Boosting the concordance index for survival data - a unified framework to derive and evaluate biomarker combinations

The development of molecular signatures for the prediction of time-to-event outcomes is a methodologically challenging task in bioinformatics and biostatistics. Although there are numerous approaches for the derivation of marker combinations and their evaluation, the underlying methodology often suffers from the problem that different optimization criteria are mixed during the feature selection, estimation and evaluation steps. This might result in marker combinations that are only suboptimal regarding the evaluation criterion of interest. To address this issue, we propose a unified framework to derive and evaluate biomarker combinations. Our approach is based on the concordance index for time-to-event data, which is a non-parametric measure to quantify the discrimatory power of a prediction rule. Specifically, we propose a component-wise boosting algorithm that results in linear biomarker combinations that are optimal with respect to a smoothed version of the concordance index. We investigate the performance of our algorithm in a large-scale simulation study and in two molecular data sets for the prediction of survival in breast cancer patients. Our numerical results show that the new approach is not only methodologically sound but can also lead to a higher discriminatory power than traditional approaches for the derivation of gene signatures.Comment: revised manuscript - added simulation study, additional result

Empirical Likelihood Confidence Intervals for ROC Curves Under Right Censorship

In this thesis, we apply smoothed empirical likelihood method to investigate confidence intervals for the receiver operating characteristic (ROC) curve with right censoring. As a particular application of comparison of distributions from two populations, the ROC curve is constructed by the combination of cumulative distribution function and quantile function. Under mild conditions, the smoothed empirical likelihood ratio converges to chi-square distribution, which is the well-known Wilks\u27s theorem. Furthermore, the performances of the empirical likelihood method are also illustrated by simulation studies in terms of coverage probability and average length of confidence intervals. Finally, a primary biliary cirrhosis data is used to illustrate the proposed empirical likelihood procedure

Jackknife Emperical Likelihood Method and its Applications

In this dissertation, we investigate jackknife empirical likelihood methods motivated by recent statistics research and other related fields. Computational intensity of empirical likelihood can be significantly reduced by using jackknife empirical likelihood methods without losing computational accuracy and stability. We demonstrate that proposed jackknife empirical likelihood methods are able to handle several challenging and open problems in terms of elegant asymptotic properties and accurate simulation result in finite samples. These interesting problems include ROC curves with missing data, the difference of two ROC curves in two dimensional correlated data, a novel inference for the partial AUC and the difference of two quantiles with one or two samples. In addition, empirical likelihood methodology can be successfully applied to the linear transformation model using adjusted estimation equations. The comprehensive simulation studies on coverage probabilities and average lengths for those topics demonstrate the proposed jackknife empirical likelihood methods have a good performance in finite samples under various settings. Moreover, some related and attractive real problems are studied to support our conclusions. In the end, we provide an extensive discussion about some interesting and feasible ideas based on our jackknife EL procedures for future studies

Novel Nonparametric Methods For ROC Curves

The receiver operating characteristic (ROC) curve is a widely used graphical method for evaluating the discriminating power of a diagnostic test or a statistical model in various areas such as epidemiology, industrial quality control and material testing, etc. One important quantitative measure summarizing the ROC curve is the area under the ROC curve (AUC). The accuracy of two diagnostic tests with right censored data can be compared using the difference of two ROC curves and the difference of two AUC\u27s. Moreover, the difference of two volumes under surfaces (VUS) is investigated to compare two treatments for the discrimination of three-category classification data, extending the ROC curve to the ROC surface in the three-dimensional case. A few scientific progresses have been achieved in ROC curves and its related fields over the past decades. In this dissertation, we propose a plug-in empirical likelihood (EL) procedure combining placement values and weighting of inverse probability techniques, to construct stable and precise confidence intervals of the ROC curves, the difference of two ROC curves, the AUC\u27s and the difference of two AUC\u27s with right censoring. We proved that the limiting distribution of the EL ratio is a weighted $\chi^2$ distribution. Furthermore, we introduce a jackknife empirical likelihood (JEL) procedure to explore the difference of two correlated VUS\u27s with complete data. We proved that the limiting distribution of the proposed JEL ratio is a $\chi^2$ distribution, i.e., the Wilk\u27s theorem holds. Extensive simulation studies demonstrate that the new methods have better performance than the existing methods in terms of coverage probability of confidence intervals in most cases. Finally, the proposed methods are applied to analyze data sets of Primary Biliary Cirrhosis (PBC), Alzheimer\u27s disease, etc

Semi-Empirical Likelihood Confidence Intervals for the ROC Curve with Missing Data

The receiver operating characteristic (ROC) curve is one of the most commonly used methods to compare the diagnostic performances of two or more laboratory or diagnostic tests. In this thesis, we propose semi-empirical likelihood based confidence intervals for ROC curves of two populations, where one population is parametric while the other one is non-parametric and both populations have missing data. After imputing missing values, we derive the semi-empirical likelihood ratio statistic and the corresponding likelihood equations. It has been shown that the log-semi-empirical likelihood ratio statistic is asymptotically chi-square distributed. The estimating equations are solved simultaneously to obtain the estimated lower and upper bounds of semi-empirical likelihood confidence intervals. Simulation studies are conducted to evaluate the finite sample performance of the proposed empirical likelihood confidence intervals with various sample sizes and different missing rates

Empirical Likelihood Inferences in Survival Analysis

In survival analysis, different regression models are used to estimate the effects of covariates on the survival time. The proportional hazards model is commonly applied. However, the proportional hazards model does not always give good fit in the real life. Other models, such as proportional odds models, additive hazards models are useful alternative. Motivated by this limitation, we investigate empirical likelihood method and make inference for semiparametric transformation models and accelerated failure time models in this dissertation. The proposed empirical likelihood methods can solve several challenging and open problems. These interesting problems include semiparametirc transformation model with length-biased sampling, semiparametric analysis based on weighted estimating equations with missing covariates. In addition, a more computationally efficient method called jackknife empirical likelihood (JEL) is proposed, which can be applied to make statistical inference for the accelerated failure time model without computing the limiting variance. We show that under certain regularity conditions, the empirical log-likelihood ratio test statistic converges to a standard chi-squared distribution. Finally, computational algorithms are developed for utilizing the proposed empirical likelihood and jackknife empirical likelihood methods. Extensive simulation studies on coverage probabilities and average lengths of confidence intervals for the regression parameters for those topics indicate good finite samples performance under various settings. Furthermore, for each model, real data sets are analyzed for illustration of the proposed methods

Empirical Likelihood Confidence Intervals for ROC Curves with Missing Data

The receiver operating characteristic, or the ROC curve, is widely utilized to evaluate the diagnostic performance of a test, in other words, the accuracy of a test to discriminate normal cases from diseased cases. In the biomedical studies, we often meet with missing data, which the regular inference procedures cannot be applied to directly. In this thesis, the random hot deck imputation is used to obtain a \u27complete\u27 sample. Then empirical likelihood (EL) confidence intervals are constructed for ROC curves. The empirical log-likelihood ratio statistic is derived whose asymptotic distribution isproved to be a weighted chi-square distribution. The results of simulation study show that the EL confidence intervals perform well in terms of the coverage probability and the average length for various sample sizes and response rates