Biometrika

Prentice, R. L.

doi: 10.1093/biomet/asu010pmid: 25632162

Abstract As usually formulated the nonparametric likelihood for the bivariate survivor function is overparameterized, resulting in uniqueness problems for the corresponding nonparametric maximum likelihood estimator. Here the estimation problem is redefined to include parameters for marginal hazard rates, and for double failure hazard rates only at informative uncensored failure time grid points where there is pertinent empirical information. Double failure hazard rates at other grid points in the risk region are specified rather than estimated. With this approach the nonparametric maximum likelihood estimator is unique, and can be calculated using a two-step procedure. The first step involves setting aside all doubly censored observations that are interior to the risk region. The nonparametric maximum likelihood estimator from the remaining data turns out to be the Dabrowska (1988) estimator. The omitted doubly censored observations are included in the procedure in the second stage using self-consistency, resulting in a noniterative nonparametric maximum likelihood estimator for the bivariate survivor function. Simulation evaluation and asymptotic distributional results are provided. Moderate sample size efficiency for the survivor function nonparametric maximum likelihood estimator is similar to that for the Dabrowska estimator as applied to the entire dataset, while some useful efficiency improvement arises for the corresponding distribution function estimator, presumably due to the avoidance of negative mass assignments. References Akritas M. G. , Van Keilegom I. . Estimation of the bivariate and marginal distributions with censored data , J. R. Statist. Soc. B , 2003 , vol. 65 (pg. 457 - 71 ) Google Scholar Crossref Search ADS WorldCat Campbell G. . Nonparametric bivariate estimation with randomly censored data , Biometrika , 1981 , vol. 68 (pg. 417 - 22 ) Google Scholar Crossref Search ADS WorldCat Clayton D. G. . Model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence , Biometrika , 1978 , vol. 65 (pg. 141 - 51 ) Google Scholar Crossref Search ADS WorldCat Dabrowska D. M. . Kaplan–Meier estimate on the plane , Ann. Statist. , 1988 , vol. 16 (pg. 1475 - 89 ) Google Scholar Crossref Search ADS WorldCat Efron B. . The two sample problem with censored data , Proc. 5th Berkeley Symp. Math. Statist. Prob. , 1967 Prentice-Hall, vol. 4 Google Scholar OpenURL Placeholder Text WorldCat Gill R. , Johansen S. . A survey of product integration with a view towards application in survival analysis , Ann. Statist. , 1990 , vol. 18 (pg. 1501 - 55 ) Google Scholar Crossref Search ADS WorldCat Gill R. D. , van der Laan M. J. , Wellner J. A. . Inefficient estimators of the bivariate survival function for three models , Ann. Inst. H. Poincaré Prob. Statist. , 1995 , vol. 31 (pg. 545 - 97 ) Google Scholar OpenURL Placeholder Text WorldCat Hanley J. A. , Parnes M. N. . Nonparametric estimation of a multivariate distribution in the presence of censoring , Biometrics , 1983 , vol. 39 (pg. 129 - 39 ) Google Scholar Crossref Search ADS PubMed WorldCat Moodie F. Z. , Prentice R. L. . An adjustment to improve the bivariate survivor function repaired npmle , Lifetime Data Anal. , 2005 , vol. 11 (pg. 291 - 307 ) Google Scholar Crossref Search ADS PubMed WorldCat Prentice R. L. , Cai J. . Covariance and survivor function estimation using censored multivariate failure time data , Biometrika , 1992 , vol. 79 (pg. 495 - 512 ) Google Scholar Crossref Search ADS WorldCat Tsai W.-Y. , Crowley J. . A large sample study of generalized maximum likelihood estimators from incomplete data via self-consistency , Ann. Statist. , 1985 , vol. 13 (pg. 1317 - 34 ) Google Scholar Crossref Search ADS WorldCat Tsai W.-Y. , Leurgans S. , Crowley J. . Nonparametric estimation of a bivariate survival function in the presence of censoring , Ann. Statist. , 1986 , vol. 14 (pg. 1351 - 65 ) Google Scholar Crossref Search ADS WorldCat van der Laan M. J. . Efficient estimation in the bivariate censoring model and repairing NPMLE , Ann. Statist. , 1996 , vol. 24 (pg. 596 - 627 ) Google Scholar Crossref Search ADS WorldCat This content is only available as a PDF. © 2014 Biometrika Trust This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. © 2014 Biometrika Trust

Zhu, Hong; Wang, Mei-Cheng

doi: 10.1093/biomet/asu005pmid: 32390662

In many biomedical applications, interest focuses on the occurrence of two or more consecutive failure events and the relationship between event times, such as age of disease onset and residual lifetime. Bivariate survival data with interval sampling arise frequently when disease registries or surveillance systems collect data based on disease incidence occurring within a specific calendar time interval. The initial event is then retrospectively confirmed and the subsequent failure event may be observed during follow-up. In life history studies, the initial and two consecutive failure events could correspond to birth, disease onset and death. The statistical features and bias of observed data in relation to interval sampling were discussed by Zhu & Wang (2012). Here we propose nonparametric estimation of the association between bivariate failure times based on Kendalls tau for data collected with interval sampling. A nonparametric estimator is given, where the contribution of each comparable and orderable pair is weighted by the inverse of the associated selection probability. Analysis methods for bivariate survival data with interval sampling rely on the assumption of quasi-independence, i.e., that bivariate failure times and the time of the initial event are independent in the observable region. This paper develops a nonparametric test of quasi-independence based on a bivariate conditional Kendalls tau for such data. Simulation studies demonstrate that the association estimator and testing procedure perform well with moderate sample sizes. Illustrations with two real datasets are provided.

Chauvel, C.; O'Quigley, J.

doi: 10.1093/biomet/asu015pmid: N/A

We describe a class of statistical tests for the comparison of two or more survival curves, typically estimated using the KaplanMeier method. The class is based on the construction of OQuigley (2003), and some special cases are of particular interest. Underlying the inferential development are the arguments of Efron & Hinkley (1978), leading to a theoretical sampling model that is in some sense closer to the observed data. The log-rank and weighted log-rank tests arise as special members of the class. In practice the log-rank test will often be a suboptimal, even poor, test due to the presence of non-proportional hazards. The proposed test maintains good power and, in all the cases considered, has greater power than the log-rank test under non-proportional hazards. The power will depend on the alternatives being considered, and under reasonable assumptions on the alternatives, we conclude that the proposed test is more powerful than the log-rank test. Simulations support these conclusions. An example is given as an illustration.

Wang, Mei-Cheng; Huang, Chiung-Yu

doi: 10.1093/biomet/asu016pmid: 26412863

This paper proposes a unified framework to characterize the rate function of a recurrent event process through shape and size parameters. In contrast to the intensity function, which is the event occurrence rate conditional on the event history, the rate function is the occurrence rate unconditional on the event history, and thus it can be interpreted as a population-averaged count of events in unit time. In this paper, shape and size parameters are introduced and used to characterize the association between the rate function () and a random variable X. Measures of association between X and () are defined via shape- and size-based coefficients. Rate-independence of X and () is studied through tests of shape-independence and size-independence, where the shape- and size-based test statistics can be used separately or in combination. These tests can be applied when X is a covariable possibly correlated with the recurrent event process through () or, in the one-sample setting, when X is the censoring time at which the observation of N() is terminated. The proposed tests are shape- and size-based, so when a null hypothesis is rejected, the test results can serve to distinguish the source of violation.

Delaigle, A.; Hall, P.; Wishart, J. R.

doi: 10.1093/biomet/asu025pmid: N/A

We consider nonparametric and semiparametric estimation of a conditional probability curve in the case of group testing data, where the individuals are pooled randomly into groups and only the pooled data are available. We derive a nonparametric weighted estimator that has optimality properties accounting for group sizes, and show how to extend it to multivariate settings, including the partially linear model. In the group testing context, it is natural to assume that the probability curve depends on the covariates only through a linear combination of them. Motivated by this condition, we develop a nonparametric estimator based on the single-index model. We study theoretical properties of the proposed estimators and derive data-driven procedures. Practical properties of the methods are demonstrated via real and simulated examples, and our estimators are shown to have smaller median integrated square error than existing competitors.

Wang, D.; McMahan, C. S.; Gallagher, C. M.; Kulasekera, K. B.

doi: 10.1093/biomet/asu007pmid: N/A

Group testing, through the use of pooling, has proven to be an efficient method of reducing the time and cost associated with screening for a binary characteristic of interest, such as infection status. A topic of key interest in the statistical literature involves the development of regression models that relate individual-level covariates to testing responses observed from pooled specimens. In this article, we propose a general semiparametric framework that allows for the inclusion of multi-dimensional covariates, decoding information, and imperfect testing. The asymptotic properties of our estimators are presented and guidance on finite sample implementation is provided. We illustrate the performance of our methods through simulation and by applying them to chlamydia and gonorrhea data collected by the Nebraska Public Health Laboratory as a part of the Infertility Prevention Project.

Jia, Xiaoyu; Lee, Shing M.; Cheung, Ying Kuen

doi: 10.1093/biomet/asu012pmid: N/A

This paper deals with the design of the likelihood continual reassessment method, which is an increasingly widely used model-based method for dose-finding studies. It is common to implement the method in a two-stage approach, whereby the model-based stage is activated after an initial sequence of patients has been treated. While this two-stage approach is practically appealing, it lacks a theoretical framework, and it is often unclear how the design components should be specified. This paper develops a general framework based on the coherence principle, from which we derive a design calibration process. A real clinical-trial example is used to demonstrate that the proposed process can be implemented in a timely and reproducible manner, while offering competitive operating characteristics. We explore the operating characteristics of different models within this framework and show the performance to be insensitive to the choice of dose-toxicity model.

Hu, Zonghui; Follmann, Dean A.; Wang, Naisyin

doi: 10.1093/biomet/asu022pmid: 25797955

We introduce the effective balancing score for estimation of the mean response under a missing-at-random mechanism. Unlike conventional balancing scores, the proposed score is constructed via dimension reduction free of model specification. Three types of such scores are introduced, distinguished by whether they carry the covariate information about the missingness, the response, or both. The effective balancing score leads to consistent estimation with little or no loss in efficiency. Compared to existing estimators, it reduces the burden of model specification and is more robust. It is a near-automatic procedure which is most appealing when high-dimensional covariates are involved. We investigate its asymptotic and numerical properties, and illustrate its application with an HIV disease study.

Zhang, Chong; Liu, Yufeng

doi: 10.1093/biomet/asu017pmid: 26538663

Large-margin classifiers are popular methods for classification. Among existing simultaneous multicategory large-margin classifiers, a common approach is to learn k different functions for a k-class problem with a sum-to-zero constraint. Such a formulation can be inefficient. We propose a new multicategory angle-based large-margin classification framework. The proposed angle-based classifiers consider a simplex-based prediction rule without the sum-to-zero constraint, and enjoy more efficient computation. Many binary large-margin classifiers can be naturally generalized for multicategory problems through the angle-based framework. Theoretical and numerical studies demonstrate the usefulness of the angle-based methods.

Kundu, S.; Dunson, D. B.

doi: 10.1093/biomet/asu019pmid: N/A

Although discrete mixture modelling has formed the backbone of the literature on Bayesian density estimation, there are some well-known disadvantages. As an alternative to discrete mixtures, we propose a class of priors based on random nonlinear functions of a uniform latent variable with an additive residual. The induced prior for the density is shown to have desirable properties, including ease of centring on an initial guess, large support, posterior consistency and straightforward computation via Gibbs sampling. Some advantages over discrete mixtures, such as Dirichlet process mixtures of Gaussian kernels, are discussed and illustrated via simulations and an application.