Diagnostic measures for empirical likelihood of general estimating equations

We develop diagnostic measures for assessing the influence ofindividual observations when using empirical likelihood withgeneral estimating equations, and we use these measures to constructgoodness-of-fit statistics for testing possible misspecificationin the estimating equations. Our diagnostics include case-deletionmeasures, local influence measures and pseudo-residuals. Ourgoodness-of-fit statistics include the sum of local influencemeasures and the processes of pseudo-residuals. Simulation studiesare conducted to evaluate our methods, and real datasets areanalyzed to illustrate the use of our diagnostic measures andgoodness-of-fit statistics.     

Empirical likelihood inference for the area under the ROC curve

For a continuous-scale diagnostic test, the most commonly used summary index of the receiver operating characteristic curve (ROC) is the area under the curve (AUC) that measures the accuracy of the diagnostic test. In this article, we propose an empirical likelihood (EL) approach for the inference on the AUC. First we define an EL ratio for the AUC and show that its limiting distribution is a scaled chi-square distribution. We then obtain an EL-based confidence interval for the AUC using the scaled chi-square distribution. This EL inference for the AUC can be extended to stratified samples, and the resulting limiting distribution is a weighted sum of independent chi-square distributions. Additionally we conduct simulation studies to compare the relative performance of the proposed EL-based interval with the existing normal approximation-based intervals and bootstrap intervals for the AUC.     

Empirical likelihood for cumulative hazard ratio estimation with covariate adjustment

In medical studies, it is often of scientific interest to evaluate the treatment effect via the ratio of cumulative hazards, especially when those hazards may be nonproportional. To deal with nonproportionality in the Cox regression model, investigators usually assume that the treatment effect has some functional form. However, to do so may create a model misspecification problem because it is generally difficult to justify the specific parametric form chosen for the treatment effect. In this article, we employ empirical likelihood (EL) to develop a nonparametric estimator of the cumulative hazard ratio with covariate adjustment under two nonproportional hazard models, one that is stratified, as well as a less restrictive framework involving group-specific treatment adjustment. The asymptotic properties of the EL ratio statistic are derived in each situation and the finite-sample properties of EL-based estimators are assessed via simulation studies. Simultaneous confidence bands for all values of the adjusted cumulative hazard ratio in a fixed interval of interest are also developed. The proposed methods are illustrated using two different datasets concerning the survival experience of patients with non-Hodgkin's lymphoma or ovarian cancer.     

On the finite sample distribution of the likelihood ratio statistic for testing heterogeneity in meta-analysis

In meta-analysis, hypothesis testing is one of the commonly used approaches for assessing whether heterogeneity exists in effects between studies. The literature concluded that the Q-statistic is clearly the best choice and criticized the performance of the likelihood ratio test in terms of the type I error control and power. However, all the criticism for the likelihood ratio test is based on the use of a mixture of two chi-square distributions with 0 and 1 degrees of freedom, which is justified only asymptotically. In this study, we develop a novel method to derive the finite sample distribution of the likelihood ratio test and restricted likelihood ratio test statistics for testing the zero variance component in the random effects model for meta-analysis. We also extend this result to the heterogeneity test when metaregression is applied. A numerical study shows that the proposed statistics have superior performance to the Q-statistic, especially when the number of studies collected for meta-analysis is small to moderate.     

Pairwise rank-based likelihood for estimation and inference on the mixture proportion

We consider the problem of estimation and inference on the mixture parameter in the two-sample problem when sample data from the two distributions as well as from a third population consisting of a mixture of the two are used. Under a general nonparametric model, where the relationship between the two populations is unspecified, we develop a pairwise rank-based likelihood. Simultaneous inference on the mixture proportion and a parameter representing the probability an observation from one population is greater than an observation from the other population is based on this likelihood. Under some regularity conditions, it is shown that the maximum pairwise rank likelihood estimator is consistent and has an asymptotic normal distribution. Simulation results indicate that the performance of this statistic is satisfactory. The methodology is demonstrated on a data set in prostate cancer.     

Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information

This article presents a new method for maximum likelihood estimation of logistic regression models with incomplete covariate data where auxiliary information is available. This auxiliary information is extraneous to the regression model of interest but predictive of the covariate with missing data. Ibrahim (1990, Journal of the American Statistical Association 85, 765-769) provides a general method for estimating generalized linear regression models with missing covariates using the EM algorithm that is easily implemented when there is no auxiliary data. Vach (1997, Statistics in Medicine 16, 57-72) describes how the method can be extended when the outcome and auxiliary data are conditionally independent given the covariates in the model. The method allows the incorporation of auxiliary data without making the conditional independence assumption. We suggest tests of conditional independence and compare the performance of several estimators in an example concerning mental health service utilization in children. Using an artificial dataset, we compare the performance of several estimators when auxiliary data are available.     

Efficient Estimators for the verianoe of a Finite Population

For the estimation of finite population variance of a character, DAS and TRIPATHI (1978) proposed some efficient estimators when population mean of variance of some other character (associated with main character) is known. We propose here some estimators for population variance using same information and compare with those given by DAS and TRIPATHI (1978). Some examples are given for illustration.