Semiparametric Maximum Likelihood Estimation Research Articles

The Fine-Gray Model Under Interval Censored Competing Risks Data.

We consider semiparametric analysis of competing risks data subject to mixed case interval censoring. The Fine-Gray model (Fine & Gray, 1999) is used to model the cumulative incidence function and is coupled with sieve semiparametric maximum likelihood estimation based on univariate or multivariate likelihood. The univariate likelihood of cause-specific data enables separate estimation of cumulative incidence function for each competing risk, in contrast with the multivariate likelihood of full data which estimates cumulative incidence functions for multiple competing risks jointly. Under both likelihoods and certain regularity conditions, we show that the regression parameter estimator is asymptotically normal and semiparametrically efficient, although the spline-based sieve estimator of the baseline cumulative subdistribution hazard converges at a rate slower than root-n. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated by a competing risk analysis of data from an dementia cohort study.

Semiparametric maximum likelihood estimation of stochastic frontier model with errors-in-variables

This paper presents a new technique to analyze a stochastic frontier model when covariates are incorporated with measurement errors. We propose a semiparametric mixture likelihood method to estimate the stochastic frontier model which is free from any erroneous specification of the distribution of latent covariates. Some numerical studies including a real data analysis were done, which highly support the proposed approach.

Joint analysis of current count and current status data

We consider joint analysis of event times to a recurrent and a non-recurrent event, with the event time data subject to type I interval censoring. The motivation arises from a survey study, which collected current count data for time to occurrences of fracture (recurrent event), and current status (i.e., type I interval censored) data for time to osteoporosis (non-recurrent event). The aim of the study is to examine risk factors for, and levels of association between, the recurrent and non-recurrent events. We propose a joint analysis of current count and current status data based on a joint modeling for recurrent and non-recurrent events. In the proposed framework, a non-homogeneous Poisson process is assumed for the recurrent event, a proportional hazards model is assumed for failure time of the non-recurrent event, and the two event time processes share a common gamma frailty. A semiparametric maximum likelihood estimator, together with a stable computation algorithm, is developed for the joint model. The parametric (covariate effects and frailty) and nonparametric (baseline mean and cumulative hazard functions) components of the estimator are consistent at rates of square root and cubic root of the sample size, respectively. The asymptotic normality for the parametric component of the estimator is established. The application to the survey data mentioned above shows that, female is a common strong risk factor for both fracture and osteoporosis, and times to fracture and osteoporosis are highly associated.

Semiparametric estimation with missing covariates

This paper considers estimation in semiparametric models when some of the covariates are missing at random. The paper proposes an iterative estimator based on inverse probability weighting and local linear estimation of the nonparametric component. The resulting estimator is very general and can be used in the context of semiparametric maximum likelihood, quasi likelihood and robust estimation. The paper establishes the asymptotic normality of the estimator using both nonparametric and parametric estimation of the unknown probability weights. Two general examples illustrate the theory and Monte Carlo simulations show that the proposed estimator has good finite sample properties.

Semi-Parametric Maximum Likelihood Method for Interaction in Case-Mother Control-Mother Designs: PackageSPmlficmcm

The analysis of interaction effects involving genetic variants and environmental exposures on the risk of adverse obstetric and early-life outcomes is generally performed using standard logistic regression in the case-mother and control-mother design. However such an analysis is inefficient because it does not take into account the natural family-based constraints present in the parent-child relationship. Recently, a new approach based on semi-parametric maximum likelihood estimation was proposed. The advantage of this approach is that it takes into account the parental relationship between the mother and her child in estimation. But a package implementing this method has not been widely available. In this paper, we present SPmlficmcm, an R package implementing this new method and we propose an extension of the method to handle missing offspring genotype data by maximum likelihood estimation. Our choice to treat missing data of the offspring genotype was motivated by the fact that in genetic association studies where the genetic data of mother and child are available, there are usually more missing data on the genotype of the offspring than that of the mother. The package builds a non-linear system from the data and solves and computes the estimates from the gradient and the Hessian matrix of the log profile semi-parametric likelihood function. Finally, we analyze a simulated dataset to show the usefulness of the package.

Analysis of case-cohort designs with binary outcomes: Improving efficiency using whole-cohort auxiliary information.

The case-cohort design has been widely adopted for reducing the cost of covariate measurements in large prospective cohort studies. Under the case-cohort design, complete covariate data are collected only on randomly sampled cases and a subcohort randomly selected from the whole cohort. For the analysis of case-cohort studies with binary outcomes, logistic regression analysis has been routinely used. However, in many applications, certain covariates are readily measured on all samples from the whole cohort, and the case-cohort design may be regarded as a two-phase sampling design. Using this auxiliary covariate information, estimators for the regression parameters can be substantially improved. In this article, we discuss the theoretical basis of the case-cohort design derived from the formulation of the two-phase design and the improved estimators using whole-cohort auxiliary variable information. In particular, we show that the sampling scheme of the case-cohort design is substantially equivalent to that of conventional two-phase case-control studies (also known as two-stage case-control studies for epidemiologists), i.e., the methodologies of two-phase case-control studies can be directly applied to case-cohort data. Under this framework, we review and apply the following improved estimators to the case-cohort design with binary outcomes: (i) weighted estimators, (ii) a semiparametric maximum likelihood estimator, and (iii) a multiple imputation estimator. In addition, based on the framework of the two-phase design, we can obtain risk ratio and risk difference estimators without the rare-disease assumption. We illustrate these methodologies via simulations and the National Wilms Tumor Study data.

Computationally simple estimation and improved efficiency for special cases of double truncation

Doubly truncated survival data arise when event times are observed only if they occur within subject specific intervals of times. Existing iterative estimation procedures for doubly truncated data are computationally intensive (Turnbull 38:290-295, 1976; Efron and Petrosian 94:824-825, 1999; Shen 62:835-853, 2010a). These procedures assume that the event time is independent of the truncation times, in the sample space that conforms to their requisite ordering. This type of independence is referred to as quasi-independence. In this paper we identify and consider two special cases of quasi-independence: complete quasi-independence and complete truncation dependence. For the case of complete quasi-independence, we derive the nonparametric maximum likelihood estimator in closed-form. For the case of complete truncation dependence, we derive a closed-form nonparametric estimator that requires some external information, and a semi-parametric maximum likelihood estimator that achieves improved efficiency relative to the standard nonparametric maximum likelihood estimator, in the absence of external information. We demonstrate the consistency and potentially improved efficiency of the estimators in simulation studies, and illustrate their use in application to studies of AIDS incubation and Parkinson's disease age of onset.

The educational consequences of teen childbearing.

A huge literature shows that teen mothers face a variety of detriments across the life course, including truncated educational attainment. To what extent is this association causal? The estimated effects of teen motherhood on schooling vary widely, ranging from no discernible difference to 2.6 fewer years among teen mothers. The magnitude of educational consequences is therefore uncertain, despite voluminous policy and prevention efforts that rest on the assumption of a negative and presumably causal effect. This study adjudicates between two potential sources of inconsistency in the literature—methodological differences or cohort differences—by using a single, high-quality data source: namely, The National Longitudinal Study of Adolescent Health. We replicate analyses across four different statistical strategies: ordinary least squares regression; propensity score matching; and parametric and semiparametric maximum likelihood estimation. Results demonstrate educational consequences of teen childbearing, with estimated effects between 0.7 and 1.9 fewer years of schooling among teen mothers. We select our preferred estimate (0.7), derived from semiparametric maximum likelihood estimation, on the basis of weighing the strengths and limitations of each approach. Based on the range of estimated effects observed in our study, we speculate that variable statistical methods are the likely source of inconsistency in the past. We conclude by discussing implications for future research and policy, and recommend that future studies employ a similar multimethod approach to evaluate findings.

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.

Efficient estimation for semiparametric cure models with interval-censored data

This paper is concerned with the analysis of interval-censored survival data in the presence of a non-negligible cure fraction using semiparametric non-mixture cure models. We propose a spline-based sieve estimation method which overcomes numerical difficulties encountered in the existing semiparametric maximum likelihood estimation for the unknown nonparametric component in models. This method is easy to implement using the sequential quadratic programming technique. Under certain regularity conditions, we show the consistency, asymptotic normality and semiparametric efficiency of the proposed estimators for parameters. For the nonparametric component, our estimator has an explicit convergence rate, higher than that conjectured by Liu and Shen (2009) [16]. We conduct extensive simulation studies to evaluate the finite-sample performance of the method proposed. The results suggest that our method produces generally more efficient estimators than the existing method. The application of the method is illustrated with data from a study of smoking cessation.

A FRAILTY MODEL APPROACH FOR REGRESSION ANALYSIS OF BIVARIATE INTERVAL-CENSORED SURVIVAL DATA

Owing to the fact that general semiparametric inference procedures are still underdeveloped for multivariate interval-censored event time data, we pro- pose semiparametric maximum likelihood estimation for the gamma-frailty Cox model under mixed-case interval censoring. We establish the consistency of the semiparametric maximum likelihood estimator (SPMLE) for the model parame- ters, including the regression coefficients and the cumulative hazard functions in the Cox model, and the variance of the gamma frailty. The SPMLEs of the cumu- lative hazard functions are shown to have a n 1=3 -rate of convergence, while those of the regression coefficients and the frailty variance have a n 1=2 -rate of convergence; here n denotes the number of study units. The asymptotic normality of the regres- sion coefficients and the frailty variance is also established, with the asymptotic variance given by the inverse of the efficient Fisher information matrix. A profile- likelihood approach is proposed for estimating the asymptotic variance. Based on the self-consistency equations and the contraction principle, we propose a stable and efficient computation algorithm. Simulation results reveal that the large sam- ple theories work quite well in finite samples. We analyze a dataset from an AIDS clinical trial by the proposed methods to assess the effects of the baseline CD4 cell counts on the times to CMV shedding in blood and urine.

Intra-Household Effects on Demand for Telephone Service: Empirical Evidence

I present a game-theoretical model to empirically analyze the consumption interac-tions among household members. Although multiple Nash equilibria of consumption decisions may exist in a household, model parameters are pointwise identified for households with only two members. I propose a semiparametric maximum likelihood estimator and apply it to study the subscription decision for cellular phone service in Taiwan. On average, a consumer’s probability of subscribing to cellular service rises 14 percentage points when the other household member chooses to subscribe, suggesting the existence of intra-household network effects. Moreover, intra-household effect is heterogeneous among households. It increases in household income, but decreases in the number of children and the age difference in a household.

Intra-household effects on demand for telephone service: Empirical evidence

Knowing the determinants of consumption decisions is important for targeted marketing. This paper proposes a method to study household adoption of network goods. I empirically investigate the factors which affect the interactions among household members in the choice of telephone service using a game-theoretical model. The proposed semiparametric maximum likelihood estimator accounts for potential multiple Nash equilibria. I apply the method to study the subscription decisions for cellular phone service in Taiwan. On average, a consumer’s probability of subscribing to cellular service rises 13 percentage points when the other household member chooses to subscribe, suggesting the existence of intra-household network effects. Moreover, the intra-household effect is heterogeneous among households. It increases in household income, but decreases in the number of children and the age difference within a household. Firms can substantially increase the effectiveness of a marketing strategy by targeting on the observed consumer characteristics.

A semiparametric approach to source separation using independent component analysis

Data processing and source identification using lower dimensional hidden structure plays an essential role in many fields of applications, including image processing, neural networks, genome studies, signal processing and other areas where large datasets are often encountered. One of the common methods for source separation using lower dimensional structure involves the use of Independent Component Analysis, which is based on a linear representation of the observed data in terms of independent hidden sources. The problem thus involves the estimation of the linear mixing matrix and the densities of the independent hidden sources. However, the solution to the problem depends on the identifiability of the sources. This paper first presents a set of sufficient conditions to establish the identifiability of the sources and the mixing matrix using moment restrictions of the hidden source variables. Under such sufficient conditions a semi-parametric maximum likelihood estimate of the mixing matrix is obtained using a class of mixture distributions. The consistency of our proposed estimate is established under additional regularity conditions. The proposed method is illustrated and compared with existing methods using simulated and real datasets.

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.

Composite Partial Likelihood Estimation Under Length-Biased Sampling, With Application to a Prevalent Cohort Study of Dementia

The Canadian Study of Health and Aging (CSHA) employed a prevalent cohort design to study survival after onset of dementia, where patients with dementia were sampled and the onset time of dementia was determined retrospectively. The prevalent cohort sampling scheme favors individuals who survive longer. Thus, the observed survival times are subject to length bias. In recent years, there has been a rising interest in developing estimation procedures for prevalent cohort survival data that not only account for length bias but also actually exploit the incidence distribution of the disease to improve efficiency. This article considers semiparametric estimation of the Cox model for the time from dementia onset to death under a stationarity assumption with respect to the disease incidence. Under the stationarity condition, the semiparametric maximum likelihood estimation is expected to be fully efficient yet difficult to perform for statistical practitioners, as the likelihood depends on the baseline hazard function in a complicated way. Moreover, the asymptotic properties of the semiparametric maximum likelihood estimator are not well-studied. Motivated by the composite likelihood method (Besag 1974), we develop a composite partial likelihood method that retains the simplicity of the popular partial likelihood estimator and can be easily performed using standard statistical software. When applied to the CSHA data, the proposed method estimates a significant difference in survival between the vascular dementia group and the possible Alzheimer's disease group, while the partial likelihood method for left-truncated and right-censored data yields a greater standard error and a 95% confidence interval covering 0, thus highlighting the practical value of employing a more efficient methodology. To check the assumption of stable disease for the CSHA data, we also present new graphical and numerical tests in the article. The R code used to obtain the maximum composite partial likelihood estimator for the CSHA data is available in the online Supplementary Material, posted on the journal web site.

Pseudo semiparametric maximum likelihood estimation exploiting gene environment independence for population-based case-control studies with complex samples

Advances in human genetics have led to epidemiological investigations not only of the effects of genes alone but also of gene-environment (G-E) interaction. A widely accepted design strategy in the study of how G-E relate to disease risks is the population-based case-control study (PBCCS). For simple random samples, semiparametric methods for testing G-E have been developed by Chatterjee and Carroll in 2005. The use of complex sampling in PBCCS that involve differential probabilities of sample selection of cases and controls and possibly cluster sampling is becoming more common. Two complexities, weighting for selection probabilities and intracluster correlation of observations, are induced by the complex sampling. We develop pseudo-semiparametric maximum likelihood estimators (pseudo-SPMLE) that apply to PBCCS with complex sampling. We study the finite sample performance of the pseudo-SPMLE using simulations and illustrate the pseudo-SPMLE with a US case-control study of kidney cancer.

Analysis of Family Data Collected under A General Proband Sampling Scheme

The case-control sampling scheme commonly used in familial aggregation studies assumes that each family is ascertained through a single proband. Tosteson et al. generalized this sampling scheme to allow for more than one proband in a family. They proposed a parametric approach to the analysis of family data collected under the expanded sampling scheme. In this article, we derive a semiparametric maximum likelihood estimator for the parameter vector in the prospective model of all family members’ data. This semiparameteric estimator is more efficient because our method uses a likelihood component containing information about the correlation of familial covariates and probands’ response. The estimator can be obtained by iteratively maximizing a pseudo-likelihood and updating the estimates of a set of nuisance parameters. Estimates of the asymptotic variance-covariance matrix are also given. Our simulations show a gain in efficiency using our estimator as compared with Tosteson et al.'s parametric maximum likelihood estimator.

Estimation of finite mixtures with symmetric components

It may sometimes be clear from background knowledge that a population under investigation proportionally consists of a known number of subpopulations, whose distributions belong to the same, yet unknown, family. While a parametric family is commonly used in practice, one can also consider some nonparametric families to avoid distributional misspecification. In this article, we propose a solution using a mixture-based nonparametric family for the component distribution in a finite mixture model as opposed to some recent research that utilizes a kernel-based approach. In particular, we present a semiparametric maximum likelihood estimation procedure for the model parameters and tackle the bandwidth parameter selection problem via some popular means for model selection. Empirical comparisons through simulation studies and three real data sets suggest that estimators based on our mixture-based approach are more efficient than those based on the kernel-based approach, in terms of both parameter estimation and overall density estimation.

Maximum Likelihood Estimations and EM Algorithms With Length-Biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, and 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 estimation 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 semiparametric 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.