Missing covariates in longitudinal data with informative dropouts: bias analysis and inference

We consider estimation in generalized linear mixed models (GLMM) for longitudinal data with informative dropouts. At the time a unit drops out, time-varying covariates are often unobserved in addition to the missing outcome. However, existing informative dropout models typically require covariates to be completely observed. This assumption is not realistic in the presence of time-varying covariates. In this article, we first study the asymptotic bias that would result from applying existing methods, where missing time-varying covariates are handled using naive approaches, which include: (1) using only baseline values; (2) carrying forward the last observation; and (3) assuming the missing data are ignorable. Our asymptotic bias analysis shows that these naive approaches yield inconsistent estimators of model parameters. We next propose a selection/transition model that allows covariates to be missing in addition to the outcome variable at the time of dropout. The EM algorithm is used for inference in the proposed model. Data from a longitudinal study of human immunodeficiency virus (HIV)-infected women are used to illustrate the methodology.     

Multiple imputation methods for handling incomplete longitudinal and clustered data where the target analysis is a linear mixed effects model

Multiple imputation (MI) is increasingly popular for handling multivariate missing data. Two general approaches are available in standard computer packages: MI based on the posterior distribution of incomplete variables under a multivariate (joint) model, and fully conditional specification (FCS), which imputes missing values using univariate conditional distributions for each incomplete variable given all the others, cycling iteratively through the univariate imputation models. In the context of longitudinal or clustered data, it is not clear whether these approaches result in consistent estimates of regression coefficient and variance component parameters when the analysis model of interest is a linear mixed effects model (LMM) that includes both random intercepts and slopes with either covariates or both covariates and outcome contain missing information. In the current paper, we compared the performance of seven different MI methods for handling missing values in longitudinal and clustered data in the context of fitting LMMs with both random intercepts and slopes. We study the theoretical compatibility between specific imputation models fitted under each of these approaches and the LMM, and also conduct simulation studies in both the longitudinal and clustered data settings. Simulations were motivated by analyses of the association between body mass index (BMI) and quality of life (QoL) in the Longitudinal Study of Australian Children (LSAC). Our findings showed that the relative performance of MI methods vary according to whether the incomplete covariate has fixed or random effects and whether there is missingnesss in the outcome variable. We showed that compatible imputation and analysis models resulted in consistent estimation of both regression parameters and variance components via simulation. We illustrate our findings with the analysis of LSAC data.     

Use of Covariates and Survey Wave to Adjust for Nonresponse

Using covariates available on all sampled individuals along with an intensive follow‐up of nonrespondents is a useful approach to reduce the impact of nonresponse. We examined the performance of several different nonresponse models to account for such covariates and multiple survey waves to estimate the prevalence of birth control use in 106463 women taking the teratogenic drug Accutane. Nonrespondents to a first mailing were followed with two more mailings and then a telephone call. The first mailing had a response rate of 59.7 percent; the overall nonresponse rate was 15.2 percent. Age was available on all women and was strongly associated with both response and birth control use. We considered both multiple imputation and weighting strategies using information on age and time to response assuming that nonrespondents were: (i) like all respondents conditional on age, (ii) like reluctant respondents conditional on age, and (iii) projected from a model which includes trend. Simulation studies were used to examine the potential bias in this setting and we compared models under different assumptions about the missing data mechanism. Results describe the extent to which availability of covariate information and use of multiple waves improves estimation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimating treatment effects with partially observed covariates using outcome regression with missing indicators

Missing data is a common issue in research using observational studies to investigate the effect of treatments on health outcomes. When missingness occurs only in the covariates, a simple approach is to use missing indicators to handle the partially observed covariates. The missing indicator approach has been criticized for giving biased results in outcome regression. However, recent papers have suggested that the missing indicator approach can provide unbiased results in propensity score analysis under certain assumptions. We consider assumptions under which the missing indicator approach can provide valid inferences, namely, (1) no unmeasured confounding within missingness patterns; either (2a) covariate values of patients with missing data were conditionally independent of treatment or (2b) these values were conditionally independent of outcome; and (3) the outcome model is correctly specified: specifically, the true outcome model does not include interactions between missing indicators and fully observed covariates. We prove that, under the assumptions above, the missing indicator approach with outcome regression can provide unbiased estimates of the average treatment effect. We use a simulation study to investigate the extent of bias in estimates of the treatment effect when the assumptions are violated and we illustrate our findings using data from electronic health records. In conclusion, the missing indicator approach can provide valid inferences for outcome regression, but the plausibility of its assumptions must first be considered carefully.     

Accounting for post-randomization variables in meta-analysis: A joint meta-regression approach

Meta-regression is widely used in systematic reviews to investigate sources of heterogeneity and the association of study-level covariates with treatment effectiveness. Existing meta-regression approaches are successful in adjusting for baseline covariates, which include real study-level covariates (e.g., publication year) that are invariant within a study and aggregated baseline covariates (e.g., mean age) that differ for each participant but are measured before randomization within a study. However, these methods have several limitations in adjusting for post-randomization variables. Although post-randomization variables share a handful of similarities with baseline covariates, they differ in several aspects. First, baseline covariates can be aggregated at the study level presumably because they are assumed to be balanced by the randomization, while post-randomization variables are not balanced across arms within a study and are commonly aggregated at the arm level. Second, post-randomization variables may interact dynamically with the primary outcome. Third, unlike baseline covariates, post-randomization variables are themselves often important outcomes under investigation. In light of these differences, we propose a Bayesian joint meta-regression approach adjusting for post-randomization variables. The proposed method simultaneously estimates the treatment effect on the primary outcome and on the post-randomization variables. It takes into consideration both between- and within-study variability in post-randomization variables. Studies with missing data in either the primary outcome or the post-randomization variables are included in the joint model to improve estimation. Our method is evaluated by simulations and a real meta-analysis of major depression disorder treatments.     

Bayesian latent multi‐state modeling for nonequidistant longitudinal electronic health records

Large amounts of longitudinal health records are now available for dynamic monitoring of the underlying processes governing the observations. However, the health status progression across time is not typically observed directly: records are observed only when a subject interacts with the system, yielding irregular and often sparse observations. This suggests that the observed trajectories should be modeled via a latent continuous‐time process potentially as a function of time‐varying covariates. We develop a continuous‐time hidden Markov model to analyze longitudinal data accounting for irregular visits and different types of observations. By employing a specific missing data likelihood formulation, we can construct an efficient computational algorithm. We focus on Bayesian inference for the model: this is facilitated by an expectation‐maximization algorithm and Markov chain Monte Carlo methods. Simulation studies demonstrate that these approaches can be implemented efficiently for large data sets in a fully Bayesian setting. We apply this model to a real cohort where patients suffer from chronic obstructive pulmonary disease with the outcome being the number of drugs taken, using health care utilization indicators and patient characteristics as covariates.     

Extended generalized estimating equations for binary familial data with incomplete families

In this article, we assess the performance of two standard, but naive, methods for handling incomplete familial data in GEE2 analyses when the outcome is binary. We also propose a new method for analyzing such data using GEE2 when explanatory variables are discrete. Unlike the naive methods, the new method does not require the missing data process to be ignorable. We illustrate our method with an example that examines the familial aggregation of obesity.     

Likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates

We consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical. With no missing responses or covariates, one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters. Unfortunately, incomplete data in the responses and covariates are a common occurrence in longitudinal studies. Here we assume the missing data are missing at random (Rubin, 1976, Biometrika 63, 581-592). Since all of the missing data (responses and covariates) are categorical, a useful technique for obtaining maximum likelihood parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). In using the EM algorithm with missing responses and covariates, one specifies the joint distribution of the responses and covariates. Here we consider the parameters of the covariate distribution as a nuisance. In data sets where the percentage of missing data is high, the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest. We propose a conditional model for the covariate distribution that has several modeling advantages for the EM algorithm and provides a reduction in the number of nuisance parameters, thus providing more stable estimates in finite samples.     

Estimation with correlated censored survival data with missing covariates

Incomplete covariate data are a common occurrence in studies in which the outcome is survival time. Further, studies in the health sciences often give rise to correlated, possibly censored, survival data. With no missing covariate data, if the marginal distributions of the correlated survival times follow a given parametric model, then the estimates using the maximum likelihood estimating equations, naively treating the correlated survival times as independent, give consistent estimates of the relative risk parameters Lipsitz et al. 1994 50, 842-846. Now, suppose that some observations within a cluster have some missing covariates. We show in this paper that if one naively treats observations within a cluster as independent, that one can still use the maximum likelihood estimating equations to obtain consistent estimates of the relative risk parameters. This method requires the estimation of the parameters of the distribution of the covariates. We present results from a clinical trial Lipsitz and Ibrahim (1996b) 2, 5-14 with five covariates, four of which have some missing values. In the trial, the clusters are the hospitals in which the patients were treated.     

Estimating Propensity Scores and Causal Survival Functions Using Prevalent Survival Data

Summary This article develops semiparametric approaches for estimation of propensity scores and causal survival functions from prevalent survival data. The analytical problem arises when the prevalent sampling is adopted for collecting failure times and, as a result, the covariates are incompletely observed due to their association with failure time. The proposed procedure for estimating propensity scores shares interesting features similar to the likelihood formulation in case‐control study, but in our case it requires additional consideration in the intercept term. The result shows that the corrected propensity scores in logistic regression setting can be obtained through standard estimation procedure with specific adjustments on the intercept term. For causal estimation, two different types of missing sources are encountered in our model: one can be explained by potential outcome framework; the other is caused by the prevalent sampling scheme. Statistical analysis without adjusting bias from both sources of missingness will lead to biased results in causal inference. The proposed methods were partly motivated by and applied to the Surveillance, Epidemiology, and End Results (SEER)‐Medicare linked data for women diagnosed with breast cancer.     

Exact two-sample inference with missing data

When comparing follow-up measurements from two independent populations, missing records may arise due to censoring by events whose occurrence is associated with baseline covariates. In these situations, inferences based only on the completely followed observations may be biased if the follow-up measurements and the covariates are correlated. This article describes exact inference for a class of modified U-statistics under covariate-dependent dropouts. The method involves weighing each permutation according to the retention probabilities, and thus requires estimation of the missing data mechanism. The proposed procedure is nonparametric in that no distributional assumption is necessary for the outcome variables and the missingness patterns. Monte Carlo approximation by the Gibbs sampler is proposed, and is shown to be fast and accurate via simulation. The method is illustrated in two small data sets for which asymptotic inferential procedures may not be appropriate.     

Likelihood methods for treatment noncompliance and subsequent nonresponse in randomized trials

While several new methods that account for noncompliance or missing data in randomized trials have been proposed, the dual effects of noncompliance and nonresponse are rarely dealt with simultaneously. We construct a maximum likelihood estimator (MLE) of the causal effect of treatment assignment for a two-armed randomized trial assuming all-or-none treatment noncompliance and allowing for subsequent nonresponse. The EM algorithm is used for parameter estimation. Our likelihood procedure relies on a latent compliance state covariate that describes the behavior of a subject under all possible treatment assignments and characterizes the missing data mechanism as in Frangakis and Rubin (1999, Biometrika 86, 365-379). Using simulated data, we show that the MLE for normal outcomes compares favorably to the method-of-moments (MOM) and the standard intention-to-treat (ITT) estimators under (1) both normal and non-normal data, and (2) departures from the latent ignorability and compound exclusion restriction assumptions. We illustrate methods using data from a trial to compare the efficacy of two antipsychotics for adults with refractory schizophrenia.     

Generalizing treatment effects with incomplete covariates: Identifying assumptions and multiple imputation algorithms

We focus on the problem of generalizing a causal effect estimated on a randomized controlled trial (RCT) to a target population described by a set of covariates from observational data. Available methods such as inverse propensity sampling weighting are not designed to handle missing values, which are however common in both data sources. In addition to coupling the assumptions for causal effect identifiability and for the missing values mechanism and to defining appropriate estimation strategies, one difficulty is to consider the specific structure of the data with two sources and treatment and outcome only available in the RCT. We propose three multiple imputation strategies to handle missing values when generalizing treatment effects, each handling the multisource structure of the problem differently (separate imputation, joint imputation with fixed effect, joint imputation ignoring source information). As an alternative to multiple imputation, we also propose a direct estimation approach that treats incomplete covariates as semidiscrete variables. The multiple imputation strategies and the latter alternative rely on different sets of assumptions concerning the impact of missing values on identifiability. We discuss these assumptions and assess the methods through an extensive simulation study. This work is motivated by the analysis of a large registry of over 20,000 major trauma patients and an RCT studying the effect of tranexamic acid administration on mortality in major trauma patients admitted to intensive care units. The analysis illustrates how the missing values handling can impact the conclusion about the effect generalized from the RCT to the target population.     

Multiple Imputation of Missing Composite Outcomes in Longitudinal Data

In longitudinal randomised trials and observational studies within a medical context, a composite outcome—which is a function of several individual patient-specific outcomes—may be felt to best represent the outcome of interest. As in other contexts, missing data on patient outcome, due to patient drop-out or for other reasons, may pose a problem. Multiple imputation is a widely used method for handling missing data, but its use for composite outcomes has been seldom discussed. Whilst standard multiple imputation methodology can be used directly for the composite outcome, the distribution of a composite outcome may be of a complicated form and perhaps not amenable to statistical modelling. We compare direct multiple imputation of a composite outcome with separate imputation of the components of a composite outcome. We consider two imputation approaches. One approach involves modelling each component of a composite outcome using standard likelihood-based models. The other approach is to use linear increments methods. A linear increments approach can provide an appealing alternative as assumptions concerning both the missingness structure within the data and the imputation models are different from the standard likelihood-based approach. We compare both approaches using simulation studies and data from a randomised trial on early rheumatoid arthritis patients. Results suggest that both approaches are comparable and that for each, separate imputation offers some improvement on the direct imputation of a composite outcome.     

A nonproportional hazards Weibull accelerated failure time regression model

We present a study of risk factors measured in mean before age 50 and subsequent incidence of heart disease over 32 years of follow-up. The data are from the Framingham Heart Study. The standard accelerated failure time model assumes the logarithm of time until an event has a constant dispersion parameter and a location parameter that is a linear function of covariates. Parameters are estimated by maximum likelihood. We reject a standard Weibull model for these data in favor of a model with the dispersion parameter depending on the location parameter. This model suggests that the cumulative hazard ratio for two individuals shrinks towards unity over the follow-up period. Thus, not only the standard Weibull, but also the semiparametric proportional hazards (Cox) model is inadequate for this data. The model improvement appears particularly valuable when estimating the difference in predicted outcome probabilities for two individuals.     

A pattern-mixture model for longitudinal binary responses with nonignorable nonresponse

Longitudinal studies frequently incur outcome-related nonresponse. In this article, we discuss a likelihood-based method for analyzing repeated binary responses when the mechanism leading to missing response data depends on unobserved responses. We describe a pattern-mixture model for the joint distribution of the vector of binary responses and the indicators of nonresponse patterns. Specifically, we propose an extension of the multivariate logistic model to handle nonignorable nonresponse. This method yields estimates of the mean parameters under a variety of assumptions regarding the distribution of the unobserved responses. Because these models make unverifiable identifying assumptions, we recommended conducting sensitivity analyses that provide a range of inferences, each of which is valid under different assumptions for nonresponse. The methodology is illustrated using data from a longitudinal study of obesity in children.     

Bias Reduction and a Solution for Separation of Logistic Regression with Missing Covariates

Summary Logistic regression is an important statistical procedure used in many disciplines. The standard software packages for data analysis are generally equipped with this procedure where the maximum likelihood estimates of the regression coefficients are obtained iteratively. It is well known that the estimates from the analyses of small‐ or medium‐sized samples are biased. Also, in finding such estimates, often a separation is encountered in which the likelihood converges but at least one of the parameter estimates diverges to infinity. Standard approaches of finding such estimates do not take care of these problems. Moreover, the missingness in the covariates adds an extra layer of complexity to the whole process. In this article, we address these three practical issues—bias, separation, and missing covariates by means of simple adjustments. We have applied the proposed technique using real and simulated data. The proposed method always finds a solution and the estimates are less biased. A SAS macro that implements the proposed method can be obtained from the authors.     

Nonignorable missingness in matched case-control data analyses

Matched case-control data analysis is often challenged by a missing covariate problem, the mishandling of which could cause bias or inefficiency. Satten and Carroll (2000, Biometrics56, 384-388) and other authors have proposed methods to handle missing covariates when the probability of missingness depends on the observed data, i.e., when data are missing at random. In this article, we propose a conditional likelihood method to handle the case when the probability of missingness depends on the unobserved covariate, i.e., when data are nonignorably missing. When the missing covariate is binary, the proposed method can be implemented using standard software. Using the Northern Manhattan Stroke Study data, we illustrate the method and discuss how sensitivity analysis can be conducted.     

Semiparametric methods in the proportional odds model for ordinal response data with missing covariates

Summary We consider the estimation problem of a proportional odds model with missing covariates. Based on the validation and nonvalidation data sets, we propose a joint conditional method that is an extension of Wang et al. (2002, Statistica Sinica 12, 555–574). The proposed method is semiparametric since it requires neither an additional model for the missingness mechanism, nor the specification of the conditional distribution of missing covariates given observed variables. Under the assumption that the observed covariates and the surrogate variable are categorical, we derived the large sample property. The simulation studies show that in various situations, the joint conditional method is more efficient than the conditional estimation method and weighted method. We also use a real data set that came from a survey of cable TV satisfaction to illustrate the approaches.