Marginal Correlation in Longitudinal Binary Data Based on Generalized Linear Mixed Models (original) (raw)
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Biometrical Journal, 2009
Longitudinal trials can yield outcomes that are continuous, binary (yes/no), or are realizations of counts. In this setting we compare three approaches that have been proposed for estimation of the correlation in the framework of generalized estimating equations (GEE): quasi-least squares (QLS), pseudo-likelihood (PL), and an approach we refer to as Wang-Carey (WC). We prove that WC and QLS are identical for the first-order autoregressive AR(1) correlation structure. Using simulations, we then develop guidelines for selection of an appropriate method for analysis of data from a longitudinal trial. In particular, we argue that no method is uniformly superior for analysis of unbalanced and unequally spaced data with a Markov correlation structure. Choice of the best approach will depend on the degree of imbalance and variability in the temporal spacing of measurements, value of the correlation, and type of outcome, e.g. binary or continuous. Finally, we contrast the methods in analysis of a longitudinal study of obesity following renal transplantation in children. ; . In this note, we extend their comparisons to stage two (bias corrected) QLS that was developed in . We also present comparisons for a wider range of values for a. In addition, we consider a higher degree of variability in the temporal spacing of measurements.
Statistical Methods in Medical Research, 2015
Different types of outcomes (e.g. binary, count, continuous) can be simultaneously modeled with multivariate generalized linear mixed models by assuming: (1) same or different link functions, (2) same or different conditional distributions, and (3) conditional independence given random subject effects. Others have used this approach for determining simple associations between subject-specific parameters (e.g. correlations between slopes). We demonstrate how more complex associations (e.g. partial regression coefficients between slopes adjusting for intercepts, time lags of maximum correlation) can be estimated. Reparameterizing the model to directly estimate coefficients allows us to compare standard errors based on the inverse of the Hessian matrix with more usual standard errors approximated by the delta method; a mathematical proof demonstrates their equivalence when the gradient vector approaches zero. Reparameterization also allows us to evaluate significance of coefficients with likelihood ratio tests and to compare this approach with more usual Wald-type t-tests and Fisher's z transformations. Simulations indicate that the delta method and inverse Hessian standard errors are nearly equivalent and consistently overestimate the true standard error. Only the likelihood ratio test based on the reparameterized model has an acceptable type I error rate and is therefore recommended for testing associations between stochastic parameters. Online supplementary materials include our medical data example, annotated code, and simulation details.
Journal of Biopharmaceutical Statistics, 2008
This work investigates how generalizability, an extension of reliability, can be defined and estimated based on longitudinal data sequences resulting from, for example, clinical studies. Useful and intuitive approximate expressions are derived based on generalized linear mixed models. Data from four double-blind randomized clinical trials in schizophrenia motivate the research and are used to estimate generalizability for a binary response parameter.
Pharmaceutical Statistics, 2016
There are various settings in which researchers are interested in the assessment of the correlation between repeated measurements that are taken within the same subject (i.e., reliability). For example, the same rating scale may be used to assess the symptom severity of the same patients by multiple physicians, or the same outcome may be measured repeatedly over time in the same patients. Reliability can be estimated in various ways, e.g., using the classical Pearson correlation or the intra-class correlation in clustered data. However, contemporary data often have a complex structure that goes well beyond the restrictive assumptions that are needed with the more conventional methods to estimate reliability. In the current paper, we propose a general and exible modeling approach that allows for the derivation of reliability estimates, standard errors, and condence intervals appropriately taking hierarchies and covariates in the data into account. Our methodology is developed for continuous outcomes together with covariates of an arbitrary type. The methodology is illustrated in a case study, and a Web Appendix is provided which details the computations using the R package CorrMixed and the SAS software.
Controlled Clinical Trials, 2004
Repeated measures are exploited to study reliability in the context of psychiatric health sciences. It is shown how test -retest reliability can be derived using linear mixed models when the scale is continuous or quasi-continuous. The advantage of this approach is that the full modeling power of mixed models can be used. Repeated measures with a different mean structure can be used to usefully study reliability, correction for covariate effects is possible, and a complicated variance -covariance structure between measurements is allowed. In case the variance structure reduces to a random intercept (compound symmetry), classical methods are recovered. With more complex variance structures (e.g., including random slopes of time and/or serial correlation), time-dependent reliability functions are obtained. The methodology is motivated by and applied to data from five double-blind randomized clinical trials comparing the effects of risperidone to conventional antipsychotic agents for the treatment of chronic schizophrenia. Model assumptions are investigated through residual plots and by investigating the effect of influential observations. D
The Estimation of Reliability in Longitudinal Models
International Journal of Behavioral Development, 1998
Despite the increasing attention devoted to the study and analysis of longitudinal data, relatively little consideration has been directed toward understanding the issues of reliability and measurement error. Perhaps one reason for this neglect has been that traditional methods of estimation (e.g. generalisability theory) require assumptions that are often not tenable in longitudinal designs. This paper first examines applications of generalisability theory to the estimation of m easurement error and reliability in longitudinal research, and notes how factors such as missing data, correlated errors, and true score instability prohibit traditional variance com ponent estimation. Next, we discuss how estimation methods using restricted maximum likelihood can account for these factors, thereby providing m any advantages over traditional estimation methods. Finally, we provide a substantive exam ple illustrating these advantages, and include brief discussions of programming and software...
Statistics in Medicine, 2009
The method of generalized estimating equations (GEE) models the association between the repeated observations on a subject with a patterned correlation matrix. Correct specification of the underlying structure is a potentially beneficial goal, in terms of improving efficiency and enhancing scientific understanding. We consider two sets of criteria that have previously been suggested, respectively, for selecting an appropriate working correlation structure, and for ruling out a particular structure(s), in the GEE analysis of longitudinal studies with binary outcomes. The first selection criterion chooses the structure for which the model-based and the sandwich-based estimator of the covariance matrix of the regression parameter estimator are closest, while the second selection criterion chooses the structure that minimizes the weighted error sum of squares. The rule out criterion deselects structures for which the estimated correlation parameter violates standard constraints for binary data that depend on the marginal means. In addition, we remove structures from consideration if their estimated parameter values yield an estimated correlation structure that is not positive definite. We investigate the performance of the two sets of criteria using both simulated and real data, in the context of a longitudinal trial that compares two treatments for major depressive episode. Practical recommendations are also given on using these criteria to aid in the efficient selection of a working correlation structure in GEE analysis of longitudinal binary data.
2006
It is well-known that the correlation among binary outcomes is constrained by the marginal means, yet approaches such as generalized estimating equations (GEE) do not check that the constraints for the correlations are satisfied. We explore this issue for Markovian dependence in the context of a GEE analysis of a clinical trial that compares Venlafaxine with Lithium in the prevention of major depressive episode. We obtain simplified expressions for the constraints for the logistic model and the equicorrelated and first-order autoregressive correlation structures. We then obtain the limiting values of the GEE and quasi-least squares (QLS) estimates of the correlation parameter when the working structure has been misspecified and prove that misidentification can lead to a severe violation of bounds. As a result, we suggest that violation of bounds can provide additional evidence in ruling out application of a particular working correlation structure. For a structure that is otherwise plausible and results in only a minor violation, we propose an iterative algorithm that yields an estimate that satifies the constraints. We compare our algorithm with two other approaches for estimation of the correlation that have been proposed to avoid a violation of bounds and demonstrate that it estimates the correlation parameter and bivariate probabilities with smaller mean square error and bias, especially when the correlation is large.
Biometrical Journal
Patient-reported outcomes (PROs) are currently being increasingly used as primary outcome measures in observational and experimental studies since they inform clinicians and researchers about the health-status of patients and generate data to facilitate improved care. PROs usually appear as discrete and bounded with U, J or inverse J-shapes and hence, exponential family members offer inadequate distributional fits. The beta-binomial distribution has been proposed in the literature to fit PROs. However, the fact that the beta-binomial distribution does not belong to the exponential family limits its applicability in the regression model context, and classical estimation approaches are not straightforward. Moreover, PROs are usually measured in a longitudinal framework in which individuals are followed up for a certain period. Hence, each individual obtains several scores of the PRO over time, which leads to the repeated-measures and defines the correlation structure in the data. In this work, we have developed and proposed an estimation procedure for the analysis of correlated discrete and bounded outcomes, particularly PROs, by a beta-binomial mixed-effects model. Additionally, we have implemented the methodology in the PROreg package in R. Because there are similar approaches in the literature to address the same issue, this work also incorporates a comparison study between our proposal and alternative methodologies commonly implemented in R and shows the superior performance of our estimation procedure. This paper was motivated by the analysis of the health-status of patients with chronic obstructive pulmonary disease, where the main objective is the assessment of risk factors that may affect the evolution of the disease. The application of the proposed approach in the study leads to clinically relevant results.
A Measure for the Reliability of a Rating Scale Based on Longitudinal Clinical Trial Data
Psychometrika, 2007
A new measure for reliability of a rating scale is introduced, based on the classical definition of reliability, as the ratio of the true score variance and the total variance. Clinical trial data can be employed to estimate the reliability of the scale in use, whenever repeated measurements are taken. The reliability is estimated from the covariance parameters obtained from a linear mixed model. The method provides a single number to express the reliability of the scale, but allows for the study of the reliability’s time evolution. The method is illustrated using a case study in schizophrenia.