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.     

Multiple Imputation Methods for Estimating Regression Coefficients in the Competing Risks Model with Missing Cause of Failure

We propose a method to estimate the regression coefficients in a competing risks model where the cause-specific hazard for the cause of interest is related to covariates through a proportional hazards relationship and when cause of failure is missing for some individuals. We use multiple imputation procedures to impute missing cause of failure, where the probability that a missing cause is the cause of interest may depend on auxiliary covariates, and combine the maximum partial likelihood estimators computed from several imputed data sets into an estimator that is consistent and asymptotically normal. A consistent estimator for the asymptotic variance is also derived. Simulation results suggest the relevance of the theory in finite samples. Results are also illustrated with data from a breast cancer study.     

Improving estimation efficiency for regression with MNAR covariates

For regression with covariates missing not at random where the missingness depends on the missing covariate values, complete-case (CC) analysis leads to consistent estimation when the missingness is independent of the response given all covariates, but it may not have the desired level of efficiency. We propose a general empirical likelihood framework to improve estimation efficiency over the CC analysis. We expand on methods in Bartlett et al. (2014, Biostatistics 15 , 719–730) and Xie and Zhang (2017, Int J Biostat 13 , 1–20) that improve efficiency by modeling the missingness probability conditional on the response and fully observed covariates by allowing the possibility of modeling other data distribution-related quantities. We also give guidelines on what quantities to model and demonstrate that our proposal has the potential to yield smaller biases than existing methods when the missingness probability model is incorrect. Simulation studies are presented, as well as an application to data collected from the US National Health and Nutrition Examination Survey.     

A new proposal for multivariable modelling of time-varying effects in survival data based on fractional polynomial time-transformation

The Cox proportional hazards model has become the standard for the analysis of survival time data in cancer and other chronic diseases. In most studies, proportional hazards (PH) are assumed for covariate effects. With long-term follow-up, the PH assumption may be violated, leading to poor model fit. To accommodate non-PH effects, we introduce a new procedure, MFPT, an extension of the multivariable fractional polynomial (MFP) approach, to do the following: (1) select influential variables; (2) determine a sensible dose-response function for continuous variables; (3) investigate time-varying effects; (4) model such time-varying effects on a continuous scale. Assuming PH initially, we start with a detailed model-building step, including a search for possible non-linear functions for continuous covariates. Sometimes a variable with a strong short-term effect may appear weak or non-influential if 'averaged' over time under the PH assumption. To protect against omitting such variables, we repeat the analysis over a restricted time-interval. Any additional prognostic variables identified by this second analysis are added to create our final time-fixed multivariable model. Using a forward-selection algorithm we search for possible improvements in fit by adding time-varying covariates. The first part to create a final time-fixed model does not require the use of MFP. A model may be given from 'outside' or a different strategy may be preferred for this part. This broadens the scope of the time-varying part. To motivate and illustrate the methodology, we create prognostic models from a large database of patients with primary breast cancer. Non-linear time-fixed effects are found for progesterone receptor status and number of positive lymph nodes. Highly statistically significant time-varying effects are present for progesterone receptor status and tumour size.     

Monte Carlo EM for missing covariates in parametric regression models

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.     

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.     

Maximum likelihood methods for nonignorable missing responses and covariates in random effects models

This article analyzes quality of life (QOL) data from an Eastern Cooperative Oncology Group (ECOG) melanoma trial that compared treatment with ganglioside vaccination to treatment with high-dose interferon. The analysis of this data set is challenging due to several difficulties, namely, nonignorable missing longitudinal responses and baseline covariates. Hence, we propose a selection model for estimating parameters in the normal random effects model with nonignorable missing responses and covariates. Parameters are estimated via maximum likelihood using the Gibbs sampler and a Monte Carlo expectation maximization (EM) algorithm. Standard errors are calculated using the bootstrap. The method allows for nonmonotone patterns of missing data in both the response variable and the covariates. We model the missing data mechanism and the missing covariate distribution via a sequence of one-dimensional conditional distributions, allowing the missing covariates to be either categorical or continuous, as well as time-varying. We apply the proposed approach to the ECOG quality-of-life data and conduct a small simulation study evaluating the performance of the maximum likelihood estimates. Our results indicate that a patient treated with the vaccine has a higher QOL score on average at a given time point than a patient treated with high-dose interferon.     

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.     

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.     

Just-identified versus overidentified two-level hierarchical linear models with missing data

The development of model-based methods for incomplete data has been a seminal contribution to statistical practice. Under the assumption of ignorable missingness, one estimates the joint distribution of the complete data for thetainTheta from the incomplete or observed data y(obs). Many interesting models involve one-to-one transformations of theta. For example, with y(i) approximately N(mu, Sigma) for i= 1, ... , n and theta= (mu, Sigma), an ordinary least squares (OLS) regression model is a one-to-one transformation of theta. Inferences based on such a transformation are equivalent to inferences based on OLS using data multiply imputed from f(y(mis) | y(obs), theta) for missing y(mis). Thus, identification of theta from y(obs) is equivalent to identification of the regression model. In this article, we consider a model for two-level data with continuous outcomes where the observations within each cluster are dependent. The parameters of the hierarchical linear model (HLM) of interest, however, lie in a subspace of Theta in general. This identification of the joint distribution overidentifies the HLM. We show how to characterize the joint distribution so that its parameters are a one-to-one transformation of the parameters of the HLM. This leads to efficient estimation of the HLM from incomplete data using either the transformation method or the method of multiple imputation. The approach allows outcomes and covariates to be missing at either of the two levels, and the HLM of interest can involve the regression of any subset of variables on a disjoint subset of variables conceived as covariates.     

Variable Selection in the Cox Regression Model with Covariates Missing at Random

Summary .  We consider variable selection in the Cox regression model ( Cox, 1975 ,  Biometrika   362, 269–276) with covariates missing at random. We investigate the smoothly clipped absolute deviation penalty and adaptive least absolute shrinkage and selection operator (LASSO) penalty, and propose a unified model selection and estimation procedure. A computationally attractive algorithm is developed, which simultaneously optimizes the penalized likelihood function and penalty parameters. We also optimize a model selection criterion, called the   IC _Q    statistic ( Ibrahim, Zhu, and Tang, 2008 ,  Journal of the American Statistical Association   103, 1648–1658), to estimate the penalty parameters and show that it consistently selects all important covariates. Simulations are performed to evaluate the finite sample performance of the penalty estimates. Also, two lung cancer data sets are analyzed to demonstrate the proposed methodology.     

A robust method using propensity score stratification for correcting verification bias for binary tests

Sensitivity and specificity are common measures of the accuracy of a diagnostic test. The usual estimators of these quantities are unbiased if data on the diagnostic test result and the true disease status are obtained from all subjects in an appropriately selected sample. In some studies, verification of the true disease status is performed only for a subset of subjects, possibly depending on the result of the diagnostic test and other characteristics of the subjects. Estimators of sensitivity and specificity based on this subset of subjects are typically biased; this is known as verification bias. Methods have been proposed to correct verification bias under the assumption that the missing data on disease status are missing at random (MAR), that is, the probability of missingness depends on the true (missing) disease status only through the test result and observed covariate information. When some of the covariates are continuous, or the number of covariates is relatively large, the existing methods require parametric models for the probability of disease or the probability of verification (given the test result and covariates), and hence are subject to model misspecification. We propose a new method for correcting verification bias based on the propensity score, defined as the predicted probability of verification given the test result and observed covariates. This is estimated separately for those with positive and negative test results. The new method classifies the verified sample into several subsamples that have homogeneous propensity scores and allows correction for verification bias. Simulation studies demonstrate that the new estimators are more robust to model misspecification than existing methods, but still perform well when the models for the probability of disease and probability of verification are correctly specified.     

Generalized additive models for cancer mapping with incomplete covariates

Maps depicting cancer incidence rates have become useful tools in public health research, giving valuable information about the spatial variation in rates of disease. Typically, these maps are generated using count data aggregated over areas such as counties or census blocks. However, with the proliferation of geographic information systems and related databases, it is becoming easier to obtain exact spatial locations for the cancer cases and suitable control subjects. The use of such point data allows us to adjust for individual-level covariates, such as age and smoking status, when estimating the spatial variation in disease risk. Unfortunately, such covariate information is often subject to missingness. We propose a method for mapping cancer risk when covariates are not completely observed. We model these data using a logistic generalized additive model. Estimates of the linear and non-linear effects are obtained using a mixed effects model representation. We develop an EM algorithm to account for missing data and the random effects. Since the expectation step involves an intractable integral, we estimate the E-step with a Laplace approximation. This framework provides a general method for handling missing covariate values when fitting generalized additive models. We illustrate our method through an analysis of cancer incidence data from Cape Cod, Massachusetts. These analyses demonstrate that standard complete-case methods can yield biased estimates of the spatial variation of cancer risk.     

Maximum likelihood estimation in random effects cure rate models with nonignorable missing covariates

We introduce a method of parameter estimation for a random effects cure rate model. We also propose a methodology that allows us to account for nonignorable missing covariates in this class of models. The proposed method corrects for possible bias introduced by complete case analysis when missing data are not missing completely at random and is motivated by data from a pair of melanoma studies conducted by the Eastern Cooperative Oncology Group in which clustering by cohort or time of study entry was suspected. In addition, these models allow estimation of cure rates, which is desirable when we do not wish to assume that all subjects remain at risk of death or relapse from disease after sufficient follow-up. We develop an EM algorithm for the model and provide an efficient Gibbs sampling scheme for carrying out the E-step of the algorithm.     

A Semiparametric Missing‐Data‐Induced Intensity Method for Missing Covariate Data in Individually Matched Case–Control Studies

Summary In individually matched case–control studies, when some covariates are incomplete, an analysis based on the complete data may result in a large loss of information both in the missing and completely observed variables. This usually results in a bias and loss of efficiency. In this article, we propose a new method for handling the problem of missing covariate data based on a missing‐data‐induced intensity approach when the missingness mechanism does not depend on case–control status and show that this leads to a generalization of the missing indicator method. We derive the asymptotic properties of the estimates from the proposed method and, using an extensive simulation study, assess the finite sample performance in terms of bias, efficiency, and 95% confidence coverage under several missing data scenarios. We also make comparisons with complete‐case analysis (CCA) and some missing data methods that have been proposed previously. Our results indicate that, under the assumption of predictable missingness, the suggested method provides valid estimation of parameters, is more efficient than CCA, and is competitive with other, more complex methods of analysis. A case–control study of multiple myeloma risk and a polymorphism in the receptor Inter‐Leukin‐6 (IL‐6‐α) is used to illustrate our findings.     

A simulation-based marginal method for longitudinal data with dropout and mismeasured covariates

Longitudinal data often contain missing observations and error-prone covariates. Extensive attention has been directed to analysis methods to adjust for the bias induced by missing observations. There is relatively little work on investigating the effects of covariate measurement error on estimation of the response parameters, especially on simultaneously accounting for the biases induced by both missing values and mismeasured covariates. It is not clear what the impact of ignoring measurement error is when analyzing longitudinal data with both missing observations and error-prone covariates. In this article, we study the effects of covariate measurement error on estimation of the response parameters for longitudinal studies. We develop an inference method that adjusts for the biases induced by measurement error as well as by missingness. The proposed method does not require the full specification of the distribution of the response vector but only requires modeling its mean and variance structures. Furthermore, the proposed method employs the so-called functional modeling strategy to handle the covariate process, with the distribution of covariates left unspecified. These features, plus the simplicity of implementation, make the proposed method very attractive. In this paper, we establish the asymptotic properties for the resulting estimators. With the proposed method, we conduct sensitivity analyses on a cohort data set arising from the Framingham Heart Study. Simulation studies are carried out to evaluate the impact of ignoring covariate measurement error and to assess the performance of the proposed method.     

Factors influencing soay sheep survival: a Bayesian analysis

This article presents a Bayesian analysis of mark-recapture-recovery data on Soay sheep. A reversible jump Markov chain Monte Carlo technique is used to determine age classes of common survival, and to model the survival probabilities in those classes using logistic regression. This involves environmental and individual covariates, as well as random effects. Auxiliary variables are used to impute missing covariates measured on individual sheep. The Bayesian approach suggests different models from those previously obtained using classical statistical methods. Following model averaging, features that were not previously detected, and which are of ecological importance, are identified.