Estimation of a Semiparametric Recursive Bivariate Probit Model with Nonparametric Mixing

We consider an extension of the recursive bivariate probit model for estimating the effect of a binary variable on a binary outcome in the presence of unobserved confounders, nonlinear covariate effects and overdispersion. Specifically, the model consists of a system of two binary outcomes with a binary endogenous regressor which includes smooth functions of covariates, hence allowing for flexible functional dependence of the responses on the continuous regressors, and arbitrary random intercepts to deal with overdispersion arising from correlated observations on clusters or from the omission of non‐confounding covariates. We fit the model by maximizing a penalized likelihood using an Expectation‐Maximisation algorithm. The issues of automatic multiple smoothing parameter selection and inference are also addressed. The empirical properties of the proposed algorithm are examined in a simulation study. The method is then illustrated using data from a survey on health, aging and wealth.     

Semiparametric Quantile Regression Analysis of Right‐censored and Length‐biased Failure Time Data with Partially Linear Varying Effects

Right‐censored and length‐biased failure time data arise in many fields including cross‐sectional prevalent cohort studies, and their analysis has recently attracted a great deal of attention. It is well‐known that for regression analysis of failure time data, two commonly used approaches are hazard‐based and quantile‐based procedures, and most of the existing methods are the hazard‐based ones. In this paper, we consider quantile regression analysis of right‐censored and length‐biased data and present a semiparametric varying‐coefficient partially linear model. For estimation of regression parameters, a three‐stage procedure that makes use of the inverse probability weighted technique is developed, and the asymptotic properties of the resulting estimators are established. In addition, the approach allows the dependence of the censoring variable on covariates, while most of the existing methods assume the independence between censoring variables and covariates. A simulation study is conducted and suggests that the proposed approach works well in practical situations. Also, an illustrative example is provided.     

A dynamic model for double‐bounded time series with chaotic‐driven conditional averages

In this work, we introduce a class of dynamic models for time series taking values on the unit interval. The proposed model follows a generalized linear model approach where the random component, conditioned on the past information, follows a beta distribution, while the conditional mean specification may include covariates and also an extra additive term given by the iteration of a map that can present chaotic behavior. The resulting model is very flexible and its systematic component can accommodate short‐ and long‐range dependence, periodic behavior, laminar phases, etc. We derive easily verifiable conditions for the stationarity of the proposed model, as well as conditions for the law of large numbers and a Birkhoff‐type theorem to hold. A Monte Carlo simulation study is performed to assess the finite sample behavior of the partial maximum likelihood approach for parameter estimation in the proposed model. Finally, an application to the proportion of stored hydroelectrical energy in Southern Brazil is presented.     

Likelihood Inference for Multivariate Extreme Value Distributions Whose Spectral Vectors have known Conditional Distributions

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component‐wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max‐stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.     

Testing for state dependence in binary panel data with individual covariates by a modified quadratic exponential model

We propose a test for state dependence in binary panel data with individual covariates. For this aim, we rely on a quadratic exponential model in which the association between the response variables is accounted for in a different way with respect to more standard formulations. The level of association is measured by a single parameter that may be estimated by a Conditional Maximum Likelihood (CML) approach. Under the dynamic logit model, the conditional estimator of this parameter converges to zero when the hypothesis of absence of state dependence is true. Therefore, it is possible to implement a t-test for this hypothesis which may be very simply performed and attains the nominal significance level under several structures of the individual covariates. Through an extensive simulation study, we find that our test has good finite sample properties and it is more robust to the presence of (autocorrelated) covariates in the model specification in comparison with other existing testing procedures for state dependence. The proposed approach is illustrated by two empirical applications: the first is based on data coming from the Panel Study of Income Dynamics and concerns employment and fertility; the second is based on the Health and Retirement Study and concerns the self reported health status.     

Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses

Abstract. A right‐censored version of a U ‐statistic with a kernel of degree m 1 is introduced by the principle of a mean preserving reweighting scheme which is also applicable when the dependence between failure times and the censoring variable is explainable through observable covariates. Its asymptotic normality and an expression of its standard error are obtained through a martingale argument. We study the performances of our U ‐statistic by simulation and compare them with theoretical results. A doubly robust version of this reweighted U ‐statistic is also introduced to gain efficiency under correct models while preserving consistency in the face of model mis‐specifications. Using a Kendall's kernel, we obtain a test statistic for testing homogeneity of failure times for multiple failure causes in a multiple decrement model. The performance of the proposed test is studied through simulations. Its usefulness is also illustrated by applying it to a real data set on graft‐versus‐host‐disease.     

Balanced covariates with response adaptive randomization

Response adaptive randomization (RAR) methods for clinical trials are susceptible to imbalance in the distribution of influential covariates across treatment arms. This can make the interpretation of trial results difficult, because observed differences between treatment groups may be a function of the covariates and not necessarily because of the treatments themselves. We propose a method for balancing the distribution of covariate strata across treatment arms within RAR. The method uses odds ratios to modify global RAR probabilities to obtain stratum‐specific modified RAR probabilities. We provide illustrative examples and a simple simulation study to demonstrate the effectiveness of the strategy for maintaining covariate balance. The proposed method is straightforward to implement and applicable to any type of RAR method or outcome.     

Feature screening for case‐cohort studies with failure time outcome

Case‐cohort design has been demonstrated to be an economical and efficient approach in large cohort studies when the measurement of some covariates on all individuals is expensive. Various methods have been proposed for case‐cohort data when the dimension of covariates is smaller than sample size. However, limited work has been done for high‐dimensional case‐cohort data which are frequently collected in large epidemiological studies. In this paper, we propose a variable screening method for ultrahigh‐dimensional case‐cohort data under the framework of proportional model, which allows the covariate dimension increases with sample size at exponential rate. Our procedure enjoys the sure screening property and the ranking consistency under some mild regularity conditions. We further extend this method to an iterative version to handle the scenarios where some covariates are jointly important but are marginally unrelated or weakly correlated to the response. The finite sample performance of the proposed procedure is evaluated via both simulation studies and an application to a real data from the breast cancer study.     

A Copula‐Based Non‐parametric Measure of Regression Dependence

Abstract. This article presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO). In addition, we define a new non‐parametric measure of regression dependence and study its properties. Besides being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. A consistent non‐parametric estimator of the new measure is constructed and its asymptotic properties are investigated. Finally, the finite sample properties of the estimate are studied by means of a small simulation study.     

Inference in Semi‐Parametric Dynamic Models for Repeated Count Data

This paper deals with a longitudinal semi‐parametric regression model in a generalised linear model setup for repeated count data collected from a large number of independent individuals. To accommodate the longitudinal correlations, we consider a dynamic model for repeated counts which has decaying auto‐correlations as the time lag increases between the repeated responses. The semi‐parametric regression function involved in the model contains a specified regression function in some suitable time‐dependent covariates and a non‐parametric function in some other time‐dependent covariates. As far as the inference is concerned, because the non‐parametric function is of secondary interest, we estimate this function consistently using the independence assumption‐based well‐known quasi‐likelihood approach. Next, the proposed longitudinal correlation structure and the estimate of the non‐parametric function are used to develop a semi‐parametric generalised quasi‐likelihood approach for consistent and efficient estimation of the regression effects in the parametric regression function. The finite sample performance of the proposed estimation approach is examined through an intensive simulation study based on both large and small samples. Both balanced and unbalanced cluster sizes are incorporated in the simulation study. The asymptotic performances of the estimators are given. The estimation methodology is illustrated by reanalysing the well‐known health care utilisation data consisting of counts of yearly visits to a physician by 180 individuals for four years and several important primary and secondary covariates.     

Boosted coefficient models

Regression methods typically construct a mapping from the covariates into the real numbers. Here, however, we consider regression problems where the task is to form a mapping from the covariates into a set of (univariate) real-valued functions. Examples are given by conditional density estimation, hazard regression and regression with a functional response. Our approach starts by modeling the function of interest using a sum of B-spline basis functions. To model dependence on the covariates, the coefficients of this expansion are each modeled as functions of the covariates. We propose to estimate these coefficient functions using boosted tree models. Algorithms are provided for the above three situations, and real data sets are used to investigate their performance. The results indicate that the proposed methodology performs well. In addition, it is both straightforward, and capable of handling a large number of covariates.     

Non parametric hazard estimation with dependent censoring using penalized likelihood and an assumed copula

This article introduces a novel non parametric penalized likelihood hazard estimation when the censoring time is dependent on the failure time for each subject under observation. More specifically, we model this dependence using a copula, and the method of maximum penalized likelihood (MPL) is adopted to estimate the hazard function. We do not consider covariates in this article. The non negatively constrained MPL hazard estimation is obtained using a multiplicative iterative algorithm. The consistency results and the asymptotic properties of the proposed hazard estimator are derived. The simulation studies show that our MPL estimator under dependent censoring with an assumed copula model provides a better accuracy than the MPL estimator under independent censoring if the sign of dependence is correctly specified in the copula function. The proposed method is applied to a real dataset, with a sensitivity analysis performed over various values of correlation between failure and censoring times.     

Locally Efficient Semiparametric Estimators for Proportional Hazards Models with Measurement Error

We propose a new class of semiparametric estimators for proportional hazards models in the presence of measurement error in the covariates, where the baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are considered as unknown infinite dimensional parameters. We estimate the model components by solving estimating equations based on the semiparametric efficient scores under a sequence of restricted models where the logarithm of the hazard functions are approximated by reduced rank regression splines. The proposed estimators are locally efficient in the sense that the estimators are semiparametrically efficient if the distribution of the error‐prone covariates is specified correctly and are still consistent and asymptotically normal if the distribution is misspecified. Our simulation studies show that the proposed estimators have smaller biases and variances than competing methods. We further illustrate the new method with a real application in an HIV clinical trial.     

A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

A Non‐parametric Test of Exchangeability for Extreme‐Value and Left‐Tail Decreasing Bivariate Copulas

Abstract. A non‐parametric rank‐based test of exchangeability for bivariate extreme‐value copulas is first proposed. The two key ingredients of the suggested approach are the non‐parametric rank‐based estimators of the Pickands dependence function recently studied by Genest and Segers, and a multiplier technique for obtaining approximate p‐values for the derived statistics. The proposed approach is then extended to left‐tail decreasing dependence structures that are not necessarily extreme‐value copulas. Large‐scale Monte Carlo experiments are used to investigate the level and power of the various versions of the test and show that the proposed procedure can be substantially more powerful than tests of exchangeability derived directly from the empirical copula. The approach is illustrated on well‐known financial data.     

Nonparametric regression estimation of conditional tails: the random covariate case

We present families of nonparametric estimators for the conditional tail index of a Pareto-type distribution in the presence of random covariates. These families are constructed from locally weighted sums of power transformations of excesses over a high threshold. The asymptotic properties of the proposed estimators are derived under some assumptions on the conditional response distribution, the weight function and the density function of the covariates. We also introduce bias-corrected versions of the estimators for the conditional tail index, and propose in this context a consistent estimator for the second-order tail parameter. The finite sample performance of some specific examples from our classes of estimators is illustrated with a small simulation experiment.     

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

Motivated 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.     

A Quantile Survival Model for Censored Data

In this paper we propose a quantile survival model to analyze censored data. This approach provides a very effective way to construct a proper model for the survival time conditional on some covariates. Once a quantile survival model for the censored data is established, the survival density, survival or hazard functions of the survival time can be obtained easily. For illustration purposes, we focus on a model that is based on the generalized lambda distribution (GLD). The GLD and many other quantile function models are defined only through their quantile functions, no closed‐form expressions are available for other equivalent functions. We also develop a Bayesian Markov Chain Monte Carlo (MCMC) method for parameter estimation. Extensive simulation studies have been conducted. Both simulation study and application results show that the proposed quantile survival models can be very useful in practice.     

Estimation of dependence between paired correlated failure times in the presence of covariate measurement error

Summary. In many biomedical studies, covariates are subject to measurement error. Although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated, there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times. We show that the dependence parameter estimator in the Clayton–Oakes model can be considerably biased if the measurement error in the covariate is not accommodated. In contrast with the typical bias towards the null for marginal regression coefficients, the dependence parameter can be biased in either direction. We introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates, and we study the large and small sample properties of the dependence parameter estimator proposed.     

An efficient analysis of covariance model for crossover thorough QT studies with period‐specific pre‐dose baselines

Baseline adjustment is an important consideration in thorough QT studies for non‐antiarrhythmic drugs. For crossover studies with period‐specific pre‐dose baselines, we propose a by‐time‐point analysis of covariance model with change from pre‐dose baseline as response, treatment as a fixed effect, pre‐dose baseline for current treatment and pre‐dose baseline averaged across treatments as covariates, and subject as a random effect. Additional factors such as period and sex should be included in the model as appropriate. Multiple pre‐dose measurements can be averaged to obtain a pre‐dose‐averaged baseline and used in the model. We provide conditions under which the proposed model is more efficient than other models. We demonstrate the efficiency and robustness of the proposed model both analytically and through simulation studies. The advantage of the proposed model is also illustrated using the data from a real clinical trial. Copyright © 2014 John Wiley & Sons, Ltd.