Empirical likelihood inference for a semiparametric hazards regression model

ABSTRACT

We investigated the empirical likelihood inference approach under a general class of semiparametric hazards regression models with survival data subject to right-censoring. An empirical likelihood ratio for the full 2p regression parameters involved in the model is obtained. We showed that it converged weakly to a random variable which could be written as a weighted sum of 2p independent chi-squared variables with one degree of freedom. Using this, we could construct a confidence region for parameters. We also suggested an adjusted version for the preceding statistic, whose limit followed a standard chi-squared distribution with 2p degrees of freedom.     

Semiparametric estimation method for accelerated failure time model with dependent censoring

Independent censoring is commonly assumed in survival analysis. However, it may be questionable when censoring is related to event time. We model the event and censoring time marginally through accelerated failure time models, and model their association by a known copula. An iteration algorithm is proposed to estimate the regression parameters. Simulation results show the improvement of the proposed method compared to the naive method under independent censoring. Sensitivity analysis gives the evidences that the proposed method can obtain reasonable estimates even when the forms of copula are misspecified. We illustrate its application by analyzing prostate cancer data.     

Tests of independence for censored bivariate failure time data

Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of chi2-type tests for independence between pairs of failure times after adjusting for covariates. A bivariate accelerated failure time model is proposed for the joint distribution of bivariate failure times while leaving the dependence structures for related failure times completely unspecified. Theoretical properties of the proposed tests are derived and variance estimates of the test statistics are obtained using a resampling technique. Simulation studies show that the proposed tests are appropriate for practical use. Two examples including the study of infection in catheters for patients on dialysis and the diabetic retinopathy study are also given to illustrate the methodology.     

Block empirical likelihood for longitudinal partially linear regression models

The authors propose a block empirical likelihood procedure to accommodate the within‐group correlation in longitudinal partially linear regression models. This leads them to prove a nonparametric version of the Wilks theorem. In comparison with normal approximations, their method does not require a consistent estimator for the asymptotic covariance matrix, which makes it easier to conduct inference on the parametric component of the model. An application to a longitudinal study on fluctuations of progesterone level in a menstrual cycle is used to illustrate the procedure developed here.     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Empirical likelihood inferences for varying coefficient partially nonlinear models

In this article, empirical likelihood inferences for the varying coefficient partially nonlinear models are investigated. An empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function part and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component are proposed. The corresponding Wilks phenomena are proved and the confidence regions for parametric component and nonparametric component are constructed. Simulation studies indicate that, in terms of coverage probabilities and average areas of the confidence regions, the empirical likelihood method performs better than the normal approximation-based method. Furthermore, a real data set application is also provided to illustrate the proposed empirical likelihood estimation technique.     

Empirical likelihood for nonlinear regression models with nonignorable missing responses

This article develops three empirical likelihood (EL) approaches to estimate parameters in nonlinear regression models in the presence of nonignorable missing responses. These are based on the inverse probability weighted (IPW) method, the augmented IPW (AIPW) method and the imputation technique. A logistic regression model is adopted to specify the propensity score. Maximum likelihood estimation is used to estimate parameters in the propensity score by combining the idea of importance sampling and imputing estimating equations. Under some regularity conditions, we obtain the asymptotic properties of the maximum EL estimators of these unknown parameters. Simulation studies are conducted to investigate the finite sample performance of our proposed estimation procedures. Empirical results provide evidence that the AIPW procedure exhibits better performance than the other two procedures. Data from a survey conducted in 2002 are used to illustrate the proposed estimation procedure. The Canadian Journal of Statistics 48: 386–416; 2020 © 2020 Statistical Society of Canada     

Empirical likelihood dimension reduction inference for partially non-linear error-in-responses models with validation data

ABSTRACT

In this article, partially non linear models when the response variable is measured with error and explanatory variables are measured exactly are considered. Without specifying any error structure equation, a semiparametric dimension reduction technique is employed. Two estimators of unknown parameter in non linear function are obtained and asymptotic normality is proved. In addition, empirical likelihood method for parameter vector is provided. It is shown that the estimated empirical log-likelihood ratio has asymptotic Chi-square distribution. A simulation study indicates that, compared with normal approximation method, empirical likelihood method performs better in terms of coverage probabilities and average length of the confidence intervals.     

Empirical likelihood confidence intervals for nonparametric functional data analysis

We consider the problem of constructing confidence intervals for nonparametric functional data analysis using empirical likelihood. In this doubly infinite-dimensional context, we demonstrate the Wilk's phenomenon and propose a bias-corrected construction that requires neither undersmoothing nor direct bias estimation. We also extend our results to partially linear regression models involving functional data. Our numerical results demonstrate improved performance of the empirical likelihood methods over normal approximation-based methods.     

Maximum likelihood estimation for the proportional odds model with mixed interval-censored failure time data

This article discusses regression analysis of mixed interval-censored failure time data. Such data frequently occur across a variety of settings, including clinical trials, epidemiologic investigations, and many other biomedical studies with a follow-up component. For example, mixed failure times are commonly found in the two largest studies of long-term survivorship after childhood cancer, the datasets that motivated this work. However, most existing methods for failure time data consider only right-censored or only interval-censored failure times, not the more general case where times may be mixed. Additionally, among regression models developed for mixed interval-censored failure times, the proportional hazards formulation is generally assumed. It is well-known that the proportional hazards model may be inappropriate in certain situations, and alternatives are needed to analyze mixed failure time data in such cases. To fill this need, we develop a maximum likelihood estimation procedure for the proportional odds regression model with mixed interval-censored data. We show that the resulting estimators are consistent and asymptotically Gaussian. An extensive simulation study is performed to assess the finite-sample properties of the method, and this investigation indicates that the proposed method works well for many practical situations. We then apply our approach to examine the impact of age at cranial radiation therapy on risk of growth hormone deficiency in long-term survivors of childhood cancer.     

Empirical likelihood based diagnostics for heteroscedasticity in partially linear errors-in-variables models

A standard assumption in regression analysis is homogeneity of the error variance. Violation of this assumption can have adverse consequences for the efficiency of estimators. In this paper, we propose an empirical likelihood based diagnostic technique for heteroscedasticity in the partially linear errors-in-variables models. Under mild conditions, a nonparametric version of Wilk's theorem is derived. Simulation results reveal that our test performs well in both size and power.     

Generalized likelihood ratio tests for the structure of semiparametric additive models

Semiparametric additive models (SAMs) are very useful in multivariate nonparametric regression. In this paper, the authors study nonparametric testing problems for the nonparametric components of SAMs. Using the backfitting algorithm and the local polynomial smoothing technique, they extend to SAMs the generalized likelihood ratio tests of Fan &Jiang (2005). The authors show that the proposed tests possess the Wilks‐type property and that they can detect alternatives nearing the null hypothesis with a rate arbitrarily close to root‐n while error distributions are unspecified. They report simulations which demonstrate the Wilks phenomenon and the powers of their tests. They illustrate the performance of their approach by simulation and using the Boston housing data set.     

Empirical Likelihood Method for Censored Median Regression Models

Based on the semiparametric median regression analysis for the right-censored data developed by Ying et al. (1995 Ying , Z. , Jung , S. H. , Wei , L. J. ( 1995 ). Survival analysis with median regression models . J. Amer. Statist. Assoc. 90 : 178 – 184 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), an empirical likelihood based inferential procedure for the regression coefficients is proposed. The limiting distribution of the proposed log-empirical likelihood ratio test statistic follows a chi-squared distribution, which corresponds to the standard asymptotic results of the empirical likelihood method. The inference about the subsets of the entire regression coefficients vector is discussed. The proposed method is illustrated by some simulation studies.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

Empirical likelihood for nonlinear models with missing responses

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ² asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical log-likelihood on the response variables associated with the imputed data is also asymptotically χ². The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.     

Empirical likelihood based diagnostics for heteroscedasticity in semiparametric varying-coefficient partially linear errors-in-variables models

In this paper, we propose an empirical likelihood based diagnostic technique for heteroscedasticity in the semiparametric varying-coefficient partially linear errors-in-variables models. Under mild conditions, a nonparametric version of Wilk’s theorem is derived. Simulation results reveal that our test performs well in both size and power.     

Empirical likelihood inference for error density estimators in first-order autoregression models

In this paper we apply empirical likelihood method to the error density estimators in first-order autoregressive models under some mild conditions. The log-likelihood ratio statistic is shown to be asymptotically chi-squared distributed at a fixed point. In simulation, we show that the empirical likelihood produces confidence intervals having theoretical coverage accuracy which is better than normal approximation.     

Chi-Squared Type Test for the AFT-Generalized Inverse Weibull Distribution

The generalized inverse Weibull distribution is a newlife time probability distribution which can be used to model a variety of failure characteristics. It has several desirable properties and nice physical interpretations which enable them to be used frequently. In this article, we present a chi-squared goodness-of-fit test for an accelerated failure time (AFT) model with generalized inverse Weibull distribution (GIW) as the baseline distribution, in both of complete and censored data. This test is based on a modification of the NRR (Nikulin-Rao-Robson) statistic Y², proposed by Bagdonavicius and Nikulin (2011 Bagdonavicius, V., Nikulin, M. (2011). Chi-squared tests for general composite hypotheses from censored samples. Comptes Rendus Mathematique, Ser. I, 349(3–4): 219–223.[Crossref], [Web of Science ®] , [Google Scholar]), for censored data. Two applications of real data are given to illustrate the potentiality of the proposed test.     

Restricted likelihood ratio lack-of-fit tests using mixed spline models

Summary.  Penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non-linearity in the smooth function estimates. This motivates the study of lack-of-fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values. For this one-sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis. Conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies, and finite sample approximations to the asymptotic distribution are provided. Test statistics are studied for simple as well as multiple-regression models.