An investigation of penalization and data augmentation to improve convergence of generalized estimating equations for clustered binary outcomes

Correlated response data often arise in longitudinal and familial studies. The marginal regression model and its associated generalized estimating equation (GEE) method are becoming more and more popular in handling such data. Pepe and Anderson pointed out that there is an important yet implicit assumption behind the marginal model and GEE. If the assumption is violated and a nondiagonal working correlation matrix is used in GEE, biased estimates of regression coefficients may result. On the other hand, if a diagonal correlation matrix is used, irrespective of whether the assumption is violated, the resulting estimates are (nearly) unbiased. A straightforward interpretation of this phenomenon is lacking, in part due to the unavailability of a closed form for the resulting GEE estimates. In this note, we show how the bias may arise in the context of linear regression, where the GEE estimates of regression coefficients are the ordinary or generalized least squares (LS) estimates. Also we explain why the generalized LS estimator may be biased, in contrast to the well-known result that it is usually unbiased. In addition, we discuss the bias properties of the sandwich variance estimator of the ordinary LS estimate.

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.

Statistical Methods to Study Timing of Vulnerability with Sparsely Sampled Data on Environmental Toxicants

BackgroundPrevalence of depression in older persons was a leading cause of disability. This group has the lowest access to service and retention in care compared to other age groups. This study aimed to explore continuing mental health service use and examined the predictive power of the mental health service delivery system and individual factors on mental health service use among older persons diagnosed with depressive disorders.MethodsWe employed an analytic cross-sectional study design of individual and organizational variables in 12 general hospitals selected using multi-stratified sampling. There were 3 clusters comprising community hospitals, advanced and standard hospitals, and university hospitals. Participants in each group were 150 persons selected by purposive sampling. We included older persons with a first or recurring diagnosis of a depressive disorder in the last 6 to 12 months of the data collection date. Data at the individual level included socio-demographic characteristics, Charlson Comorbidity Index, Attitude toward Depression and its treatment, and perceived social support. Data at the organizational level had hospital level, nurse competency, nurse-patient ratio, and appointment reminders. Descriptive statistics, Pearson chi-square test, latent class analysis (LCA), and marginal logistic regression model using generalized estimating equation (GEE) were used to analyze the data.ResultsThe continuing mental health service use among older persons diagnosed with depressive disorders was 54%. The latent class analysis of four variables in the mental health services delivery organization yielded distinct and interpretable findings in two groups: high and low resource organization. The marginal logistic multivariable regression model using GEE found that organizational group and attitude toward depression and its treatment were significantly associated with mental health service use (p-value = 0.046; p-value = 0.003).ConclusionsThe findings suggest that improving continuing mental health services use in older persons diagnosed with depressive disorders should emphasize specialty resources of the mental health services delivery system and attitude toward depression and its treatment.

Flexible models and methods for longitudinal and multilevel functional data

Chapter 9 - Generalized estimating equations (GEEs) models

SummaryCorrelated data are ubiquitous in ecological and evolutionary research, and appropriate statistical analysis requires that these correlations are taken into account. For regressions with correlated, non‐normal outcomes, two main approaches are used: conditional and marginal modelling. The former leads to generalized linear mixed models (GLMMs), while the latter are estimated using generalized estimating equations (GEEs), or marginalized multilevel regression models. Differences, advantages and drawbacks of conditional and marginal models have been discussed extensively in the statistical and applied literature, and there is some agreement that the choice of the model must depend on the question under study. Yet, there still appears to be a lot of confusion and disagreement over when to choose which model.We start with a review of conditional and marginal models, and the differences in the interpretation of the resulting parameter estimates. We highlight that the two types of models propagate different linear relations between the covariates and the response. Moreover, while conditional models explicitly account for heterogeneity among clustered observations, marginal models yield averages over such heterogeneities and are therefore often interpreted as population‐averaged models.We point out theoretically and with an example that when modelling non‐normal outcomes no unambiguous definition of a marginal model generally exists. Instead, marginal model parameters are marginal only with respect to unaccounted differences among clusters and thus depend on the fixed effects in the model. Therefore, marginal model parameters should not be loosely interpreted as population‐averaged parameters. In addition, we explain how marginal modelling is mathematically analogous to deliberately omitting covariates with explanatory power, and to deliberately introducing a Berkson measurement error into covariates. We also reiterate that marginal modelling is related to a well‐known statistical phenomenon, the Simpson's paradox.In most cases, therefore, we regard the conditional model as the more powerful choice to explain how covariates are associated with a non‐normal response. Still, marginal models can be useful, given that the scientific question explicitly requires such a model formulation.

The quadratic inference functions (QIF) method proposed by Qu et al. (2000) and the generalized method of moments (GMM) for marginal regression analysis of longitudinal data with time-dependent covariates proposed by Lai and Small (2007) both are the methods based on generalized method of moment (GMM) introduced by Hansen (1982) and both use generalized estimating equations (GEE). Lai and Small (2007) divided time-dependent covariates into three types such as: Type I, Type II and Type III. In this paper, we compared these methods in the case of Type II and Type III in which full covariates conditional mean assumption (FCCM) is violated and interested in whether they can improve the results of GEE with independence working correlation. We show that in the marginal regression model with Type II time-dependent covariates, GMM Type II of Lai and Small (2007) provides more ecient result than QIF and for the Type III time-dependent covariates, QIF with independence working correlation and GMM Type III methods provide the same results. Our simulation study showed the same results.

In health services research, researchers often use clustered data to estimate the independent association between individual outcomes and cluster-level covariates after adjusting for individual-level characteristics. Marginal generalized linear models estimated using generalized estimating equation (GEE) methods or hierarchical (or multilevel) regression models can be used when there is a single source of clustering (e.g., patients nested within hospitals). Hierarchical regression models can also be used when there are multiple sources of clustering (e.g., patients nested within surgeons who in turn are nested within hospitals). Methods for estimating marginal regression models are less well-developed when there are multiple sources of non-nested clustering (e.g., patients are clustered both within hospitals and within in neighborhoods, but neither neighborhoods or hospitals are nested in the other). Miglioretti and Heagerty developed a GEE-type variance estimator for use when fitting marginal generalized linear models to non-nested multilevel data. We propose a variance estimator for a marginal Cox regression model fit to non-nested multilevel data that combined their approach with Lin and Wei's robust variance estimator for the Cox model. We evaluated the performance of the proposed variance estimator using an extensive set of Monte Carlo simulations. We illustrated the use of the variance estimator in a case study consisting of patients hospitalized with an acute myocardial infarction who were clustered within hospitals and who were also clustered in neighborhoods. In summary, a variance estimator motivated by that proposed by Miglioretti and Heagerty can be used with marginal Cox regression models fit to non-nested multilevel data.

25 Estimation of Marginal Regression Models with Multiple Source Predictors

Using a generalized estimating equation (GEE) can lead to a bias in regression coefficients for a small sample or sparse data. The bias-corrected GEE (BCGEE) and penalized GEE (PGEE) were proposed to resolve the small-sample bias. Moreover, the standard sandwich covariance estimator leads to a bias of standard error for small samples; several modified covariance estimators have been proposed to address this issue. We review the modified GEEs and modified covariance estimators, and evaluate their performance in sparse binary data from small-sample longitudinal studies. The simulation results showed that GEE and BCGEE often failed to achieve convergence, whereas the convergence proportion for PGEE was quite high. The bias for the regression coefficients was generally in the ascending order of PGEE BCGEE GEE. However, PGEE and BCGEE did not sufficiently remove the bias involving 20-30 subjects with unequal exposure levels with a 5% response rate. The coverage probability (CP) of the confidence interval for BCGEE was relatively poor compared with GEE and PGEE. The CP with the sandwich covariance estimator deteriorated regardless of the GEE methods under the small sample size and low response rate, whereas the CP with the modified covariance estimators-such as Morel's method-was relatively acceptable. PGEE will be the reasonable way for analyzing sparse binary data in small-sample studies. Instead of using the standard sandwich covariance estimator, one should always apply the modified covariance estimators for analyzing these data.

For the marginal model and generalized estimating equations (GEE) method there is important full covariates conditional mean (FCCM) assumption which is pointed out by Pepe and Anderson (1994). With longitudinal data with time-varying stochastic covariates, this assumption may not necessarily hold. If this assumption is violated, the biased estimates of regression coefficients may result. But if a diagonal working correlation matrix is used, irrespective of whether the assumption is violated, the resulting estimates are (nearly) unbiased (Pan et al., 2000).The quadratic inference functions (QIF) method proposed by Qu et al. (2000) is the method based on generalized method of moment (GMM) using GEE. The QIF yields a substantial improvement in efficiency for the estimator of <TEX>${\beta}$</TEX> when the working correlation is misspecified, and equal efficiency to the GEE when the working correlation is correct (Qu et al., 2000).In this paper, we interest in whether the QIF can improve the results of the GEE method in the case of FCCM is violated. We show that the QIF with exchangeable and AR(1) working correlation matrix cannot be consistent and asymptotically normal in this case. Also it may not be efficient than GEE with independence working correlation. Our simulation studies verify the result.

Data are analysed from a longitudinal psychiatric study in which there are no dropouts that do not occur completely at random. A marginal proportional odds model is fitted that relates the response (severity of side effects) to various covariates. Two methods of estimation are used: generalized estimating equations (GEE) and maximum likelihood (ML). Both the complete set of data and the data from only those subjects completing the study are analysed. For the completers-only data, the GEE and ML analyses produce very similar results. These results differ considerably from those obtained from the analyses of the full data set. There are also marked differences between the results obtained from the GEE and ML analysis of the full data set. The occurrence of such differences is consistent with the presence of a non-completely-random dropout process and it can be concluded in this example that both the analyses of the completers only and the GEE analysis of the full data set produce misleading conclusions about the relationships between the response and covariates.

Generalized estimating equations (GEE) methodology as proposed by Liang and Zeger has received widespread use in the analysis of correlated binary data. Miller et al. and Lipsitz et al. extended GEE to correlated nominal and ordinal categorical data; in particular, they used GEE for fitting McCullagh's proportional odds model. In this paper, we consider robust (that is, empirically corrected) and model-based versions of both a score test and a Wald test for assessing the assumption of proportional odds in the proportional odds model fitted with GEE. The Wald test is based on fitting separate multiple logistic regression models for each dichotomization of the response variable, whereas the score test requires fitting just the proportional odds model. We evaluate the proposed tests in small to moderate samples by simulating data from a series of simple models. We illustrate the use of the tests on three data sets from medical studies.

9614 Background: Gefitinib is a potential first-line treatment option for patients with advanced non-small cell lung cancer (NSCLC), especially for patients with activating mutations in the EGFR gene. However, little is known about patient-reported health-related quality of life (HRQOL) in this patient population. The aims of this study were to explore the prognostic values of baseline HRQOL for time-to-treatment failure (TTF), as well as the predictors of repeatedly measured posttreatment HRQOL, in advanced NSCLC patients receiving first-line gefitinib. Methods: A total of 106 chemonaive patients with advanced NSCLC were enrolled in a phase II trial. Gefitinib was given at a dose of 250 mg/d. HRQOL was assessed monthly with the EuroQoL instrument (EQ-5D) and the Lung Cancer Symptom Scale (LCSS) questionnaire. Baseline HRQOL and clinical/molecular predictors of TTF were jointly examined by multiple Cox's proportional hazards model. The associations between the clinical/molecular factors and repeatedly measured posttreatment HRQOL were analyzed by fitting marginal linear regression model using the generalized estimating equations (GEE) method. Results: In this prospective study, HRQOL data were obtained from 94 patients. Baseline EQ-5D index (estimated hazard ratio = 0.286, 95% C.I.: 0.135–0.603, p = 0.001) and the presence of L858R EGFR mutation in adenocarcinoma (estimated hazard ratio = 0.520, 95% C.I.: 0.307–0.880, p = 0.015) were retained as independent prognostic factors in the final multiple Cox's proportional hazards model for TTF. According to preliminary GEE analysis of repeatedly measured posttreatment HRQOL, the patients with wild-type EGFR consistently had worse HRQOL in EQ-5D index (p &lt; 0.0001), EQ-5D VAS score (p = 0.0002), and LCSS global score (p &lt; 0.0001), respectively. Conclusions: In advanced NSCLC patients receiving first-line gefitinib, better baseline EQ-5D index and L858R EGFR mutation in adenocarcinoma predict longer TTF. In addition, patients with wild-type EGFR had worse posttreatment HRQOL. No significant financial relationships to disclose.