Generalized estimating equation model for binary outcomes with missing covariates

This paper presents an approach to handling missing covariates in the generalized estimating equation (GEE) model for binary outcomes when the probability of missingness depends on the observed outcomes and covariates. The proposed method is to replace the missing quantities in the estimating function with consistent estimates. In special cases, the proposed model reduces to a weighted GEE model for the completely observed units, where the weight is the inverse of the probability of missingness. Our method can be viewed as an extension of the mean score method by Reilly and Pepe (1995, Biometrika 82, 299-314) to the GEE context. Under certain regularity conditions, the estimates of the regression coefficients obtained by the proposed method are consistent and asymptotically normally distributed. The finite sample properties of the estimates are illustrated via computer simulations. An application to the study of dementia among stroke patients is presented.     

Meta-analysis of multiple outcomes by regression with random effects

Earlier work showed how to perform fixed-effects meta-analysis of studies or trials when each provides results on more than one outcome per patient and these multiple outcomes are correlated. That fixed-effects generalized-least-squares approach analyzes the multiple outcomes jointly within a single model, and it can include covariates, such as duration of therapy or quality of trial, that may explain observed heterogeneity of results among the trials. Sometimes the covariates explain all the heterogeneity, and the fixed-effects regression model is appropriate. However, unexplained heterogeneity may often remain, even after taking into account known or suspected covariates. Because fixed-effects models do not make allowance for this remaining unexplained heterogeneity, the potential exists for bias in estimated coefficients, standard errors and p-values. We propose two random-effects approaches for the regression meta-analysis of multiple correlated outcomes. We compare their use with fixed-effects models and with separate-outcomes models in a meta-analysis of periodontal clinical trials. A simulation study shows the advantages of the random-effects approach. These methods also facilitate meta-analysis of trials that compare more than two treatments.     

Comparison of population-averaged and subject-specific approaches for analyzing repeated binary outcomes

Several approaches have been proposed to model binary outcomes that arise from longitudinal studies. Most of the approaches can be grouped into two classes: the population-averaged and subject-specific approaches. The generalized estimating equations (GEE) method is commonly used to estimate population-averaged effects, while random-effects logistic models can be used to estimate subject-specific effects. However, it is not clear to many epidemiologists how these two methods relate to one another or how these methods relate to more traditional stratified analysis and standard logistic models. The authors address these issues in the context of a longitudinal smoking prevention trial, the Midwestern Prevention Project. In particular, the authors compare results from stratified analysis, standard logistic models, conditional logistic models, the GEE models, and random-effects models by analyzing a binary outcome from two and seven repeated measurements, respectively. In the comparison, the authors focus on the interpretation of both time-varying and time-invariant covariates under different models. Implications of these methods for epidemiologic research are discussed.     

Multiple-outcome meta-analysis of clinical trials

When several clinical trials report multiple outcomes, meta-analyses ordinarily analyse each outcome separately. Instead, by applying generalized-least-squares (GLS) regression, Raudenbush et al. showed how to analyse the multiple outcomes jointly in a single model. A variant of their GLS approach, discussed here, can incorporate correlations among the outcomes within treatment groups and thus provide more accurate estimates. Also, it facilitates adjustment for covariates. In our approach, each study need not report all outcomes nor evaluate all treatments. For example, a meta-analysis may evaluate two or more treatments (one 'treatment' may be a control) and include all randomized controlled trials that report on any subset (of one or more) of the treatments of interest. The analysis omits other treatments that these trials evaluated but that are not of interest to the meta-analyst. In the proposed fixed-effects GLS regression model, study-level and treatment-arm-level covariates may be predictors of one or more of the outcomes. An analysis of rheumatoid arthritis data from trials of second-line drug treatments (used after initial standard therapies prove unsatisfactory for a patient) motivates and applies the method. Data from 44 randomized controlled trials were used to evaluate the effectiveness of injectable gold and auranofin on the three outcomes tender joint count, grip strength, and erythrocyte sedimentation rate. The covariates in the regression model were quality and duration of trial and baseline measures of the patients' disease severity and disease activity in each trial. The meta-analysis found that gold was significantly more effective than auranofin on all three treatment outcomes. For all estimated coefficients, the multiple-outcomes model produced moderate changes in their values and slightly smaller standard errors, to the three separate outcome models.     

Analysis of prevention program effectiveness with clustered data using generalized estimating equations.

Experimental studies of prevention programs often randomize clusters of individuals rather than individuals to treatment conditions. When the correlation among individuals within clusters is not accounted for in statistical analysis, the standard errors are biased, potentially resulting in misleading conclusions about the significance of treatment effects. This study demonstrates the generalized estimating equations (GEE) method, focusing specifically on the GEE-independent method, to control for within-cluster correlation in regression models with either continuous or binary outcomes. The GEE-independent method yields consistent and robust variance estimates. Data from Project DARE, a youth substance abuse prevention program, are used for illustration. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Handling covariates in population pharmacokinetics

Population models were developed to analyze processes, described by parametric models, from measurements obtained in a sample of individuals. In order to analyze the sources of interindividual variability, covariates may be incorporated in the population analysis. The exploratory analyses and the two-stage approaches which use standard non-linear regression techniques are simple tools to select meaningful covariates. The global population approaches may be divided into two classes within which the covariates are handled differently: the parametric and the non-parametric methods. The power as well as the limitations of each approach regarding handling of covariates are illustrated and compared using the same data set which concerns the pharmacokinetics of gentamicin in neonates. With parametric approaches a second-stage model between structural parameters and covariates has to be defined. In the non-parametric method the joint distribution of parameters and covariates is estimated without parametric assumptions; however, it is assumed that covariates are observed with some error and parameters involved in functional relationships are not estimated. The important results concerning gentamicin in neonates were found by the two methods.     

Awareness of interpersonal style and self-evaluation

Multiple regression models are commonly used to control for confounding in epidemiologic research. Parametric regression models, such as multiple logistic regression, are powerful tools to control for multiple covariates provided that the covariate-risk associations are correctly specified. Residual confounding may result, however, from inappropriate specification of the confounder-risk association. In this paper, we illustrate the order of magnitude of residual confounding that may occur with traditional approaches to control for continuous confounders in multiple logistic regression, such as inclusion of a single linear term or categorization of the confounder, under a variety of assumptions on the confounder-risk association. We show that inclusion of the confounder as a single linear term often provides satisfactory control for confounding even in situations in which the model assumptions are clearly violated. In contrast, categorization of the confounder may often lead to serious residual confounding if the number of categories is small. Alternative strategies to control for confounding, such as polynomial regression or linear spline regression, are a useful supplement to the more traditional approaches.     

Between- and within-cluster covariate effects in the analysis of clustered data

Standard methods for the regression analysis of clustered data postulate models relating covariates to the response without regard to between- and within-cluster covariate effects. Implicit in these analyses is the assumption that these effects are identical. Example data show that this is frequently not the case and that analyses that ignore differential between- and within-cluster covariate effects can be misleading. Consideration of between- and within-cluster effects also helps to explain observed and theoretical differences between mixture model analyses and those based on conditional likelihood methods. In particular, we show that conditional likelihood methods estimate purely within-cluster covariate effects, whereas mixture model approaches estimate a weighted average of between- and within-cluster covariate effects.     

An application of maximum likelihood and generalized estimating equations to the analysis of ordinal data from a longitudinal study with cases missing at random

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.     

Gain in efficiency from using generalized least squares in the Haseman-Elston test

Techniques that test for linkage between a marker and a trait locus based on the regression methods proposed by Haseman and Elston [1972] involve testing a null hypothesis of no linkage by examination of the regression coefficient. Modified Haseman-Elston methods accomplish this using ordinary least squares (OLS), weighted least squares (WLS), in which weights are reciprocals of estimated variances, and generalized estimating equations (GEE). Methods implementing the WLS and GEE currently use a diagonal covariance matrix, thus incorrectly treating the squared trait differences of two sib pairs within a family as uncorrelated. Correctly specifying the correlations between sib pairs in a family yields the best linear unbiased estimator of the regression coefficient [Scheffe, 1959]. This estimator will be referred to as the generalized least squares (GLS) estimator. We determined the null variance of the GLS estimator and the null variance of the WLS/OLS estimator. The correct null variance of the WLS/OLS estimate of the Haseman-Elston (H-E) regression coefficient may be either larger or smaller than the variance of the WLS/OLS estimate calculated assuming that the squared sib-pair differences are uncorrelated. For a fully informative marker locus, the gain in efficiency using GLS rather than WLS/OLS under the null hypothesis is approximately 11% in a large multifamily study with three siblings per family and 25% for families with four siblings each.     

rhDNase as an example of recurrent event analysis

We consider counting process methods for analysing time-to-event data with multiple or recurrent outcomes, using the models developed by Anderson and Gill, Wei, Lin and Weissfeld and Prentice, Williams and Peterson. We compare the methods, and show how to implement them using popular statistical software programs. By analysing three data sets, we illustrate the strengths and pitfalls of each method. The first example is simulated and involves the effect of a hidden covariate. The second is based on a trial of gamma interferon, and behaves remarkably like the first. The third and most interesting example involves both multiple events and discontinuous intervals at risk, and the three approaches give dissimilar answers. We recommend the AG and marginal models for the analysis of this type of data.     

Median and ulnar nerve conduction studies among workers: normative values

To determine normative values for nerve conduction studies among workers, we selected a subset of 326 workers from 955 subjects who participated in medical surveys in the workplace. The reference cohort was composed exclusively of active workers, in contrast to the typical convenience samples. Nerve conduction measures included bilateral median and ulnar sensory amplitude and latency (onset and peak). Workers with upper extremity symptoms, medical conditions that could adversely affect peripheral nerve function, low hand temperature, or highly repetitive jobs were excluded from the "normal" cohort. Linear regression models explained between 21% and 51% of the variance in nerve function, with covariates of age, sex, hand temperature, and anthropometric factors. The most robust models were fitted for sensory amplitudes in the median and ulnar nerves for dominant and nondominant hands. The median-ulnar difference was least sensitive to adjustment, indicating it is the best measure to use if corrections are not made to account for relevant covariates. A key point was that the magnitude of variance increased with age and anthropometric factors. These findings provide strong evidence that to improve diagnostic accuracy, electrodiagnostic testing should control for relevant covariates, particularly age, sex, hand temperature, and anthropometric factors.     

Comparison of possible covariates for use in a random regression model for analyses of test day yields

Random regression models have been proposed for the genetic evaluation of dairy cattle using test day records. Random regression models contain linear functions of fixed and random coefficients and a set of covariates to describe the shapes of lactation curves for groups of cows and for individual cows. Previous work has used a linear function of five covariates to describe lactation shape. This study compared the function of five covariates with a function of only three covariates in three random regression models. Comparisons of estimates of components of variances and covariances, as well as comparisons of EBV and their prediction errors for milk yield, were made among models. Small practical differences existed between models in all respects. The model using regressions with five covariates had a slight advantage for comparison of prediction error variances of daily yields.     

Covariate measurement error and the estimation of random effect parameters in a mixed model for longitudinal data

We explore the effects of measurement error in a time-varying covariate for a mixed model applied to a longitudinal study of plasma levels and dietary intake of beta-carotene. We derive a simple expression for the bias of large sample estimates of the variance of random effects in a longitudinal model for plasma levels when dietary intake is treated as a time-varying covariate subject to measurement error. In general, estimates for these variances made without consideration of measurement error are biased positively, unlike estimates for the slope coefficients which tend to be 'attenuated'. If we can assume that the residuals from a longitudinal fit for the time-varying covariate behave like measurement errors, we can estimate the original parameters without the need for additional validation or reliability studies. We propose a method to test this assumption and show that the assumption is reasonable for the example data. We then use a likelihood-based method of estimation that involves a simple extension of existing methods for fitting mixed models. Simulations illustrate the properties estimators.     

Associations of types of lens opacities between and within eyes of individuals: an application of second-order generalized estimating equations. The Framingham Eye Studies Group

The lens opacity characteristics of individuals constitute multivariate data. Our goal was to estimate the associations between the three main types of age-related lens opacities (nuclear, cortical, posterior subcapsular) both between and within eyes of individuals using cross-sectional data from the Framingham (Massachusetts) Eye Studies. We describe use of a recently proposed extension of the generalized estimating equations approach to marginal logistic models (GEE2), and we demonstrate that a variety of research problems can be investigated with this methodology. For example, in our data, there were strong associations of the same opacity types between the two eyes of individuals and weak associations between different types of opacities. We also note that estimation of such associations may be limited in other epidemiologic settings.     

The effect of self-reported and observed job conditions on depression and anxiety symptoms: A comparison of theoretical models.

The demand/control/support and effort/reward imbalance models have relied on self-reported methods to describe how poor psychosocial working conditions lead to harmful health outcomes. The hindrance/utilization model uses an observational methodology to assess these relationships. Cross-sectional observational and self-reported data from 98 civil servants participating in the Whitehall II Study of British civil servants were used to test whether work conditions measured by each of the three theoretical models explained a significant amount of the variance in depression and anxiety symptoms. Observational measures were also used to assess potential common methods variance bias between the self-reported job conditions and the outcomes. Results showed that the demand/control/support model explained the most variance in depression and anxiety symptoms and the associations were not wholly due to common methods variance. Moreover, measures associated with job resources (e.g., skill discretion, social support and skill utilization) had a protective effect on depression and anxiety symptoms. Exertion-related conditions (e.g., demands, effort, over commitment) were not consistently associated with depression or anxiety symptoms. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Inference using conditional logistic regression with missing covariates

When there are many nuisance parameters in a logistic regression model, a popular method for eliminating these nuisance parameters is conditional logistic regression. Unfortunately, another common problem in a logistic regression analysis is missing covariate data. With many nuisance parameters to eliminate and missing covariates, many investigators exclude any subject with missing covariates and then use conditional logistic regression, often called a complete-case analysis. In this article, we derive a modified conditional logistic regression that is appropriate with covariates that are missing at random. Performing a conditional logistic regression with only the complete cases is convenient with existing statistical packages, but it may give bias if missingness is not completely at random.     

Genetic basis of variation in carotid artery wall thickness

BACKGROUND AND PURPOSE: Other than the documented associations of risk factors and carotid artery wall thickness, the genetic basis of variation in carotid artery intimal-medial thickness (IMT) is unknown. The purpose of this study was to examine the extent to which variation in common carotid artery (CCA) IMT and internal carotid artery (ICA) IMT are under genetic control. METHODS: The sibship data used for this analysis were part of an epidemiological survey in Mexico City. The CCA and ICA analyses were based on 46 and 44 sibships of various sizes, respectively. The CCA and ICA IMTs were measured with carotid ultrasonography. Using a robust variance decomposition method, we performed genetic analyses of CCA IMT and ICA IMT measurements with models incorporating several cardiovascular risk factors (eg, lipids, diabetes, blood pressure, and smoking) as covariates. RESULTS: After accounting for the effects of covariates, we detected high heritabilities for CCA IMT (h2 = 0.92 +/- 0.05, P = .001) and ICA IMT (h2 = 0.86 +/- 0.13, P = .029). Genes accounted for 66.0% of the total variation in CCA IMT, whereas 27.7% of variation was attributable to covariates. For ICA IMT, genes explained a high proportion (74.9%) of total phenotypic variation. The covariates accounted for 11.5% of variation in ICA IMT. CONCLUSIONS: Our results suggest that substantial proportions of phenotypic variance in CCA IMT and ICA IMT are attributable to shared genetic factors.     

Re-using data from case-control studies

Despite its ability to maximize statistical power while keeping data collection costs to a minimum, case-control sampling provides a non-representative sample of the population. When fitting a logistic regression model to data obtained in such a study, using the variable stratifying the population as the response, it is well known that the estimate of the constant term will be biased, but those of the coefficients of the covariates will not. However, subsequent to the case-control study, it is often desired to conduct a secondary analysis, using a variable that was previously a covariate in the main study as the response. If this new response is associated with the original variable used to stratify the population into cases and controls, a conventional logistic regression analysis will usually result in biased estimates of all the regression coefficients, not just the constant. This situation has recently been studied by Nagelkerke et al. who describe some situations where no bias occurs. In this paper we discuss how to calculate maximum likelihood estimates of all the regression coefficients, in the situation where the sampling rates for cases and controls are known. An example using data from the New Zealand Cot Death Study is presented.     

Generalized additive models for current status data

Current status data arise in studies where the target measurement is the time of occurrence of some event, but observations are limited to indicators of whether or not the event has occurred at the time the sample is collected--only the current status of each individual with respect to event occurrence is observed. Examples of such data arise in several fields, including demography, epidemiology, econometrics and bioassay. Although estimation of the marginal distribution of times of event occurrence is well understood, techniques for incorporating covariate information are not well developed. This paper proposes a semiparametric approach to estimation for regression models of current status data, using techniques from generalized additive modeling and isotonic regression. This procedure provides simultaneous estimates of the baseline distribution of event times and covariate effects. No parametric assumptions about the form of the baseline distribution are required. The results are illustrated using data from a demographic survey of breastfeeding practices in developing countries, and from an epidemiological study of heterosexual Human Immunodeficiency Virus (HIV) transmission.