Study of Cardiovascular Disease and the Baltimore Longitudinal Study of Aging (BLSA). The latter deserves to be better known. Hopefully, this volume, a bargain at $18, will serve as an introduction to the BLSA. Started in 1958 with a group of volunteers, the BLSA admittedly lacks the breadth one would like to see in an epidemiologic study. The volunteers were all male (females were not admitted until 1978) and mostly middle-class and professional, and thus the study has a restricted socioeconomic base. However, with this restricted study population, the BLSA has served as a useful antidote to the many studies that have been based on such skewed samples as persons in nursing homes or old patients from hospitals mostly serving the indigent. It is apparent that only a select group would have volunteered for this kind of study
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paper_title: Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters

paper_content:

0 and ai. The parameter 0, upon which all the distributions depend, is called "structural"; the parameters {aiI} are called "incidental". Throughout this paper we shall assume that the Xi, are independently distributed when 0, a1, *** a., are given, and shall consider the problem of consisteently estimating 0 (as n --* X ). The chance variables {Xij} and the parameters 0 and {fa} may be vectors. However, for simplicity of exposition we shall throughout this paper, except in Example 2, assume that they are scalars. Obvious changes will suffice to treat the vector case. Very many interesting problems are subsumed under the above formulation. Among these is the following:
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paper_title: Nonparametric Maximum Likelihood Estimation of a Mixing Distribution

paper_content:

Abstract The nonparametric maximum likelihood estimate of a mixing distribution is shown to be self-consistent, a property which characterizes the nonparametric maximum likelihood estimate of a distribution function in incomplete data problems. Under various conditions the estimate is a step function, with a finite number of steps. Its computation is illustrated with a small example.
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paper_title: Flexible Modelling of the Covariance Matrix in a Linear Random Effects Model

paper_content:

A flexible approach is proposed for modelling the covariance matrix of a linear mixed model for longitudinal data. The method combines parametric modelling of the random effects part with flexible modelling of the serial correlation component. The approach is exemplified on weight gain data and on the evolution of height of children in their first year of life of the Jimma Infant Survival Study, an Ethiopian cohort study. The analyses show the usefulness of the approach.
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paper_title: A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population

paper_content:

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.
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paper_title: Semiparametric Estimation in the Rasch Model and Related Exponential Response Models, Including a Simple Latent Class Model for Item Analysis

paper_content:

Abstract The Rasch model for item analysis is an important member of the class of exponential response models in which the number of nuisance parameters increases with the number of subjects, leading to the failure of the usual likelihood methodology. Both conditional-likelihood methods and mixture-model techniques have been used to circumvent these problems. In this article, we show that these seemingly unrelated analyses are in fact closely linked to each other, despite dramatic structural differences between the classes of models implied by each approach. We show that the finite-mixture model for J dichotomous items having T latent classes gives the same estimates of item parameters as conditional likelihood on a set whose probability approaches one if T ≥ (J + 1)/2. Unconditional maximum likelihood estimators for the finite-mixture model can be viewed as Keifer-Wolfowitz estimators for the random-effects version of the Rasch model. Latent-class versions of the model are especially attractive when T is...
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paper_title: A Smooth Nonparametric Estimate of a Mixing Distribution Using Mixtures of Gaussians

paper_content:

Abstract We propose a method of estimating mixing distributions using maximum likelihood over the class of arbitrary mixtures of Gaussians subject to the constraint that the component variances be greater than or equal to some minimum value h. This approach can lead to estimates of many shapes, with smoothness controlled by parameter h. We show that the resulting estimate will always be a finite mixture of Gaussians, each having variance h. The nonparametric maximum likelihood estimate can be viewed as a special case, with h = 0. The method can be extended to estimate multivariate mixing distributions. Examples and the results of a simulation study are presented.
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paper_title: Mixed-effects regression models for studying the natural history of prostate disease

paper_content:

Although prostate cancer and benign prostatic hyperplasia are major health problems in U.S. men, little is known about the early stages of the natural history of prostate disease. A molecular biomarker called prostate specific antigen (PSA), together with a unique longitudinal bank of frozen serum, now allows a historic prospective study of changes in PSA levels for decades prior to the diagnosis of prostate disease. Linear mixed-effects regression models were used to test whether rates of change in PSA were different in men with and without prostate disease. In addition, since the prostate cancer cases developed their tumours at different (and unknown) times during their periods of follow-up, a piece-wise non-linear mixed-effects regression model was used to estimate the time when rapid increases in PSA were first observable beyond the background level of PSA change. These methods have a wide range of applications in biomedical research utilizing repeated measures data such as pharmacokinetic studies, crossover trials, growth and development studies, aging studies, and disease detection.
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paper_title: The detection of residual serial correlation in linear mixed models

paper_content:

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. © 1998 John Wiley & Sons, Ltd.
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paper_title: Conditional Linear Mixed Models

paper_content:

The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences among subjects at the start of the study (cross-sectional effects). In observational studies, however, longitudinal changes need to be studied after correction for potential important cross-sectional differences between subjects. It will be shown that, in the context of linear mixed models, the estimation of longitudinal effects may be highly influenced by the assumptions about cross-sectional effects. Furthermore, aspects from conditional and mixture inference will be combined, yielding so-called conditional linear mixed models that allow estimation of longitudinal effects (average trends as well as subject-specific trends), independent of any cross-sectional assumptions. These models will be introduced and justified, and extensively illustrated in the analysis of longitudinal data from 680 participants in the Baltimore Longitudinal Study of Aging.
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paper_title: The Effect of Drop-Out on the Efficiency of Longitudinal Experiments

paper_content:

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.
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paper_title: Small sample inference for fixed effects from restricted maximum likelihood.

paper_content:

Restricted maximum likelihood (REML) is now well established as a method for estimating the parameters of the general Gaussian linear model with a structured covariance matrix, in particular for mixed linear models. Conventionally, estimates of precision and inference for fixed effects are based on their asymptotic distribution, which is known to be inadequate for some small-sample problems. In this paper, we present a scaled Wald statistic, together with an F approximation to its sampling distribution, that is shown to perform well in a range of small sample settings. The statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias. This approach has the advantage that it reproduces both the statistics and F distributions in those settings where the latter is exact, namely for Hotelling T2 type statistics and for analysis of variance F-ratios. The performance of the modified statistics is assessed through simulation studies of four different REML analyses and the methods are illustrated using three examples.
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paper_title: Recovery of inter-block information when block sizes are unequal

paper_content:

SUMMARY A method is proposed for estimating intra-block and inter-block weights in the analysis of incomplete block designs with block sizes not necessarily equal. The method consists of maximizing the likelihood, not of all the data, but of a set of selected error contrasts. When block sizes are equal results are identical with those obtained by the method of Nelder (1968) for generally balanced designs. Although mainly concerned with incomplete block designs the paper also gives in outline an extension of the modified maximum likelihood procedure to designs with a more complicated block structure. In this paper we consider the estimation of weights to be used in the recovery of interblock information in incomplete block designs with possibly unequal block sizes. The problem can also be thought of as one of estimating constants and components of variance from data arranged in a general two-way classification when the effects of one classification are regarded as fixed and the effects of the second classification are regarded as random. Nelder (1968) described the efficient estimation of weights in generally balanced designs, in which the blocks are usually, although not always, of equal size. Lack of balance resulting from unequal block sizes is, however, common in some experimental work, for example in animal breeding experiments. The maximum likelihood procedure described by Hartley & Rao (1967) can be used but does not give the same estimates as Nelder's method in the balanced case. As will be shown, the two methods in effect use the same weighted sums of squares of residuals but assign different expectations. In the maximum likelihood approach, expectations are taken over a conditional distribution with the treatment effects fixed at their estimated values. In contrast Nelder uses unconditional expectations. The difference between the two methods is analogous to the well-known difference between two methods of estimating the variance o2 of a normal distribution, given a random sample of n values. Both methods use the same total sum of squares of deviations. But
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paper_title: Synthesis of Variance

paper_content:

The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student “t” and the Fisher “z” tests to a wider range of problems.
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paper_title: Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems

paper_content:

Abstract Recent developments promise to increase greatly the popularity of maximum likelihood (ml) as a technique for estimating variance components. Patterson and Thompson (1971) proposed a restricted maximum likelihood (reml) approach which takes into account the loss in degrees of freedom resulting from estimating fixed effects. Miller (1973) developed a satisfactory asymptotic theory for ml estimators of variance components. There are many iterative algorithms that can be considered for computing the ml or reml estimates. The computations on each iteration of these algorithms are those associated with computing estimates of fixed and random effects for given values of the variance components.
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paper_title: Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood

paper_content:

Likelihood ratio tests for fixed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation. Bartlett‐type adjustments, using an approximate decomposition of the data, are developed for the test statistics. A simulation study is used to compare properties of the test statistics proposed, with or without adjustment, with a Wald test. A proposed test statistic constructed by dropping fixed terms from the full fixed model is shown to give a better approximation to the asymptotic χ2‐distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the χ2‐approximation for the proposed tests substantially.
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paper_title: Bayesian Data Analysis

paper_content:

FUNDAMENTALS OF BAYESIAN INFERENCE Probability and Inference Single-Parameter Models Introduction to Multiparameter Models Asymptotics and Connections to Non-Bayesian Approaches Hierarchical Models FUNDAMENTALS OF BAYESIAN DATA ANALYSIS Model Checking Evaluating, Comparing, and Expanding Models Modeling Accounting for Data Collection Decision Analysis ADVANCED COMPUTATION Introduction to Bayesian Computation Basics of Markov Chain Simulation Computationally Efficient Markov Chain Simulation Modal and Distributional Approximations REGRESSION MODELS Introduction to Regression Models Hierarchical Linear Models Generalized Linear Models Models for Robust Inference Models for Missing Data NONLINEAR AND NONPARAMETRIC MODELS Parametric Nonlinear Models Basic Function Models Gaussian Process Models Finite Mixture Models Dirichlet Process Models APPENDICES A: Standard Probability Distributions B: Outline of Proofs of Asymptotic Theorems C: Computation in R and Stan Bibliographic Notes and Exercises appear at the end of each chapter.
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paper_title: Bayesian Inference in Statistical Analysis

paper_content:

Nature of Bayesian Inference Standard Normal Theory Inference Problems Bayesian Assessment of Assumptions: Effect of Non-Normality on Inferences About a Population Mean with Generalizations Bayesian Assessment of Assumptions: Comparison of Variances Random Effect Models Analysis of Cross Classification Designs Inference About Means with Information from More than One Source: One-Way Classification and Block Designs Some Aspects of Multivariate Analysis Estimation of Common Regression Coefficients Transformation of Data Tables References Indexes.
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paper_title: Bayesian Data Analysis

paper_content:

FUNDAMENTALS OF BAYESIAN INFERENCE Probability and Inference Single-Parameter Models Introduction to Multiparameter Models Asymptotics and Connections to Non-Bayesian Approaches Hierarchical Models FUNDAMENTALS OF BAYESIAN DATA ANALYSIS Model Checking Evaluating, Comparing, and Expanding Models Modeling Accounting for Data Collection Decision Analysis ADVANCED COMPUTATION Introduction to Bayesian Computation Basics of Markov Chain Simulation Computationally Efficient Markov Chain Simulation Modal and Distributional Approximations REGRESSION MODELS Introduction to Regression Models Hierarchical Linear Models Generalized Linear Models Models for Robust Inference Models for Missing Data NONLINEAR AND NONPARAMETRIC MODELS Parametric Nonlinear Models Basic Function Models Gaussian Process Models Finite Mixture Models Dirichlet Process Models APPENDICES A: Standard Probability Distributions B: Outline of Proofs of Asymptotic Theorems C: Computation in R and Stan Bibliographic Notes and Exercises appear at the end of each chapter.
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paper_title: Empirical Bayes estimation of individual growth-curve parameters and their relationship to covariates.

paper_content:

The analysis of growth curves has long been important in biostatistics. Work has focused on two problems: the estimation of individual curves based on many data points, and the estimation of the mean growth curve for a group of individuals. This paper extends a recent approach that seeks to combine data from a group of individuals in order to improve the estimates of individual growth parameters. Growth is modeled as polynomial in time, and the group model is also linear, incorporating growth-related covariates into the model. The estimation used is empirical Bayes. The estimation formulas are illustrated with a set of data on rat growth, originally presented by Box (1950, Biometrics 6, 362-389).
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paper_title: A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population

paper_content:

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.
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paper_title: Mixed-effects regression models for studying the natural history of prostate disease

paper_content:

Although prostate cancer and benign prostatic hyperplasia are major health problems in U.S. men, little is known about the early stages of the natural history of prostate disease. A molecular biomarker called prostate specific antigen (PSA), together with a unique longitudinal bank of frozen serum, now allows a historic prospective study of changes in PSA levels for decades prior to the diagnosis of prostate disease. Linear mixed-effects regression models were used to test whether rates of change in PSA were different in men with and without prostate disease. In addition, since the prostate cancer cases developed their tumours at different (and unknown) times during their periods of follow-up, a piece-wise non-linear mixed-effects regression model was used to estimate the time when rapid increases in PSA were first observable beyond the background level of PSA change. These methods have a wide range of applications in biomedical research utilizing repeated measures data such as pharmacokinetic studies, crossover trials, growth and development studies, aging studies, and disease detection.
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paper_title: The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data

paper_content:

Maximum likelihood estimators for fixed effects and variance components in linear mixed models, obtained under the assumption of normally distributed random effects, are shown to be consistent and asymptotically normally distributed, even when the random-effects distribution is not normal. However, a sandwich-type correction to the inverse Fisher information matrix is then needed in order to get the correct asymptotic covariance matrix. Extensive simulations show that the so-obtained corrected standard errors are clearly superior to the naive uncorrected ones, especially for the parameters in the random-effects covariance matrix, even in moderate samples.
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paper_title: The analysis of incomplete data.

paper_content:

A non-woven mesh reflector for radio waves comprises two or more parallelly positioned layers of electrically conductive high modulus, high yield strength, inextensible fibers. The fibers in each layer extend parallel to each other and the fibers of one layer extend in a direction different from the direction of the fibers of the other layer. Examples of fibers include wires of beryllium, aluminum, stainless steel type 304, CHROMEL R, INVAR 36, and other alloys of the stainless steel INVAR type.
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paper_title: Missing Observations in Multivariate Statistics I. Review of the Literature

paper_content:

Abstract In this paper we review the literature on the problem of handling multivariate data with observations missing on some or all of the variables under study. We examine the ways that statisticians have devised to estimate means, variances, correlations and linear regression functions from such data and refer to specific computer programs for carrying out the estimation. We show how the estimation problems can be simplified if the missing data follows certain patterns. Finally, we outline the statistical properties of the various estimators.
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paper_title: Pattern-Mixture Models for Multivariate Incomplete Data

paper_content:

Consider a random sample on variables X1, …, Xv with some values of Xv missing. Selection models specify the distribution of X1 , …, XV over respondents and nonrespondents to Xv , and the conditional distribution that Xv is missing given X1 , …, Xv . In contrast, pattern-mixture models specify the conditional distribution of X 1, …, Xv given that XV is observed or missing respectively and the marginal distribution of the binary indicator for whether or not Xv is missing. For multivariate data with a general pattern of missing values, the literature has tended to adopt the selection-modeling approach (see for example Little and Rubin); here, pattern-mixture models are proposed for this more general problem. Pattern-mixture models are chronically underidentified; in particular for the case of univariate nonresponse mentioned above, there are no data on the distribution of Xv given X1 , …, XV–1 , in the stratum with Xv missing. Thus the models require restrictions or prior information to identify the paramet...
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paper_title: Modeling the drop-out mechanism in repeated-measures studies

paper_content:

Abstract Subjects often drop out of longitudinal studies prematurely, yielding unbalanced data with unequal numbers of measures for each subject. Modern software programs for handling unbalanced longitudinal data improve on methods that discard the incomplete cases by including all the data, but also yield biased inferences under plausible models for the drop-out process. This article discusses methods that simultaneously model the data and the drop-out process within a unified model-based framework. Models are classified into two broad classes—random-coefficient selection models and random-coefficient pattern-mixture models—depending on how the joint distribution of the data and drop-out mechanism is factored. Inference is likelihood-based, via maximum likelihood or Bayesian methods. A number of examples in the literature are placed in this framework, and possible extensions outlined. Data collection on the nature of the drop-out process is advocated to guide the choice of model. In cases where the drop-...
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paper_title: The Calculation of Posterior Distributions by Data Augmentation

paper_content:

Abstract The idea of data augmentation arises naturally in missing value problems, as exemplified by the standard ways of filling in missing cells in balanced two-way tables. Thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze. This device is used to great advantage by the EM algorithm (Dempster, Laird, and Rubin 1977) in solving maximum likelihood problems. In situations when the likelihood cannot be approximated closely by the normal likelihood, maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements. From the Bayesian point of view, one must now calculate the posterior distribution of parameters of interest. If data augmentation can be used in the calculation of the maximum likelihood estimate, then in the same cases one ought to be able to use it in the computation of the posterior distribution. It is the purpose of this article to explain how this can be done. The basic idea ...
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paper_title: Parametric models for incomplete continuous and categorical longitudinal data

paper_content:

This paper reviews models for incomplete continuous and categorical longitudinal data. In terms of Rubin's classification of missing value processes we are specifically concerned with the problem of nonrandom missingness. A distinction is drawn between the classes of selection and pattern-mixture models and, using several examples, these approaches are compared and contrasted. The central roles of identifiability and sensitivity are emphasized throughout.
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paper_title: Pattern-Mixture Models for Multivariate Incomplete Data

paper_content:

Consider a random sample on variables X1, …, Xv with some values of Xv missing. Selection models specify the distribution of X1 , …, XV over respondents and nonrespondents to Xv , and the conditional distribution that Xv is missing given X1 , …, Xv . In contrast, pattern-mixture models specify the conditional distribution of X 1, …, Xv given that XV is observed or missing respectively and the marginal distribution of the binary indicator for whether or not Xv is missing. For multivariate data with a general pattern of missing values, the literature has tended to adopt the selection-modeling approach (see for example Little and Rubin); here, pattern-mixture models are proposed for this more general problem. Pattern-mixture models are chronically underidentified; in particular for the case of univariate nonresponse mentioned above, there are no data on the distribution of Xv given X1 , …, XV–1 , in the stratum with Xv missing. Thus the models require restrictions or prior information to identify the paramet...
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paper_title: MIXTURE MODELS FOR THE JOINT DISTRIBUTION OF REPEATED MEASURES AND EVENT TIMES

paper_content:

Many long-term clinical trials collect both a vector of repeated measurements and an event time on each subject; often, the two outcomes are dependent. One example is the use of surrogate markers to predict disease onset or survival. Another is longitudinal trials which have outcome-related dropout. We describe a mixture model for the joint distribution which accommodates incomplete repeated measures and right-censored event times, and provide methods for full maximum likelihood estimation. The methods are illustrated through analysis of data from a clinical trial for a new schizophrenia therapy; in the trial, dropout time is closely related to outcome, and the dropout process differs between treatments. The parameter estimates from the model are used to make a treatment comparison after adjusting for the effects of dropout. An added benefit of the analysis is that it permits using the repeated measures to increase efficiency of estimates of the event time distribution. © 1997 by John Wiley & Sons, Ltd.
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paper_title: The Muscatine children’s obesity data reanalysed using pattern mixture models

paper_content:

A set of longitudinal binary, partially incomplete, data on obesity among children in the USA is reanalysed. The multivariate Bernoulli distribution is parameterized by the univariate marginal probabilities and dependence ratios of all orders, which together support maximum likelihood inference. The temporal association of obesity is strong and complex but stationary. We fit a saturated model for the distribution of response patterns and find that non-response is missing completely at random for boys but that the probability of obesity is consistently higher among girls who provided incomplete records than among girls who provided complete records. We discuss the statistical and substantive features of, respectively, pattern mixture and selection models for this data set.
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paper_title: Modeling the drop-out mechanism in repeated-measures studies

paper_content:

Abstract Subjects often drop out of longitudinal studies prematurely, yielding unbalanced data with unequal numbers of measures for each subject. Modern software programs for handling unbalanced longitudinal data improve on methods that discard the incomplete cases by including all the data, but also yield biased inferences under plausible models for the drop-out process. This article discusses methods that simultaneously model the data and the drop-out process within a unified model-based framework. Models are classified into two broad classes—random-coefficient selection models and random-coefficient pattern-mixture models—depending on how the joint distribution of the data and drop-out mechanism is factored. Inference is likelihood-based, via maximum likelihood or Bayesian methods. A number of examples in the literature are placed in this framework, and possible extensions outlined. Data collection on the nature of the drop-out process is advocated to guide the choice of model. In cases where the drop-...
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paper_title: Modelling progression of CD4-lymphocyte count and its relationship to survival time.

paper_content:

The purpose of this article is to model the progression of CD4-lymphocyte count and the relationship between different features of this progression and survival time. The complicating factors in this analysis are that the CD4-lymphocyte count is observed only at certain fixed times and with a high degree of measurement error, and that the length of the vector of observations is determined, in part, by the length of survival. If probability of death depends on the true, unobserved CD4-lymphocyte count, then the survival process must be modelled. Wu and Carroll (1988, Biometrics 44, 175-188) proposed a random effects model for two-sample longitudinal data in the presence of informative censoring, in which the individual effects included only slopes and intercepts. We propose methods for fitting a broad class of models of this type, in which both the repeated CD4-lymphocyte counts and the survival time are modelled using random effects. These methods permit us to estimate parameters describing the progression of CD4-lymphocyte count as well as the effect of differences in the CD4 trajectory on survival. We apply these methods to results of AIDS clinical trials.
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paper_title: Inference from Nonrandomly Missing Categorical Data: An Example from a Genetic Study on Turner's Syndrome

paper_content:

Abstract The process leading to partial classification with categorical data is sometimes nonrandom. A particular model accounting for incomplete data, which allows the probability of uncertain classification to depend on category identity, is utilized for an analysis of data obtained from a genetic study on Turner's syndrome. Estimates of population proportions are obtained from maximum likelihood. A method for handling nonrandomly missing data arrayed in contingency tables is discussed. Sensitivity analyses incorporating parameters related to the missing-data mechanism are recommended for estimation and testing.
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paper_title: Pattern-Mixture Models for Multivariate Incomplete Data

paper_content:

Consider a random sample on variables X1, …, Xv with some values of Xv missing. Selection models specify the distribution of X1 , …, XV over respondents and nonrespondents to Xv , and the conditional distribution that Xv is missing given X1 , …, Xv . In contrast, pattern-mixture models specify the conditional distribution of X 1, …, Xv given that XV is observed or missing respectively and the marginal distribution of the binary indicator for whether or not Xv is missing. For multivariate data with a general pattern of missing values, the literature has tended to adopt the selection-modeling approach (see for example Little and Rubin); here, pattern-mixture models are proposed for this more general problem. Pattern-mixture models are chronically underidentified; in particular for the case of univariate nonresponse mentioned above, there are no data on the distribution of Xv given X1 , …, XV–1 , in the stratum with Xv missing. Thus the models require restrictions or prior information to identify the paramet...
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paper_title: Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process

paper_content:

Abstract : In estimating and comparing the rates of change of a continuous variable between two groups, the unweighted averages of individual simple least squares estimates from each group are often used. Under a linear random effects model, when all individuals have completed observations at identical time points these statistics are maximum likelihood estimates for the expected rates of change. However, with censored of missing data, these estimates are no longer efficient when compared to generalized least squares estimates. When, in addition, the right censoring process is dependent upon the individual rates of change (i.e., informative right censoring), the generalized least squares estimates will be biased. Likelihood ratio test for informativeness of the censoring process and maximum likelihood estimates for the expected rates of change and the parameters of the right censoring process are developed under a linear random effect models with a probit model for the right censoring process. In realistic situations, we illustrate that the bias in estimating group rate of change and the reduction of power in comparing group difference could be substantial when strong dependency of the right censoring process on individual rates of change is ignored. (Author)
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paper_title: Sensitivity Analysis for Nonrandom Dropout: A Local Influence Approach

paper_content:

Diggle and Kenward (1994, Applied Statistics 43, 49-93) proposed a selection model for continuous longitudinal data subject to nonrandom dropout. It has provoked a large debate about the role for such models. The original enthusiasm was followed by skepticism about the strong but untestable assumptions on which this type of model invariably rests. Since then, the view has emerged that these models should ideally be made part of a sensitivity analysis. This paper presents a formal and flexible approach to such a sensitivity assessment based on local influence (Cook, 1986, Journal of the Royal Statistical Society, Series B 48, 133-169). The influence of perturbing a missing-at-random dropout model in the direction of nonrandom dropout is explored. The method is applied to data from a randomized experiment on the inhibition of testosterone production in rats.
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paper_title: Strategies to fit pattern-mixture models.

paper_content:

Whereas most models for incomplete longitudinal data are formulated within the selection model framework, pattern-mixture models have gained considerable interest in recent years (Little, 1993, 1994). In this paper, we outline several strategies to fit pattern-mixture models, including the so-called identifying restrictions strategy. Multiple imputation is used to apply this strategy to realistic settings, such as quality-of-life data from a longitudinal study on metastatic breast cancer patients.
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paper_title: Logistic regression with incompletely observed categorical covariates--investigating the sensitivity against violation of the missing at random assumption.

paper_content:

Missing values in the covariates are a widespread complication in the statistical inference of regression models. The maximum likelihood principle requires specification of the distribution of the covariates, at least in part. For categorical covariates, log-linear models can be used. Additionally, the missing at random assumption is necessary, which excludes a dependence of the occurrence of missing values on the unobserved covariate values. This assumption is often highly questionable. We present a framework to specify alternative missing value mechanisms such that maximum likelihood estimation of the regression parameters under a specified alternative is possible. This allows investigation of the sensitivity of a single estimate against violations of the missing at random assumption. The possible results of a sensitivity analysis are illustrated by artificial examples. The practical application is demonstrated by the analysis of two case-control studies.
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paper_title: Modeling the drop-out mechanism in repeated-measures studies

paper_content:

Abstract Subjects often drop out of longitudinal studies prematurely, yielding unbalanced data with unequal numbers of measures for each subject. Modern software programs for handling unbalanced longitudinal data improve on methods that discard the incomplete cases by including all the data, but also yield biased inferences under plausible models for the drop-out process. This article discusses methods that simultaneously model the data and the drop-out process within a unified model-based framework. Models are classified into two broad classes—random-coefficient selection models and random-coefficient pattern-mixture models—depending on how the joint distribution of the data and drop-out mechanism is factored. Inference is likelihood-based, via maximum likelihood or Bayesian methods. A number of examples in the literature are placed in this framework, and possible extensions outlined. Data collection on the nature of the drop-out process is advocated to guide the choice of model. In cases where the drop-...
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paper_title: Regression Models for Longitudinal Binary Responses with Informative Drop‐Outs

paper_content:

This paper reviews both likelihood-based and non-likelihood (generalized estimating equations) regression models for longitudinal binary responses when there are drop-outs. Throughout, it is assumed that the regression parameters for the marginal expectations of the binary responses are of primary scientific interest. The association or time dependence between the responses is largely regarded as a nuisance characteristic of the data. The performance of the methods is compared, in terms of asymptotic bias, under mis-specification of the association between the responses and the missing data mechanism or drop-out process.
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paper_title: Parametric models for incomplete continuous and categorical longitudinal data

paper_content:

This paper reviews models for incomplete continuous and categorical longitudinal data. In terms of Rubin's classification of missing value processes we are specifically concerned with the problem of nonrandom missingness. A distinction is drawn between the classes of selection and pattern-mixture models and, using several examples, these approaches are compared and contrasted. The central roles of identifiability and sensitivity are emphasized throughout.
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paper_title: Local influence in linear mixed models.

paper_content:

The linear mixed model has become an important tool in modelling, partially due to the introduction of the SAS procedure MIXED, which made the method widely available to practising statisticians. Its growing popularity calls for data-analytic methods to check the underlying assumptions and robustness. Here, the problem of detecting influential subjects in the context of longitudinal data is considered, following the approach of local influence proposed by Cook (1986, Journal of the Royal Statistical Society, Series B 48, 133-169). The diagnostics are illustrated on a prostate cancer data set.
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paper_title: Sensitivity Analysis for Nonrandom Dropout: A Local Influence Approach

paper_content:

Diggle and Kenward (1994, Applied Statistics 43, 49-93) proposed a selection model for continuous longitudinal data subject to nonrandom dropout. It has provoked a large debate about the role for such models. The original enthusiasm was followed by skepticism about the strong but untestable assumptions on which this type of model invariably rests. Since then, the view has emerged that these models should ideally be made part of a sensitivity analysis. This paper presents a formal and flexible approach to such a sensitivity assessment based on local influence (Cook, 1986, Journal of the Royal Statistical Society, Series B 48, 133-169). The influence of perturbing a missing-at-random dropout model in the direction of nonrandom dropout is explored. The method is applied to data from a randomized experiment on the inhibition of testosterone production in rats.
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paper_title: Assessment of Local Influence

paper_content:

SUMMARY Statistical models usually involve some degree of approximation and therefore are nearly always wrong. Because of this inexactness, an assessment of the influence of minor perturbations of the model is important. We discuss a method for carrying out such an assessment. The method is not restricted to linear regression models, and it seems to provide a relatively simple, unified approach for handling a variety of problems.
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paper_title: Selection models for repeated measurements with non-random dropout: an illustration of sensitivity

paper_content:

The outcome-based selection model of Diggle and Kenward for repeated measurements with non-random dropout is applied to a very simple example concerning the occurrence of mastitis in dairy cows, in which the occurrence of mastitis can be modelled as a dropout process. It is shown through sensitivity analysis how the conclusions concerning the dropout mechanism depend crucially on untestable distributional assumptions. This example is exceptional in that from a simple plot of the data two outlying observations can be identified that are the source of the apparent evidence for non-random dropout and also provide an explanation of the behaviour of the sensitivity analysis. It is concluded that a plausible model for the data does not require the assumption of non-random dropout.
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paper_title: Local influence in linear mixed models.

paper_content:

The linear mixed model has become an important tool in modelling, partially due to the introduction of the SAS procedure MIXED, which made the method widely available to practising statisticians. Its growing popularity calls for data-analytic methods to check the underlying assumptions and robustness. Here, the problem of detecting influential subjects in the context of longitudinal data is considered, following the approach of local influence proposed by Cook (1986, Journal of the Royal Statistical Society, Series B 48, 133-169). The diagnostics are illustrated on a prostate cancer data set.
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paper_title: The Analysis of Designed Experiments and Longitudinal Data by Using Smoothing Splines

paper_content:

In designed experiments and in particular longitudinal studies, the aim may be to assess the effect of a quantitative variable such as time on treatment effects. Modelling treatment effects can be complex in the presence of other sources of variation. Three examples are presented to illustrate an approach to analysis in such cases. The first example is a longitudinal experiment on the growth of cows under a factorial treatment structure where serial correlation and variance heterogeneity complicate the analysis. The second example involves the calibration of optical density and the concentration of a protein DNase in the presence of sampling variation and variance heterogeneity. The final example is a multienvironment agricultural field experiment in which a yield-seeding rate relationship is required for several varieties of lupins. Spatial variation within environments, heterogeneity between environments and variation between varieties all need to be incorporated in the analysis. In this paper, the cubic smoothing spline is used in conjunction with fixed and random effects, random coefficients and variance modelling to provide simultaneous modelling of trends and covariance structure. The key result that allows coherent and flexible empirical model building in complex situations is the linear mixed model representation of the cubic smoothing spline. An extension is proposed in which trend is partitioned into smooth and nonsmooth components. Estimation and inference, the analysis of the three examples and a discussion of extensions and unresolved issues are also presented.
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paper_title: A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population

paper_content:

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.
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paper_title: Assessment of Local Influence

paper_content:

SUMMARY Statistical models usually involve some degree of approximation and therefore are nearly always wrong. Because of this inexactness, an assessment of the influence of minor perturbations of the model is important. We discuss a method for carrying out such an assessment. The method is not restricted to linear regression models, and it seems to provide a relatively simple, unified approach for handling a variety of problems.
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paper_title: A Smooth Nonparametric Estimate of a Mixing Distribution Using Mixtures of Gaussians

paper_content:

Abstract We propose a method of estimating mixing distributions using maximum likelihood over the class of arbitrary mixtures of Gaussians subject to the constraint that the component variances be greater than or equal to some minimum value h. This approach can lead to estimates of many shapes, with smoothness controlled by parameter h. We show that the resulting estimate will always be a finite mixture of Gaussians, each having variance h. The nonparametric maximum likelihood estimate can be viewed as a special case, with h = 0. The method can be extended to estimate multivariate mixing distributions. Examples and the results of a simulation study are presented.
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paper_title: The detection of residual serial correlation in linear mixed models

paper_content:

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. © 1998 John Wiley & Sons, Ltd.
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paper_title: Conditional Linear Mixed Models

paper_content:

The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences among subjects at the start of the study (cross-sectional effects). In observational studies, however, longitudinal changes need to be studied after correction for potential important cross-sectional differences between subjects. It will be shown that, in the context of linear mixed models, the estimation of longitudinal effects may be highly influenced by the assumptions about cross-sectional effects. Furthermore, aspects from conditional and mixture inference will be combined, yielding so-called conditional linear mixed models that allow estimation of longitudinal effects (average trends as well as subject-specific trends), independent of any cross-sectional assumptions. These models will be introduced and justified, and extensively illustrated in the analysis of longitudinal data from 680 participants in the Baltimore Longitudinal Study of Aging.
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<format>
Title: A Review on Linear Mixed Models for Longitudinal Data, Possibly Subject to Dropout

Section 1: Introduction
Description 1: Introduce the use of longitudinal studies in medical science and present examples; explain the motivation behind the review and outline the paper's structure.

Section 2: The Linear Mixed Model
Description 2: Provide a detailed explanation of the linear mixed model formulation, including fixed effects, random effects, and model assumptions.

Section 3: Inference for the Marginal Model
Description 3: Discuss methods for deriving inferences for the marginal model, including maximum likelihood estimation and restricted maximum likelihood estimation (REML).

Section 4: Inference for the Random Effects
Description 4: Explore the estimation of random effects and their interpretation, including empirical Bayes estimates and the concept of shrinkage.

Section 5: The Missing Data Problem
Description 5: Address general issues related to missing data in longitudinal studies, classify different types of missing data, and introduce conceptual approaches for handling them.

Section 6: Selection Models
Description 6: Review selection models for handling dropout in longitudinal data, focusing on methods such as the Tobit model and the Diggle and Kenward selection model.

Section 7: Pattern-Mixture Models
Description 7: Explain pattern-mixture models as an alternative framework to handle dropout; illustrate with applications to practical data sets.

Section 8: Sensitivity Analysis
Description 8: Highlight the importance of sensitivity analysis when dealing with incomplete longitudinal data; outline different strategies and tools for conducting such analyses.

Section 9: Local Influence
Description 9: Discuss the concept of local influence and how small model perturbations can impact parameter estimates; introduce methods for detecting influential data points.

Section 10: Computational Issues
Description 10: Discuss computational challenges in fitting linear mixed models, including starting values, convergence problems, and the need for advanced numerical procedures.

Section 11: Conclusion
Description 11: Summarize the key points discussed in the paper; emphasize the practical importance and current research directions related to linear mixed models for longitudinal data.
</format>
