A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

Semiparametric inference for estimating equations with nonignorably missing covariates

We consider statistical inference of unknown parameters in estimating equations (EEs) when some covariates have nonignorably missing values, which is quite common in practice but has rarely been discussed in the literature. When an instrument, a fully observed covariate vector that helps identifying parameters under nonignorable missingness, is available, the conditional distribution of the missing covariates given other covariates can be estimated by the pseudolikelihood method of Zhao and Shao [(2015), ‘Semiparametric pseudo likelihoods in generalised linear models with nonignorable missing data’, Journal of the American Statistical Association, 110, 1577–1590)] and be used to construct unbiased EEs. These modified EEs then constitute a basis for valid inference by empirical likelihood. Our method is applicable to a wide range of EEs used in practice. It is semiparametric since no parametric model for the propensity of missing covariate data is assumed. Asymptotic properties of the proposed estimator and the empirical likelihood ratio test statistic are derived. Some simulation results and a real data analysis are presented for illustration.     

Likelihood methods in randomized trials with noncompliance and subsequent nonresponse

This article considers likelihood methods for estimating the causal effect of treatment assignment for a two-armed randomized trial assuming all-or-none treatment noncompliance and allowing for subsequent nonresponse. We first derive the observed data likelihood function as a closed form expression of the parameter given the observed data where both response and compliance state are treated as variables with missing values. Then we describe an iterative procedure which maximizes the observed data likelihood function directly to compute a maximum likelihood estimator (MLE) of the causal effect of treatment assignment. Closed form expressions at each iterative step are provided. Finally we compare the MLE with an alternative estimator where the probability distribution of the compliance state is estimated independent of the response and its missingness mechanism. Our work indicates that direct maximum likelihood inference is straightforward for this problem. Extensive simulation studies are provided to examine the finite sample performance of the proposed methods.     

Cox Regression with Covariate Measurement Error

This article deals with parameter estimation in the Cox proportional hazards model when covariates are measured with error. We consider both the classical additive measurement error model and a more general model which represents the mis-measured version of the covariate as an arbitrary linear function of the true covariate plus random noise. Only moment conditions are imposed on the distributions of the covariates and measurement error. Under the assumption that the covariates are measured precisely for a validation set, we develop a class of estimating equations for the vector-valued regression parameter by correcting the partial likelihood score function. The resultant estimators are proven to be consistent and asymptotically normal with easily estimated variances. Furthermore, a corrected version of the Breslow estimator for the cumulative hazard function is developed, which is shown to be uniformly consistent and, upon proper normalization, converges weakly to a zero-mean Gaussian process. Simulation studies indicate that the asymptotic approximations work well for practical sample sizes. The situation in which replicate measurements (instead of a validation set) are available is also studied.     

Missing covariates in generalized linear models when the missing data mechanism is non-ignorable

We propose a method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. We use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one-dimensional conditional distributions, with each one-dimensional conditional distribution consisting of a logistic regression. We allow the covariates to be either categorical or continuous. The joint covariate distribution is also modelled via a sequence of one-dimensional conditional distributions, and the response variable is assumed to be completely observed. We derive the E- and M-steps of the EM algorithm with non-ignorable missing covariate data. For categorical covariates, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For continuous covariates, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. Computational techniques for Gibbs sampling are proposed and implemented. The parametric form of the assumed missing data mechanism itself is not `testable' from the data, and thus the non-ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model. Therefore, although a model may have `passed' the tests for a certain missing data mechanism, this does not mean that we have captured, even approximately, the correct missing data mechanism. Hence, model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail. Several simulations are given to demonstrate the methodology. In addition, a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed.     

The Lehmann Model with Time-dependent Covariates

Consider the Lehmann model with time-dependent covariates, which is different from Cox’s model. We find out that (1) the parameter space for β under the Lehmann model is restricted, and the maximum point of the parametric likelihood for β may lie outside the parameter space; (2) for some particular time-dependent covariate, under the standard generalized likelihood the semiparametric maximum likelihood estimator (SMLE) is inconsistent and we propose a modified generalized likelihood which leads to the consistent SMLE.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Semiparametric Regression with Kernel Error Model

Abstract.  We propose and study a class of regression models, in which the mean function is specified parametrically as in the existing regression methods, but the residual distribution is modelled non-parametrically by a kernel estimator, without imposing any assumption on its distribution. This specification is different from the existing semiparametric regression models. The asymptotic properties of such likelihood and the maximum likelihood estimate (MLE) under this semiparametric model are studied. We show that under some regularity conditions, the MLE under this model is consistent (when compared with the possibly pseudo-consistency of the parameter estimation under the existing parametric regression model), is asymptotically normal with rate and efficient. The non-parametric pseudo-likelihood ratio has the Wilks property as the true likelihood ratio does. Simulated examples are presented to evaluate the accuracy of the proposed semiparametric MLE method.     

Cox Regression with Incomplete Covariate Measurements using the EM-algorithm

Ibrahim (1990) used the EM-algorithm to obtain maximum likelihood estimates of the regression parameters in generalized linear models with partially missing covariates. The technique was termed EM by the method of weights. In this paper, we generalize this technique to Cox regression analysis with missing values in the covariates. We specify a full model letting the unobserved covariate values be random and then maximize the observed likelihood. The asymptotic covariance matrix is estimated by the inverse information matrix. The missing data are allowed to be missing at random but also the non-ignorable non-response situation may in principle be considered. Simulation studies indicate that the proposed method is more efficient than the method suggested by Paik & Tsai (1997). We apply the procedure to a clinical trials example with six covariates with three of them having missing values.     

Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing 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 assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

Maximum likelihood estimation for the proportional hazards model with partly interval-censored data

Summary. The maximum likelihood estimator (MLE) for the proportional hazards model with partly interval-censored data is studied. Under appropriate regularity conditions, the MLEs of the regression parameter and the cumulative hazard function are shown to be consistent and asymptotically normal. Two methods to estimate the variance–covariance matrix of the MLE of the regression parameter are considered, based on a generalized missing information principle and on a generalized profile information procedure. Simulation studies show that both methods work well in terms of the bias and variance for samples of moderate size. An example illustrates the methods.     

Doubly robust empirical likelihood inference in covariate-missing data problems

Missing covariate data occurs often in regression analysis. We study methods for estimating the regression coefficients in an assumed conditional mean function when some covariates are completely observed but other covariates are missing for some subjects. We adopt the semiparametric perspective of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866] on regression analyses with missing covariates, in which they pioneered the use of two working models, the working propensity score model and the working conditional score model. A recent approach to missing covariate data analysis is the empirical likelihood method of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503], which effectively combines unbiased estimating equations. In this paper, we consider an alternative likelihood approach based on the full likelihood of the observed data. This full likelihood-based method enables us to generate estimators for the vector of the regression coefficients that are (a) asymptotically equivalent to those of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the working propensity score model is correctly specified, and (b) doubly robust, like the augmented inverse probability weighting (AIPW) estimators of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Am Statist Assoc. 1994;89:846–866]. Thus, the proposed full likelihood-based estimators improve on the efficiency of the AIPW estimators when the working propensity score model is correct but the working conditional score model is possibly incorrect, and also improve on the empirical likelihood estimators of Qin, Zhang and Leung [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the reverse is true, that is, the working conditional score model is correct but the working propensity score model is possibly incorrect. In addition, we consider a regression method for estimation of the regression coefficients when the working conditional score model is correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866]. Finally, we compare the finite-sample performance of various estimators in a simulation study.     

Missing Data Mechanisms for Analysing Longitudinal Data with Incomplete Observations in Both Responses and Covariates

Missing observations in both responses and covariates arise frequently in longitudinal studies. When missing data are missing not at random, inferences under the likelihood framework often require joint modelling of response and covariate processes, as well as missing data processes associated with incompleteness of responses and covariates. Specification of these four joint distributions is a nontrivial issue from the perspectives of both modelling and computation. To get around this problem, we employ pairwise likelihood formulations, which avoid the specification of third or higher order association structures. In this paper, we consider three specific missing data mechanisms which lead to further simplified pairwise likelihood (SPL) formulations. Under these missing data mechanisms, inference methods based on SPL formulations are developed. The resultant estimators are consistent, and enjoy better robustness and computation convenience. The performance is evaluated empirically though simulation studies. Longitudinal data from the National Population Health Survey and Waterloo Smoking Prevention Project are analysed to illustrate the usage of our methods.     

A Cox-type regression model with change-points in the covariates

We consider a Cox-type regression model with change-points in the covariates. A change-point specifies the unknown threshold at which the influence of a covariate shifts smoothly, i.e., the regression parameter may change over the range of a covariate and the underlying regression function is continuous but not differentiable. The model can be used to describe change-points in different covariates but also to model more than one change-point in a single covariate. Estimates of the change-points and of the regression parameters are derived and their properties are investigated. It is shown that not only the estimates of the regression parameters are [Formula: see text] -consistent but also the estimates of the change-points in contrast to the conjecture of other authors. Asymptotic normality is shown by using results developed for M-estimators. At the end of this paper we apply our model to an actuarial dataset, the PBC dataset of Fleming and Harrington (Counting processes and survival analysis, 1991) and to a dataset of electric motors.     

A risk set calibration method for failure time regression by using a covariate reliability sample

Regression parameter estimation in the Cox failure time model is considered when regression variables are subject to measurement error. Assuming that repeat regression vector measurements adhere to a classical measurement model, we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements. However, since the rate of withdrawal from the risk set across the time axis, due to failure or censoring, will typically depend on covariates, we may improve the regression parameter estimator by recalibrating within each risk set. The asymptotic and small sample properties of such a risk set regression calibration estimator are studied. A simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances. Corresponding asymptotic distribution theory is developed, small sample properties are studied using computer simulations and an illustration is provided.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

Constrained inference for generalized linear models with incomplete covariate data

Missing data are common in many experiments, including surveys, clinical trials, epidemiological studies, and environmental studies. Unconstrained likelihood inferences for generalized linear models (GLMs) with nonignorable missing covariates have been studied extensively in the literature. However, parameter orderings or constraints may occur naturally in practice, and thus the efficiency of a statistical method may be improved by incorporating parameter constraints into the likelihood function. In this paper, we consider constrained inference for analysing GLMs with nonignorable missing covariates under linear inequality constraints on the model parameters. Specifically, constrained maximum likelihood (ML) estimation is based on the gradient projection expectation maximization approach. Further, we investigate the asymptotic null distribution of the constrained likelihood ratio test (LRT). Simulations study the empirical properties of the constrained ML estimators and LRTs, which demonstrate improved precision of these constrained techniques. An application to contaminant levels in an environmental study is also presented.     

Incomplete covariates in the Cox model with applications to biological marker data

A common occurrence in clinical trials with a survival end point is missing covariate data. With ignorably missing covariate data, Lipsitz and Ibrahim proposed a set of estimating equations to estimate the parameters of Cox's proportional hazards model. They proposed to obtain parameter estimates via a Monte Carlo EM algorithm. We extend those results to non-ignorably missing covariate data. We present a clinical trials example with three partially observed laboratory markers which are used as covariates to predict survival.     

Information attainable in some randomly incomplete data models

The Fisher information is intricately linked to the asymptotic (first-order) optimality of maximum likelihood estimators for parametric complete-data models. When data are missing completely at random in a multivariate setup, it is shown that information in a single observation is well-defined and it plays the same role as in the complete-data model in characterizing the first-order asymptotic optimality properties of associated maximum likelihood estimators; computational aspects are also thoroughly appraised. As an illustration, the logistic regression model with incomplete binary responses and an incomplete categorical covariate is worked out.     

Information attainable in some randomly incomplete multivariate response models

In a general parametric setup, a multivariate regression model is considered when responses may be missing at random while the explanatory variables and covariates are completely observed. Asymptotic optimality properties of maximum likelihood estimators for such models are linked to the Fisher information matrix for the parameters. It is shown that the information matrix is well defined for the missing-at-random model and that it plays the same role as in the complete-data linear models. Applications of the methodologic developments in hypothesis-testing problems, without any imputation of missing data, are illustrated. Some simulation results comparing the proposed method with Rubin's multiple imputation method are presented.