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.     

Optimal estimation in surrogate outcome regression problems

The authors consider a double robust estimation of the regression parameter defined by an estimating equation in a surrogate outcome set‐up. Under a correct specification of the propensity score, the proposed estimator has smallest trace of asymptotic covariance matrix whether the “working outcome regression model” involved is specified correct or not, and it is particularly meaningful when it is incorrectly specified. Simulations are conducted to examine the finite sample performance of the proposed procedure. Data on obesity and high blood pressure are analyzed for illustration. The Canadian Journal of Statistics 38: 633–646; 2010 © 2010 Statistical Society of Canada     

Survival analysis with long‐term survivors and partially observed covariates

The authors describe a method for fitting failure time mixture models that postulate the existence of both susceptibles and long‐term survivors when covariate data are only partially observed. Their method is based on a joint model that combines a Weibull regression model for the susceptibles, a logistic regression model for the probability of being a susceptible, and a general location model for the distribution of the covariates. A Bayesian approach is taken, and Gibbs sampling is used to fit the model to the incomplete data. An application to clinical data on tonsil cancer and a small Monte Carlo study indicate potential large gains in efficiency over standard complete‐case analysis as well as reasonable performance in a variety of situations.     

Influence Function Based Variance Estimation and Missing Data Issues in Case-Cohort Studies

Recognizing that the efficiency in relative risk estimation for the Cox proportional hazards model is largely constrained by the total number of cases, Prentice (1986) proposed the case-cohort design in which covariates are measured on all cases and on a random sample of the cohort. Subsequent to Prentice, other methods of estimation and sampling have been proposed for these designs. We formalize an approach to variance estimation suggested by Barlow (1994), and derive a robust variance estimator based on the influence function. We consider the applicability of the variance estimator to all the proposed case-cohort estimators, and derive the influence function when known sampling probabilities in the estimators are replaced by observed sampling fractions. We discuss the modifications required when cases are missing covariate information. The missingness may occur by chance, and be completely at random; or may occur as part of the sampling design, and depend upon other observed covariates. We provide an adaptation of S-plus code that allows estimating influence function variances in the presence of such missing covariates. Using examples from our current case-cohort studies on esophageal and gastric cancer, we illustrate how our results our useful in solving design and analytic issues that arise in practice.     

Planned and Unplanned Births and Conceptions in Italy, 1970–1995

Using FFS data on births andofficial data on abortions, focus is firstgiven to trends in planned and unplannedpregnancies during 1970–1995, comparing Italywith the USA and France. The next step is topinpoint groups where unplanned events are morecommon. Finally, the relationship betweenunplanned births and changes in contraceptivebehaviour is examined.The planned fertility decline distinguishesItaly from other Western countries. Unplannedbirths and abortions have also declinedconsiderably during the last twenty years.However, some caution is urged before it may beassumed that Italians are the masters of theirown fertility as, during the first half of1990s, 37% of conceptions were unplanned (atthe moment of the event), reaching 45% amongunder 20 and over 35 years old women.     

Bayesian latent variable model for mixed continuous and ordinal responses with possibility of missing responses

A general framework is proposed for joint modelling of mixed correlated ordinal and continuous responses with missing values for responses, where the missing mechanism for both kinds of responses is also considered. Considering the posterior distribution of unknowns given all available information, a Markov Chain Monte Carlo sampling algorithm via winBUGS is used for estimating the posterior distribution of the parameters. For sensitivity analysis to investigate the perturbation from missing at random to not missing at random, it is shown how one can use some elements of covariance structure. These elements associate responses and their missing mechanisms. Influence of small perturbation of these elements on posterior displacement and posterior estimates is also studied. The model is illustrated using data from a foreign language achievement study.     

Random change point models: investigating cognitive decline in the presence of missing data

With the aim of identifying the age of onset of change in the rate of cognitive decline while accounting for the missing observations, we considered a selection modelling framework. A random change point model was fitted to data from a population-based longitudinal study of ageing (the Cambridge City over 75 Cohort Study) to model the longitudinal process. A missing at random mechanism was modelled using logistic regression. Random effects such as initial cognitive status, rate of decline before and after the change point, and the age of onset of change in rate of decline were estimated after adjustment for risk factors for cognitive decline. Among other possible predictors, the last observed cognitive score was used to adjust the probability of death and dropout. Individuals who experienced less variability in their cognitive scores experienced a change in their rate of decline at older ages than individuals whose cognitive scores varied more.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

A note on dealing with missing standard errors in meta-analyses of continuous outcome measures in WinBUGS

A meta-analysis of a continuous outcome measure may involve missing standard errors. This is not a problem depending on assumptions made about the population standard deviation. Multiple imputation can be used to impute missing values while allowing for uncertainty in the imputation. Markov chain Monte Carlo simulation is a multiple imputation technique for generating posterior predictive distributions for missing data. We present an example of imputing missing variances using WinBUGS. The example highlights the importance of checking model assumptions, whether for missing or observed data.     

缺失数据的单一插补方法

插补是处理缺失数据的一种常用的方法。本文介绍了单一插补的最常见的几种方法。