Bivariate Survival Function Research Articles

Time-dependent diagnostic accuracy analysis with censored outcome and censored predictor

We consider a unified approach for estimating time-dependent diagnostic accuracy measures, including time-dependent sensitivity, specificity, positive predictive value, negative predictive value, receiver operating characteristic (ROC) curve, area under the ROC curve (AUC) and integrated AUC across time. In particular, our estimation method incorporates the double censoring setting, i.e. a censored outcome and a censored marker. Our unified approach greatly broadens the application of time-dependent diagnostic measures and allows for comparison between event-type predictors and/or completely observed continuous markers. More specifically, we express these time-dependent diagnostic accuracy measures in terms of bivariate and univariate survival functions. Hence they can be estimated by simply plugging in the Kaplan–Meier estimator for univariate survival functions and the Dabrowska estimator for the bivariate survival function. Asymptotic properties for our proposed estimators and bootstrap validity are established by using empirical processes techniques. Our simulation studies show that our proposed estimators and test procedures perform well for samples of moderate size. We apply our methods to an econometric example as well as a diabetic study.

Self-Consistent Nonparametric Maximum Likelihood Estimator of the Bivariate Survivor Function.

As usually formulated the nonparametric likelihood for the bivariate survivor function is over-parameterized, 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 non-iterative nonpara-metric 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 corresponding distribution function estimator, presumably due to the avoidance of negative mass assignments.

Simple nonparametric estimators of the bivariate survival function under random left truncation and right censoring

We consider estimating the bivariate survival function when both components are subject to random left truncation and right censoring. Using the idea of Sankran and Antony (Sankhya 69:425---447, 2007) in the competing risks set up, we propose two types of estimators as generalizations of the Dabrowska (Ann Stat 18:1475---1489, 1988) and Campbell and Foldes (Nonparametric statistical inference, North-Holland, Amsterdam 1982) estimators. The proposed estimators are easy to implement and do not require iteration. The consistency of the proposed estimators is established. Simulation results indicate that the proposed estimators can outperform the estimators of Shen and Yan (J Stat Plan Inference 138:4041---4054, 2008), which require complex iteration.

Large sample properties for a class of copulas in bivariate survival analysis

This work is concerned with asymptotic properties of the bivariate survival function estimator using the functional relationship between marginal survival functions and a class of copulas for the dependence structure. Specifically, we study consistency and weak convergence of the bivariate survival function estimator obtained considering a two-step procedure of estimation. The obtained results are found from a key decomposition of the bivariate survival function in quantities that can be studied separately. In particular, we use relating results to almost sure and weak convergence of estimators, almost sure convergence of uniformly equicontinuous functions, and the delta method for functionals.

Nonparametric estimators for a survivor function of paired recurrent events

Recurrent event data arise in longitudinal studies where each study subject may experience multiple events during the follow-up. In many situations in survival studies, pairs of individuals can potentially experience recurrent events. The analysis of such data is not straightforward as it involves two kinds of dependences, namely, dependence between the individuals in the same pair and dependence among a sequence of pairs. In the present paper, we introduce a new stochastic model for the analysis of such recurrent event data. Nonparametric estimators for a bivariate survivor function are developed. Asymptotic properties of the estimators are discussed. Simulation studies are carried out to assess the finite sample properties of the estimator. We illustrate the procedure with real life data on eye disease.

A method of moments to estimate bivariate survival functions: the copula approach

In this paper we discuss the problem on parametric and non parametric estimation of the distributions generated by the Marshall-Olkin copula. This copula comes from the Marshall-Olkin bivariate exponential distribution used in reliability analysis. We generalize this model by the copula and different marginal distributions to construct several bivariate survival functions. The cumulative distribution functions are not absolutely continuous and they unknown parameters are often not be obtained in explicit form. In order to estimate the parameters we propose an easy procedure based on the moments. This method consist in two steps: in the first step we estimate only the parameters of marginal distributions and in the second step we estimate only the copula parameter. This procedure can be used to estimate the parameters of complex survival functions in which it is difficult to find an explicit expression of the mixed moments. Moreover it is preferred to the maximum likelihood one for its simplex mathematic form; in particular for distributions whose maximum likelihood parameters estimators can not be obtained in explicit form.

Nonparametric estimation of the bivariate survival function for one modified form of current-status data

In this article, the estimation of the bivariate survival function for one modified form of current-status data is considered. Two types of estimators, which are generalizations of the estimators by Campbell and Földes [G. Campbell and A. Földes, Large sample properties of nonparametric statistical inference, in Nonparametric Statistical Inference, B.V. Gnredenko, M.L. Puri, and I. Vineze, eds., North-Holland, Amsterdam, 1982, pp. 103–122] and Dabrowska [D.M. Dabrowska, Kaplan-Meier estimate on the plane, Ann. Stat. 18 (1988), pp. 1475–1489; D.M. Dabrowska, Kaplan-Meier estimate on the plane: weak convergence, LIL, and the bootstrap, J. Multivariate Anal. 29 (1989), pp. 308–325], are proposed. The consistency of the proposed estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators.

A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.

Nonparametric estimation of the bivariate survival function for one modified form of doubly censored data

In this article, we consider estimating the bivariate distribution function when both components are subject to double censoring. We propose three types of estimators, the first two are generalizations of the Dabrowska and Campbell and Foldes estimators, and the third is an inverse-probability-weighted estimator. The consistency of the proposed estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators.

An inverse probability weighted estimator for the bivariate distribution function under right censoring

An inverse probability weighted estimator is proposed for the joint distribution function of bivariate random vectors under right censoring. The new estimator is based on the idea of transformation of bivariate survival functions and bivariate random vectors to univariate survival functions and univariate random variables. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Numerical studies show that the new estimator is more efficient than some existing inverse probability weighted estimators.

Score test for age at onset genetic linkage analysis in selected sibling pairs

A new score statistic is derived, which uses information from registries (age-specific incidences) and family studies (sib-sib marginal correlation) to weight affected sibling pairs according to their age at onset. Age at onset of sibling pairs is modelled by a gamma frailty model. From this model we derive a bivariate survival function, which depends on the marginal survival and on the marginal correlation. The score statistic for linkage is a classical nonparametric linkage (NPL) statistic where the identical by descent sharing is weighted by a particular function of the age at onset data. Since the statistic is based on survival models, it can also be applied to discordant and healthy sibling pairs. Simulation studies show that the proposed method is robust and more powerful than standard NPL methods. As illustration we apply the new score statistic to data from a breast cancer study.

Kendall distributions and level sets in bivariate exchangeable survival models

For a given bivariate survival function F ¯ , we study the relations between the set of the level curves of F ¯ and the Kendall distribution. Then we characterize the survival models simultaneously admitting a specified Kendall distribution and a specified set of level curves. Attention will be restricted to exchangeable survival models.

Nonparametric estimation of the bivariate survival function with left-truncated and right-censored data

In this note, we consider estimating the bivariate survival function when both components are subject to left truncation and right censoring. We propose two types of estimators as generalizations of the Dabrowska and Campbell and Földes estimators. The consistency of the proposed estimators is established. A simple bootstrap method is used for obtaining precision estimation of the proposed estimators. A simulation study is conducted to investigate the performance of the proposed estimators.

Nonparametric estimation of bivariate survivor function under masked causes of failure

Consider a system that consists of k components. Each component is subject to more than one cause of failure. Due to inadequacy in the diagnostic mechanism or reluctance to report any specific cause of failure (disease), the exact cause of failure cannot be identified easily. In such situations, where the cause of failure is masked, test procedures restrict the cause of failure to a set of possible types containing the true failure cause. In this paper, we develop a nonparametric estimator for the bivariate survivor function of competing risk models under masked causes of failure based on the vector hazard rate. Asymptotic properties of the estimator are established. A simulation study is carried out to assess the performance of the estimator. We also illustrate the method with a data set.

Some general results for moments in bivariate distributions

The purpose of this paper is to present a closed formula to compute the moments of a general function from the knowledge of its bivariate survival function. The result is derived by utilizing an integration by parts formula for two variables, which is not readily available in the literature. Many of the existing results are obtained as special cases. Finally, two examples are presented to illustrate the results. In both the examples, mixed moments as well as moments for the series system and parallel system are obtained. The integration by parts formula in two variables, derived here, is of interest in its own right and we hope that it will be useful in other investigations. The integration by parts formula in two variables is derived as a special case of a general formula in n variables.

Nonparametric Estimation of the Concordance Correlation Coefficient under Univariate Censoring

Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. Lin's (1989, Biometrics 45, 255-268) concordance correlation coefficient (CCC) has become a popular measure of agreement for correlated continuous outcomes. However, commonly used estimation methods for the CCC do not accommodate censored observations and are, therefore, not applicable for survival outcomes. In this article, we estimate the CCC nonparametrically through the bivariate survival function. The proposed estimator of the CCC is proven to be strongly consistent and asymptotically normal, with a consistent bootstrap variance estimator. Furthermore, we propose a time-dependent agreement coefficient as an extension of Lin's (1989) CCC for measuring the agreement between survival times among subjects who survive beyond a specified time point. A nonparametric estimator is developed for the time-dependent agreement coefficient as well. It has the same asymptotic properties as the estimator of the CCC. Simulation studies are conducted to evaluate the performance of the proposed estimators. A real data example from a prostate cancer study is used to illustrate the method.

Survival Instantaneous Log-odds Ratio from Empirical Functions

The objective of this work is to introduce a new method called the Survivorship Instantaneous Log-odds Ratios (SILOR); to illustrate the creation of SILOR from empirical bivariate survival functions; to also derive standard errors of estimation; to compare results with those derived from logistic regression. Hip fracture, AGE and BMI from the Third National Health and Nutritional Examination Survey (NHANES III) were used to calculate empirical survival functions for the adverse health outcome (AHO) and non-AHO. A stable copula was used to create a parametric bivariate survival function, that was fitted to the empirical bivariate survival function. The bivariate survival function had SILOR contours which are not constant. The proposed method has better advantages than logistic regression by following two reasons. The comparison deals with (i) the shapes of the survival surfaces, S(X1, X2), and (ii) the isobols of the log-odds ratios. When using logistic regression the survival surface is either a hyper plane or at most a conic section. Our approach preserves the shape of the survival surface in two dimensions, and the isobols are observed in every detail instead of being overly smoothed by a regression with no more than a second degree polynomial. The present method is straightforward, and it captures all but random variability of the data.

A computationally simple bivariate survival estimator for efficacy and safety

Both treatment efficacy and safety are typically the primary endpoints in Phase II, and even in some Phase III, clinical trials. Efficacy is frequently measured by time to response, death, or some other milestone event and thus is a continuous, possibly censored, outcome. Safety, however, is frequently measured on a discrete scale; in Eastern Cooperative Oncology Group clinical trial E2290, it was measured as the number of weekly rounds of chemotherapy that were tolerable to colorectal cancer patients. For the joint analysis of efficacy and safety, we propose a non-parametric, computationally simple estimator for the bivariate survival function when one time-to-event is continuous, one is discrete, and both are subject to right-censoring. The bivariate censoring times may depend on each other, but they are assumed to be independent of both event times. We derive a closed-form covariance estimator for the survivor function which allows for inference to be based on any of several possible statistics of interest. In addition, we derive its covariance with respect to calendar time of analysis, allowing for its use in sequential studies.

Characterizations and Modelling of Multivariate Lack of Memory Property

In this article we establish characterizations of multivariate lack of memory property in terms of the hazard gradient (whenever exists), the survival function and the cumulative hazard function. Based on one of these characterizations we establish a method of generating bivariate lifetime distributions possessing bivariate lack of memory property (BLMP) with specified marginals. It is observed that the marginal distributions have to satisfy certain conditions to be stated. The method generates absolutely continuous bivariate distributions as well as those containing a singular component. Bivariate exponential distributions due to Proschan and Sullo (Reliability and biometry, pp 423–440, 1974), Freund (in J Am Stat Assoc 56:971–977, 1961), Block and Basu (J Am Stat Assoc 89:1091–1097, 1974) and Marshall and Olkin (J Am Math Assoc 62:30–44, 1967) are generated as particular cases among others using the proposed method. Some other distributions generated using the method may be of practical importance. Shock models leading to bivariate distributions possessing BLMP are given. Some closure properties of a class of univariate failure rate functions that can generate distributions possessing BLMP and of the class of bivariate survival functions having BLMP are studied.

An Adjustment to Improve the Bivariate Survivor Function Repaired NPMLE

We recently proposed a representation of the bivariate survivor function as a mapping of the hazard function for truncated failure time variates. The representation led to a class of estimators that includes van der Laan's repaired nonparametric maximum likelihood estimator (NPMLE) as an important special case. We proposed a Greenwood-like variance estimator for the repaired NPMLE but found somewhat poor agreement between the empirical variance estimates and these analytic estimates for the sample sizes and bandwidths considered in our simulation study. The simulation results also confirmed those of others in showing slightly inferior performance for the repaired NPMLE compared to other competing estimators as well as a sensitivity to bandwidth choice in moderate sized samples. Despite its attractive asymptotic properties, the repaired NPMLE has drawbacks that hinder its practical application. This paper presents a modification of the repaired NPMLE that improves its performance in moderate sized samples and renders it less sensitive to the choice of bandwidth. Along with this modified estimator, more extensive simulation studies of the repaired NPMLE and Greenwood-like variance estimates are presented. The methods are then applied to a real data example.