M-estimation in regression models for censored data

In this paper, we study M-estimators of regression parameters in semiparametric linear models for censored data. A class of consistent and asymptotically normal M-estimators is constructed. A resampling method is developed for the estimation of the asymptotic covariance matrix of the estimators.     

Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.     

Semiparametric Analysis of Truncated Data

Randomly truncated data are frequently encountered in many studies where truncation arises as a result of the sampling design. In the literature, nonparametric and semiparametric methods have been proposed to estimate parameters in one-sample models. This paper considers a semiparametric model and develops an efficient method for the estimation of unknown parameters. The model assumes that K populations have a common probability distribution but the populations are observed subject to different truncation mechanisms. Semiparametric likelihood estimation is studied and the corresponding inferences are derived for both parametric and nonparametric components in the model. The method can also be applied to two-sample problems to test the difference of lifetime distributions. Simulation results and a real data analysis are presented to illustrate the methods.     

Using Adjacent-category Logits Procedure for Estimating Receiver Operating Characteristic Surface

This article presents a semiparametric method for estimating receiver operating characteristic surface under density ratio model. The construction of the proposed method is based on the adjacent-category logit model and the empirical likelihood approach. A bootstrap approach for the VUS estimator inference is presented. In a simulation study, the proposed estimator is compared with the existing parametric and nonparametric estimators in terms of bias, standard error, and mean square error. Finally, a real data example and some discussions on the proposed method are provided.     

Bias-corrected maximum semiparametric likelihood estimation under logistic regression models based on case–control data

We consider the corrective approach (Theoretical Statistics, Chapman & Hall, London, 1974, p. 310) and preventive approach (Biometrica 80 (1993) 27) to bias reduction of maximum likelihood estimators under the logistic regression model based on case–control data. The proposed bias-corrected maximum likelihood estimators are based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the prospective and retrospective analyses on the basis of the corrective approach to bias reduction produce identical bias-corrected maximum likelihood estimators of the odds ratio parameter, but this does not hold when using the preventive approach unless the case and control sample sizes are identical. We present some results on simulation and on the analysis of two real data sets.     

Semiparametric Estimation for Two-Sample Location-Scale Models under Type I Censorship

To compare two samples under Type I censorship, this article proposes a method of semiparametric inference for the two-sample location-scale problem when the model for two samples is characterized by an unknown distribution and two unknown parameters. Simultaneous estimators for both the location shift and scale change parameters are given. It is shown that the two estimators are strongly consistent and asymptotically normal. The approach in this article can also be used for scale-shape models. Monte Carlo studies indicate that the proposed estimation procedure performs well in finite and heavily censored samples, maintains high relative efficiencies for a wide range of censoring proportions and is robust to the model misspecification, and also outperforms other competitive estimators.     

Two transformation models for estimating an ROC curve derived from continuous data

A receiver operating characteristic (ROC) curve is a plot of two survival functions, derived separately from the diseased and healthy samples. A special feature is that the ROC curve is invariant to any monotone transformation of the measurement scale. We propose and analyse semiparametric and parametric transformation models for this two-sample problem. Following an unspecified or specified monotone transformation, we assume that the healthy and diseased measurements have two normal distributions with different means and variances. Maximum likelihood algorithms for estimating ROC curve parameters are developed. The proposed methods are illustrated on the marker CA125 in the diagnosis of gastric cancer.     

Efficient Estimation of Data Combination Models by the Method of Auxiliary-to-Study Tilting (AST)

We propose a locally efficient estimator for a class of semiparametric data combination problems. A leading estimand in this class is the average treatment effect on the treated (ATT). Data combination problems are related to, but distinct from, the class of missing data problems with data missing at random (of which the average treatment effect (ATE) estimand is a special case). Our estimator also possesses a double robustness property. Our procedure may be used to efficiently estimate, among other objects, the ATT, the two-sample instrumental variables model (TSIV), counterfactual distributions, poverty maps, and semiparametric difference-in-differences. In an empirical application, we use our procedure to characterize residual Black–White wage inequality after flexibly controlling for “premarket” differences in measured cognitive achievement. Supplementary materials for this article are available online.     

Semiparametric ROC surface estimation for continuous diagnostic tests via polytomous logistic regression procedures

In this paper, we propose a semiparametric method of estimating receiver operating characteristic (ROC) surfaces for continuous diagnostic tests under density ratio models. Implementation of our method is easy since the usual polytomous logistic regression procedures in many statistical software packages can be employed. A simulated example is provided to facilitate the implementation of our method. Simulation results show that the proposed semiparametric ROC surface estimator is more efficient than the nonparametric counterpart and the parametric counterpart whether the normality assumption of data holds or not. Moreover, some simulation results on the underlying semiparametric distribution function estimators are also reported. In addition, some discussions on the proposed method as well as analysis of a real data set are provided.     

Maximum Likelihood Estimation in a Semiparametric Logistic/Proportional-Hazards Mixture Model

Abstract.  We consider large sample inference in a semiparametric logistic/proportional-hazards mixture model. This model has been proposed to model survival data where there exists a positive portion of subjects in the population who are not susceptible to the event under consideration. Previous studies of the logistic/proportional-hazards mixture model have focused on developing point estimation procedures for the unknown parameters. This paper studies large sample inferences based on the semiparametric maximum likelihood estimator. Specifically, we establish existence, consistency and asymptotic normality results for the semiparametric maximum likelihood estimator. We also derive consistent variance estimates for both the parametric and non-parametric components. The results provide a theoretical foundation for making large sample inference under the logistic/proportional-hazards mixture model.     

Multiple robustness estimation in causal inference

Abstract

Estimation of average treatment effect is crucial in causal inference for evaluation of treatments or interventions in biostatistics, epidemiology, econometrics, sociology. However, existing estimators require either a propensity score model, an outcome vector model, or both is correctly specified, which is difficult to verify in practice. In this paper, we allow multiple models for both the propensity score models and the outcome models, and then construct a weighting estimator based on observed data by using two-sample empirical likelihood. The resulting estimator is consistent if any one of those multiple models is correctly specified, and thus provides multiple protection on consistency. Moreover, the proposed estimator can attain the semiparametric efficiency bound when one propensity score model and one outcome vector model are correctly specified, without requiring knowledge of which models are correct. Simulations are performed to evaluate the finite sample performance of the proposed estimators. As an application, we analyze the data collected from the AIDS Clinical Trials Group Protocol 175.     

A test for the linearity of the nonparametric part of a semiparametric logistic regression model

A semiparametric logistic regression model is proposed in which its nonparametric component is approximated with fixed-knot cubic B-splines. To assess the linearity of the nonparametric component, we construct a penalized likelihood ratio test statistic. When the number of knots is fixed, the null distribution of the test statistic is shown to be asymptotically the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. We set the asymptotic null expectation of this test statistic equal to a value to determine the smoothing parameter value. Monte Carlo experiments are conducted to investigate the performance of the proposed test. Its practical use is illustrated with a real-life example.     

Assessing goodness-of-fit of categorical regression models based on case-control data

Demonstrated equivalence between a categorical regression model based on case‐control data and an I‐sample semiparametric selection bias model leads to a new goodness‐of‐fit test. The proposed test statistic is an extension of an existing Kolmogorov–Smirnov‐type statistic and is the weighted average of the absolute differences between two estimated distribution functions in each response category. The paper establishes an optimal property for the maximum semiparametric likelihood estimator of the parameters in the I‐sample semiparametric selection bias model. It also presents a bootstrap procedure, some simulation results and an analysis of two real datasets.     

A new parametric family for modelling cumulative incidence functions: application to breast cancer data

Summary.  Competing risks situations can be encountered in many research areas such as medicine, social science and engineering. The main stream of analyses of those competing risks data has been nonparametric or semiparametric in the statistical literature. We propose a new parametric family to parameterize the cumulative incidence function completely. The new distribution is sufficiently flexible to fit various shapes of hazard patterns in survival data and increases the efficiency of the cumulative incidence estimates over the distribution-free approaches. A simple two-sample parametric test statistic is also proposed to compare the cumulative incidence functions between two groups at a given time point. The new parametric approach is illustrated by using breast cancer data sets from the National Surgical Adjuvant Breast and Bowel Project.     

On empirical likelihood for semiparametric two-sample density ratio models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a dual form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ$\phi$-divergences and duality techniques. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and approximation of the power function of the test is deduced.     

Goodness-of-Fit Methods for Probabilistic Index Models

A class of semiparametric regression models, called probabilistic index models, has been recently proposed. Because these models are semiparametric, inference is only valid when the proposed model is consistent with the underlying data-generating model. However, no formal goodness-of-fit methods for these probabilistic index models exist yet. We propose a test and a graphical tool for assessing the model adequacy. Simulation results indicate that both methods succeed in detecting lack-of-fit. The methods are also illustrated on a case study.     

Likelihood methods for missing covariate data in highly stratified studies

Summary. The paper considers canonical link generalized linear models with stratum-specific nuisance intercepts and missing covariate data. This family includes the conditional logistic regression model. Existing methods for this problem, each of which uses a conditioning argu- ment to eliminate the nuisance intercept, model either the missing covariate data or the missingness process. The paper compares these methods under a common likelihood framework. The semiparametric efficient estimator is identified, and a new estimator, which reduces dependence on the model for the missing covariate, is proposed. A simulation study compares the methods with respect to efficiency and robustness to model misspecification.     

Semiparametric estimation of the cumulative incidence function under competing risks and left-truncated sampling

In this article, we propose semiparametric methods to estimate the cumulative incidence function of two dependent competing risks for left-truncated and right-censored data. The proposed method is based on work by Huang and Wang (1995). We extend previous model by allowing for a general parametric truncation distribution and a third competing risk before recruitment. Based on work by Vardi (1989), several iterative algorithms are proposed to obtain the semiparametric estimates of cumulative incidence functions. The asymptotic properties of the semiparametric estimators are derived. Simulation results show that a semiparametric approach assuming the parametric truncation distribution is correctly specified produces estimates with smaller mean squared error than those obtained in a fully nonparametric model.     

Robust M-estimators of multivariate location and scatter in the presence of asymmetry

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived.     

A Semiparametric Approach for Accelerated Failure Time Models with Covariates Subject to Measurement Error

There are relatively few discussions about measurement error in the accelerated failure time (AFT) model, particularly for the semiparametric AFT model. In this article, we propose an adjusted estimation procedure for the semiparametric AFT model with covariates subject to measurement error, based on the profile likelihood approach and simulation and exploration (SIMEX) method. The simulation studies show that the proposed semiparametric SIMEX approach performs well. The proposed approach is applied to a coronary heart disease dataset from the Busselton Health study for illustration.