Using conditional logistic regression to fit proportional odds models to interval censored data

An easily implemented approach to fitting the proportional odds regression model to interval-censored data is presented. The approach is based on using conditional logistic regression routines in standard statistical packages. Using conditional logistic regression allows the practitioner to sidestep complications that attend estimation of the baseline odds ratio function. The approach is applicable both for interval-censored data in settings in which examinations continue regardless of whether the event of interest has occurred and for current status data. The methodology is illustrated through an application to data from an AIDS study of the effect of treatment with ZDV+ddC versus ZDV alone on 50% drop in CD4 cell count from baseline level. Simulations are presented to assess the accuracy of the procedure.     

Local bandwidth selection for kernel estimation of population densities with line transect sampling

Seber (1986, Biometrics 42, 267-292) suggested an approach to biological population density estimation using kernel estimates of the probability density of detection distances in line transect sampling. Chen (1996a, Applied Statistics 45, 135-150) and others have employed cross validation to choose a global bandwidth for the kernel estimator or have suggested adaptive kernel estimation (Chen, 1996b, Biometrics 52, 1283-1294). Because estimation of the density is required at only a single point, we investigate a local bandwidth selection procedure that is a modification of the method of Schucany (1995, Journal of the American Statistical Association 90, 535-540) for nonparametric regression. We report on simulation results comparing the proposed method and a local normal scale rule with cross validation and adaptive estimation. The local bandwidths and normal scale rule produce estimates with mean squares that are half the size of the others in most cases. Consistency results are also provided.     

Nonparametric estimation of the survival function for ordered multivariate failure time data: A comparative study

In longitudinal studies of disease, patients may experience several events through a follow‐up period. In these studies, the sequentially ordered events are often of interest and lead to problems that have received much attention recently. Issues of interest include the estimation of bivariate survival, marginal distributions, and the conditional distribution of gap times. In this work, we consider the estimation of the survival function conditional to a previous event. Different nonparametric approaches will be considered for estimating these quantities, all based on the Kaplan–Meier estimator of the survival function. We explore the finite sample behavior of the estimators through simulations. The different methods proposed in this article are applied to a dataset from a German Breast Cancer Study. The methods are used to obtain predictors for the conditional survival probabilities as well as to study the influence of recurrence in overall survival.     

Stochastic functional data analysis: a diffusion model-based approach

This article presents a new modeling strategy in functional data analysis. We consider the problem of estimating an unknown smooth function given functional data with noise. The unknown function is treated as the realization of a stochastic process, which is incorporated into a diffusion model. The method of smoothing spline estimation is connected to a special case of this approach. The resulting models offer great flexibility to capture the dynamic features of functional data, and allow straightforward and meaningful interpretation. The likelihood of the models is derived with Euler approximation and data augmentation. A unified Bayesian inference method is carried out via a Markov chain Monte Carlo algorithm including a simulation smoother. The proposed models and methods are illustrated on some prostate-specific antigen data, where we also show how the models can be used for forecasting.     

Smoothing spline ANOVA frailty model for recurrent event data

Gap time hazard estimation is of particular interest in recurrent event data. This article proposes a fully nonparametric approach for estimating the gap time hazard. Smoothing spline analysis of variance (ANOVA) decompositions are used to model the log gap time hazard as a joint function of gap time and covariates, and general frailty is introduced to account for between-subject heterogeneity and within-subject correlation. We estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the Newton-Raphson procedure, the stochastic approximation algorithm (SAA), and the Markov chain Monte Carlo (MCMC) method. The convergence of the algorithm is guaranteed by decreasing the step size of parameter update and/or increasing the MCMC sample size along iterations. Model selection procedure is also developed to identify negligible components in a functional ANOVA decomposition of the log gap time hazard. We evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data.     

Analyzing multiple spike trains with nonparametric granger causality

Simultaneous recordings of spike trains from multiple single neurons are becoming commonplace. Understanding the interaction patterns among these spike trains remains a key research area. A question of interest is the evaluation of information flow between neurons through the analysis of whether one spike train exerts causal influence on another. For continuous-valued time series data, Granger causality has proven an effective method for this purpose. However, the basis for Granger causality estimation is autoregressive data modeling, which is not directly applicable to spike trains. Various filtering options distort the properties of spike trains as point processes. Here we propose a new nonparametric approach to estimate Granger causality directly from the Fourier transforms of spike train data. We validate the method on synthetic spike trains generated by model networks of neurons with known connectivity patterns and then apply it to neurons simultaneously recorded from the thalamus and the primary somatosensory cortex of a squirrel monkey undergoing tactile stimulation.     

Estimating time to pregnancy from current durations in a cross-sectional sample

A new design for estimating the distribution of time to pregnancy is proposed and investigated. The design is based on recording current durations in a cross-sectional sample of women, leading to statistical problems similar to estimating renewal time distributions from backward recurrence times. Non-parametric estimation is studied in some detail and a parametric approach is indicated. The results are illustrated on Monte Carlo simulations and on data from a recent European collaborative study. The role and applicability of this approach is discussed.     

Nonparametric estimation of a Markov 'illness-death' process from interval-censored observations, with application to diabetes survival data

The nonparametric estimation of the cumulative transition intensityfunctions in a threestate time-nonhomogeneous Markov processwith irreversible transitions, an ‘illness-death’model, is considered when times of the intermediate transition,e.g. onset of a disease, are interval-censored. The times of‘death’ are assumed to be known exactly or to beright-censored. In addition the observed process may be left-truncated.Data of this type arise when the process is sampled periodically.For example, when the patients are monitored through periodicexaminations the observations on times of change in their diseasestatus will be interval-censored. Under the sampling schemeconsidered here the Nelson–Aalen estimator (Aalen, 1978)for a cumulative transition intensity is not applicable. Inthe proposed method the maximum likelihood estimators of someof the transition intensities are derived from the estimatorsof the corresponding subdistribution functions. The maximumlikelihood estimators are shown to have a self-consistency property.The self-consistency algorithm is developed for the computationof the estimators. This approach generalises the results fromTurnbull (1976) and Frydman (1992). The methods are illustratedwith diabetes survival data.     

Identifying latent clusters of variability in longitudinal data

Means or other central tendency measures are by far the most common focus of statistical analyses. However, as Carroll (2003) noted, "systematic dependence of variability on known factors" may be "fundamental to the proper solution of scientific problems" in certain settings. We develop a latent cluster model that relates underlying "clusters" of variability to baseline or outcome measures of interest. Because estimation of variability is inextricably linked to estimation of trend, assumptions about underlying trends are minimized by using nonparametric regression estimates. The resulting residual errors are then clustered into unobserved clusters of variability that are in turn related to subject-level predictors of interest. An application is made to psychological affect data.     

Subset clustering of binary sequences, with an application to genomic abnormality data

This article develops a model-based approach to clustering multivariate binary data, in which the attributes that distinguish a cluster from the rest of the population may depend on the cluster being considered. The clustering approach is based on a multivariate Dirichlet process mixture model, which allows for the estimation of the number of clusters, the cluster memberships, and the cluster-specific parameters in a unified way. Such a clustering approach has applications in the analysis of genomic abnormality data, in which the development of different types of tumors may depend on the presence of certain abnormalities at subsets of locations along the genome. Additionally, such a mixture model provides a nonparametric estimation scheme for dependent sequences of binary data.     

Multiple imputation for missing values through conditional Semiparametric odds ratio models

Multiple imputation is a practically useful approach to handling incompletely observed data in statistical analysis. Parameter estimation and inference based on imputed full data have been made easy by Rubin's rule for result combination. However, creating proper imputation that accommodates flexible models for statistical analysis in practice can be very challenging. We propose an imputation framework that uses conditional semiparametric odds ratio models to impute the missing values. The proposed imputation framework is more flexible and robust than the imputation approach based on the normal model. It is a compatible framework in comparison to the approach based on fully conditionally specified models. The proposed algorithms for multiple imputation through the Markov chain Monte Carlo sampling approach can be straightforwardly carried out. Simulation studies demonstrate that the proposed approach performs better than existing, commonly used imputation approaches. The proposed approach is applied to imputing missing values in bone fracture data.     

Locally Weighted Score Estimation for Quantile Classification in Binary Regression Models

One common use of binary response regression methods is classification based on an arbitrary probability threshold dictated by the particular application. Since this is given to us a priori, it is sensible to incorporate the threshold into our estimation procedure. Specifically, for the linear logistic model, we solve a set of locally weighted score equations, using a kernel-like weight function centered at the threshold. The bandwidth for the weight function is selected by cross validation of a novel hybrid loss function that combines classification error and a continuous measure of divergence between observed and fitted values; other possible cross-validation functions based on more common binary classification metrics are also examined. This work has much in common with robust estimation, but differs from previous approaches in this area in its focus on prediction, specifically classification into high- and low-risk groups. Simulation results are given showing the reduction in error rates that can be obtained with this method when compared with maximum likelihood estimation, especially under certain forms of model misspecification. Analysis of a melanoma dataset is presented to illustrate the use of the method in practice.     

A semiparametric approach for the nonparametric transformation survival model with multiple covariates

The nonparametric transformation model makes no parametric assumptions on the forms of the transformation function and the error distribution. This model is appealing in its flexibility for modeling censored survival data. Current approaches for estimation of the regression parameters involve maximizing discontinuous objective functions, which are numerically infeasible to implement with multiple covariates. Based on the partial rank (PR) estimator (Khan and Tamer, 2004), we propose a smoothed PR estimator which maximizes a smooth approximation of the PR objective function. The estimator is shown to be asymptotically equivalent to the PR estimator but is much easier to compute when there are multiple covariates. We further propose using the weighted bootstrap, which is more stable than the usual sandwich technique with smoothing parameters, for estimating the standard error. The estimator is evaluated via simulation studies and illustrated with the Veterans Administration lung cancer data set.     

Comparing three methods for variance estimation with duplicated high density oligonucleotide arrays

Microarray experiments are being increasingly used in molecular biology. A common task is to detect genes with differential expression across two experimental conditions, such as two different tissues or the same tissue at two time points of biological development. To take proper account of statistical variability, some statistical approaches based on the t-statistic have been proposed. In constructing the t-statistic, one needs to estimate the variance of gene expression levels. With a small number of replicated array experiments, the variance estimation can be challenging. For instance, although the sample variance is unbiased, it may have large variability, leading to a large mean squared error. For duplicated array experiments, a new approach based on simple averaging has recently been proposed in the literature. Here we consider two more general approaches based on nonparametric smoothing. Our goal is to assess the performance of each method empirically. The three methods are applied to a colon cancer data set containing 2,000 genes. Using two arrays, we compare the variance estimates obtained from the three methods. We also consider their impact on the t-statistics. Our results indicate that the three methods give variance estimates close to each other. Due to its simplicity and generality, we recommend the use of the smoothed sample variance for data with a small number of replicates. Electronic Publication     

Polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form

The Cox proportional hazards model usually assumes an exponential form for the dependence of the hazard function on covariate variables. However, in practice this assumption may be violated and other relative risk forms may be more appropriate. In this article, we consider the proportional hazards model with an unknown relative risk form. Issues in model interpretation are addressed. We propose a method to estimate the relative risk form and the regression parameters simultaneously by first approximating the logarithm of the relative risk form by a spline, and then employing the maximum partial likelihood estimation. An iterative alternating optimization procedure is developed for efficient implementation. Statistical inference of the regression coefficients and of the relative risk form based on parametric asymptotic theory is discussed. The proposed methods are illustrated using simulation and an application to the Veteran's Administration lung cancer data.     

Small area estimation for semicontinuous skewed spatial data: An application to the grape wine production in Tuscany

Linear‐mixed models are frequently used to obtain model‐based estimators in small area estimation (SAE) problems. Such models, however, are not suitable when the target variable exhibits a point mass at zero, a highly skewed distribution of the nonzero values and a strong spatial structure. In this paper, a SAE approach for dealing with such variables is suggested. We propose a two‐part random effects SAE model that includes a correlation structure on the area random effects that appears in the two parts and incorporates a bivariate smooth function of the geographical coordinates of units. To account for the skewness of the distribution of the positive values of the response variable, a Gamma model is adopted. To fit the model, to get small area estimates and to evaluate their precision, a hierarchical Bayesian approach is used. The study is motivated by a real SAE problem. We focus on estimation of the per‐farm average grape wine production in Tuscany, at subregional level, using the Farm Structure Survey data. Results from this real data application and those obtained by a model‐based simulation experiment show a satisfactory performance of the suggested SAE approach.     

Nonparametric estimation in a Markov "illness-death" process from interval censored observations with missing intermediate transition status

Summary .  In many clinical trials patients are intermittently assessed for the transition to an intermediate state, such as occurrence of a disease-related nonfatal event, and death. Estimation of the distribution of nonfatal event free survival time, that is, the time to the first occurrence of the nonfatal event or death, is the primary focus of the data analysis. The difficulty with this estimation is that the intermittent assessment of patients results in two forms of incompleteness: the times of occurrence of nonfatal events are interval censored and, when a nonfatal event does not occur by the time of the last assessment, a patient's nonfatal event status is not known from the time of the last assessment until the end of follow-up for death. We consider both forms of incompleteness within the framework of an "illness–death" model. We develop nonparametric maximum likelihood (ML) estimation in an "illness–death" model from interval-censored observations with missing status of intermediate transition. We show that the ML estimators are self-consistent and propose an algorithm for obtaining them. This work thus provides new methodology for the analysis of incomplete data that arise from clinical trials. We apply this methodology to the data from a recently reported cancer clinical trial ( Bonner et al., 2006 , New England Journal of Medicine 354, 567–578) and compare our estimation results with those obtained using a Food and Drug Administration recommended convention.     

Bayesian nonparametric centered random effects models with variable selection

In a linear mixed effects model, it is common practice to assume that the random effects follow a parametric distribution such as a normal distribution with mean zero. However, in the case of variable selection, substantial violation of the normality assumption can potentially impact the subset selection and result in poor interpretation and even incorrect results. In nonparametric random effects models, the random effects generally have a nonzero mean, which causes an identifiability problem for the fixed effects that are paired with the random effects. In this article, we focus on a Bayesian method for variable selection. We characterize the subject‐specific random effects nonparametrically with a Dirichlet process and resolve the bias simultaneously. In particular, we propose flexible modeling of the conditional distribution of the random effects with changes across the predictor space. The approach is implemented using a stochastic search Gibbs sampler to identify subsets of fixed effects and random effects to be included in the model. Simulations are provided to evaluate and compare the performance of our approach to the existing ones. We then apply the new approach to a real data example, cross‐country and interlaboratory rodent uterotrophic bioassay.