Takeshi Emura

Gene selection for survival data under dependent censoring: A copula-based approach.

Dependent censoring arises in biomedical studies when the survival outcome of interest is censored by competing risks. In survival data with microarray gene expressions, gene selection based on the univariate Cox regression analyses has been used extensively in medical research, which however, is only valid under the independent censoring assumption. In this paper, we first consider a copula-based framework to investigate the bias caused by dependent censoring on gene selection. Then, we utilize the copula-based dependence model to develop an alternative gene selection procedure. Simulations show that the proposed procedure adjusts for the effect of dependent censoring and thus outperforms the existing method when dependent censoring is indeed present. The non-small-cell lung cancer data are analyzed to demonstrate the usefulness of our proposal. We implemented the proposed method in an R “compound.Cox” package.

Testing quasi-independence for truncation data

Quasi-independence is a common assumption for analyzing truncated data. To verify this condition, we propose a class of weighted log-rank type statistics that include existing tests proposed by Tsai (1990) and Martin and Betensky (2005) as special cases. To choose an appropriate weight function that may lead to a more power test, we derive a score test when the dependence structure under the alternative hypothesis is modeled via the odds ratio function proposed by Chaieb, Rivest and Abdous (2006). Asymptotic properties of the proposed tests are established based on the functional delta method which can handle more general situations than results based on rank-statistics or U-statistics. Extension of the proposed methodology under two different censoring settings is also discussed. Simulations are performed to examine finite-sample performances of the proposed method and its competitors. Two datasets are analyzed for illustrative purposes.

Survival Prediction Based on Compound Covariate under Cox Proportional Hazard Models

Survival prediction from a large number of covariates is a current focus of statistical and medical research. In this paper, we study a methodology known as the compound covariate prediction performed under univariate Cox proportional hazard models. We demonstrate via simulations and real data analysis that the compound covariate method generally competes well with ridge regression and Lasso methods, both already well-studied methods for predicting survival outcomes with a large number of covariates. Furthermore, we develop a refinement of the compound covariate method by incorporating likelihood information from multivariate Cox models. The new proposal is an adaptive method that borrows information contained in both the univariate and multivariate Cox regression estimators. We show that the new proposal has a theoretical justification from a statistical large sample theory and is naturally interpreted as a shrinkage-type estimator, a popular class of estimators in statistical literature. Two datasets, the primary biliary cirrhosis of the liver data and the non-small-cell lung cancer data, are used for illustration. The proposed method is implemented in R package “compound.Cox” available in CRAN at http://cran.r-project.org/.

A joint frailty-copula model between tumour progression and death for meta-analysis

Dependent censoring often arises in biomedical studies when time to tumour progression (e.g., relapse of cancer) is censored by an informative terminal event (e.g., death). For meta-analysis combining existing studies, a joint survival model between tumour progression and death has been considered under semicompeting risks, which induces dependence through the study-specific frailty. Our paper here utilizes copulas to generalize the joint frailty model by introducing additional source of dependence arising from intra-subject association between tumour progression and death. The practical value of the new model is particularly evident for meta-analyses in which only a few covariates are consistently measured across studies and hence there exist residual dependence. The covariate effects are formulated through the Cox proportional hazards model, and the baseline hazards are nonparametrically modeled on a basis of splines. The estimator is then obtained by maximizing a penalized log-likelihood function. We also show that the present methodologies are easily modified for the competing risks or recurrent event data, and are generalized to accommodate left-truncation. Simulations are performed to examine the performance of the proposed estimator. The method is applied to a meta-analysis for assessing a recently suggested biomarker CXCL12 for survival in ovarian cancer patients. We implement our proposed methods in R joint.Cox package.

Personalized dynamic prediction of death according to tumour progression and high-dimensional genetic factors: Meta-analysis with a joint model.

Developing a personalized risk prediction model of death is fundamental for improving patient care and touches on the realm of personalized medicine. The increasing availability of genomic information and large-scale meta-analytic data sets for clinicians has motivated the extension of traditional survival prediction based on the Cox proportional hazards model. The aim of our paper is to develop a personalized risk prediction formula for death according to genetic factors and dynamic tumour progression status based on meta-analytic data. To this end, we extend the existing joint frailty-copula model to a model allowing for high-dimensional genetic factors. In addition, we propose a dynamic prediction formula to predict death given tumour progression events possibly occurring after treatment or surgery. For clinical use, we implement the computation software of the prediction formula in the joint.Cox R package. We also develop a tool to validate the performance of the prediction formula by assessing the prediction error. We illustrate the method with the meta-analysis of individual patient data on ovarian cancer patients.

Nonparametric maximum likelihood estimation for dependent truncation data based on copulas

Truncation occurs when the variable of interest can be observed only if its value satisfies certain selection criteria. Most existing methods for analyzing such data critically rely on the assumption that the truncation variable is quasi-independent of the variable of interest. In this article, the authors propose a likelihood-based inference approach under the assumption that the dependence structure of the two variables follows a general form of copula model. They develop a model selection method for choosing the best-fitted copula among a broad class of model alternatives, and they derive large-sample properties of the proposed estimators, including the inverse Fisher information matrix. The treatment of ties is also discussed. They apply their methods to the analysis of a transfusion-related AIDS data set and compare the results with existing methods. Simulation results are also provided to evaluate the finite-sample performances of all the competing methods.

A copula-based inference to piecewise exponential models under dependent censoring, with application to time to metamorphosis of salamander larvae

In ecology and evolutionary biology, controlled animal experiments are often conducted to measure time to metamorphosis which is possibly censored by the competing risk of death and the follow-up end. This paper considers the problem of estimating the survival function of time-to-event when it is subject to dependent censoring. When the censorship is due to competing risks, the traditional assumption of independent censorship may not be satisfied, and hence, the usual application of the Kaplan–Meier estimator yields a biased estimation for the survival function of the event time. This paper follows an assumed copula approach (Zheng and Klein in Biometrika 82(1):127–138, 1995) to adjust for dependence between the event time of interest and the competing event time. While the literature on an assumed copula approach has mostly focused on semiparametric settings, we alternatively consider a parametric approach with piecewise exponential models for fitting the survival function. We develop maximum likelihood estimation under the piecewise exponential models with an assumed copula. A goodness-of-fit procedure is also developed, which touches upon the identifiability issue of the copula. We conduct simulations to examine the performance of the proposed method and compare it with an existing semiparametric method. The method is applied to real data analysis on time to metamorphosis for salamander larvae living in Hokkaido, Japan (Michimae et al. in Evol Ecol Res 16:617–629, 2014).

Correlated evolution of phenotypic plasticity in metamorphic timing

Phenotypic plasticity has long been a focus of research, but the mechanisms of its evolution remain controversial. Many amphibian species exhibit a similar plastic response in metamorphic timing in response to multiple environmental factors; therefore, more than one environmental factor has likely influenced the evolution of plasticity. However, it is unclear whether the plastic responses to different factors have evolved independently. In this study, we examined the relationship between the plastic responses to two experimental factors (water level and food type) in larvae of the salamander Hynobius retardatus, using a cause‐specific Cox proportional hazards model on the time to completion of metamorphosis. Larvae from ephemeral ponds metamorphosed earlier than those from permanent ponds when kept at a low water level or fed conspecific larvae instead of larval Chironomidae. This acceleration of metamorphosis depended only on the permanency of the larvae’s pond of origin, but not on the conspecific larval density (an indicator of the frequency of cannibalism) in the ponds. The two plastic responses were significantly correlated, indicating that they may evolve correlatively. Once plasticity evolved as an adaptation to habitat desiccation, it might have relatively easily become a response to other ecological factors, such as food type via the pre‐existing developmental pathway.

A Comparison of Normal Approximation Rules for Attribute Control Charts

Control charts, known for more than 80 years, have been important tools for business and industrial manufactures. Among many different types of control charts, the attribute control chart (np-chart or p-chart) is one of the most popular methods to monitor the number of observed defects in products, such as semiconductor chips, automobile engines, and loan applications. The attribute control chart requires that the sample size n is sufficiently large and the defect rate p is not too small so that the normal approximation to the binomial works well. Some rules for the required values for n and p are available in the textbooks of quality control and mathematical statistics. However, these rules are considerably different and hence it is less clear which rule is most appropriate in practical applications. In this paper, we perform a comparison of five frequently used rules for n and p required for the normal approximation to the binomial. Based on this result, we also refine the existing rules to develop a new rule that has a reliable performance. Datasets are analyzed for illustration.

Statistical inference based on the nonparametric maximum likelihood estimator under double-truncation

Doubly truncated data consist of samples whose observed values fall between the right- and left- truncation limits. With such samples, the distribution function of interest is estimated using the nonparametric maximum likelihood estimator (NPMLE) that is obtained through a self-consistency algorithm. Owing to the complicated asymptotic distribution of the NPMLE, the bootstrap method has been suggested for statistical inference. This paper proposes a closed-form estimator for the asymptotic covariance function of the NPMLE, which is computationally attractive alternative to bootstrapping. Furthermore, we develop various statistical inference procedures, such as confidence interval, goodness-of-fit tests, and confidence bands to demonstrate the usefulness of the proposed covariance estimator. Simulations are performed to compare the proposed method with both the bootstrap and jackknife methods. The methods are illustrated using the childhood cancer dataset.