Nonparametric regression models for right-censored data using Bernstein polynomials

In some applications of survival analysis with covariates, the commonly used semiparametric assumptions (e.g., proportional hazards) may turn out to be stringent and unrealistic, particularly when there is scientific background to believe that survival curves under different covariate combinations will cross during the study period. We present a new nonparametric regression model for the conditional hazard rate using a suitable sieve of Bernstein polynomials. The proposed nonparametric methodology has three key features: (i) the smooth estimator of the conditional hazard rate is shown to be a unique solution of a strictly convex optimization problem for a wide range of applications; making it computationally attractive, (ii) the model is shown to encompass a proportional hazards structure, and (iii) large sample properties including consistency and convergence rates are established under a set of mild regularity conditions. Empirical results based on several simulated data scenarios indicate that the proposed model has reasonably robust performance compared to other semiparametric models particularly when such semiparametric modeling assumptions are violated. The proposed method is further illustrated on the gastric cancer data and the Veterans Administration lung cancer data.     

Dimension-reduced nonparametric maximum likelihood computation for interval-censored data

A general technique is proposed for efficient computation of the nonparametric maximum likelihood estimate (NPMLE) of a survival function. The main idea is to include a new support interval that has the largest gradient value between inclusively every two neighbouring support intervals in the support set at each iteration. It is thus able to expand the support set exponentially fast during the initial stage of computation and tends to produce the same support set of the NPMLE afterward. The use of the proposed technique needs to be combined with an algorithm that can effectively find and remove redundant support intervals, for example, the constrained Newton method, the iterative convex minorant algorithm and the subspace-based Newton method. Numerical studies show that the dimension-reducing technique works very well, especially for purely interval-censored data, where a significant computational improvement via dimension reduction is possible. Strengths and weaknesses of various algorithms are also discussed and demonstrated.     

Nonparametric maximum likelihood analysis of clustered current status data with the gamma-frailty Cox model

The Cox model with frailties has been popular for regression analysis of clustered event time data under right censoring. However, due to the lack of reliable computation algorithms, the frailty Cox model has been rarely applied to clustered current status data, where the clustered event times are subject to a special type of interval censoring such that we only observe for each event time whether it exceeds an examination (censoring) time or not. Motivated by the cataract dataset from a cross-sectional study, where bivariate current status data were observed for the occurrence of cataracts in the right and left eyes of each study subject, we develop a very efficient and stable computation algorithm for nonparametric maximum likelihood estimation of gamma-frailty Cox models with clustered current status data. The algorithm proposed is based on a set of self-consistency equations and the contraction principle. A convenient profile-likelihood approach is proposed for variance estimation. Simulation and real data analysis exhibit the nice performance of our proposal.     

Progressively Type-II censored competing risks data from Lomax distributions

A competing risks model based on Lomax distributions is considered under progressive Type-II censoring. Maximum likelihood estimates for the distribution parameters are established. Moreover, the expected Fisher information matrix is computed and optimal Fisher information based censoring plans are discussed. In particular, it turns out that the optimal censoring scheme depends on the particular parametrization of the Lomax distributions.     

Nonparametric analysis of fingerprint data on large data sets

By executing different fingerprint-image matching algorithms on large data sets, it reveals that the match and non-match similarity scores have no specific underlying distribution function. Thus, it requires a nonparametric analysis for fingerprint-image matching algorithms on large data sets without any assumption about such irregularly discrete distribution functions. A precise receiver operating characteristic (ROC) curve based on the true accept rate (TAR) of the match similarity scores and the false accept rate (FAR) of the non-match similarity scores can be constructed. The area under such an ROC curve computed using the trapezoidal rule is equivalent to the Mann-Whitney statistic directly formed from the match and non-match similarity scores. Thereafter, the Z statistic formulated using the areas under ROC curves along with their variances and the correlation coefficient is applied to test the significance of the difference between two ROC curves. Four examples from the extensive testing of commercial fingerprint systems at the National Institute of Standards and Technology are provided. The nonparametric approach presented in this article can also be employed in the analysis of other large biometric data sets.     

Causal effects in longitudinal studies: Definition and maximum likelihood estimation

Recently, a nonparametric marginal structural model (NPMSM) approach to Causal Inference has been proposed [Neugebauer, R., van der Laan, M., 2006. Nonparametric causal effects based on marginal structural models. J. Statist. Plann. Inference (in press), 〈www http://www.sciencedirect.com/science/journal/03783758〉.] as an appealing practical alternative to the original parametric MSM (PMSM) approach introduced by Robins [Robins, J., 1998a. Marginal structural models. In: 1997 Proceedings of the American Statistical Association, American Statistical Association, Alexandria, VA, pp. 1-10]. The new MSM-based causal inference methodology generalizes the concept of causal effects: the proposed nonparametric causal effects are interpreted as summary measures of the causal effects defined with PMSMs. In addition, causal inference with NPMSM does not rely on the assumed correct specification of a parametric MSM but instead defines causal effects based on a user-specified working causal model which can be willingly misspecified. The NPMSM approach was developed for studies with point treatment data or with longitudinal data where the outcome is not time-dependent (typically collected at the end of data collection). In this paper, we generalize this approach to longitudinal studies where the outcome is time-dependent, i.e. collected throughout the span of the studies, and address the subsequent estimation inconsistency which could easily arise from a hasty generalization of the algorithm for maximum likelihood estimation. More generally, we provide an overview of the multiple causal effect representations which have been developed based on MSMs in longitudinal studies.     

Robust estimation for circular data

The problems arising when there are outliers in a data set that follow a circular distribution are considered. A robust estimation of the unknown parameters is obtained using the methods of weighted likelihood and minimum disparity, each of which is defined for a general parametric family of circular data. The class of power divergence and the related residual adjustment function is investigated in order to improve the performance of the two methods which are studied for the Von Mises (circular normal) and the Wrapped Normal distributions. The techniques are illustrated via two examples based on a real data set and a Monte Carlo study, which also enables the discussion of various computational aspects.     

The Lindley distribution applied to competing risks lifetime data

Competing risks data usually arises in studies in which the death or failure of an individual or an item may be classified into one of k ≥ 2 mutually exclusive causes. In this paper a simple competing risks distribution is proposed as a possible alternative to the Exponential or Weibull distributions usually considered in lifetime data analysis. We consider the case when the competing risks have a Lindley distribution. Also, we assume that the competing events are uncorrelated and that each subject can experience only one type of event at any particular time.     

Inference for a simple step-stress model with competing risks for failure from the exponential distribution under time constraint

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under time constraint when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. Since it is found that the MLEs do not exist when there is no failure by any particular risk factor within the specified time frame, the exact sampling distributions of the MLEs are derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions, the parametric bootstrap method, and the Bayesian posterior distribution, we discuss the construction of confidence intervals and credible intervals for the parameters. Their performance is assessed through Monte Carlo simulations and finally, we illustrate the methods of inference discussed here with an example.     

Nonparametric identification of population models via Gaussian processes

Population models are used to describe the dynamics of different subjects belonging to a population and play an important role in drug pharmacokinetics. A nonparametric identification scheme is proposed in which both the average impulse response of the population and the individual ones are modelled as Gaussian stochastic processes. Assuming that the average curve is an integrated Wiener process, it is shown that its estimate is a cubic spline. An empirical Bayes algorithm for estimating both the average and the individual curves is worked out. The model is tested on simulated data sets as well as on xenobiotics pharmacokinetic data.     

Nonparametric estimation for normalized coprime factors of a MIMO system

This paper extends to multi-input multi-output (MIMO) systems the results on frequency response estimation for normalized coprime factors (NCF) under a stochastic framework from closed-loop frequency domain experimental data. Under the condition that the covariance matrices are available for the external disturbances and measurement errors, an analytic solution has been obtained for the maximum likelihood estimate (MLE), on the basis of a linear fractional transformation (LFT) representation for all the plant possible normalized right coprime factors (NRCF). It is proved that the estimate can be expressed by a linear combination of a normalized random matrix with all its columns having independent complex normal distributions. Some methods are suggested to reduce the estimation bias when high-quality experimental data can be obtained. A numerical example is included that confirms the theoretical results.     

Efficient semiparametric estimation via Cholesky decomposition for longitudinal data

Semiparametric methods for longitudinal data with dependence within subjects have recently received considerable attention. Existing approaches that focus on modeling the mean structure require a correct specification of the covariance structure as misspecified covariance structures may lead to inefficient or biased mean parameter estimates. Besides, computation and estimation problems arise when the repeated measurements are taken at irregular and possibly subject-specific time points, the dimension of the covariance matrix is large, and the positive definiteness of the covariance matrix is required. In this article, we propose a profile kernel approach based on semiparametric partially linear regression models for the mean and model covariance structures simultaneously, motivated by the modified Cholesky decomposition. We also study the large-sample properties of the parameter estimates. The proposed method is evaluated through simulation and applied to a real dataset. Both theoretical and empirical results indicate that properly taking into account the within-subject correlation among the responses using our method can substantially improve efficiency.     

Statistical analysis of water-quality data containing multiple detection limits: S-language software for regression on order statistics

Trace contaminants in water, including metals and organics, often are measured at sufficiently low concentrations to be reported only as values below the instrument detection limit. Interpretation of these “less thans” is complicated when multiple detection limits occur. Statistical methods for multiply censored, or multiple-detection limit, datasets have been developed for medical and industrial statistics, and can be employed to estimate summary statistics or model the distributions of trace-level environmental data.We describe S-language-based software tools that perform robust linear regression on order statistics (ROS). The ROS method has been evaluated as one of the most reliable procedures for developing summary statistics of multiply censored data. It is applicable to any dataset that has 0 to 80% of its values censored. These tools are a part of a software library, or add-on package, for the R environment for statistical computing. This library can be used to generate ROS models and associated summary statistics, plot modeled distributions, and predict exceedance probabilities of water-quality standards.     

Full and conditional likelihood approaches for hazard change-point estimation with truncated and censored data

Hazard function plays an important role in reliability and survival analysis. In some real life applications, abrupt changes in the hazard function may be observed and it is of interest to detect the location and the size of the change. Hazard models with a change-point are considered when the observations are subject to random left truncation and right censoring. For a piecewise constant hazard function with a single change-point, two estimation methods based on the maximum likelihood ideas are considered. The first method assumes parametric families of distributions for the censoring and truncation variables, whereas the second one is based on conditional likelihood approaches. A simulation study is carried out to illustrate the performances of the proposed estimators. The results indicate that the fully parametric method performs better especially for estimating the size of the change, however the difference between the two methods vanish as the sample size increases. It is also observed that the full likelihood approach is not robust to model misspecification.     

SAS macros for estimation of the cumulative incidence functions based on a Cox regression model for competing risks survival data

When considering competing risks survival data, the cause specific hazard functions are often modelled by the proportional hazards Cox regression model. First, we present how to estimate the parameters in this model when some of the covariates are allowed to have exactly the same effect on several causes of failure. In many cases, the focus is not on the parameter estimates, but rather on the probability of observing a failure from a specific cause for individuals with specified covariate values. These probabilities, the cumulative incidences, are not simple functions of the parameters and they are, so far, not provided by the standard statistical software packages. We present two SAS macros: a SAS macro named CumInc for estimation of the cumulative incidences and a SAS macro named CumIncV for estimation of the cumulative incidences and the variances of the estimated cumulative incidences. The use of the macros is demonstrated through an example.     

多来源数据的概率融合方法

在数据融合过程中，针对数据来源相互独立，而数据融合目的是要对研究对象进行总体类型判别的问题，通过对数据获取过程的统计分析，给出了一种建立在极大似然思想基础上的概率融合方法，该方法较好地解决了当测量数据具有不同特征时，数据所含信息的融合问题，并且具有计算简便的特点。     

A comparative study of nonparametric estimation in Weibull regression: A penalized likelihood approach

The Weibull distribution is popularly used to model lifetime distributions in many areas of applied statistics. This paper employs a penalized likelihood method to estimate the shape parameter and an unknown regression function simultaneously in a nonparametric Weibull regression. Four methods were considered: two cross-validation methods, a corrected Akaike information criterion, and a Bayesian information criterion. Each method was evaluated based on shape parameter estimation as well as selecting the smoothing parameter in a penalized likelihood model through a simulation study. Adapting a lower-dimensional approximation and deriving confidence intervals from Bayes models of the penalized likelihood, the comparative performances of methods using both censored and uncensored data were examined for various censoring rates. The methods are applied to a real data example of lung cancer.     

Local likelihood regression in generalized linear single-index models with applications to microarray data

Searching for an effective dimension reduction space is an important problem in regression, especially for high-dimensional data such as microarray data. A major characteristic of microarray data consists in the small number of observations n and a very large number of genes p. This “large p, small n” paradigm makes the discriminant analysis for classification difficult. In order to offset this dimensionality problem a solution consists in reducing the dimension. Supervised classification is understood as a regression problem with a small number of observations and a large number of covariates. A new approach for dimension reduction is proposed. This is based on a semi-parametric approach which uses local likelihood estimates for single-index generalized linear models. The asymptotic properties of this procedure are considered and its asymptotic performances are illustrated by simulations. Applications of this method when applied to binary and multiclass classification of the three real data sets Colon, Leukemia and SRBCT are presented.     

Local polynomial estimation in partial linear regression models under dependence

A regression model whose regression function is the sum of a linear and a nonparametric component is presented. The design is random and the response and explanatory variables satisfy mixing conditions. A new local polynomial type estimator for the nonparametric component of the model is proposed and its asymptotic normality is obtained. Specifically, this estimator works on a prewhitening transformation of the dependent variable, and the results show that it is asymptotically more efficient than the conventional estimator (which works on the original dependent variable) when the errors of the model are autocorrelated. A simulation study and an application to a real data set give promising results.     

SPHERE: A contouring program for spherical data

This paper describes a method for displaying a sample of spherical data, by computing an “optimally” smoothed estimate of the underlying distribution and making a stereographic projection of the contours of this estimate. An interactive FORTRAN program which applies this method is supplied and described and examples given of its use.