Sample size estimation for a two-group comparison of repeated count outcomes using GEE

Randomized clinical trials with count measurements as the primary outcome are common in various medical areas such as seizure counts in epilepsy trials, or relapse counts in multiple sclerosis trials. Controlled clinical trials frequently use a conventional parallel-group design that assigns subjects randomly to one of two treatment groups and repeatedly evaluates them at baseline and intervals across a treatment period of a fixed duration. The primary interest is to compare the rates of change between treatment groups. Generalized estimating equations (GEEs) have been widely used to compare rates of change between treatment groups because of its robustness to misspecification of the true correlation structure. In this paper, we derive a sample size formula for comparing the rates of change between two groups in a repeatedly measured count outcome using GEE. The sample size formula incorporates general missing patterns such as independent missing and monotone missing, and general correlation structures such as AR(1) and compound symmetry (CS). The performance of the sample size formula is evaluated through simulation studies. Sample size estimation is illustrated by a clinical trial example from epilepsy.     

Sample size calculations for time-averaged difference of longitudinal binary outcomes

In clinical trials with repeated measurements, the responses from each subject are measured multiple times during the study period. Two approaches have been widely used to assess the treatment effect, one that compares the rate of change between two groups and the other that tests the time-averaged difference (TAD). While sample size calculations based on comparing the rate of change between two groups have been reported by many investigators, the literature has paid relatively little attention to the sample size estimation for time-averaged difference (TAD) in the presence of heterogeneous correlation structure and missing data in repeated measurement studies. In this study, we investigate sample size calculation for the comparison of time-averaged responses between treatment groups in clinical trials with longitudinally observed binary outcomes. The generalized estimating equation (GEE) approach is used to derive a closed-form sample size formula, which is flexible enough to account for arbitrary missing patterns and correlation structures. In particular, we demonstrate that the proposed sample size can accommodate a mixture of missing patterns, which is frequently encountered by practitioners in clinical trials. To our knowledge, this is the first study that considers the mixture of missing patterns in sample size calculation. Our simulation shows that the nominal power and type I error are well preserved over a wide range of design parameters. Sample size calculation is illustrated through an example.     

Power and sample size for GEE analysis of incomplete paired outcomes in 2 × 2 crossover trials

The 2 × 2 crossover trial uses subjects as their own control to reduce the intersubject variability in the treatment comparison, and typically requires fewer subjects than a parallel design. The generalized estimating equations (GEE) methodology has been commonly used to analyze incomplete discrete outcomes from crossover trials. We propose a unified approach to the power and sample size determination for the Wald Z-test and t-test from GEE analysis of paired binary, ordinal and count outcomes in crossover trials. The proposed method allows misspecification of the variance and correlation of the outcomes, missing outcomes, and adjustment for the period effect. We demonstrate that misspecification of the working variance and correlation functions leads to no or minimal efficiency loss in GEE analysis of paired outcomes. In general, GEE requires the assumption of missing completely at random. For bivariate binary outcomes, we show by simulation that the GEE estimate is asymptotically unbiased or only minimally biased, and the proposed sample size method is suitable under missing at random (MAR) if the working correlation is correctly specified. The performance of the proposed method is illustrated with several numerical examples. Adaption of the method to other paired outcomes is discussed.     

Comparing alternating logistic regressions to other approaches to modelling correlated binary data

Alternating logistic regressions (ALRs) seem to offer some of the advantages of marginal models estimated via generalized estimating equations (GEE) and generalized linear mixed models (GLMMs). Via simulation study we compared ALRs to marginal models estimated via GEE and subject-specific models estimated via GLMMs, with a focus on estimation of the correlation structure in three-level data sets (e.g. students in classes in schools). Data set size and structure, and amount of correlation in the data sets were varied. For simple correlation structures, ALRs performed well. For three-level correlation structures, all approaches, but especially ALRs, had difficulty assigning the correlation to the correct level, though sample sizes used were small. In addition, ALRs and GEEs had trouble attaching correct inference to the mean effects, though this improved as overall sample size improved. ALRs are a valuable addition to the data analyst's toolkit, though care should be taken when modelling data with three-level structures.     

Bias from the use of generalized estimating equations to analyze incomplete longitudinal binary data

Patient dropout is a common problem in studies that collect repeated binary measurements. Generalized estimating equations (GEE) are often used to analyze such data. The dropout mechanism may be plausibly missing at random (MAR), i.e. unrelated to future measurements given covariates and past measurements. In this case, various authors have recommended weighted GEE with weights based on an assumed dropout model, or an imputation approach, or a doubly robust approach based on weighting and imputation. These approaches provide asymptotically unbiased inference, provided the dropout or imputation model (as appropriate) is correctly specified. Other authors have suggested that, provided the working correlation structure is correctly specified, GEE using an improved estimator of the correlation parameters (‘modified GEE’) show minimal bias. These modified GEE have not been thoroughly examined. In this paper, we study the asymptotic bias under MAR dropout of these modified GEE, the standard GEE, and also GEE using the true correlation. We demonstrate that all three methods are biased in general. The modified GEE may be preferred to the standard GEE and are subject to only minimal bias in many MAR scenarios but in others are substantially biased. Hence, we recommend the modified GEE be used with caution.     

Comparison of methods for analyzing binary repeated measures data: A simulation-based study (comparison of methods for binary repeated measures)

In this study, some methods suggested for binary repeated measures, namely, Weighted Least Squares (WLS), Generalized Estimating Equations (GEE), and Generalized Linear Mixed Models (GLMM) are compared with respect to power, type 1 error, and properties of estimates. The results indicate that with adequate sample size, no missing data, the only covariate being time effect, and a relatively limited number of time points, the WLS method performs well. The GEE approach performs well only for large sample sizes. The GLMM method is satisfactory with respect to type I error, but its estimates have poorer properties than the other methods.     

Learning causal structure from mixed data with missing values using Gaussian copula models

We consider the problem of causal structure learning from data with missing values, assumed to be drawn from a Gaussian copula model. First, we extend the ‘Rank PC’ algorithm, designed for Gaussian copula models with purely continuous data (so-called nonparanormal models), to incomplete data by applying rank correlation to pairwise complete observations and replacing the sample size with an effective sample size in the conditional independence tests to account for the information loss from missing values. When the data are missing completely at random (MCAR), we provide an error bound on the accuracy of ‘Rank PC’ and show its high-dimensional consistency. However, when the data are missing at random (MAR), ‘Rank PC’ fails dramatically. Therefore, we propose a Gibbs sampling procedure to draw correlation matrix samples from mixed data that still works correctly under MAR. These samples are translated into an average correlation matrix and an effective sample size, resulting in the ‘Copula PC’ algorithm for incomplete data. Simulation study shows that: (1) ‘Copula PC’ estimates a more accurate correlation matrix and causal structure than ‘Rank PC’ under MCAR and, even more so, under MAR and (2) the usage of the effective sample size significantly improves the performance of ‘Rank PC’ and ‘Copula PC.’ We illustrate our methods on two real-world datasets: riboflavin production data and chronic fatigue syndrome data.

     

A Two-Latent-Class Model for Smoking Cessation Data with Informative Dropouts

Non ignorable missing data is a common problem in longitudinal studies. Latent class models are attractive for simplifying the modeling of missing data when the data are subject to either a monotone or intermittent missing data pattern. In our study, we propose a new two-latent-class model for categorical data with informative dropouts, dividing the observed data into two latent classes; one class in which the outcomes are deterministic and a second one in which the outcomes can be modeled using logistic regression. In the model, the latent classes connect the longitudinal responses and the missingness process under the assumption of conditional independence. Parameters are estimated by the method of maximum likelihood estimation based on the above assumptions and the tetrachoric correlation between responses within the same subject. We compare the proposed method with the shared parameter model and the weighted GEE model using the areas under the ROC curves in the simulations and the application to the smoking cessation data set. The simulation results indicate that the proposed two-latent-class model performs well under different missing procedures. The application results show that our proposed method is better than the shared parameter model and the weighted GEE model.     

The impact of dichotomization in longitudinal data analysis: a simulation study

In this paper, a simulation study is conducted to systematically investigate the impact of dichotomizing longitudinal continuous outcome variables under various types of missing data mechanisms. Generalized linear models (GLM) with standard generalized estimating equations (GEE) are widely used for longitudinal outcome analysis, but these semi‐parametric approaches are only valid under missing data completely at random (MCAR). Alternatively, weighted GEE (WGEE) and multiple imputation GEE (MI‐GEE) were developed to ensure validity under missing at random (MAR). Using a simulation study, the performance of standard GEE, WGEE and MI‐GEE on incomplete longitudinal dichotomized outcome analysis is evaluated. For comparisons, likelihood‐based linear mixed effects models (LMM) are used for incomplete longitudinal original continuous outcome analysis. Focusing on dichotomized outcome analysis, MI‐GEE with original continuous missing data imputation procedure provides well controlled test sizes and more stable power estimates compared with any other GEE‐based approaches. It is also shown that dichotomizing longitudinal continuous outcome will result in substantial loss of power compared with LMM. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison of GEE1 and GEE2 estimation applied to clustered logistic regression

Generalized estimating equations (GEE) have become a popular method for marginal regression modelling of data that occur in clusters. Features of the GEE methodology are the use of a ‘working covariance’, an approximation to the underlying covariance, which is used to improve the efficiency in estimating the regression coefficients, and the ‘sandwich’ estimate of variance, which provides a way of consistently estimating their standard errors. These techniques have been extended to include estimating equations for the underlying correlation structure, both to improve the efficiency of the regression coefficient estimates and to provide estimates of correlations between units in a cluster, when these are of interest. If the mean structure is of primary interest, then a simpler set of equations (GEE1) can be used, whereas if the underlying covariance structure is of interest in its own right, the use of the more complex GEE2 estimating equations is often recommended. In this paper, we compare the effect of increasing the complexity of the ‘working covariances’ on the variance of the parameter estimates, as well as the mean-squared error of the ‘sandwich’ estimate of variance. We give asymptotic expressions for these variances and mean-squared error terms. We use these to study the behaviour of different variants of GEE1 and GEE2 when we change the number of clusters, the cluster size, and the within-cluster correlation. We conclude that the extra complexity of the full GEE2 approach is not usually justified if the mean structure is of primary interest.     

Marginal regression models with a time to event outcome and discrete multiple source predictors

Information from multiple informants is frequently used to assess psychopathology. We consider marginal regression models with multiple informants as discrete predictors and a time to event outcome. We fit these models to data from the Stirling County Study; specifically, the models predict mortality from self report of psychiatric disorders and also predict mortality from physician report of psychiatric disorders. Previously, Horton et al. found little relationship between self and physician reports of psychopathology, but that the relationship of self report of psychopathology with mortality was similar to that of physician report of psychopathology with mortality. Generalized estimating equations (GEE) have been used to fit marginal models with multiple informant covariates; here we develop a maximum likelihood (ML) approach and show how it relates to the GEE approach. In a simple setting using a saturated model, the ML approach can be constructed to provide estimates that match those found using GEE. We extend the ML technique to consider multiple informant predictors with missingness and compare the method to using inverse probability weighted (IPW) GEE. Our simulation study illustrates that IPW GEE loses little efficiency compared with ML in the presence of monotone missingness. Our example data has non-monotone missingness; in this case, ML offers a modest decrease in variance compared with IPW GEE, particularly for estimating covariates in the marginal models. In more general settings, e.g., categorical predictors and piecewise exponential models, the likelihood parameters from the ML technique do not have the same interpretation as the GEE. Thus, the GEE is recommended to fit marginal models for its flexibility, ease of interpretation and comparable efficiency to ML in the presence of missing data.     

Modification of GEE1 and linear mixed-effects models for heteroscedastic longitudinal Gaussian data

A characterization of GLMs is given. Modification of the Gaussian GEE1, modified GEE1, was applied to heteroscedastic longitudinal data, to which linear mixed-effects models are usually applied. The modified GEE1 models scale multivariate data to homoscedastic data maintaining the correlation structure and apply usual GEE1 to homoscedastic data, which needs no-diagnostics for diagonal variances. Relationships among multivariate linear regression methods, ordinary/generalized LS, naïve/modified GEE1, and linear mixed-effects models were discussed. An application showed modified GEE1 gave most efficient parameter estimation. Correct specification of the main diagonals of heteroscedastic data variance appears to be more important for efficient mean parameter estimation.     

An R package for model fitting,model selection and the simulation for longitudinal data with dropout missingness

Abstract

Missing data arise frequently in clinical and epidemiological fields, in particular in longitudinal studies. This paper describes the core features of an R package wgeesel, which implements marginal model fitting (i.e., weighted generalized estimating equations, WGEE; doubly robust GEE) for longitudinal data with dropouts under the assumption of missing at random. More importantly, this package comprehensively provide existing information criteria for WGEE model selection on marginal mean or correlation structures. Also, it can serve as a valuable tool for simulating longitudinal data with missing outcomes. Lastly, a real data example and simulations are presented to illustrate and validate our package.     

Optimal model averaging estimation for correlation structure in generalized estimating equations

Longitudinal data analysis requires a proper estimation of the within-cluster correlation structure in order to achieve efficient estimates of the regression parameters. When applying likelihood-based methods one may select an optimal correlation structure by the AIC or BIC. However, such information criteria are not applicable for estimating equation based approaches. In this paper we develop a model averaging approach to estimate the correlation matrix by a weighted sum of a group of patterned correlation matrices under the GEE framework. The optimal weight is determined by minimizing the difference between the weighted sum and a consistent yet inefficient estimator of the correlation structure. The computation of our proposed approach only involves a standard quadratic programming on top of the standard GEE procedure and can be easily implemented in practice. We provide theoretical justifications and extensive numerical simulations to support the application of the proposed estimator. A couple of well-known longitudinal data sets are revisited where we implement and illustrate our methodology.     

A Distribution-free Approach in Statistical Modelling with Repeated Measurements and Missing Values

Mixed effect models, which contain both fixed effects and random effects, are frequently used in dealing with correlated data arising from repeated measurements (made on the same statistical units). In mixed effect models, the distributions of the random effects need to be specified and they are often assumed to be normal. The analysis of correlated data from repeated measurements can also be done with GEE by assuming any type of correlation as initial input. Both mixed effect models and GEE are approaches requiring distribution specifications (likelihood, score function). In this article, we consider a distribution-free least square approach under a general setting with missing value allowed. This approach does not require the specifications of the distributions and initial correlation input. Consistency and asymptotic normality of the estimation are discussed.     

The pseudo‐GEE approach to the analysis of longitudinal surveys

Longitudinal surveys have emerged in recent years as an important data collection tool for population studies where the primary interest is to examine population changes over time at the individual level. Longitudinal data are often analyzed through the generalized estimating equations (GEE) approach. The vast majority of existing literature on the GEE method; however, is developed under non‐survey settings and are inappropriate for data collected through complex sampling designs. In this paper the authors develop a pseudo‐GEE approach for the analysis of survey data. They show that survey weights must and can be appropriately accounted in the GEE method under a joint randomization framework. The consistency of the resulting pseudo‐GEE estimators is established under the proposed framework. Linearization variance estimators are developed for the pseudo‐GEE estimators when the finite population sampling fractions are small or negligible, a scenario often held for large‐scale surveys. Finite sample performances of the proposed estimators are investigated through an extensive simulation study using data from the National Longitudinal Survey of Children and Youth. The results show that the pseudo‐GEE estimators and the linearization variance estimators perform well under several sampling designs and for both continuous and binary responses. The Canadian Journal of Statistics 38: 540–554; 2010 © 2010 Statistical Society of Canada     

Robust Small Sample Inference for Generalised Estimating Equations: An Application of the Anova‐type Test

Asymptotically, the Wald‐type test for generalised estimating equations (GEE) models can control the type I error rate at the nominal level. However in small sample studies, it may lead to inflated type I error rates. Even with currently available small sample corrections for the GEE Wald‐type test, the type I error rate inflation is still serious when the tested contrast is multidimensional. This paper extends the ANOVA‐type test for heteroscedastic factorial designs to GEE and shows that the proposed ANOVA‐type test can also control the type I error rate at the nominal level in small sample studies while still maintaining robustness with respect to mis‐specification of the working correlation matrix. Differences of inference between the Wald‐type test and the proposed test are observed in a two‐way repeated measures ANOVA model for a diet‐induced obesity study and a two‐way repeated measures logistic regression for a collagen‐induced arthritis study. Simulation studies confirm that the proposed test has better control of the type I error rate than the Wald‐type test in small sample repeated measures models. Additional simulation studies further show that the proposed test can even achieve larger power than the Wald‐type test in some cases of the large sample repeated measures ANOVA models that were investigated.     

Model selection information criteria in latent class models with missing data and contingency question

Latent class analysis (LCA) has been found to have important applications in social and behavioural sciences for modelling categorical response variables, and non-response is typical when collecting data. In this study, the non-response mainly included ‘contingency questions’ and real ‘missing data’. The primary objective of this study was to evaluate the effects of some potential factors on model selection indices in LCA with non-response data. We simulated missing data with contingency question and evaluated the accuracy rates of eight information criteria for selecting the correct models. The results showed that the main factors are latent class proportions, conditional probabilities, sample size, the number of items, the missing data rate and the contingency data rate. Interactions of the conditional probabilities with class proportions, sample size and the number of items are also significant. From our simulation results, the impact of missing data and contingency questions can be amended by increasing the sample size or the number of items.     

A Note on Sample Size Determination for the Estimation of the Mean Vector of a Multivariate Population

In this article, we present a straightforward Bonferroni approach for determining sample size for estimating the mean vector of a multivariate population under two scenarios: (1) a pre-specified overall confidence level is desired; and (2) a pre-specified confidence level needs to be guaranteed for each individual variable. It is demonstrated that correlation between variables helps reduce the sample size. The formula to calculate the reduced sample size is derived. A binormal example is presented to illustrate the effect of correlation on sample size reduction for various values of the correlation coefficient.     

Methods for repeated measures data analysis with missing values

There are various techniques for dealing with incomplete data; some are computationally highly intensive and others are not as computationally intensive, while all may be comparable in their efficiencies. In spite of these developments, analysis using only the complete data subset is performed when using popular statistical software. In an attempt to demonstrate the efficiencies and advantages of using all available data, we compared several approaches that are relatively simple but efficient alternatives to those using the complete data subset for analyzing repeated measures data with missing values, under the assumption of a multivariate normal distribution of the data. We also assumed that the missing values occur in a monotonic pattern and completely at random. The incomplete data procedure is demonstrated to be more powerful than the procedure of using the complete data subset, generally when the within-subject correlation gets large. One other principal finding is that even with small sample data, for which various covariance models may be indistinguishable, the empirical size and power are shown to be sensitive to misspecified assumptions about the covariance structure. Overall, the testing procedures that do not assume any particular covariance structure are shown to be more robust in keeping the empirical size at the nominal level than those assuming a special structure.