A reduced-bias nonparametric estimator of the cumulative distribution function (CDF) and the survival function is proposed using infinite-order kernels. Fourier transform theory on generalized functions is utilized to obtain the improved bias estimates. The new estimators are analyzed in terms of their relative deficiency to the empirical distribution function and Kaplan-Meier estimator, and even improvements in terms of asymptotic relative efficiency (ARE) are present under specified assumptions on the data. The deficiency analysis introduces a deficiency rate which provides a continuum between the classical deficiency analysis and an efficiency analysis. Additionally, an automatic bandwidth selection algorithm, specially tailored to the infinite-order kernels, is incorporated into the estimators. In small sample sizes these estimators can significantly improve the estimation of the CDF and survival function as is illustrated through the deficiency analysis and computer simulations.
The strata-specific treatment effect or so-called blip for a randomly drawn strata of confounders defines a random variable and a corresponding cumulative distribution function. However, the CDF is not pathwise differentiable, necessitating a kernel smoothing approach to estimate it at a given point or perhaps many points. Assuming the CDF is continuous, we derive the efficient influence curve of the kernel smoothed version of the blip CDF and a CV-TMLE estimator. The estimator is asymptotically efficient under two conditions, one of which involves a second order remainder term which, in this …
We study the comparison problem of distribution equality between two random samples under a right censoring scheme. To address this problem, we design a series of tests based on energy distance and kernel mean embeddings. We calibrate our tests using permutation methods and prove that they are consistent against all fixed continuous alternatives. To evaluate our proposed tests, we simulate survival curves from previous clinical trials. Additionally, we provide practitioners with a set of recommendations on how to select parameters/distances for the delay effect problem. Based on the method for…
Predicting patient survival probabilities based on observed covariates is an important assessment in clinical practice. These patient-specific covariates are often measured over multiple follow-up appointments. It is then of interest to predict survival based on the history of these longitudinal measurements, and to update predictions as more observations become available. The standard approaches to these so-called `dynamic prediction' assessments are joint models and landmark analysis. Joint models involve high-dimensional parametrisations, and their computational complexity often prohibits i…
In this article a new family of tests is proposed for the comparison problem of the equality of distribution of two-sample under right censoring scheme. The tests are based on energy distance and kernels mean embedding, are calibrated by permutations and are consistent against all alternatives. The good performance of the new tests in real situations with finite samples is established with a simulation study.
Based on the theories of sliced inverse regression (SIR) and reproducing kernel Hilbert space (RKHS), a new approach RDSIR (RKHS-based Double SIR) to nonlinear dimension reduction for survival data is proposed and discussed. An isometrically isomorphism is constructed based on RKHS property, then the nonlinear function in the RKHS can be represented by the inner product of two elements which reside in the isomorphic feature space. Due to the censorship of survival data, double slicing is used to estimate weight function or conditional survival function to adjust for the censoring bias. The suf…
Otto-von-Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学）；Universidad de Valparaíso（瓦尔帕莱索大学）
In this paper we propose a Multiple kernel testing procedure to infer survival data when several factors (e.g. different treatment groups, gender, medical history) and their interaction are of interest simultaneously. Our method is able to deal with complex data and can be seen as an alternative to the omnipresent Cox model when assumptions such as proportionality cannot be justified. Our methodology combines well-known concepts from Survival Analysis, Machine Learning and Multiple Testing: differently weighted log-rank tests, kernel methods and multiple contrast tests. By that, complex hazard…
Non-parametric maximum likelihood estimation encompasses a group of classic methods to estimate distribution-associated functions from potentially censored and truncated data, with extensive applications in survival analysis. These methods, including the Kaplan-Meier estimator and Turnbull's method, often result in overfitting, especially when the sample size is small. We propose an improvement to these methods by applying kernel smoothing to their raw estimates, based on a BIC-type loss function that balances the trade-off between optimizing model fit and controlling model complexity. In the …
University of Warwick（华威大学）；Nanjing University（南京大学）；London School of Economics and Political Science（伦敦政治经济学院）；University of Cambridge（剑桥大学）；Yale School of Public Health（耶鲁公共卫生学院）
We propose the interval censored recursive forests (ICRF) which is an iterative tree ensemble method for interval censored survival data. This nonparametric regression estimator makes the best use of censored information by iteratively updating the survival estimate, and can be viewed as a self-consistent estimator with convergence monitored using out-of-bag samples. Splitting rules optimized for interval censored data are developed and kernel-smoothing is applied. The ICRF displays the highest prediction accuracy among competing nonparametric methods in most of the simulations and in an appli…
Nonparametric Estimation of Event-Free Survival for Data with Left-Truncated Death and Intermittently Assessed Nonfatal Events
University of Minnesota（明尼苏达大学）；Columbia University（哥伦比亚大学）；University of Texas（德克萨斯大学）；University of Texas Southwestern Medical Center（德克萨斯大学西南医学中心）；University of Mississippi Medical Center（密西西比大学医学中心）；University of North Carolina（北卡罗来纳大学）；Wake Forest University School of Medicine（维克森林大学医学院）
We introduce a general non-parametric independence test between right-censored survival times and covariates, which may be multivariate. Our test statistic has a dual interpretation, first in terms of the supremum of a potentially infinite collection of weight-indexed log-rank tests, with weight functions belonging to a reproducing kernel Hilbert space (RKHS) of functions; and second, as the norm of the difference of embeddings of certain finite measures into the RKHS, similar to the Hilbert-Schmidt Independence Criterion (HSIC) test-statistic. We study the asymptotic properties of the test, f…
De-biased Distribution Estimation for Locally Differential Private Survival Data
The State University of New York (SUNY) at Buffalo（纽约州立大学布法罗分校）；National Chengchi University（国立政治大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
We study inference for censored survival data where some covariates are distorted by some unknown functions of an observable confounding variable in a multiplicative form. Example of this kind of data in medical studies is the common practice to normalizing some important observed exposure variables by patients' body mass index (BMI), weight or age. Such phenomenon also appears frequently in environmental studies where ambient measure is used for normalization, and in genomic studies where library size needs to be normalized for next generation sequencing data. We propose a new covariate-adjus…
