Nonparametric estimation in a cure model with random cure times

Acute respiratory distress syndrome (ARDS) is a life-threatening acute condition that sometimes follows pneumonia or surgery. Patients who recover and leave the hospital are considered to have been cured at the time they leave the hospital. These data differ from typical data in which cure is a possibility: death times are not observed for patients who are cured and cure times are observed and vary among patients. Here we apply a competing risks model to these data and show it to be equivalent to a mixture model, the more common approach for cure data. Further, we derive an estimator for the variance of the cumulative incidence function from the competing risks model, and thus for the cure rate, based on elementary calculations. We compare our variance estimator to Gray's (1988, Annals of Statistics 16, 1140-1154) estimator, which is based on counting process theory. We find our estimator to be slightly more accurate in small samples. We apply these results to data from an ARDS clinical trial.     

Nonparametric and parametric estimation for a linear germination-growth model

Seeds are planted on the interval [0, L] at various locations. Each seed has a location x and a potential germination time t epsilon [0, infinity), and it is assumed that the collection of such (x, t) pairs forms a Poisson process in [0, L] x [0, infinity) with intensity measure dxd lambda(t). From each seed that germinates, an inhibiting region grows bidirectionally at rate 2v. These regions inhibit germination of any seed in the region with a later potential germination time. Thus, seeds only germinate in the uninhibited part of [0, L]. We want to estimate lambda on the basis of one or more realizations of the process, the data being the locations and germination times of the germinated seeds. We derive the maximum likelihood estimator of v and a nonparametric estimator of lambda and describe methods of obtaining parametric estimates from it, illustrating these with reference to gamma densities. Simulation results are described and the methods applied to some neurobiological data. An Appendix outlines the S-PLUS code used.     

Nonparametric estimation for the three-stage irreversible illness-death model

In this paper, we present new nonparametric estimators of the stage-occupation probabilities in the three-stage irreversible illness-death model. These estimators use a fractional risk set and a reweighting approach and are valid under stage-dependent censoring. Using a simulated data set, we compare the behavior of our estimators with previously proposed estimators. We also apply our estimators to data on time to Pneumocystis pneumonia and death obtained from an AIDS cohort study.     

A nonparametric mixture model for cure rate estimation

Nonparametric methods have attracted less attention than their parametric counterparts for cure rate analysis. In this paper, we study a general nonparametric mixture model. The proportional hazards assumption is employed in modeling the effect of covariates on the failure time of patients who are not cured. The EM algorithm, the marginal likelihood approach, and multiple imputations are employed to estimate parameters of interest in the model. This model extends models and improves estimation methods proposed by other researchers. It also extends Cox's proportional hazards regression model by allowing a proportion of event-free patients and investigating covariate effects on that proportion. The model and its estimation method are investigated by simulations. An application to breast cancer data, including comparisons with previous analyses using a parametric model and an existing nonparametric model by other researchers, confirms the conclusions from the parametric model but not those from the existing nonparametric model.     

Nonparametric estimation of stage occupation probabilities in a multistage model with current status data

Multistage models are used to describe individuals (or experimental units) moving through a succession of "stages" corresponding to distinct states (e.g., healthy, diseased, diseased with complications, dead). The resulting data can be considered to be a form of multivariate survival data containing information about the transition times and the stages occupied. Traditional survival analysis is the simplest example of a multistage model, where individuals begin in an initial stage (say, alive) and move irreversibly to a second stage (death). In this article, we consider general multistage models with a directed tree structure (progressive models) in which individuals traverse through stages in a possibly non-Markovian manner. We construct nonparametric estimators of stage occupation probabilities and marginal cumulative transition hazards. Empirical calculations of these quantities are not possible due to the lack of complete data. We consider current status information which represents a more severe form of censoring than the commonly used right censoring. Asymptotic validity of our estimators can be justified using consistency results for nonparametric regression estimators. Finite-sample behavior of our estimators is studied by simulation, in which we show that our estimators based on these limited data compare well with those based on complete data. We also apply our method to a real-life data set arising from a cardiovascular diseases study in Taiwan.     

Nonparametric estimation of the size-metastasis relationship in solid cancers.

This paper is concerned with the relationship between the occurrence of metastases and the size of primary cancers. We consider two probabilistic characterizations of this relationship. First is the distribution function of tumor sizes at the point of metastatic transition; second is the probability that detectable metastases are present when the cancer comes to medical attention. The equation relating these two functions is developed and conditions for their being identical are explored. Since the tumor size at the point of metastasis is not usually observable, estimation of the first distribution requires the use of the EM algorithm. Nonparametric methods of estimating both functions are explored, with attention to the fact that tumors often fail to be measured, particularly those that are known to be metastatic. The methods are applied to the estimation of primary tumor size at the point of distant metastasis in lung cancer (epidermoid and adenocarcinoma) and colorectal cancer and at the point of nodal metastasis in breast cancer. Monte Carlo experiments confirm that the bias inherent in the methodology is acceptably small.     

Group testing regression model estimation when case identification is a goal

Group testing is frequently used to reduce the costs of screening a large number of individuals for infectious diseases or other binary characteristics in small prevalence situations. In many applications, the goals include both identifying individuals as positive or negative and estimating the probability of positivity. The identification aspect leads to additional tests being performed, known as “retests”, beyond those performed for initial groups of individuals. In this paper, we investigate how regression models can be fit to estimate the probability of positivity while also incorporating the extra information from these retests. We present simulation evidence showing that significant gains in efficiency occur by incorporating retesting information, and we further examine which testing protocols are the most efficient to use. Our investigations also demonstrate that some group testing protocols can actually lead to more efficient estimates than individual testing when diagnostic tests are imperfect. The proposed methods are applied retrospectively to chlamydia screening data from the Infertility Prevention Project. We demonstrate that significant cost savings could occur through the use of particular group testing protocols.     

Nonparametric estimation of a multivariate distribution in the presence of censoring

This paper presents examples of situations in which one wishes to estimate a multivariate distribution from data that may be right-censored. A distinction is made between what we term 'homogeneous' and 'heterogeneous' censoring. It is shown how a multivariate empirical survivor function must be constructed in order to be considered a (nonparametric) maximum likelihood estimate of the underlying survivor function. A closed-form solution, similar to the product-limit estimate of Kaplan and Meier, is possible with homogeneous censoring, but an iterative method, such as the EM algorithm, is required with heterogeneous censoring. An example is given in which an anomaly is produced if censored multivariate data are analyzed as a series of univariate variables; this anomaly is shown to disappear if the methods of this paper are used.     

Nonparametric estimation of a delay distribution based on left-censored and right-truncated data

This paper proposes a nonparametric method for estimating a delay distribution based on left-censored and right-truncated data. A variance-covariance estimator is provided. The method is applied to the Australian AIDS data in which some data are left censored and some data are not left censored. This situation arises with AIDS case-reporting data in Australia because reporting delays were recorded only from November 1990 rather than from the beginning of the epidemic there. It is shown that inclusion of the left-censored data, as opposed to analyzing only the uncensored data, improves the precision of the estimate.     

Nonparametric estimation of ROC curves in the absence of a gold standard

In the evaluation of diagnostic accuracy of tests, a gold standard on the disease status is required. However, in many complex diseases, it is impossible or unethical to obtain such a gold standard. If an imperfect standard is used, the estimated accuracy of the tests would be biased. This type of bias is called imperfect gold standard bias. In this article we develop a nonparametric maximum likelihood method for estimating ROC curves and their areas of ordinal-scale tests in the absence of a gold standard. Our simulation study shows that the proposed estimators for the ROC curve areas have good finite-sample properties in terms of bias and mean squared error. Further simulation studies show that our nonparametric approach is comparable to the binormal parametric method, and is easier to implement. Finally, we illustrate the application of the proposed method in a real clinical study on assessing the accuracy of seven specific pathologists in detecting carcinoma in situ of the uterine cervix.