Sample size estimation for GEE method for comparing slopes in repeated measurements data

Sample size calculation is an important component at the design stage of clinical trials. Controlled clinical trials often use a repeated measurement design in which individuals are randomly assigned to treatment groups and followed-up for measurements at intervals across a treatment period of fixed duration. In studies with repeated measurements, one of the popular primary interests is the comparison of the rates of change in a response variable between groups. Statistical models for calculating sample sizes for repeated measurement designs often fail to take into account the impact of missing data correctly. In this paper we propose to use the generalized estimating equation (GEE) method in comparing the rates of change in repeated measurements and introduce closed form formulae for sample size and power that can be calculated using a scientific calculator. Since the sample size formula is based on asymptotic theory, we investigate the performance of the estimated sample size in practical settings through simulations.     

Analysis of incomplete longitudinal binary data using multiple imputation

We propose a propensity score-based multiple imputation (MI) method to tackle incomplete missing data resulting from drop-outs and/or intermittent skipped visits in longitudinal clinical trials with binary responses. The estimation and inferential properties of the proposed method are contrasted via simulation with those of the commonly used complete-case (CC) and generalized estimating equations (GEE) methods. Three key results are noted. First, if data are missing completely at random, MI can be notably more efficient than the CC and GEE methods. Second, with small samples, GEE often fails due to 'convergence problems', but MI is free of that problem. Finally, if the data are missing at random, while the CC and GEE methods yield results with moderate to large bias, MI generally yields results with negligible bias. A numerical example with real data is provided for illustration.     

Missing data and sensitivity analysis for binary data with implications for sample size and power of randomized clinical trials

Despite our best efforts, missing outcomes are common in randomized controlled clinical trials. The National Research Council's Committee on National Statistics panel report titled The Prevention and Treatment of Missing Data in Clinical Trials noted that further research is required to assess the impact of missing data on the power of clinical trials and how to set useful target rates and acceptable rates of missing data in clinical trials. In this article, using binary responses for illustration, we establish that conclusions based on statistical analyses that include only complete cases can be seriously misleading, and that the adverse impact of missing data grows not only with increasing rates of missingness but also with increasing sample size. We illustrate how principled sensitivity analysis can be used to assess the robustness of the conclusions. Finally, we illustrate how sample sizes can be adjusted to account for expected rates of missingness. We find that when sensitivity analyses are considered as part of the primary analysis, the required adjustments to the sample size are dramatically larger than those that are traditionally used. Furthermore, in some cases, especially in large trials with small target effect sizes, it is impossible to achieve the desired power.     

A new approach for sizing trials with composite binary endpoints using anticipated marginal values and accounting for the correlation between components

Composite binary endpoints are increasingly used as primary endpoints in clinical trials. When designing a trial, it is crucial to determine the appropriate sample size for testing the statistical differences between treatment groups for the primary endpoint. As shown in this work, when using a composite binary endpoint to size a trial, one needs to specify the event rates and the effect sizes of the composite components as well as the correlation between them. In practice, the marginal parameters of the components can be obtained from previous studies or pilot trials; however, the correlation is often not previously reported and thus usually unknown. We first show that the sample size for composite binary endpoints is strongly dependent on the correlation and, second, that slight deviations in the prior information on the marginal parameters may result in underpowered trials for achieving the study objectives at a pre-specified significance level. We propose a general strategy for calculating the required sample size when the correlation is not specified and accounting for uncertainty in the marginal parameter values. We present the web platform CompARE to characterize composite endpoints and to calculate the sample size just as we propose in this paper. We evaluate the performance of the proposal with a simulation study and illustrate it by means of a real case study using CompARE.     

Correlation structure and variable selection in generalized estimating equations via composite likelihood information criteria

The method of generalized estimating equations (GEE) is popular in the biostatistics literature for analyzing longitudinal binary and count data. It assumes a generalized linear model for the outcome variable, and a working correlation among repeated measurements. In this paper, we introduce a viable competitor: the weighted scores method for generalized linear model margins. We weight the univariate score equations using a working discretized multivariate normal model that is a proper multivariate model. Because the weighted scores method is a parametric method based on likelihood, we propose composite likelihood information criteria as an intermediate step for model selection. The same criteria can be used for both correlation structure and variable selection. Simulations studies and the application example show that our method outperforms other existing model selection methods in GEE. From the example, it can be seen that our methods not only improve on GEE in terms of interpretability and efficiency but also can change the inferential conclusions with respect to GEE. Copyright © 2016 John Wiley & Sons, Ltd.     

广义估计方程与多水平logistic回归模型在临床纵向数据分析中的应用及比较

目的探讨广义估计方程和多水平模型的应用与临床纵向研究以解决个体重复观测数据内部的相关性问题。方法根据临床纵向实例数据的特点,拟合因变量为二分类的广义估计方程和多水平模型,并与一般logistic模型比较。结果广义估计方程和多水平模型的分析结果与一般logistic模型不同。由于未能考虑个体内重复观测数据的相关性,一般logistic模型错误显示临床分期与近期疗效相关,而广义估计方程和多水平模型分析结果则显示相关无统计学意义。经分层分析也未发现临床分期与近期疗效的关联。结论广义估计方程和多水平模型都能有效地考虑重复观测数据内部相关性并能处理有缺失值的资料。与多水平模型相比,广义估计方程的参数估计较为稳定,可有效的估计各解释变量的效应。     

Performance of weighted estimating equations for longitudinal binary data with drop-outs missing at random

The generalized estimating equations (GEE) approach is commonly used to model incomplete longitudinal binary data. When drop-outs are missing at random through dependence on observed responses (MAR), GEE may give biased parameter estimates in the model for the marginal means. A weighted estimating equations approach gives consistent estimation under MAR when the drop-out mechanism is correctly specified. In this approach, observations or person-visits are weighted inversely proportional to their probability of being observed. Using a simulation study, we compare the performance of unweighted and weighted GEE in models for time-specific means of a repeated binary response with MAR drop-outs. Weighted GEE resulted in smaller finite sample bias than GEE. However, when the drop-out model was misspecified, weighted GEE sometimes performed worse than GEE. Weighted GEE with observation-level weights gave more efficient estimates than a weighted GEE procedure with cluster-level weights.     

Small sample performance of bias‐corrected sandwich estimators for cluster‐randomized trials with binary outcomes

The sandwich estimator in generalized estimating equations (GEE) approach underestimates the true variance in small samples and consequently results in inflated type I error rates in hypothesis testing. This fact limits the application of the GEE in cluster‐randomized trials (CRTs) with few clusters. Under various CRT scenarios with correlated binary outcomes, we evaluate the small sample properties of the GEE Wald tests using bias‐corrected sandwich estimators. Our results suggest that the GEE Wald z‐test should be avoided in the analyses of CRTs with few clusters even when bias‐corrected sandwich estimators are used. With t‐distribution approximation, the Kauermann and Carroll (KC)‐correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes. However, in cases with large variations in cluster sizes, the Fay and Graubard (FG)‐correction should be used instead. Furthermore, we derive a formula to calculate the power and minimum total number of clusters one needs using the t‐test and KC‐correction for the CRTs with binary outcomes. The power levels as predicted by the proposed formula agree well with the empirical powers from the simulations. The proposed methods are illustrated using real CRT data. We conclude that with appropriate control of type I error rates under small sample sizes, we recommend the use of GEE approach in CRTs with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure. Copyright © 2014 John Wiley & Sons, Ltd.     

Sample size and power determination in joint modeling of longitudinal and survival data

Owing to the rapid development of biomarkers in clinical trials, joint modeling of longitudinal and survival data has gained its popularity in the recent years because it reduces bias and provides improvements of efficiency in the assessment of treatment effects and other prognostic factors. Although much effort has been put into inferential methods in joint modeling, such as estimation and hypothesis testing, design aspects have not been formally considered. Statistical design, such as sample size and power calculations, is a crucial first step in clinical trials. In this paper, we derive a closed-form sample size formula for estimating the effect of the longitudinal process in joint modeling, and extend Schoenfeld's sample size formula to the joint modeling setting for estimating the overall treatment effect. The sample size formula we develop is quite general, allowing for p-degree polynomial trajectories. The robustness of our model is demonstrated in simulation studies with linear and quadratic trajectories. We discuss the impact of the within-subject variability on power and data collection strategies, such as spacing and frequency of repeated measurements, in order to maximize the power. When the within-subject variability is large, different data collection strategies can influence the power of the study in a significant way. Optimal frequency of repeated measurements also depends on the nature of the trajectory with higher polynomial trajectories and larger measurement error requiring more frequent measurements.     

On assessing model fit for distribution‐free longitudinal models under missing data

The generalized estimating equation (GEE), a distribution‐free, or semi‐parametric, approach for modeling longitudinal data, is used in a wide range of behavioral, psychotherapy, pharmaceutical drug safety, and healthcare‐related research studies. Most popular methods for assessing model fit are based on the likelihood function for parametric models, rendering them inappropriate for distribution‐free GEE. One rare exception is a score statistic initially proposed by Tsiatis for logistic regression (1980) and later extended by Barnhart and Willamson to GEE (1998). Because GEE only provides valid inference under the missing completely at random assumption and missing values arising in most longitudinal studies do not follow such a restricted mechanism, this GEE‐based score test has very limited applications in practice. We propose extensions of this goodness‐of‐fit test to address missing data under the missing at random assumption, a more realistic model that applies to most studies in practice. We examine the performance of the proposed tests using simulated data and demonstrate the utilities of such tests with data from a real study on geriatric depression and associated medical comorbidities. Copyright © 2013 John Wiley & Sons, Ltd.     

An appraisal of methods for the analysis of longitudinal categorical data with MAR drop-outs

A number of methods for analysing longitudinal ordinal categorical data with missing-at-random drop-outs are considered. Two are maximum-likelihood methods (MAXLIK) which employ marginal global odds ratios to model associations. The remainder use weighted or unweighted generalized estimating equations (GEE). Two of the GEE use Cholesky-decomposed standardized residuals to model the association structure, while another three extend methods developed for longitudinal binary data in which the association structures are modelled using either Gaussian estimation, multivariate normal estimating equations or conditional residuals. Simulated data sets were used to discover differences among the methods in terms of biases, variances and convergence rates when the association structure is misspecified. The methods were also applied to a real medical data set. Two of the GEE methods, referred to as Cond and ML-norm in this paper and by their originators, were found to have relatively good convergence rates and mean squared errors for all sample sizes (80, 120, 300) considered, and one more, referred to as MGEE in this paper and by its originators, worked fairly well for all but the smallest sample size, 80.     

Allowing for uncertainty due to missing continuous outcome data in pairwise and network meta‐analysis

Missing outcome data are commonly encountered in randomized controlled trials and hence may need to be addressed in a meta‐analysis of multiple trials. A common and simple approach to deal with missing data is to restrict analysis to individuals for whom the outcome was obtained (complete case analysis). However, estimated treatment effects from complete case analyses are potentially biased if informative missing data are ignored. We develop methods for estimating meta‐analytic summary treatment effects for continuous outcomes in the presence of missing data for some of the individuals within the trials. We build on a method previously developed for binary outcomes, which quantifies the degree of departure from a missing at random assumption via the informative missingness odds ratio. Our new model quantifies the degree of departure from missing at random using either an informative missingness difference of means or an informative missingness ratio of means, both of which relate the mean value of the missing outcome data to that of the observed data. We propose estimating the treatment effects, adjusted for informative missingness, and their standard errors by a Taylor series approximation and by a Monte Carlo method. We apply the methodology to examples of both pairwise and network meta‐analysis with multi‐arm trials. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

GEE estimation of a misspecified time-varying covariate: an example with the effect of alcoholism treatment on medical utilization

The generalized estimation equation (GEE) method is widely used in longitudinal data analysis, particularly when the outcome variable is non-Gaussian distributed. Under mild regulatory conditions, the parameter estimates are consistent and their asymptotic variances are efficient. In an observational study focusing on alcoholism patients, we applied the GEE method to longitudinal count data from medical utilization records from a large national managed care organization. The health services research question was whether there was a change in medical utilization for patients after engaging in alcoholism treatment as compared to before treatment. Thus, the main effect of interest was a time-varying covariate indicating whether the patient had undergone treatment yet or not. GEE under five different working correlations was employed and mixed results regarding the significance of the treatment effect were found. Because of the large sample size, i.e. 8485 patients with an average of 46 repeated measurements per patient, differences across the estimates produced by the different working correlation structures was suspicious. It is shown that these differences are maybe caused by the fact that the time-varying covariate in the marginal mean model is misspecified. A simulation study is performed to demonstrate that misspecification of the time-varying covariate in the marginal mean structure can cause differences in GEE results across various choices of working correlation structure.     

Concordance correlation coefficients estimated by generalized estimating equations and variance components for longitudinal repeated measurements

The concordance correlation coefficient (CCC) is a commonly accepted measure of agreement between two observers for continuous responses. This paper proposes a generalized estimating equations (GEE) approach allowing dependency between repeated measurements over time to assess intra‐agreement for each observer and inter‐ and total agreement among multiple observers simultaneously. Furthermore, the indices of intra‐, inter‐, and total agreement through variance components (VC) from an extended three‐way linear mixed model (LMM) are also developed with consideration of the correlation structure of longitudinal repeated measurements. Simulation studies are conducted to compare the performance of the GEE and VC approaches for repeated measurements from longitudinal data. An application of optometric conformity study is used for illustration. In conclusion, the GEE approach allowing flexibility in model assumptions and correlation structures of repeated measurements gives satisfactory results with small mean square errors and nominal 95% coverage rates for large data sets, and when the assumption of the relationship between variances and covariances for the extended three‐way LMM holds, the VC approach performs outstandingly well for all sample sizes. Copyright © 2017 John Wiley & Sons, Ltd.     

Incorporating pragmatic features into power analysis for cluster randomized trials with a count outcome

Cluster randomized designs are frequently employed in pragmatic clinical trials which test interventions in the full spectrum of everyday clinical settings in order to maximize applicability and generalizability. In this study, we propose to directly incorporate pragmatic features into power analysis for cluster randomized trials with count outcomes. The pragmatic features considered include arbitrary randomization ratio, overdispersion, random variability in cluster size, and unequal lengths of follow-up over which the count outcome is measured. The proposed method is developed based on generalized estimating equation (GEE) and it is advantageous in that the sample size formula retains a closed form, facilitating its implementation in pragmatic trials. We theoretically explore the impact of various pragmatic features on sample size requirements. An efficient Jackknife algorithm is presented to address the problem of underestimated variance by the GEE sandwich estimator when the number of clusters is small. We assess the performance of the proposed sample size method through extensive simulation and an application example to a real clinical trial is presented.     

A Bayesian dose‐finding design for phase I/II clinical trials with nonignorable dropouts

Phase I/II trials utilize both toxicity and efficacy data to achieve efficient dose finding. However, due to the requirement of assessing efficacy outcome, which often takes a long period of time to be evaluated, the duration of phase I/II trials is often longer than that of the conventional dose‐finding trials. As a result, phase I/II trials are susceptible to the missing data problem caused by patient dropout, and the missing efficacy outcomes are often nonignorable in the sense that patients who do not experience treatment efficacy are more likely to drop out of the trial. We propose a Bayesian phase I/II trial design to accommodate nonignorable dropouts. We treat toxicity as a binary outcome and efficacy as a time‐to‐event outcome. We model the marginal distribution of toxicity using a logistic regression and jointly model the times to efficacy and dropout using proportional hazard models to adjust for nonignorable dropouts. The correlation between times to efficacy and dropout is modeled using a shared frailty. We propose a two‐stage dose‐finding algorithm to adaptively assign patients to desirable doses. Simulation studies show that the proposed design has desirable operating characteristics. Our design selects the target dose with a high probability and assigns most patients to the target dose. Copyright © 2015 John Wiley & Sons, Ltd.     

Accounting for expected attrition in the planning of community intervention trials

Trials in which intact communities are the units of randomization are increasingly being used to evaluate interventions which are more naturally administered at the community level, or when there is a substantial risk of treatment contamination. In this article we focus on the planning of community intervention trials in which k communities (for example, medical practices, worksites, or villages) are to be randomly allocated to each of an intervention and a control group, and fixed cohorts of m individuals enrolled in each community prior to randomization. Formulas to determine k or m may be obtained by adjusting standard sample size formulas to account for the intracluster correlation coefficient rho. In the presence of individual-level attrition however, observed cohort sizes are likely to vary. We show that conventional approaches of accounting for potential attrition, such as dividing standard sample size formulas by the anticipated follow-up rate pi or using the average anticipated cohort size m pi, may, respectively, overestimate or underestimate the required sample size when cluster follow-up rates are highly variable, and m or rho are large. We present new sample size estimation formulas for the comparison of two means or two proportions, which appropriately account for variation among cluster follow-up rates. These formulas are derived by specifying a model for the binary missingness indicators under the population-averaged approach, assuming an exchangeable intracluster correlation coefficient, denoted by tau. To aid in the planning of future trials, we recommend that estimates for tau be reported in published community intervention trials.     

广义估计方程在临床试验重复测量资料中的应用

目的：探讨广义估计方程在临床试验重复测量资料分析中的应用。方法：利用广义估计方程分析结果指标为分类变量的重复测量资料，通过参数和标准误的估计得出统计学结论。结果：对于临床试验重复测量资料，广义估计方程能有效的考虑组内相关性，处理有缺失值的资料，可以获得中心效应的参数及其标准误的估计值，以及在考虑了中心效应之后，可以有效估计处理因素有无作用及其作用大小。结论：采用广义估计方程对临床试验重复测量资料进行统计分析，可以使药物疗效评价更为客观。     

Coping with missing data in clinical trials: a model-based approach applied to asthma trials

In most clinical trials, some patients do not complete their intended follow-up according to protocol, for a variety of reasons, and are often described as having 'dropped out' before the conclusion of the trial. Their subsequent measurements are missing, and this makes the analysis of the trial's repeated measures data more difficult. In this paper we briefly review the reasons for patient drop-out, and their implications for some commonly used methods of analysis. We then propose a class of models for modelling both the response to treatment and the drop-out process. Such models are readily fitted in a Bayesian framework using non-informative priors with the software BUGS. The results from such models are then compared with the results of standard methods for dealing with missing data in clinical trials, such as last observation carried forward. We further propose the use of a time transformation to linearize an asymptotic pattern of repeated measures over time and therefore simplify the modelling. All these ideas are illustrated using data from a five-arm asthma clinical trial.     

The analysis of cross-over trials with baseline measurements

The statistical power of cross-over trials can be increased by taking 'baseline' measurements of the outcome variable at the start of each treatment period. Analysis of covariance (ANCOVA), rather than analysis of change scores, takes best advantage of this. However, ANCOVA can give biased treatment effect estimates in observational studies with true baseline imbalance. While in truth balanced, chance baseline imbalance is possible in individual randomized cross-over studies due to their typically small sample size. Although such chance imbalance does not cause biased estimation on average over repeated trials, this simulation study will aim to confirm the appropriateness of ANCOVA when faced with the analysis of data from an individual trial in which chance baseline imbalance is clearly apparent. Randomized cross-over trials were simulated, varying in sample size and the pattern and strength of correlation between repeated measures. Estimates from ANCOVA, change scores, and post-treatment difference were unbiased on average across each set of simulated data sets. ANCOVA and change scores could use baseline information to improve precision, but change scores could also reduce precision if baseline measures were uninformative. Change scores only were correlated with chance within-subject baseline imbalance. All three estimators could be correlated with chance between-subjects imbalance in the first period baseline measurements, the strongest associations being with the post-treatment difference. Consistent results were obtained from a real data example. In conclusion, ANCOVA took best advantage of baseline measures to improve precision, and avoided bias in the widest set of circumstances with chance imbalance in those baseline measures.