Joint modeling of longitudinal proportional measurements and survival time with a cure fraction

In cancer clinical trials and other medical studies, both longitudinal measurements and data on a time to an event (survival time) are often collected from the same patients. Joint analyses of these data would improve the efficiency of the statistical inferences. We propose a new joint model for the longitudinal proportional measurements which are restricted in a finite interval and survival times with a potential cure fraction. A penalized joint likelihood is derived based on the Laplace approximation and a semiparametric procedure based on this likelihood is developed to estimate the parameters in the joint model. A simulation study is performed to evaluate the statistical properties of the proposed procedures. The proposed model is applied to data from a clinical trial on early breast cancer.     

A weighted estimator of conditional hazard rate with left-truncated and dependent data

Based on empirical likelihood method, we construct new weighted estimators of conditional density and conditional survival functions when the interest random variable is subject to random left-truncation; further, we define a plug-in weighted estimator of the conditional hazard rate. Under strong mixing assumptions, we derive asymptotic normality of the proposed estimators which permit to built a confidence interval for the conditional hazard rate. The finite sample behavior of the estimators is investigated via simulations too.     

Estimation and confidence bands of a conditional survival function with censoring indicators missing at random

The nonparametric estimator of the conditional survival function proposed by Beran is a useful tool to evaluate the effects of covariates in the presence of random right censoring. However, censoring indicators of right censored data may be missing for different reasons in many applications. We propose some estimators of the conditional cumulative hazard and survival functions which allow to handle this situation. We also construct the likelihood ratio confidence bands for them and obtain their asymptotic properties. Simulation studies are used to evaluate the performances of the estimators and their confidence bands.     

An alternative approach to nonlinear filtering

The structure of a nonlinear filter with observation process having continuous and discontinuous components is considered. The approach is based on the so-called “Bayes” formula for conditional expectations. “Fubini” type theorems for stochastic integrals are given and used to obtain the representations of an optimal estimate and of the conditional likelihood ratio. A linear unnormalized filtering equation for controlled system process is derived.     

The efficiency criteria problem for stochastic processes

This paper is concerned with the problem of finding a suitable (asymptotic) efficiency criterion for inference concerning parameters of stochastic processes. Special attention is aid to conditional exponential families of stochastic processes and to three tests based on the maximum likelihood estimate as well as to the likelihood ratio test. A contiguity calculation is used to show that a previously suggested criterion is inadequate and itself provides a partial solution to the problem. A heuristic argument is also put forward to support a proposition implying the optimality of the maximum likelihood estimate in a certain sense. Two examples which illustrate the theory are discussed.     

An analytic proof of the preservation of the up-shifted likelihood ratio order under convolutions

The closure property of the up-shifted likelihood ratio order under convolutions was first proved by Shanthikumar and Yao (Stochastic Process. Appl. 23 (1986) 259) by establishing a stochastic monotonicity property of birth–death processes. Lillo et al. (Recent Advances in Reliability Theory: Methodology, Practice, and Inference. Birkhäuser, Boston, 2000, p. 85) made a slight extension of this closure property for any random variables with interval supports by using the result of Shanthikumar and Yao. A new analytic proof of the closure property is given, and the method is applied to establish another result involving the up-shifted hazard rate and reversed hazard rate orders.     

Conditional Logistic Regression With Longitudinal Follow-up and Individual-Level Random Coefficients: A Stable and Efficient Two-Step Estimation Method

The analysis of data generated by animal habitat selection studies, by family studies of genetic diseases, or by longitudinal follow-up of households often involves fitting a mixed conditional logistic regression model to longitudinal data composed of clusters of matched case-control strata. The estimation of model parameters by maximum likelihood is especially difficult when the number of cases per stratum is greater than one. In this case, the denominator of each cluster contribution to the conditional likelihood involves a complex integral in high dimension, which leads to convergence problems in the numerical maximization. In this article we show how these computational complexities can be bypassed using a global two-step analysis for nonlinear mixed effects models. The first step estimates the cluster-specific parameters and can be achieved with standard statistical methods and software based on maximum likelihood for independent data. The second step uses the EM-algorithm in conjunction with conditional restricted maximum likelihood to estimate the population parameters. We use simulations to demonstrate that the method works well when the analysis is based on a large number of strata per cluster, as in many ecological studies. We apply the proposed two-step approach to evaluate habitat selection by pairs of bison roaming freely in their natural environment. This article has supplementary material online.     

Inference for Cox’s regression models via adjusted empirical likelihood

The Cox’s regression model is one of the most popular tools used in survival analysis. Recently, Qin and Jing (Commun Stat Simul Comput 30:79–90, 2001) applied empirical likelihood to study it with the assumption that baseline hazard function is known. However, in the Cox’s regression model the baseline hazard function is unspecified. Thus, their method suffers from severe defect. In this paper, we apply a variant of plug-in empirical likelihood by estimating the cumulative baseline hazard function. Adjusted empirical likelihood (AEL) confidence regions for the vector of regression parameters are obtained. Furthermore, we conduct a simulation study to evaluate the performance of the proposed AEL method by comparing it with normal approximation (NA) based method. The simulation studies show that both methods produce comparable coverage probabilities. The proposed AEL method outperforms the NA method based on power analysis.     

Volatility forecasting using stochastic conditional range model with leverage effect

In this paper, we propose a stochastic conditional range model with leverage effect (henceforth SCRL) for volatility forecasting. A maximum likelihood method based on the particle filters is developed to estimate the parameters of the SCRL model. Simulation results show that the proposed methodology performs well. We apply the proposed model and methodology to four stock market indices, the Shanghai Stock Exchange Composite Index of China, the Hang Seng Index of Hong Kong, the Nikkei 225 Index of Japan, and the S&P 500 Index of US. Empirical results highlight the value of incorporating leverage effect into range modeling and forecasting. In particular, the results show that our SCRL model outperforms the conditional autoregressive range model, the conditional autoregressive range model with leverage effect, and the stochastic conditional range model in both in‐sample fit and out‐of‐sample forecast.     

Sequential Order Statistics: Ageing and Stochastic Orderings

Ordered random variables play an important role in statistics, reliability theory, and many applied areas. Sequential order statistics provide a unified approach to a variety of models of ordered random variables. We investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics given some well known stochastic orderings, such as the usual stochastic, the hazard rate and the likelihood ratio orders, among others. Also, we derive sufficient conditions under which the sequential order statistics are increasing hazard rate, increasing hazard rate average or decreasing hazard rate average. Applications of the main results involving nonhomogeneous pure birth processes are also given.     

Bayesian estimation of nonlinear equilibrium models with random coefficients

In this paper, we develop a conditional likelihood based approach for estimating the equilibrium price and shares in markets with differentiated products and oligopoly supply. We model market demand using a discrete choice model with random coefficients and random utility. For most applications, the likelihood function of equilibrium prices and shares is intractable and cannot be directly analyzed. To overcome this, we develop a Markov Chain Monte Carlo simulation strategy to estimate parameters and distributions. To illustrate our methodology, we generate a dataset of prices and quantities simulated from a differentiated goods oligopoly across a number of markets. We apply our methodology to this dataset to demonstrate its attractive features as well as its accuracy and validity. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient estimation in conditional single-index regression

Semiparametric single-index regression involves an unknown finite-dimensional parameter and an unknown (link) function. We consider estimation of the parameter via the pseudo-maximum likelihood method. For this purpose we estimate the conditional density of the response given a candidate index and maximize the obtained likelihood. We show that this technique of adaptation yields an asymptotically efficient estimator: it has minimal variance among all estimators.     

Sparse Steinian Covariance Estimation

We consider a new method for sparse covariance matrix estimation which is motivated by previous results for the so-called Stein-type estimators. Stein proposed a method for regularizing the sample covariance matrix by shrinking together the eigenvalues; the amount of shrinkage is chosen to minimize an unbiased estimate of the risk (UBEOR) under the entropy loss function. The resulting estimator has been shown in simulations to yield significant risk reductions over the maximum likelihood estimator. Our method extends the UBEOR minimization problem by adding an ?₁ penalty on the entries of the estimated covariance matrix, which encourages a sparse estimate. For a multivariate Gaussian distribution, zeros in the covariance matrix correspond to marginal independences between variables. Unlike the ?₁-penalized Gaussian likelihood function, our penalized UBEOR objective is convex and can be minimized via a simple block coordinate descent procedure. We demonstrate via numerical simulations and an analysis of microarray data from breast cancer patients that our proposed method generally outperforms other methods for sparse covariance matrix estimation and can be computed efficiently even in high dimensions.     

Hazard regression with penalized spline: The smoothing parameter choice and asymptotics

In this article, we use penalized spline to estimate the hazard function from a set of censored failure time data. A new approach to estimate the amount of smoothing is provided. Under regularity conditions we establish the consistency and the asymptotic normality of the penalized likelihood estimators. Numerical studies and an example are conducted to evaluate the performances of the new procedure.     

HAZARD REGRESSION WITH PENALIZED SPLINE: THE SMOOTHING PARAMETER CHOICE AND ASYMPTOTICS

In this article, we use penalized spline to estimate the hazard function from a set of censored failure time data. A new approach to estimate the amount of smoothing is provided. Under regularity conditions we establish the consistency and the asymptotic normality of the penalized likelihood estimators. Numerical studies and an example are conducted to evaluate the performances of the new procedure.     

Standard errors for the cox proportional hazards cure model

Standard errors for the maximum likelihood estimates of the regression parameters in the logistic-proportional-hazards cure model are proposed using an approximate profile likelihood approach and a nonparametric likelihood. Two methods are given and are compared with the standard errors obtained from the inverse of the joint observed information matrix of the regression parameters and the nuisance hazard parameters. The observed information matrix is derived and is shown to be an approximation of the conditional information matrix of the regression parameters given the hazard parameters. Simulations indicate that the standard errors obtained from the inverse of the observed information matrix based on the profile likelihood and the full likelihood are comparable and appropriate. The coverage rates for the logistic regression parameter are generally good. The proportional hazards regression parameter show reasonable coverage rates under ideal conditions but lower coverage rates when the incidence proportion is low or when censoring is heavy. The three methods are applied to a data set to investigate the effects of radiation therapy on tonsil cancer.     

Multivariate Failure Times Regression with a Continuous Auxiliary Covariate

How to take advantage of the available auxiliary covariate information when the primary covariate of interest is not measured is a frequently encountered question in biomedical study. In this paper, we consider the multivariate failure times regression analysis in which the primary covariate is assessed only in a validation set, but a continuous auxiliary covariate for it is available for all subjects in the study cohort. Under the frame of marginal hazard model, we propose to estimate the induced relative risk function in the non-validation set through kernel smoothing method and then obtain an estimated pseudo-partial likelihood function. The proposed estimator which maximizes the estimated pseudo-partial likelihood is shown to be consistent and asymptotically normal. We also give an estimator of the marginal cumulative baseline hazard function. Simulation studies are conducted to evaluate the finite sample performance of our proposed estimator. The proposed method is illustrated by analyzing a heart disease data from the Study of Left Ventricular Dysfunction (SOLVD).     

Bayesian time series regression with nonparametric modeling of autocorrelation

Series models have several functions: comprehending the functional dependence of variable of interest on covariates, forecasting the dependent variable for future values of covariates and estimating variance disintegration, co-integration and steady-state relations. Although the regression function in a time series model has been extensively modeled both parametrically and nonparametrically, modeling of the error autocorrelation is mainly restricted to the parametric setup. A proper modeling of autocorrelation not only helps to reduce the bias in regression function estimate, but also enriches forecasting via a better forecast of the error term. In this article, we present a nonparametric modeling of autocorrelation function under a Bayesian framework. Moving into the frequency domain from the time domain, we introduce a Gaussian process prior to the log of the spectral density, which is then updated by using a Whittle approximation for the likelihood function (Whittle likelihood). The posterior computation is simplified due to the fact that Whittle likelihood is approximated by the likelihood of a normal mixture distribution with log-spectral density as a location shift parameter, where the mixture is of only five components with known means, variances, and mixture probabilities. The problem then becomes conjugate conditional on the mixture components, and a Gibbs sampler is used to initiate the unknown mixture components as latent variables. We present a simulation study for performance comparison, and apply our method to the two real data examples.     

Optimal Breast Biopsy Decision-Making Based on Mammographic Features and Demographic Factors

Breast cancer is the most common non-skin cancer affecting women in the United States, where every year more than 20 million mammograms are performed. Breast biopsy is commonly performed on the suspicious findings on mammograms to confirm the presence of cancer. Currently, 700,000 biopsies are performed annually in the U.S.; 55%-85% of these biopsies ultimately are found to be benign breast lesions, resulting in unnecessary treatments, patient anxiety, and expenditures. This paper addresses the decision problem faced by radiologists: When should a woman be sent for biopsy based on her mammographic features and demographic factors? This problem is formulated as a finite-horizon discrete-time Markov decision process. The optimal policy of our model shows that the decision to biopsy should take the age of patient into account; particularly, an older patient's risk threshold for biopsy should be higher than that of a younger patient. When applied to the clinical data, our model outperforms radiologists in the biopsy decision-making problem. This study also derives structural properties of the model, including sufficiency conditions that ensure the existence of a control-limit type policy and nondecreasing control-limits with age.     

On the estimation of a monotone conditional variance in nonparametric regression

A monotone estimate of the conditional variance function in a heteroscedastic, nonparametric regression model is proposed. The method is based on the application of a kernel density estimate to an unconstrained estimate of the variance function and yields an estimate of the inverse variance function. The final monotone estimate of the variance function is obtained by an inversion of this function. The method is applicable to a broad class of nonparametric estimates of the conditional variance and particularly attractive to users of conventional kernel methods, because it does not require constrained optimization techniques. The approach is also illustrated by means of a simulation study.