ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Piecewise proportional hazards models with interval-censored data

We consider the piecewise proportional hazards (PWPH) model with interval-censored (IC) relapse times under the distribution-free set-up. The partial likelihood approach is not applicable for IC data, and the generalized likelihood approach has not been studied in the literature. It turns out that under the PWPH model with IC data, the semi-parametric MLE (SMLE) of the covariate effect under the standard generalized likelihood may not be unique and may not be consistent. In fact, the parameter under the PWPH model with IC data is not identifiable unless the identifiability assumption is imposed. We propose a modification to the likelihood function so that its SMLE is unique. Under the identifiability assumption, our simulation study suggests that the SMLE is consistent. We apply the method to our cancer relapse time data and conclude that the bone marrow micrometastasis does not have a significant prognostic factor.     

Estimation Under the Lehmann Regression Model with Interval-Censored Data

We consider the estimation problem under the Lehmann model with interval-censored data, but focus on the computational issues. There are two methods for computing the semi-parametric maximum likelihood estimator (SMLE) under the Lehmann model (or called Cox model): the Newton-Raphson (NR) method and the profile likelihood (PL) method. We show that they often do not get close to the SMLE. We propose several approach to overcome the computational difficulty and apply our method to a breast cancer research data set.     

Piecewise Cox models with right-censored data

We 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.     

Asymptotic properties of the estimators of the semi-parametric spatial regression model

Spatial data and non parametric methods arise frequently in studies of different areas and it is a common practice to analyze such data with semi-parametric spatial autoregressive (SPSAR) models. We propose the estimations of SPSAR models based on maximum likelihood estimation (MLE) and kernel estimation. The estimation of spatial regression coefficient ρ was done by optimizing the concentrated log-likelihood function with respect to ρ. Furthermore, under appropriate conditions, we derive the limiting distributions of our estimators for both the parametric and non parametric components in the model.     

Parametric versus semi-parametric models for the analysis of correlated survival data: A case study in veterinary epidemiology

Correlated survival data arise frequently in biomedical and epidemiologic research, because each patient may experience multiple events or because there exists clustering of patients or subjects, such that failure times within the cluster are correlated. In this paper, we investigate the appropriateness of the semi-parametric Cox regression and of the generalized estimating equations as models for clustered failure time data that arise from an epidemiologic study in veterinary medicine. The semi-parametric approach is compared with a proposed fully parametric frailty model. The frailty component is assumed to follow a gamma distribution. Estimates of the fixed covariates effects were obtained by maximizing the likelihood function, while an estimate of the variance component ( frailty parameter) was obtained from a profile likelihood construction.     

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.     

Optimum percentile estimating equations for nonlinear random coefficient models

In nonlinear random coefficients models, the means or variances of response variables may not exist. In such cases, commonly used estimation procedures, e.g., (extended) least-squares (LS) and quasi-likelihood methods, are not applicable. This article solves this problem by proposing an estimate based on percentile estimating equations (PEE). This method does not require full distribution assumptions and leads to efficient estimates within the class of unbiased estimating equations. By minimizing the asymptotic variance of the PEE estimates, the optimum percentile estimating equations (OPEE) are derived. Several examples including Weibull regression show the flexibility of the PEE estimates. Under certain regularity conditions, the PEE estimates are shown to be strongly consistent and asymptotic normal, and the OPEE estimates have the minimal asymptotic variance. Compared with the parametric maximum likelihood estimates (MLE), the asymptotic efficiency of the OPEE estimates is more than 98%, while the LS-type of procedures can have infinite variances. When the observations have outliers or do not follow the distributions considered in model assumptions, the article shows that OPEE is more robust than the MLE, and the asymptotic efficiency in the model misspecification cases can be above 150%.     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

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.     

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.     

Maximum Likelihood Estimations and EM Algorithms with Length-biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.     

Saddlepoint-Based Bootstrap Inference for Quadratic Estimating Equations

Abstract.  We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century.     

A Partial Likelihood Estimator of Vaccine Efficacy

A partial likelihood method is proposed for estimating vaccine efficacy for a general epidemic model. In contrast to the maximum likelihood estimator (MLE) which requires complete observation of the epidemic, the suggested method only requires information on the sequence in which individuals are infected and not the exact infection times. A simulation study shows that the method performs almost as well as the MLE. The method is applied to data on the infectious disease mumps.     

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.     

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.     

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.     

A Sample Selection Model with Skew‐normal Distribution

Non‐random sampling is a source of bias in empirical research. It is common for the outcomes of interest (e.g. wage distribution) to be skewed in the source population. Sometimes, the outcomes are further subjected to sample selection, which is a type of missing data, resulting in partial observability. Thus, methods based on complete cases for skew data are inadequate for the analysis of such data and a general sample selection model is required. Heckman proposed a full maximum likelihood estimation method under the normality assumption for sample selection problems, and parametric and non‐parametric extensions have been proposed. We generalize Heckman selection model to allow for underlying skew‐normal distributions. Finite‐sample performance of the maximum likelihood estimator of the model is studied via simulation. Applications illustrate the strength of the model in capturing spurious skewness in bounded scores, and in modelling data where logarithm transformation could not mitigate the effect of inherent skewness in the outcome variable.     

Firth adjustment for Weibull current-status survival analysis

Abstract

Analysis of right-censored data is problematic due to infinite maximum likelihood estimates (MLE) and potentially biased estimates, especially for small numbers of events. Analyzing current-status data is especially troublesome because of the extreme loss of precision due to large failure intervals. We extend Firth’s method for regular parametric problems to current-status modeling with the Weibull distribution. Firth advocated a bias reduction method for MLE by systematically correcting the score equation. An advantage is that it is still applicable when the MLE does not exist. We present simulation studies and two illustrative analyses involving RFM mice lung tumor data.     

Distribution estimation with auxiliary information for missing data

There is much literature on statistical inference for distribution under missing data, but surprisingly very little previous attention has been paid to missing data in the context of estimating distribution with auxiliary information. In this article, the auxiliary information with missing data is proposed. We use Zhou, Wan and Wang's method (2008) to mitigate the effects of missing data through a reformulation of the estimating equations, imputed through a semi-parametric procedure. Whence we can estimate distribution and the τth quantile of the distribution by taking auxiliary information into account. Asymptotic properties of the distribution estimator and corresponding sample quantile are derived and analyzed. The distribution estimators based on our method are found to significantly outperform the corresponding estimators without auxiliary information. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.