Semiparametric Likelihood Research Articles

A smoothed semiparametric likelihood for estimation of nonparametric finite mixture models with a copula-based dependence structure

A smoothed semiparametric likelihood for estimation of nonparametric finite mixture models with a copula-based dependence structure

Efficient data integration under prior probability shift.

Conventional supervised learning usually operates under the premise that data are collected from the same underlying population. However, challenges may arise when integrating new data from different populations, resulting in a phenomenon known as dataset shift. This paper focuses on prior probability shift, where the distribution of the outcome varies across datasets but the conditional distribution of features given the outcome remains the same. To tackle the challenges posed by such shift, we propose an estimation algorithm that can efficiently combine information from multiple sources. Unlike existing methods that are restricted to discrete outcomes, the proposed approach accommodates both discrete and continuous outcomes. It also handles high-dimensional covariate vectors through variable selection using an adaptive least absolute shrinkage and selection operator penalty, producing efficient estimates that possess the oracle property. Moreover, a novel semiparametric likelihood ratio test is proposed to check the validity of prior probability shift assumptions by embedding the null conditional density function into Neyman's smooth alternatives (Neyman, 1937) and testing study-specific parameters. We demonstrate the effectiveness of our proposed method through extensive simulations and a real data example. The proposed methods serve as a useful addition to the repertoire of tools for dealing dataset shifts.

Efficient designs and analysis of two-phase studies with longitudinal binary data.

Researchers interested in understanding the relationship between a readily available longitudinal binary outcome and a novel biomarker exposure can be confronted with ascertainment costs that limit sample size. In such settings, two-phase studies can be cost-effective solutions that allow researchers to target informative individuals for exposure ascertainment and increase estimation precision for time-varying and/or time-fixed exposure coefficients. In this paper, we introduce a novel class of residual-dependent sampling (RDS) designs that select informative individuals using data available on the longitudinal outcome and inexpensive covariates. Together with the RDS designs, we propose a semiparametric analysis approach that efficiently uses all data to estimate the parameters. We describe a numerically stable and computationally efficient EM algorithm to maximize the semiparametric likelihood. We examine the finite sample operating characteristics of the proposed approaches through extensive simulation studies, and compare the efficiency of our designs and analysis approach with existing ones. We illustrate the usefulness of the proposed RDS designs and analysis method in practice by studying the association between a genetic marker and poor lung function among patients enrolled in the Lung Health Study (Connett et al, 1993).

Uniform and [formula omitted] convergences for nonparametric continuous time regressions with semiparametric applications

We obtain uniform and Lp convergence rates of kernel type nonparametric estimators for the instantaneous conditional mean and variance functions of continuous time regressions, where the regressor is assumed to be a general recurrent diffusion. Our asymptotics are developed under a general set-up, with a shrinking sampling interval and an increasing time span, and without the stationarity assumption. Based on our convergence results, we develop a semiparametric inferential procedure for continuous time predictive regressions. In particular, a robust semiparametric likelihood ratio test for linear predictability is proposed, with its limit distribution established. We also apply our convergence results to obtain the asymptotics of a semiparametric maximum likelihood estimator of the drift of recurrent diffusions. In our simulation study, we examine the finite sample performance of our robust test against several existing tests in the literature. An empirical illustration is presented to test the predictability of the excess returns of two major stock indices using the commonly used dividend–price ratio and earnings–price ratio as the predictor.

Methods and Software to Analyze Gene-Environment Interactions under a Case-Mother-Control-Mother Design with Partially Missing Child Genotype

Introduction: The case-mother-control-mother design allows to study fetal and maternal genetic factors together with environmental exposures on early life outcomes. Mendelian constraints and conditional independence between child genotype and environmental factors enabled semiparametric likelihood methods to estimate logistic models with greater efficiency than standard logistic regression. Difficulties in child genotype collection require methods handling missing child genotype. Methods: We review a stratified retrospective likelihood and two semiparametric likelihood approaches: a prospective one and a modified retrospective one, the latter either modeling the maternal genotype as a function of covariates or leaving their joint distribution unspecified (robust version). We also review software implementing these modeling alternatives, compare their statistical properties in a simulation study, and illustrate their application, focusing on gene-environment interactions and partially missing child genotype. Results: The robust retrospective likelihood provides generally unbiased estimates, with standard errors only slightly larger than when modeling maternal genotype based on exposure. The prospective likelihood encounters maximization problems. In the application to the association of small-for-gestational-age babies with CYP2E1 and drinking water disinfection by-products, the retrospective likelihood allowed a full array of covariates, while the prospective likelihood was limited to few covariates. Conclusion: We recommend the robust version of the modified retrospective likelihood.

Surveillance for endemic infectious disease outbreaks: Adaptive sampling using profile likelihood estimation.

Outbreaks of an endemic infectious disease can occur when the disease is introduced into a highly susceptible subpopulation or when the disease enters a network of connected individuals. For example, significant HIV outbreaks among people who inject drugs have occurred in at least half a dozen US states in recent years. This motivates the current study: how can limited testing resources be allocated across geographic regions to rapidly detect outbreaks of an endemic infectious disease? We develop an adaptive sampling algorithm that uses profile likelihood to estimate the distribution of the number of positive tests that would occur for each location in a future time period if that location were sampled. Sampling is performed in the location with the highest estimated probability of triggering an outbreak alarm in the next time period. The alarm function is determined by a semiparametric likelihood ratio test. We compare the profile likelihood sampling (PLS) method numerically to uniform random sampling (URS) and Thompson sampling (TS). TS was worse than URS when the outbreak occurred in a location with lower initial prevalence than other locations. PLS had lower time to outbreak detection than TS in some but not all scenarios, but was always better than URS even when the outbreak occurred in a location with a lower initial prevalence than other locations. PLS provides an effective and reliable method for rapidly detecting endemic disease outbreaks that is robust to this uncertainty.

FULL-SEMIPARAMETRIC-LIKELIHOOD-BASED INFERENCE FOR NON-IGNORABLE MISSING DATA

During the past few decades, missing-data problems have been studied extensively, with a focus on the ignorable missing case, where the missing probability depends only on observable quantities. By contrast, research into non-ignorable missing data problems is quite limited. The main difficulty in solving such problems is that the missing probability and the regression likelihood function are tangled together in the likelihood presentation, and the model parameters may not be identifiable even under strong parametric model assumptions. In this paper we discuss a semiparametric model for non-ignorable missing data and propose a maximum full semiparametric likelihood estimation method, which is an efficient combination of the parametric conditional likelihood and the marginal nonparametric biased sampling likelihood. The extra marginal likelihood contribution can not only produce efficiency gain but also identify the underlying model parameters without additional assumptions. We further show that the proposed estimators for the underlying parameters and the response mean are semiparametrically efficient. Extensive simulations and a real data analysis demonstrate the advantage of the proposed method over competing methods.

Semiparametric likelihood inference for heterogeneous survival data under double truncation based on a Poisson birth process

We study a selective sampling scheme in which survival data are observed during a data collection period if and only if a specific failure event is experienced. Individual units belong to one of a finite number of subpopulations, which may exhibit different survival behaviour, and thus cause heterogeneity. Based on a Poisson process model for individual emergence of population units, we derive a semiparametric likelihood model, in which the birth distribution is modeled nonparametrically and the lifetime distributions parametrically, and define maximum likelihood estimators. We propose a Newton–Raphson-type optimization method to address numerical challenges caused by the high-dimensional parameter space. The finite-sample properties and computational performance of the proposed algorithms are assessed in a simulation study. Personal insolvencies are studied as a special case of double truncation and we fit the semiparametric model to a medium-sized dataset to estimate the mean age at insolvency and the birth distribution of the underlying population.

Revisiting methods for modeling longitudinal and survival data: Framingham Heart Study

BackgroundStatistical methods for modeling longitudinal and time-to-event data has received much attention in medical research and is becoming increasingly useful. In clinical studies, such as cancer and AIDS, longitudinal biomarkers are used to monitor disease progression and to predict survival. These longitudinal measures are often missing at failure times and may be prone to measurement errors. More importantly, time-dependent survival models that include the raw longitudinal measurements may lead to biased results. In previous studies these two types of data are frequently analyzed separately where a mixed effects model is used for the longitudinal data and a survival model is applied to the event outcome.MethodsIn this paper we compare joint maximum likelihood methods, a two-step approach and a time dependent covariate method that link longitudinal data to survival data with emphasis on using longitudinal measures to predict survival. We apply a Bayesian semi-parametric joint method and maximum likelihood joint method that maximizes the joint likelihood of the time-to-event and longitudinal measures. We also implement the Two-Step approach, which estimates random effects separately, and a classic Time Dependent Covariate Model. We use simulation studies to assess bias, accuracy, and coverage probabilities for the estimates of the link parameter that connects the longitudinal measures to survival times.ResultsSimulation results demonstrate that the Two-Step approach performed best at estimating the link parameter when variability in the longitudinal measure is low but is somewhat biased downwards when the variability is high. Bayesian semi-parametric and maximum likelihood joint methods yield higher link parameter estimates with low and high variability in the longitudinal measure. The Time Dependent Covariate method resulted in consistent underestimation of the link parameter. We illustrate these methods using data from the Framingham Heart Study in which lipid measurements and Myocardial Infarction data were collected over a period of 26 years.ConclusionsTraditional methods for modeling longitudinal and survival data, such as the time dependent covariate method, that use the observed longitudinal data, tend to provide downwardly biased estimates. The two-step approach and joint models provide better estimates, although a comparison of these methods may depend on the underlying residual variance.

Statistical inference for missing data mechanisms.

In the literature of statistical analysis with missing data there is a significant gap in statistical inference for missing data mechanisms especially for nonmonotone missing data, which has essentially restricted the use of the estimation methods which require estimating the missing data mechanisms. For example, the inverse probability weighting methods (Horvitz & Thompson, 1952; Little & Rubin, 2002), including the popular augmented inverse probability weighting (Robins et al, 1994), depend on sufficient models for the missing data mechanisms to reduce estimation bias while improving estimation efficiency. This research proposes a semiparametric likelihood method for estimating missing data mechanisms where an EM algorithm with closed form expressions for both E-step and M-step is used in evaluating the estimate (Zhao et al, 2009; Zhao, 2020). The asymptotic variance of the proposed estimator is estimated from the profile score function. The methods are general and robust. Simulation studies in various missing data settings are performed to examine the finite sample performance of the proposed method. Finally, we analysis the missing data mechanism of Duke cardiac catheterization coronary artery disease diagnostic data to illustrate the method.

A unified approach to the calculation of information operators in semiparametric models.

The infinite-dimensional information operator for the nuisance parameter plays a key role in semiparametric inference, as it is closely related to the regular estimability of the target parameter. Calculation of information operators has traditionally proceeded in a case-by-case manner and has often entailed lengthy derivations with complicated arguments. We develop a unified framework for this task by exploiting commonality in the form of semiparametric likelihoods. The general formula developed allows one to derive information operators with simple calculus and, if necessary at all, a minimal amount of probabilistic evaluation. This streamlined approach shows its simplicity and versatility in application to a number of existing models as well as a new model of practical interest.

Semiparametric model for regression analysis with nonmonotone missing data

Semiparametric likelihoods for regression models with missing at random data (Chen in J Am Stat Assoc 99:1176–1189, 2004, Zhang and Rockette in J Stat Comput Simul 77(2):163–173, 2007, Zhao et al. in Biom J 51: 123–136, 2009, Zhao in Commun Stat Theory Methods 38:3736–3744, 2009) are robust as they use nonparametric models for covariate distributions and do not require modeling the missing data probabilities. Furthermore, the EM algorithms based on the semiparametric likelihoods have closed form expressions for both E-step and M-step. As far as we know the semiparametric likelihoods can only deal with the simple monotone missing data pattern. In this research we extend the semiparemetric likelihood approach to deal with regression models with arbitrary nonmonotone missing at random data. We propose a pseudo-likelihood model, which uses an empirical distribution to model the conditional distribution of missing covariates given observed covariates for each missing data pattern separately. We show that an EM algorithm with closed form updating formulas can be used for computing maximum pseudo-likelihood estimates for regression models with nonmonotone missing data. We then propose estimating the asymptotic variance of the maximum pseudo-likelihood estimator through a profile log likelihood and the EM algorithm. We examine the finite sample performance of the new methods in simulation studies and further illustrate the methods in a real data example investigating high risk gambling behavior and the associated factors.

Semiparametric Inference for the Functional Cox Model

This article studies penalized semiparametric maximum partial likelihood estimation and hypothesis testing for the functional Cox model in analyzing right-censored data with both functional and scalar predictors. Deriving the asymptotic joint distribution of finite-dimensional and infinite-dimensional estimators is a very challenging theoretical problem due to the complexity of semiparametric models. For the problem, we construct the Sobolev space equipped with a special inner product and discover a new joint Bahadur representation of estimators of the unknown slope function and coefficients. Using this key tool, we establish the asymptotic joint normality of the proposed estimators and the weak convergence of the estimated slope function, and then construct local and global confidence intervals for an unknown slope function. Furthermore, we study a penalized partial likelihood ratio test, show that the test statistic enjoys the Wilks phenomenon, and also verify the optimality of the test. The theoretical results are examined through simulation studies, and a right-censored data example from the Improving Care of Acute Lung Injury Patients study is provided for illustration. Supplementary materials for this article are available online.

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.

On High-Dimensional Constrained Maximum Likelihood Inference

Inference in a high-dimensional situation may involve regularization of a certain form to treat overparameterization, imposing challenges to inference. The common practice of inference uses either a regularized model, as in inference after model selection, or bias-reduction known as “debias.” While the first ignores statistical uncertainty inherent in regularization, the second reduces the bias inbred in regularization at the expense of increased variance. In this article, we propose a constrained maximum likelihood method for hypothesis testing involving unspecific nuisance parameters, with a focus of alleviating the impact of regularization on inference. Particularly, for general composite hypotheses, we unregularize hypothesized parameters whereas regularizing nuisance parameters through a L0-constraint controlling the degree of sparseness. This approach is analogous to semiparametric likelihood inference in a high-dimensional situation. On this ground, for the Gaussian graphical model and linear regression, we derive conditions under which the asymptotic distribution of the constrained likelihood ratio is established, permitting parameter dimension increasing with the sample size. Interestingly, the corresponding limiting distribution is the chi-square or normal, depending on if the co-dimension of a test is finite or increases with the sample size, leading to asymptotic similar tests. This goes beyond the classical Wilks phenomenon. Numerically, we demonstrate that the proposed method performs well against it competitors in various scenarios. Finally, we apply the proposed method to infer linkages in brain network analysis based on MRI data, to contrast Alzheimer’s disease patients against healthy subjects. Supplementary materials for this article are available online.

Semiparametric likelihood for estimating equations with non-ignorable non-response by non-response instrument

ABSTRACTNon-response or missing data is a common phenomenon in many areas. Non-ignorable non-response, a response mechanism that depends on the values of the variable having non-response, is the most difficult type of non-response to handle. This paper considers statistical inference of unknown parameters in estimating equations (EEs) when the variable of interest has non-ignorable non-response. By utilising the cutting edge techniques of non-response instrument, a parametric response propensity function can be identified and estimated. Then a semiparametric likelihood is constructed with the propensity function, EEs and auxiliary information being incorporated into the constraints to make the inference valid and improve the estimation efficiency. Asymptotic distributions for the resulting parameter estimates are derived. Empirical results including two simulation studies and a real example show that the proposed method gives promising results.

Statistical inference for nonignorable missing-data problems: a selective review

ABSTRACTNonignorable missing data are frequently encountered in various settings, such as economics, sociology and biomedicine. We review statistical inference for nonignorable missing-data problems, including estimation, influence analysis and model selection. For estimation of mean functionals, we review semiparametric method and empirical likelihood (EL) approach. For estimation of parameters in exponential family nonlinear structural equation models, we introduce expectation-maximisation algorithm, Bayesian approach, and Bayesian EL method. For influence analysis, we investigate the case-deletion method and local influence analysis method from the frequentist and Bayesian viewpoints. For model selection, we present the modified Akaike information criterion and penalised method.

Inferences in dynamic logit models in semi-parametric setup for repeated binary data

ABSTRACTBinary dynamic fixed and mixed logit models are extensively studied in the literature. These models are developed to examine the effects of certain fixed covariates through a parametric regression function as a part of the models. However, there are situations where one may like to consider more covariates in the model but their direct effect is not of interest. In this paper we propose a generalization of the existing binary dynamic logit (BDL) models to the semi-parametric longitudinal setup to address this issue of additional covariates. The regression function involved in such a semi-parametric BDL model contains (i) a parametric linear regression function in some primary covariates, and (ii) a non-parametric function in certain secondary covariates. We use a simple semi-parametric conditional quasi-likelihood approach for consistent estimation of the non-parametric function, and a semi-parametric likelihood approach for the joint estimation of the main regression and dynamic dependence parameters of the model. The finite sample performance of the estimation approaches is examined through a simulation study. The asymptotic properties of the estimators are also discussed. The proposed model and the estimation approaches are illustrated by reanalysing a longitudinal infectious disease data.

Semiparametric Generalized Linear Models with the gldrm Package

This paper introduces a new algorithm to estimate and perform inferences on a recently proposed and developed semiparametric generalized linear model (glm). Rather than selecting a particular parametric exponential family model, such as the Poisson distribution, this semiparametric glm assumes that the response is drawn from the more general exponential tilt family. The regression coefficients and unspecified reference distribution are estimated by maximizing a semiparametric likelihood. The new algorithm incorporates several computational stability and efficiency improvements over the algorithm originally proposed. In particular, the new algorithm performs well for either small or large support for the nonparametric response distribution. The algorithm is implemented in a new R package called gldrm.

Regression analysis for secondary response variable in a case-cohort study.

Case-cohort study design has been widely used for its cost-effectiveness. In any real study, there are always other important outcomes of interest beside the failure time that the original case-cohort study is based on. How to utilize the available case-cohort data to study the relationship of a secondary outcome with the primary exposure obtained through the case-cohort study is not well studied. In this article, we propose a non-parametric estimated likelihood approach for analyzing a secondary outcome in a case-cohort study. The estimation is based on maximizing a semiparametric likelihood function that is built jointly on both time-to-failure outcome and the secondary outcome. The proposed estimator is shown to be consistent, efficient, and asymptotically normal. Finite sample performance is evaluated via simulation studies. Data from the Sister Study is analyzed to illustrate our method.