Simultaneous inference for semiparametric nonlinear mixed-effects models with covariate measurement errors and missing responses

Semiparametric nonlinear mixed-effects (NLME) models are flexible for modeling complex longitudinal data. Covariates are usually introduced in the models to partially explain interindividual variations. Some covariates, however, may be measured with substantial errors. Moreover, the responses may be missing and the missingness may be nonignorable. We propose two approximate likelihood methods for semiparametric NLME models with covariate measurement errors and nonignorable missing responses. The methods are illustrated in a real data example. Simulation results show that both methods perform well and are much better than the commonly used naive method.     

Joint inference for nonlinear mixed-effects models and time to event at the presence of missing data

In many longitudinal studies, the individual characteristics associated with the repeated measures may be possible covariates of the time to an event of interest, and thus, it is desirable to model the time-to-event process and the longitudinal process jointly. Statistical analyses may be further complicated in such studies with missing data such as informative dropouts. This article considers a nonlinear mixed-effects model for the longitudinal process and the Cox proportional hazards model for the time-to-event process. We provide a method for simultaneous likelihood inference on the 2 models and allow for nonignorable data missing. The approach is illustrated with a recent AIDS study by jointly modeling HIV viral dynamics and time to viral rebound.     

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.     

Analysis of current status data with missing covariates

Statistical inference based on right-censored data for the proportional hazards (PH) model with missing covariates has received considerable attention, but interval-censored or current status data with missing covariates has not yet been investigated. Our study is partly motivated by the analysis of fracture data from the 2005 National Health Interview Survey Original Database in Taiwan, where the occurrence of fractures was interval censored and the covariate osteoporosis was not reported for all residents. We assume that the data are realized from a PH model. A semiparametric maximum likelihood estimate implemented by a hybrid algorithm is proposed to analyze current status data with missing covariates. A comparison of the performance of our method with full-cohort analysis, complete-case analysis, and surrogate analysis is made via simulation with moderate sample sizes. The fracture data are then analyzed.     

Identifiability and estimation of causal effects in randomized trials with noncompliance and completely nonignorable missing data

Summary .  In this article, we first study parameter identifiability in randomized clinical trials with noncompliance and missing outcomes. We show that under certain conditions the parameters of interest are identifiable even under different types of completely nonignorable missing data: that is, the missing mechanism depends on the outcome. We then derive their maximum likelihood and moment estimators and evaluate their finite-sample properties in simulation studies in terms of bias, efficiency, and robustness. Our sensitivity analysis shows that the assumed nonignorable missing-data model has an important impact on the estimated complier average causal effect (CACE) parameter. Our new method provides some new and useful alternative nonignorable missing-data models over the existing latent ignorable model, which guarantees parameter identifiability, for estimating the CACE in a randomized clinical trial with noncompliance and missing data.     

Semiparametric models for missing covariate and response data in regression models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.     

A hidden Markov model for continuous longitudinal data with missing responses and dropout

We propose a hidden Markov model for multivariate continuous longitudinal responses with covariates that accounts for three different types of missing pattern: (I) partially missing outcomes at a given time occasion, (II) completely missing outcomes at a given time occasion (intermittent pattern), and (III) dropout before the end of the period of observation (monotone pattern). The missing-at-random (MAR) assumption is formulated to deal with the first two types of missingness, while to account for the informative dropout, we rely on an extra absorbing state. Estimation of the model parameters is based on the maximum likelihood method that is implemented by an expectation-maximization (EM) algorithm relying on suitable recursions. The proposal is illustrated by a Monte Carlo simulation study and an application based on historical data on primary biliary cholangitis.     

A joint model for longitudinal measurements and survival data in the presence of multiple failure types

Summary .   In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel ( Prentice et al., 1978 , Biometrics 34, 541–554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.     

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.     

Impact of nonignorable coarsening on Bayesian inference

The coarse data model of Heitjan and Rubin (1991) generalizes the missing data model of Rubin (1976) to cover other forms of incompleteness such as censoring and grouping. The model has 2 components: an ideal data model describing the distribution of the quantity of interest and a coarsening mechanism that describes a distribution over degrees of coarsening given the ideal data. The coarsening mechanism is said to be nonignorable when the degree of coarsening depends on an incompletely observed ideal outcome, in which case failure to properly account for it can spoil inferences. A theme in recent research is to measure sensitivity to nonignorability by evaluating the effect of a small departure from ignorability on the maximum likelihood estimate (MLE) of a parameter of the ideal data model. One such construct is the "index of local sensitivity to nonignorability" (ISNI) (Troxel and others, 2004), which is the derivative of the MLE with respect to a nonignorability parameter evaluated at the ignorable model. In this paper, we adapt ISNI to Bayesian modeling by instead defining it as the derivative of the posterior expectation. We propose the application of ISNI as a first step in judging the robustness of a Bayesian analysis to nonignorable coarsening. We derive formulas for a range of models and apply the method to evaluate sensitivity to nonignorable coarsening in 2 real data examples, one involving missing CD4 counts in an HIV trial and the other involving potentially informatively censored relapse times in a leukemia trial.     

A hybrid model for nonignorable dropout in longitudinal binary responses

This article presents a likelihood-based method for handling nonignorable dropout in longitudinal studies with binary responses. The methodology developed is appropriate when the target of inference is the marginal distribution of the response at each occasion and its dependence on covariates. A "hybrid" model is formulated, which is designed to retain advantageous features of the selection and pattern-mixture model approaches. This formulation accommodates a variety of assumed forms of nonignorable dropout, while maintaining transparency of the constraints required for identifying the overall model. Once appropriate identifying constraints have been imposed, likelihood-based estimation is conducted via the EM algorithm. The article concludes by applying the approach to data from a randomized clinical trial comparing two doses of a contraceptive.     

A bivariate pseudolikelihood for incomplete longitudinal binary data with nonignorable nonmonotone missingness

For analyzing longitudinal binary data with nonignorable and nonmonotone missing responses, a full likelihood method is complicated algebraically, and often requires intensive computation, especially when there are many follow-up times. As an alternative, a pseudolikelihood approach has been proposed in the literature under minimal parametric assumptions. This formulation only requires specification of the marginal distributions of the responses and missing data mechanism, and uses an independence working assumption. However, this estimator can be inefficient for estimating both time-varying and time-stationary effects under moderate to strong within-subject associations among repeated responses. In this article, we propose an alternative estimator, based on a bivariate pseudolikelihood, and demonstrate in simulations that the proposed method can be much more efficient than the previous pseudolikelihood obtained under the assumption of independence. We illustrate the method using longitudinal data on CD4 counts from two clinical trials of HIV-infected patients.     

A Model for Adjusting for Nonignorable Verification Bias in Estimation of the ROC Curve and Its Area with Likelihood‐Based Approach

Summary In estimation of the ROC curve, when the true disease status is subject to nonignorable missingness, the observed likelihood involves the missing mechanism given by a selection model. In this article, we proposed a likelihood‐based approach to estimate the ROC curve and the area under the ROC curve when the verification bias is nonignorable. We specified a parametric disease model in order to make the nonignorable selection model identifiable. With the estimated verification and disease probabilities, we constructed four types of empirical estimates of the ROC curve and its area based on imputation and reweighting methods. In practice, a reasonably large sample size is required to estimate the nonignorable selection model in our settings. Simulation studies showed that all four estimators of ROC area performed well, and imputation estimators were generally more efficient than the other estimators proposed. We applied the proposed method to a data set from research in Alzheimer's disease.     

Parametric and semiparametric model-based estimates of the finite population mean for two-stage cluster samples with item nonresponse

This article concerns item nonresponse adjustment for two-stage cluster samples. Specifically, we focus on two types of nonignorable nonresponse: nonresponse depending on covariates and underlying cluster characteristics, and depending on covariates and the missing outcome. In these circumstances, standard weighting and imputation adjustments are liable to be biased. To obtain consistent estimates, we extend the standard random-effects model by modeling these two types of missing data mechanism. We also propose semiparametric approaches based on fitting a spline on the propensity score, to weaken assumptions about the relationship between the outcome and covariates. These new methods are compared with existing approaches by simulation. The National Health and Nutrition Examination Survey data are used to illustrate these approaches.     

A pseudo-bayesian shrinkage approach to regression with missing covariates

Summary We consider the linear regression of outcome Y on regressors W and Z with some values of W missing, when our main interest is the effect of Z on Y, controlling for W. Three common approaches to regression with missing covariates are (i) complete‐case analysis (CC), which discards the incomplete cases, and (ii) ignorable likelihood methods, which base inference on the likelihood based on the observed data, assuming the missing data are missing at random ( Rubin, 1976b ), and (iii) nonignorable modeling, which posits a joint distribution of the variables and missing data indicators. Another simple practical approach that has not received much theoretical attention is to drop the regressor variables containing missing values from the regression modeling (DV, for drop variables). DV does not lead to bias when either (i) the regression coefficient of W is zero or (ii) W and Z are uncorrelated. We propose a pseudo‐Bayesian approach for regression with missing covariates that compromises between the CC and DV estimates, exploiting information in the incomplete cases when the data support DV assumptions. We illustrate favorable properties of the method by simulation, and apply the proposed method to a liver cancer study. Extension of the method to more than one missing covariate is also discussed.     

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.     

Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information

This article presents a new method for maximum likelihood estimation of logistic regression models with incomplete covariate data where auxiliary information is available. This auxiliary information is extraneous to the regression model of interest but predictive of the covariate with missing data. Ibrahim (1990, Journal of the American Statistical Association 85, 765-769) provides a general method for estimating generalized linear regression models with missing covariates using the EM algorithm that is easily implemented when there is no auxiliary data. Vach (1997, Statistics in Medicine 16, 57-72) describes how the method can be extended when the outcome and auxiliary data are conditionally independent given the covariates in the model. The method allows the incorporation of auxiliary data without making the conditional independence assumption. We suggest tests of conditional independence and compare the performance of several estimators in an example concerning mental health service utilization in children. Using an artificial dataset, we compare the performance of several estimators when auxiliary data are available.     

Maximum likelihood methods for cure rate models with missing covariates

We propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction, in which the covariates are allowed to be missing. We allow the covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one-dimensional conditional distributions. We propose a novel EM algorithm for maximum likelihood estimation and derive standard errors by using Louis's formula (Louis, 1982, Journal of the Royal Statistical Society, Series B 44, 226-233). Computational techniques using the Monte Carlo EM algorithm are discussed and implemented. A real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology.     

Nonignorable Models for Intermittently Missing Categorical Longitudinal Responses

Summary A class of nonignorable models is presented for handling nonmonotone missingness in categorical longitudinal responses. This class of models includes the traditional selection models and shared parameter models. This allows us to perform a broader than usual sensitivity analysis. In particular, instead of considering variations to a chosen nonignorable model, we study sensitivity between different missing data frameworks. An appealing feature of the developed class is that parameters with a marginal interpretation are obtained, while algebraically simple models are considered. Specifically, marginalized mixed‐effects models ( Heagerty, 1999 , Biometrics 55, 688–698) are used for the longitudinal process that model separately the marginal mean and the correlation structure. For the correlation structure, random effects are introduced and their distribution is modeled either parametrically or non‐parametrically to avoid potential misspecifications.     

An exploration of fixed and random effects selection for longitudinal binary outcomes in the presence of nonignorable dropout

We explore a Bayesian approach to selection of variables that represent fixed and random effects in modeling of longitudinal binary outcomes with missing data caused by dropouts. We show via analytic results for a simple example that nonignorable missing data lead to biased parameter estimates. This bias results in selection of wrong effects asymptotically, which we can confirm via simulations for more complex settings. By jointly modeling the longitudinal binary data with the dropout process that possibly leads to nonignorable missing data, we are able to correct the bias in estimation and selection. Mixture priors with a point mass at zero are used to facilitate variable selection. We illustrate the proposed approach using a clinical trial for acute ischemic stroke.