Censored Responses Research Articles

Skew‐normal/independent linear mixed models for censored responses with applications to HIV viral loads

Often in biomedical studies, the routine use of linear mixed-effects models (based on Gaussian assumptions) can be questionable when the longitudinal responses are skewed in nature. Skew-normal/elliptical models are widely used in those situations. Often, those skewed responses might also be subjected to some upper and lower quantification limits (QLs; viz., longitudinal viral-load measures in HIV studies), beyond which they are not measurable. In this paper, we develop a Bayesian analysis of censored linear mixed models replacing the Gaussian assumptions with skew-normal/independent (SNI) distributions. The SNI is an attractive class of asymmetric heavy-tailed distributions that includes the skew-normal, skew-t, skew-slash, and skew-contaminated normal distributions as special cases. The proposed model provides flexibility in capturing the effects of skewness and heavy tail for responses that are either left- or right-censored. For our analysis, we adopt a Bayesian framework and develop a Markov chain Monte Carlo algorithm to carry out the posterior analyses. The marginal likelihood is tractable, and utilized to compute not only some Bayesian model selection measures but also case-deletion influence diagnostics based on the Kullback-Leibler divergence. The newly developed procedures are illustrated with a simulation study as well as an HIV case study involving analysis of longitudinal viral loads.

VARIABLE SELECTION IN PARTLY LINEAR REGRESSION MODEL WITH DIVERGING DIMENSIONS FOR RIGHT CENSORED DATA.

Recent biomedical studies often measure two distinct sets of risk factors: low-dimensional clinical and environmental measurements, and high-dimensional gene expression measurements. For prognosis studies with right censored response variables, we propose a semiparametric regression model whose covariate effects have two parts: a nonparametric part for low-dimensional covariates, and a parametric part for high-dimensional covariates. A penalized variable selection approach is developed. The selection of parametric covariate effects is achieved using an iterated Lasso approach, for which we prove the selection consistency property. The nonparametric component is estimated using a sieve approach. An empirical model selection tool for the nonparametric component is derived based on the Kullback-Leibler geometry. Numerical studies show that the proposed approach has satisfactory performance. Application to a lymphoma study illustrates the proposed method.

Bias corrected estimation for generalized probit regression with covariate measurement error and censored responses

In this paper, we propose a bias corrected estimate of the regression coefficient for the generalized probit regression model when the covariates are subject to measurement error and the responses are subject to interval censoring. The main improvement of our method is that it reduces most of the bias that the naive estimates have. The great advantage of our method is that it is baseline and censoring distribution free, in a sense that the investigator does not need to calculate the baseline or the censoring distribution to obtain the estimator of the regression coefficient, an important property of Cox regression model. A sandwich estimator for the variance is also proposed. Our procedure can be generalized to general measurement error distribution as long as the first four moments of the measurement error are known. The results of extensive simulations show that our approach is very effective in eliminating the bias when the measurement error is not too large relative to the error term of the regression model.

Quantile regression with doubly censored data

Quantile regression offers a semiparametric approach to modeling data with possible heterogeneity. It is particularly attractive for censored responses, where the conditional mean functions are unidentifiable without parametric assumptions on the distributions. A new algorithm is proposed to estimate the regression quantile process when the response variable is subject to double censoring. The algorithm distributes the probability mass of each censored point to its left or right appropriately, and iterates towards self-consistent solutions. Numerical results on simulated data and an unemployment duration study are given to demonstrate the merits of the proposed method.

Weighted estimation of single index models with right censored responses

In this paper, the unknown link function, the direction parameter, and the heteroscedastic variance in single index models are estimated by the random weight method under the random censorship, respectively. The central limit theory and the convergence rate of the law of the iterated logarithm for the estimator of the direction parameter are derived, respectively. The optimal convergence rates for the estimators of the link function and the heteroscedastic variance are obtained. Simulation results support the theoretical results of the paper.

Interrelated labor decisions of farm couples: a censored response analysis of off‐farm work

AbstractFarm couples' labor market responses are partly the discrete choice of entering the off‐farm labor market and partly the continuous choice of off‐farm working hours, given entry. Such a setting is interesting when examining the increasing occurrence of multiple job‐holdings among farmers in Western economies. Most existing analyses of off‐farm labor supply only model the decisions of the farmer, not the joint decisions of the farm couple. This article presents a framework for handling such interrelated discrete/continuous choices, involving also farm production and household consumption. The derived two‐equation sub‐model for husband/wife's censored labor responses is estimated from a 10‐year Norwegian panel data for 342 farms. The results agree to some extent with earlier studies, but are more informative because of the longer panels—which allows a more extensive examination of latent heterogeneity and behavioral persistence—because it provides cross‐effects in the spouses' labor supplies. The results show some interesting differences between how the independent variables influence the labor supply of operator and spouse. This is most evident for the cross‐effects of education, children, and wage rate. Overall, the results strongly support applying a panel‐censoring model that accounts for latent heterogeneity in this context.

Low dimensional semiparametric estimation in a censored regression model

A new estimation procedure for a partial linear additive model with censored responses is proposed. To this aim, ideas of Lewbel and Linton [A. Lewbel, O. Linton, Nonparametric censored and truncated regression, Econometrica 70 (2002) 765–779] on censored model regression are combined with those of Kim et al. [W. Kim, O. Linton, N.W. Hengartner, A computationally efficient estimator for additive nonparametric regression with bootstrap confidence intervals, Journal of Computational and Graphical Statistics, 8 (1999) 278–297] on marginal integration and those on average derivatives. This allows for dimension reduction, interpretability and — depending on the context — for weights yielding computationally attractive estimates. Asymptotic behavior is provided for all proposed estimators.

Nonparametric Estimation of the Multivariate Distribution Function in a Censored Regression Model with Applications

In a regression model with univariate censored responses, a new estimator of the joint distribution function of the covariates and response is proposed, under the assumption that the response and the censoring variable are independent conditionally to the covariates. This estimator is based on the conditional Kaplan–Meier estimator of Beran (1981), and happens to be an extension of the multivariate empirical distribution function used in the uncensored case. We derive asymptotic i.i.d. representations for the integrals with respect to the measure defined by this estimated distribution function. These representations hold even in the case where the covariates are multidimensional under some additional assumption on the censoring. Applications to censored regression and to density estimation are considered.

Non-Parametric Tests for Distributional Treatment Effect for Randomly Censored Responses

SummaryFor a binary treatment ν = 0, 1 and the corresponding ‘potential response’Y0 for the control group (ν = 0) and Y1 for the treatment group (ν = 1), one definition of no treatment effect is that Y0 and Y1 follow the same distribution given a covariate vector X. Koul and Schick have provided a non-parametric test for no distributional effect when the realized response (1−ν)Y0+νY1 is fully observed and the distribution of X is the same across the two groups. This test is thus not applicable to censored responses, nor to non-experimental (i.e. observational) studies that entail different distributions of X across the two groups. We propose ‘X-matched’ non-parametric tests generalizing the test of Koul and Schick following an idea of Gehan. Our tests are applicable to non-experimental data with randomly censored responses. In addition to these motivations, the tests have several advantages. First, they have the intuitive appeal of comparing all available pairs across the treatment and control groups, instead of selecting a number of matched controls (or treated) in the usual pair or multiple matching. Second, whereas most matching estimators or tests have a non-overlapping support (of X) problem across the two groups, our tests have a built-in protection against the problem. Third, Gehan’s idea allows the tests to make good use of censored observations. A simulation study is conducted, and an empirical illustration for a job training effect on the duration of unemployment is provided.

Penalized Estimating Functions and Variable Selection in Semiparametric Regression Models

We propose a general strategy for variable selection in semiparametric regression models by penalizing appropriate estimating functions. Important applications include semiparametric linear regression with censored responses and semiparametric regression with missing predictors. Unlike the existing penalized maximum likelihood estimators, the proposed penalized estimating functions may not pertain to the derivatives of any objective functions and may be discrete in the regression coefficients. We establish a general asymptotic theory for penalized estimating functions and present suitable numerical algorithms to implement the proposed estimators. In addition, we develop a resampling technique to estimate the variances of the estimated regression coefficients when the asymptotic variances cannot be evaluated directly. Simulation studies demonstrate that the proposed methods perform well in variable selection and variance estimation. We illustrate our methods using data from the Paul Coverdell Stroke Registry.

Efficient Estimation for the Accelerated Failure Time Model

The accelerated failure time model provides a natural formulation of the effects of covariates on potentially censored response variable. The existing semiparametric estimators are computationally intractable and statistically inefficient. In this article we propose an approximate nonparametric maximum likelihood method for the accelerated failure time model with possibly time-dependent covariates. We estimate the regression parameters by maximizing a kernel-smoothed profile likelihood function. The maximization can be achieved through conventional gradient-based search algorithms. The resulting estimators are consistent and asymptotically normal. The limiting covariance matrix attains the semiparametric efficiency bound and can be consistently estimated. We also provide a consistent estimator for the error distribution. Extensive simulation studies demonstrate that the asymptotic approximations are accurate in practical situations and the new estimators are considerably more efficient than the existing ones. Illustrations with clinical and epidemiologic studies are provided.

Imputation of Censored Response Data in a Bivariate Designed Experiment

ABSTRACT Imputation of censored response data in bivariate experiments with correlated responses has not been adequately addressed in the current literature. In this article, we show several ways of handling and imputing censored response data in bivariate designed experiments. This article introduces an imputation method for censored response data in designed experiments using the conditional expectation of the relevant truncated distribution. The method takes into consideration the dependency as well as the multivariate nature of the response variables. A numerical example is presented. Variations of the method are presented and imputed values are compared with each other and the original data.

Empirical Likelihood in Partial Linear Error-in-Covariable Model with Censored Data

This article considers the estimation issue in a partial linear model with error-in-covariable and censored response. The proposed estimator of the parametric function in the model has been proved to be asymptotically normal in the literature. In this article, we shall apply the empirical likelihood method into the censored partial linear model with error-in-covariable. In particular, an empirical log-likelihood ratio of β is proposed and shown to have a limiting distribution of a weighted sum of independent chi-square distribution, which can be used to construct an approximate confidence region for β. Some simulation studies are conducted to compare the empirical likelihood method and normal approximation-based method, which indicate that the former is better.

Bivariate Location–Scale Models for Regression Analysis, with Applications to Lifetime Data

SummaryThe literature on multivariate linear regression includes multivariate normal models, models that are used in survival analysis and a variety of models that are used in other areas such as econometrics. The paper considers the class of location–scale models, which includes a large proportion of the preceding models. It is shown that, for complete data, the maximum likelihood estimators for regression coefficients in a linear location–scale framework are consistent even when the joint distribution is misspecified. In addition, gains in efficiency arising from the use of a bivariate model, as opposed to separate univariate models, are studied. A major area of application for multivariate regression models is to clustered, ‘parallel’ lifetime data, so we also study the case of censored responses. Estimators of regression coefficients are no longer consistent under model misspecification, but we give simulation results that show that the bias is small in many practical situations. Gains in efficiency from bivariate models are also examined in the censored data setting. The methodology in the paper is illustrated by using lifetime data from the Diabetic Retinopathy Study.

Maximally Selected Rank Statistics for Dose-Response Problems

We consider the bivariate situation of some quantitative, ordinal, binary or censored response variable and some quantitative or ordinal exposure variable (dose) with a hypothetical effect on the response. Data can either be the outcome of a planned dose-response experiment with only few dose levels or of an observational study where, for example, both exposure and response variable are observed within each individual. We are interested in testing the null hypothesis of no effect of the dose variable vs. a dose-response function depending on an unknown ‘threshold’ parameter. The variety of dose-response functions considered ranges from no observed effect level (NOEL) models to umbrella alternatives. Here we discuss generalizations of the method of Lausen & Schumacher (Biometrics, 1992, 48, 73–85)which are based on combinations of two-sample rank statistics and rank statistics for trend. Our approach may be seen as a generalization of a proposal for change-point problems. Using the approach of Davies (Biometrika, 1987, 74, 33–43)we derive and approximate the asymptotic null distribution for a large number of thresholds considered. We use an improved Bonferroni inequality as approximation for a small number of thresholds considered. Moreover, we analyse the small sample behaviour by means of a Monte-Carlo study. Our paper is illustrated by examples from clinical research and epidemiology.

ESTIMATION IN PARTLY LINEAR ERROR-IN-COVARIABLE MODELS WITH CENSORED DATA

This paper studies the estimation issues in a partly linear model with error-in-covariable and censored response. The proposed estimators of the parameter and nonparametric function in the model are respectively proved to be asymptotically normal and weak consistent with an optimal rate. A simulation study is conducted to compare the proposed estimator of the parameter with its related estimators in terms of their biases and standard deviations.

Added variable plots for linear regression with censored data

In linear regression the structure of the hat matrix plays an important part in regression diagnostics. In this note we investigate the properties of the hat matrix for regression with censored responses in the presence of one or more explanatory variables observed without censoring. The censored points in the scatterplot are renovated to positions had they been observed without censoring in a renovation process based on Buckley-James censored regression estimators. This allows natural links to be established with the structure of ordinary least squares estimators. In particular, we show that the renovated hat matrix may be partitioned in a manner which assists in deciding whether further explanatory variables should be added to the linear model. The added variable plot for regression with censored data is developed as a diagnostic tool for this decision process.

Permutation Tests Using Estimated Distribution Functions

In this article we develop permutation tests for estimated distribution functions. The tests are formed by averaging a functional of estimated distribution functions that are calculated from independent sampling units, where the units may be a single response, a set of repeated responses, or a censored response. We study primarily two functionals—the difference in means functional and the Mann-Whitney functional, and two types of responses—repeated conditionally independent responses and censored responses. For repeated responses, the permutation test using the difference in means functional produces a permutation form of the corresponding mixed-effects test. A new permutation test is developed when we apply the Mann-Whitney functional to the repeated responses. This is a case in which the rank-transform method does not work. On the other hand, for right-censored or interval-censored data, we obtain permutation forms of standard rank tests using the Mann-Whitney functional (or weighted forms of the functional), and the difference in means functional gives new tests. The latter tests generalize the permutation t-test and the mean-based permutation tests to censored data. These permutation tests are valid for all sample sizes and do not need weights for stabilization like the weighted Kaplan-Meier statistics. We perform the permutation tests on two examples, one with repeated measures and one with interval-censored responses.

Permutation Tests Using Estimated Distribution Functions

Abstract In this article we develop permutation tests for estimated distribution functions. The tests are formed by averaging a functional of estimated distribution functions that are calculated from independent sampling units, where the units may be a single response, a set of repeated responses, or a censored response. We study primarily two functionals—the difference in means functional and the Mann-Whitney functional, and two types of responses—repeated conditionally independent responses and censored responses. For repeated responses, the permutation test using the difference in means functional produces a permutation form of the corresponding mixed-effects test. A new permutation test is developed when we apply the Mann-Whitney functional to the repeated responses. This is a case in which the rank-transform method does not work. On the other hand, for right-censored or interval-censored data, we obtain permutation forms of standard rank tests using the Mann-Whitney functional (or weighted forms of the functional), and the difference in means functional gives new tests. The latter tests generalize the permutation t-test and the mean-based permutation tests to censored data. These permutation tests are valid for all sample sizes and do not need weights for stabilization like the weighted Kaplan-Meier statistics. We perform the permutation tests on two examples, one with repeated measures and one with interval-censored responses.

Linear regression with doubly censored data

Linear regression with doubly censored responses is considered. Buckley-James-Ritov-type estimators are proposed. Semiparametric information and projective scores are discussed. An expansion of the estimating equations is obtained under fairly general assumptions. Sufficient conditions are given for the asymptotic consistency and normality of the estimators.