Yichuan Zhao

Simultaneous confidence bands for ratios of survival functions via empirical likelihood

We derive a simultaneous confidence band for the ratio of two survival functions based on independent right-censored data. Earlier authors have studied such bands for the difference of two survival functions, but the ratio provides a more appropriate comparison in some applications, e.g., in comparing two treatments in biomedical settings. Our approach is formulated in terms of empirical likelihood and allows us to avoid the use of simulation techniques that are often needed for Wald-type confidence bands. By the transformation-preserving property we also obtain confidence bands for the difference in the cumulative hazard functions. The approach is illustrated with a real data example.

Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.

Recursive Fuzzy Granulation for Gene Subsets Extraction and Cancer Classification

A typical microarray gene expression dataset is usually both extremely sparse and imbalanced. To select multiple highly informative gene subsets for cancer classification and diagnosis, a new fuzzy granular support vector machine-recursive feature elimination algorithm (FGSVM-RFE) is designed in this paper. As a hybrid algorithm of statistical learning, fuzzy clustering, and granular computing, the FGSVM-RFE separately eliminates irrelevant, redundant, or noisy genes in different granules at different stages and selects highly informative genes with potentially different biological functions in balance. Empirical studies on three public datasets demonstrate that the FGSVM-RFE outperforms state-of-the-art approaches. Moreover, the FGSVM-RFE can extract multiple gene subsets on each of which a classifier can be modeled with 100% accuracy. Specifically, the independent testing accuracy for the prostate cancer dataset is significantly improved. The previous best result is 86% with 16 genes and our best result is 100% with only eight genes. The identified genes are annotated by Onto-Express to be biologically meaningful.

Comparing Distribution Functions Via Empirical Likelihood

This paper develops empirical likelihood based simultaneous confidence bands for differences and ratios of two distribution functions from independent samples of right-censored survival data. The proposed confidence bands provide a flexible way of comparing treatments in biomedical settings, and bring empirical likelihood methods to bear on important target functions for which only Wald-type confidence bands have been available in the literature. The approach is illustrated with a real data example.

Empirical likelihood for linear transformation models with interval-censored failure time data

For regression analysis of interval-censored failure time data, Zhang et al. (2005) [40] proposed an estimating equation approach to fit linear transformation models. In this paper, we develop two empirical likelihood (EL) inference approaches for the regression parameters based on the generalized estimating equations. The limiting distributions of log-empirical likelihood ratios are derived and empirical likelihood confidence intervals for any specified component of regression parameters are obtained. We carry out extensive simulation studies to compare the proposed methods with the method discussed by Zhang et al. (2005) [40]. The simulation results demonstrate that the EL and jackknife EL methods for linear transformation models have better performance than the existing normal approximation method based on coverage probability of confidence intervals in most cases, and they enable us to overcome an under-coverage problem for the confidence intervals of the regression parameters using a normal approximation when sample sizes are small and right censoring is heavy. Two real data examples are provided to illustrate our procedures.

Weighted empirical likelihood estimates and their robustness properties

Maximum likelihood methods are by far the most popular methods for deriving statistical estimators. However, parametric likelihoods require distributional specifications. The empirical likelihood is a nonparametric likelihood function that does not require such distributional assumptions, but is otherwise analogous to its parametric counterpart. Both likelihoods assume that the random variables are independent with a common distribution. A nonparametric likelihood function for data that are independent, but not necessarily identically distributed is introduced. The contaminated normal density is used to compare the robustness properties of weighted empirical likelihood estimators to empirical likelihood estimators. It is shown that as the contamination level of the sample increases, the root mean squared error of the empirical likelihood estimator for the mean increases. Conversely, the root mean squared error of the weighted empirical likelihood estimator for the mean remains closer to the theoretical root mean squared error.

Regression analysis for long-term survival rate via empirical likelihood

In recent years, regression models have been shown to be useful for predicting the long-term survival probabilities of patients in clinical trials. For inference on the vector of regression parameters, there are semiparametric procedures based on normal approximations. However, the accuracy of such procedures in terms of coverage probability can be quite low when the censoring rate is heavy. In this paper, we apply an empirical likelihood ratio (ELR) method to the regression model and derive the limiting distribution of the ELR. On the basis of the result, we develop a confidence region for the vector of regression parameters. Furthermore, we use a simulation study to compare the proposed method with the normal approximation-based method proposed by Jung [Jung, S., 1996, Regression analysis for long-term survival rate. Biometrika, 83, 227–232.]. Finally, the proposed procedure is illustrated with data from a clinical trial.

Smoothed jackknife empirical likelihood inference for the difference of ROC curves

For the comparison of two diagnostic markers at a flexible specificity, people apply the difference of two correlated receiver operating characteristic (ROC) curves to identify the diagnostic test with stronger discrimination ability. In this paper, we employ the jackknife empirical likelihood (JEL) method to construct confidence intervals for the difference of two correlated continuous-scale ROC curves. Using the jackknife pseudo-sample, we can avoid estimating several nuisance variables which have to be estimated in the existing methods. We prove that the smoothed jackknife empirical log likelihood ratio is asymptotically chi-squared distributed. Furthermore, the simulation studies in terms of the coverage probability and the average length of confidence intervals show good performance in small samples with a moderate computational cost. A real data set is used to illustrate our method.

Semiparametric inference for transformation models via empirical likelihood

Recent advances in the transformation model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the regression parameters, there are semiparametric procedures based on the normal approximation. However, the accuracy of such procedures can be quite low when the censoring rate is heavy. In this paper, we apply an empirical likelihood ratio method and derive its limiting distribution via U-statistics. We obtain confidence regions for the regression parameters and compare the proposed method with the normal approximation based method in terms of coverage probability. The simulation results demonstrate that the proposed empirical likelihood method overcomes the under-coverage problem substantially and outperforms the normal approximation based method. The proposed method is illustrated with a real data example. Finally, our method can be applied to general U-statistic type estimating equations.

Inference for the Mean Residual Life Function via Empirical Likelihood

In addition to the distribution function, the mean residual life (MRL) function is the other important function which can be used to characterize a lifetime in survival analysis and reliability. For inference on the MRL function, some procedures have been proposed in the literature. However, the coverage accuracy of such procedures may be low when the sample size is small. In this article, an empirical likelihood (EL) inference procedure of MRL function is proposed and the limiting distribution of the EL ratio for MRL function is derived. Based on the result, we obtain confidence interval/band for the MRL function. The proposed method is compared with the normal approximation based method through simulation study in terms of coverage probability.