Semiparametric Maximum Likelihood Estimation Research Articles

Estimation in the Cox cure model with covariates missing not at random, with application to disease screening/prediction

In an attempt to provide a statistical tool for disease screening and prediction, we propose a semiparametric approach to analysis of the Cox proportional hazards cure model in situations where the observations on the event time are subject to right censoring and some covariates are missing not at random. To facilitate the methodological development, we begin with semiparametric maximum likelihood estimation (SPMLE) assuming that the (conditional) distribution of the missing covariates is known. A variant of the EM algorithm is used to compute the estimator. We then adapt the SPMLE to a more practical situation where the distribution is unknown and there is a consistent estimator based on available information. We establish the consistency and weak convergence of the resulting pseudo‐SPMLE, and identify a suitable variance estimator. The application of our inference procedure to disease screening and prediction is illustrated via empirical studies. The proposed approach is used to analyze the tuberculosis screening study data that motivated this research. Its finite‐sample performance is examined by simulation.

Identifiability conditions for the linear regression model under right censoring

The consistency of various estimators under the semi-parametric linear regression model and the standard right censorship model (SPLRRC model) has been studied under various assumptions since the 1970s. These assumptions are somewhat sufficient conditions for the identifiability of the parameters under the SPLRRC model. Since then, it has been a difficult open problem in survival analysis to find the necessary and sufficient condition for the identifiability of the parameters under the SPLRRC model. The open problem is solved in this paper. It is of interest to investigate whether the common estimators under this model are consistent under the identifiability condition. Under the latter condition, we show that the Buckley–James estimator and quantile regression estimator can be inconsistent and present partial results on the consistency of the semi-parametric maximum likelihood estimator.

Accelerated failure time models with log-concave errors

Summary We study accelerated failure time models in which the survivor function of the additive error term is log-concave. The log-concavity assumption covers large families of commonly used distributions and also represents the aging or wear-out phenomenon of the baseline duration. For right-censored failure time data, we construct semiparametric maximum likelihood estimates of the finite-dimensional parameter and establish the large sample properties. The shape restriction is incorporated via a nonparametric maximum likelihood estimator of the hazard function. Our approach guarantees the uniqueness of a global solution for the estimating equations and delivers semiparametric efficient estimates. Simulation studies and empirical applications demonstrate the usefulness of our method.

Asymptotic distribution theory on pseudo semiparametric maximum likelihood estimator with covariates missing not at random

Recently, Cook et al. proposed a semiparametric likelihood estimator to improve study efficiency for a kind of survival data with covariate entries missing not at random (MNAR). Readily available supplementary information on the covariate is utilized in the estimation. They assume that the conditional distributions of the covariate X that having missing entry given the completely observed covariate Z, is known. Guo et al. suggested to replace with its consistent estimator in the likelihood equation when is unknown. However, they did not derive the asymptotic theory of the resulted estimator in this case. This paper fills the gap. The theoretical development makes use of the theory of modern empirical process.

Cox regression of clustered event times with covariates missing not at random

AbstractMotivated by a recent tuberculosis (TB) study, this paper is concerned with covariates missing not at random (MNAR) and models the potential intracluster correlation by a frailty. We consider the regression analysis of right‐censored event times from clustered subjects under a Cox proportional hazards frailty model and present the semiparametric maximum likelihood estimator (SPMLE) of the model parameters. An easy‐to‐implement pseudo‐SPMLE is then proposed to accommodate more realistic situations using readily available supplementary information on the missing covariates. Algorithms are provided to compute the estimators and their consistent variance estimators. We demonstrate that both the SPMLE and the pseudo‐SPMLE are consistent and asymptotically normal by the arguments based on the theory of modern empirical processes. The proposed approach is examined numerically via simulation and illustrated with an analysis of the motivating TB study data.

Accelerated Failure Time Models with Logconcave Errors

We study accelerated failure time models in which the survivor function of the error term is log-concave. The log-concavity assumption is often implied by the underlying economic models and covers large families of commonly used distributions. For right-censored failure time data, we construct semi-parametric maximum likelihood estimates of the finite dimensional parameter subject to the shape restriction and establish the large sample properties. The shape restriction also facilitates computation, as the optimization problem has a unique global solution with probability tending to one. Simulation studies and empirical applications demonstrate the usefulness of our method.

Semiparametric transformation models for interval-censored data in the presence of a cure fraction.

Mixed case interval-censored data arise when the event of interest is known only to occur within an interval induced by a sequence of random examination times. Such data are commonly encountered in disease research with longitudinal follow-up. Furthermore, the medical treatment has progressed over the last decade with an increasing proportion of patients being cured for many types of diseases. Thus, interest has grown in cure models for survival data which hypothesize a certain proportion of subjects in the population are not expected to experience the events of interest. In this article, we consider a two-component mixture cure model for regression analysis of mixed case interval-censored data. The first component is a logistic regression model that describes the cure rate, and the second component is a semiparametric transformation model that describes the distribution of event time for the uncured subjects. We propose semiparametric maximum likelihood estimation for the considered model. We develop an EM type algorithm for obtaining the semiparametric maximum likelihood estimators (SPMLE) of regression parameters and establish their consistency, efficiency, and asymptotic normality. Extensive simulation studies indicate that the SPMLE performs satisfactorily in a wide variety of settings. The proposed method is illustrated by the analysis of the hypobaric decompression sickness data from National Aeronautics and Space Administration.

Semiparametric transformation joint models for longitudinal covariates and interval-censored failure time

In many clinical trials and epidemiology research, subjects are followed-up repeatedly, and repeated measurements on longitudinal covariates as well as an observation on a possibly censored time-to-event are collected on each subject. The longitudinal covariates are often measured intermittently with measurement errors, and the measurement process is terminated by a correlated event process, leading to informative missing data. Methods for joint modelling of longitudinal and time to-event data have received much attention in the statistics literature in recent years. Most research has focused on right-censoring mechanism for the event time. In practice, the event time is often examined at the pre-scheduled times, at which the longitudinal covariates are also measured, resulting in interval-censored survival data. To take the interval censoring into account and to provide a more general framework for studying the effects of covariates on survival time, a new class of joint models is proposed. The joint model comprises a linear mixed-effects model for the longitudinal biomarkers and a class of semiparametric transformation models for the failure time, which incorporates the underlying longitudinal biomarkers as time-dependent covariates. The likelihood approach and an EM algorithm for obtaining the semiparametric maximum likelihood estimator (SPMLE) are developed. In M-step, a hybrid algorithm combining the Newton–Raphson and self-consistency algorithms is used to compute the finite-dimensional and infinite-dimensional parameters. The existence and consistency of the SPMLE are established. The proposed method is investigated through simulation studies and illustrated using a real dataset from a Taiwanese HIV/AIDS cohort study.

Semiparametric maximum likelihood inference for nonignorable nonresponse with callbacks

AbstractWe model the nonresponse probabilities as logistic functions of the outcome variable and other covariates in the survey study with callback. The identification aspect of this callback model is investigated. Semiparametric maximum likelihood estimators of the parameters in the response probabilities are proposed and studied. As a result, an efficient estimator of the mean of the outcome variable is constructed using the estimated response probabilities. Moreover, if a regression model for the conditional mean of the outcome variable given some covariate is available, then we can obtain an even more efficient estimate of the mean of the outcome variable by fitting the regression model using an adjusted least‐squares method based on the estimated underlying distributions of the observed values. Simulation results show that the proposed method is more efficient compared with some existing competitors. The method is applied to data from the Singapore Life Panel, a survey of health spending using a population‐based sample of individuals aged 50–70 years, where nonresponse may be related to health.

404: Outcomes after non-indicated induction of labor compared to expectant management: An observational study

Despite limited evidence about the risks or benefits of non-indicated (elective) induction of labor, many healthcare systems have launched initiatives to reduce its use. Great uncertainty remains about the outcomes of elective induction, including whether the impact differs at different gestational ages. To address these research gaps, this study examined the association between elective induction of labor, compared to expectant management, and subsequent maternal and neonatal outcomes. We conducted a two-phase observational study at two integrated healthcare delivery systems, Kaiser Permanente Washington and KP Southern California. Electronic health plan and linked birth certificate data provided information about nulliparous women with a singleton birth from 2007-2013 who as of 38 weeks’ gestation did not have an indication for induction or a contraindication to vaginal delivery. We next used these electronic data to classify women’s apparent exposure and outcome status and then sampled a subgroup for medical record review to confirm eligibility, validate exposure and outcome status, and ascertain covariates. We oversampled women with apparent elective induction and study outcomes (listed in Table). Women undergoing elective induction at a given gestational age (38, 39 or 40 weeks) were compared with those managed expectantly at that age, accounting for sampling and adjusting for covariates via semiparametric maximum likelihood estimation, generating odds ratios and 95% confidence intervals. From a population of 43,378 eligible women, we reviewed medical records for 3,183. There were 36, 78, and 243 deliveries sampled with confirmed elective induction at 38, 39, and 40 weeks gestation, respectively. Results (see Table) indicated that elective induction at 39 weeks was associated with 64% higher odds of cesarean delivery than expectant management, while elective induction at 40 weeks was associated with 38% lower odds of neonatal intensive care unit admission but 48% higher odds of chorioamnionitis. Elective induction at 39 weeks gestation may increase the likelihood of cesarean delivery, while it may reduce NICU admission at 40 weeks’ gestation. Next steps to advance understanding of elective induction should include observational studies with larger sample size or a large randomized trial.

Semiparametric maximum likelihood estimation for the Cox model with length-biased survival data

Right censored survival data collected on a prevalent cohort with constant incidence rate are termed as length-biased survival data. In this paper, we consider semiparametric estimation for the Cox proportional hazards model with length-biased survival data. Utilizing the special structure of length-biased sampling, we give the semiparametric maximum likelihood estimators for the regression parameter and cumulative hazard function. The estimators are shown to be consistent and asymptotically normal, and the limiting variance of the regression parameter estimator achieves the semiparametric efficiency bound. The asymptotic variance of the estimators can be estimated consistently. Simulation studies show the proposed semiparametric maximum likelihood estimator is more efficient than existing estimators in the literature. An analysis of the prevalent data from Canadian Study of Health and Aging (CSHA) illustrates the proposed method.

Semiparametric maximum likelihood estimation with data missing not at random

AbstractNonresponse is frequently encountered in empirical studies. When the response mechanism is missing not at random (MNAR) statistical inference using the observed data is quite challenging. Handling MNAR data often requires two model assumptions: one for the outcome and the other for the response propensity. Correctly specifying these two model assumptions is challenging and difficult to verify from the responses obtained. In this article we propose a semiparametric maximum likelihood method for MNAR data in the sense that a parametric assumption is used for the response propensity part of the model and a nonparametric model is used for the outcome part. The resulting analysis is more robust than the fully parametric approach. Some asymptotic properties of our estimators are derived. Results from a simulation study are also presented.The Canadian Journal of Statistics45: 393–409; 2017 © 2017 Statistical Society of Canada

Estimation under Cox proportional hazards model with covariates missing not at random

ABSTRACTThis paper considers likelihood-based estimation under the Cox proportional hazards model in the situations where some covariate entries are missing not at random. Assuming the conditional distribution of the missing entries is known, we demonstrate the existence of the semiparametric maximum likelihood estimator of the model parameters, establish the consistency and weak convergence. By simulation, we examine the finite-sample performance of the estimation procedure, and compare the SPMLE with the one resulted from using an estimated conditional distribution of the missing entries. We analyze the data from a tuberculosis (TB) study applying the proposed approach for illustration.

Piecewise Cox models with right-censored data

ABSTRACTWe study a general class of piecewise Cox models. We discuss the computation of the semi-parametric maximum likelihood estimates (SMLE) of the parameters, with right-censored data, and a simplified algorithm for the maximum partial likelihood estimates (MPLE). Our simulation study suggests that the relative efficiency of the PMLE of the parameter to the SMLE ranges from 96% to 99.9%, but the relative efficiency of the existing estimators of the baseline survival function to the SMLE ranges from 3% to 24%. Thus, the SMLE is much better than the existing estimators.

Semiparametric Inference in a Genetic Mixture Model

ABSTRACTIn genetic backcross studies, data are often collected from complex mixtures of distributions with known mixing proportions. Previous approaches to the inference of these genetic mixture models involve parameterizing the component distributions. However, model misspecification of any form is expected to have detrimental effects. We propose a semiparametric likelihood method for genetic mixture models: the empirical likelihood under the exponential tilting model assumption, in which the log ratio of the probability (density) functions from the components is linear in the observations. An application to mice cancer genetics involves random numbers of offspring within a litter. In other words, the cluster size is a random variable. We wish to test the null hypothesis that there is no difference between the two components in the mixture model, but unfortunately we find that the Fisher information is degenerate. As a consequence, the conventional two-term expansion in the likelihood ratio statistic does not work. By using a higher-order expansion, we are able to establish a nonstandard convergence rate N− 1/4 for the odds ratio parameter estimator . Moreover, the limiting distribution of the empirical likelihood ratio statistic is derived. The underlying distribution function of each component can also be estimated semiparametrically. Analogously to the full parametric approach, we develop an expectation and maximization algorithm for finding the semiparametric maximum likelihood estimator. Simulation results and a real cancer application indicate that the proposed semiparametric method works much better than parametric methods. Supplementary materials for this article are available online.

TwoPhaseInd: an R package for estimating gene-treatment interactions and discovering predictive markers in randomized clinical trials.

The R package TwoPhaseInd is available from http://cran.r-project.org/web/packages CONTACT: jdai@fredhutch.org.

Using a Monotonic Density Ratio Model to Find the Asymptotically Optimal Combination of Multiple Diagnostic Tests

ABSTRACTWith the advent of new technology, new biomarker studies have become essential in cancer research. To achieve optimal sensitivity and specificity, one needs to combine different diagnostic tests. The celebrated Neyman–Pearson lemma enables us to use the density ratio to optimally combine different diagnostic tests. In this article, we propose a semiparametric model by directly modeling the density ratio between the diseased and nondiseased population as an unspecified monotonic nondecreasing function of a linear or nonlinear combination of multiple diagnostic tests. This method is appealing in that it is not necessary to assume separate models for the diseased and nondiseased populations. Further, the proposed method provides an asymptotically optimal way to combine multiple test results. We use a pool-adjacent-violation-algorithm to find the semiparametric maximum likelihood estimate of the receiver operating characteristic (ROC) curve. Using modern empirical process theory we show cubic root n consistency for the ROC curve and the underlying Euclidean parameter estimation. Extensive simulations show that the proposed method outperforms its competitors. We apply the method to two real-data applications. Supplementary materials for this article are available online.

Weighted composite quantile regression for single-index models

In this paper we propose a weighted composite quantile regression (WCQR) estimation for single-index models. For parametric part, the WCQR is augmented using a data-driven weighting scheme. With the error distribution unspecified, the proposed estimators share robustness from quantile regression and achieve nearly the same efficiency as the semiparametric maximum likelihood estimator for a variety of error distributions including the Normal, Student’s t, Cauchy distributions, etc. Furthermore, based on the proposed WCQR, we use the adaptive-LASSO to study variable selection for parametric part in the single-index models. For nonparametric part, the WCQR is augmented combining the equal weighted estimators with possibly different weights. Because of the use of weights, the estimation bias is eliminated asymptotically. By comparing asymptotic relative efficiency theoretically and numerically, WCQR estimation all outperforms the CQR estimation and some other estimate methods. Under regularity conditions, the asymptotic properties of the proposed estimations are established. The simulation studies and two real data applications are conducted to illustrate the finite sample performance of the proposed methods.

Semiparametric maximum likelihood estimation for a two‐sample density ratio model with right‐censored data

AbstractIn this paper we investigate a broader semiparametric two‐sample density ratio model based on two groups of right‐censored data. A semiparametric maximum likelihood estimator for the unknown finite and infinite dimensional parameters of the model is proposed and obtained by an EM algorithm. By using empirical process theory, we establish the uniform consistency and asymptotic normality of the proposed estimator. We moreover employ a Kolmogorov–Smirnov type test statistic to evaluate the model validity and a likelihood ratio test statistic to examine the treatment effects between the two groups. Simulation studies are conducted to assess the finite sample performance of the proposed estimator and to compare it with its alternatives. Finally a real data example is analyzed to illustrate its application. The Canadian Journal of Statistics 44: 58–81; 2016 © 2015 Statistical Society of Canada

A semiparametric regression cure model for interval-censored and left-truncated data

ABSTRACTMedical advancements have made it possible for patients to be cured of certain types of diseases. In follow-up studies, the disease event time can be subject to left truncation and interval censoring. In this article, we propose a semiparametric nonmixture cure model for the regression analysis of left-truncated and interval-censored (LTIC) data. We develop semiparametric maximum likelihood estimation for the nonmixture cure model with LTIC data. A simulation study is conducted to investigate the performance of the proposed estimators.