Tobit regression for modeling mean survival time using data subject to multiple sources of censoring

Mean survival time is often of inherent interest in medical and epidemiologic studies. In the presence of censoring and when covariate effects are of interest, Cox regression is the strong default, but mostly due to convenience and familiarity. When survival times are uncensored, covariate effects can be estimated as differences in mean survival through linear regression. Tobit regression can validly be performed through maximum likelihood when the censoring times are fixed (ie, known for each subject, even in cases where the outcome is observed). However, Tobit regression is generally inapplicable when the response is subject to random right censoring. We propose Tobit regression methods based on weighted maximum likelihood which are applicable to survival times subject to both fixed and random censoring times. Under the proposed approach, known right censoring is handled naturally through the Tobit model, with inverse probability of censoring weighting used to overcome random censoring. Essentially, the re‐weighting data are intended to represent those that would have been observed in the absence of random censoring. We develop methods for estimating the Tobit regression parameter, then the population mean survival time. A closed form large‐sample variance estimator is proposed for the regression parameter estimator, with a semiparametric bootstrap standard error estimator derived for the population mean. The proposed methods are easily implementable using standard software. Finite‐sample properties are assessed through simulation. The methods are applied to a large cohort of patients wait‐listed for kidney transplantation.     

Regression analysis of case-cohort studies in the presence of dependent interval censoring

The case-cohort design is widely used as a means of reducing the cost in large cohort studies, especially when the disease rate is low and covariate measurements may be expensive, and has been discussed by many authors. In this paper, we discuss regression analysis of case-cohort studies that produce interval-censored failure time with dependent censoring, a situation for which there does not seem to exist an established approach. For inference, a sieve inverse probability weighting estimation procedure is developed with the use of Bernstein polynomials to approximate the unknown baseline cumulative hazard functions. The proposed estimators are shown to be consistent and the asymptotic normality of the resulting regression parameter estimators is established. A simulation study is conducted to assess the finite sample properties of the proposed approach and indicates that it works well in practical situations. The proposed method is applied to an HIV/AIDS case-cohort study that motivated this investigation.     

Semiparametric regression for restricted mean residual life under right censoring

A mean residual life function (MRLF) is the remaining life expectancy of a subject who has survived to a certain time point. In the presence of covariates, regression models are needed to study the association between the MRLFs and covariates. If the survival time tends to be too long or the tail is not observed, the restricted mean residual life must be considered. In this paper, we propose the proportional restricted mean residual life model for fitting survival data under right censoring. For inference on the model parameters, martingale estimating equations are developed, and the asymptotic properties of the proposed estimators are established. In addition, a class of goodness-of-fit test is presented to assess the adequacy of the model. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and the approach is applied to a set of real life data collected from a randomized clinical trial.     

Semiparametric lack-of-fit tests in an additive hazard regression model

In the semiparametric additive hazard regression model of McKeague and Sasieni (Biometrika 81: 501–514), the hazard contributions of some covariates are allowed to change over time, without parametric restrictions (Aalen model), while the contributions of other covariates are assumed to be constant. In this paper, we develop tests that help to decide which of the covariate contributions indeed change over time. The remaining covariates may be modelled with constant hazard coefficients, thus reducing the number of curves that have to be estimated nonparametrically. Several bootstrap tests are proposed. The behavior of the tests is investigated in a simulation study. In a practical example, the tests consistently identify covariates with constant and with changing hazard contributions.     

Correcting the bias due to dependent censoring of the survival estimator by conditioning

We consider the conditional estimation of the survival function of the time T₂ to a second event as a function of the time T₁ to a first event when there is a censoring mechanism acting on their sum T₁+T₂. The problem has been motivated by a treatment interruption study aimed at improving the quality of life of HIV-infected patients. We base the analysis on the survival function of T₂ given that T₁∈I, where I represents a period of scientific interest (1 trimester, 1 year, 2 years, etc.) and propose a non-parametric estimator for the survival function of T₂ given that T₁∈I, which takes into account both the selection bias and the heterogeneity due to the dependent censoring. The proposed estimator for the survival function uses the risk group of T₂ conditioned on the categories of T₁ and corrects for the dependent censoring using weights defined by the observed values of T₁. The estimator, properly normalized, converges weakly to a zero-mean Gaussian process. We estimate the variance of the limiting process via a bootstrap methodology. Properties of the proposed estimator are illustrated by an extensive simulation study. The motivating data set is analysed by means of this new methodology.     

Estimation of the marginal survival time in the presence of dependent competing risks using inverse probability of censoring weighted (IPCW) methods

In medical studies, there is interest in inferring the marginal distribution of a survival time subject to competing risks. The Kyushu Lipid Intervention Study (KLIS) was a clinical study for hypercholesterolemia, where pravastatin treatment was compared with conventional treatment. The primary endpoint was time to events of coronary heart disease (CHD). In this study, however, some subjects died from causes other than CHD or were censored due to loss to follow-up. Because the treatments were targeted to reduce CHD events, the investigators were interested in the effect of the treatment on CHD events in the absence of causes of death or events other than CHD. In this paper, we present a method for estimating treatment group-specific marginal survival curves of time-to-event data in the presence of dependent competing risks. The proposed method is a straightforward extension of the Inverse Probability of Censoring Weighted (IPCW) method to settings with more than one reason for censoring. The results of our analysis showed that the IPCW marginal incidence for CHD was almost the same as the lower bound for which subjects with competing events were assumed to be censored at the end of all follow-up. This result provided reassurance that the results in KLIS were robust to competing risks.     

Maximum likelihood methods for complex sample data: logistic regression and discrete proportional hazards models

To estimate model parameters from complex sample data. we apply maximum likelihood techniques to the complex sample data from the finite population, which is treated as a sample from an i nfinite superpopulation. General asymptotic distribution theory is developed and then applied to both logistic regression and discrete proportional hazards models. Data from the Lipid Research Clinics Program areused to illustrate each model, demonstrating the effects on inference of neglecting the sampling design during parameter estimation. These empirical results also shed light on the issue of model-based vs. design-based inferences.     

The analysis of transition times in multistate stochastic processes using proportional hazard regression models

Extensions to Cox's proportional hazards regression model (Cox, 1972) for the analysis of survival data are considered for a more general multistate framework. This framework allows several transient disease states between initial entry state and death as well as incorporating possible competing causes of death. Methods for parameter and function estimation within this extension are presented and applied to the analysis of data from the Stanford Heart Transplantation Program (Crowley and Hu,1977).     

Semiparametric Efficient Estimation in the Generalized Odds-Rate Class of Regression Models for Right-Censored Time-to-Event Data

The generalized odds-rate class of regression models for time to event data is indexed by a non-negative constant and assumes thatg(S(t|Z)) = (t) + Zwhere g(s) = log(^-1(s-) for > 0, g₀(s) = log(- log s), S(t|Z) is the survival function of the time to event for an individual with qx1 covariate vector Z, is a qx1 vector of unknown regression parameters, and (t) is some arbitrary increasing function of t. When =0, this model is equivalent to the proportional hazards model and when =1, this model reduces to the proportional odds model. In the presence of right censoring, we construct estimators for and exp((t)) and show that they are consistent and asymptotically normal. In addition, we show that the estimator for is semiparametric efficient in the sense that it attains the semiparametric variance bound.     

On a measure of information gain for regression models in survival analysis

Papers dealing with measures of predictive power in survival analysis have seen their independence of censoring, or their estimates being unbiased under censoring, as the most important property. We argue that this property has been wrongly understood. Discussing the so-called measure of information gain, we point out that we cannot have unbiased estimates if all values, greater than a given time τ, are censored. This is due to the fact that censoring before τ has a different effect than censoring after τ. Such τ is often introduced by design of a study. Independence can only be achieved under the assumption of the model being valid after τ, which is impossible to verify. But if one is willing to make such an assumption, we suggest using multiple imputation to obtain a consistent estimate. We further show that censoring has different effects on the estimation of the measure for the Cox model than for parametric models, and we discuss them separately. We also give some warnings about the usage of the measure, especially when it comes to comparing essentially different models.     

Nonparametric test for checking lack of fit of the quantité regression model under random censoring

The author proposes a nonparametric test for checking the lack of fit of the quantile function of survival time given the covariates; she assumes that survival time is subjected to random right censoring. Her test statistic is a kemel‐based smoothing estimator of a moment condition. The test statistic is asymptotically Gaussian under the null hypothesis. The author investigates its behavior under local alternative sequences. She assesses its finite‐sample power through simulations and illustrates its use with the Stanford heart transplant data.     

Statistical power to detect violation of the proportional hazards assumption when using the Cox regression model

The use of the Cox proportional hazards regression model is wide-spread. A key assumption of the model is that of proportional hazards. Analysts frequently test the validity of this assumption using statistical significance testing. However, the statistical power of such assessments is frequently unknown. We used Monte Carlo simulations to estimate the statistical power of two different methods for detecting violations of this assumption. When the covariate was binary, we found that a model-based method had greater power than a method based on cumulative sums of martingale residuals. Furthermore, the parametric nature of the distribution of event times had an impact on power when the covariate was binary. Statistical power to detect a strong violation of the proportional hazards assumption was low to moderate even when the number of observed events was high. In many data sets, power to detect a violation of this assumption is likely to be low to modest.     

ANOVA decomposition of conditional Gaussian processes for sensitivity analysis with dependent inputs

Complex computer codes are widely used in science to model physical systems. Sensitivity analysis aims to measure the contributions of the inputs on the code output variability. An efficient tool to perform such analysis is the variance-based methods which have been recently investigated in the framework of dependent inputs. One of their issue is that they require a large number of runs for the complex simulators. To handle it, a Gaussian process (GP) regression model may be used to approximate the complex code. In this work, we propose to decompose a GP into a high-dimensional representation. This leads to the definition of a variance-based sensitivity measure well tailored for non-independent inputs. We give a methodology to estimate these indices and to quantify their uncertainty. Finally, the approach is illustrated on toy functions and on a river flood model.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.     

Complete convergence for weighted sums of END random variables and its application to nonparametric regression models

In this article, the complete convergence for weighted sums of extended negatively dependent (END, for short) random variables is investigated. Some sufficient conditions for the complete convergence are provided. In addition, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of END random variables is obtained. The results obtained in the article generalise and improve the corresponding one of Wang et al. [(2014b), ‘On Complete Convergence for an Extended Negatively Dependent Sequence’, Communications in Statistics-Theory and Methods, 43, 2923–2937]. As an application, the complete consistency for the estimator of nonparametric regression model is established.     

Almost sure convergence for weighted sums of WNOD random variables and its applications to non parametric regression models

In this article, some results on almost sure convergence for weighted sums of widely negative orthant dependent (WNOD) random variables are presented. The results obtained in the article generalize and improve the corresponding one of J. Lita Da Silva. [(2015), “Almost sure convergence for weighted sums of extended negatively dependent random variables.” Acta Math. Hungar. 146 (1), 56–70]. As applications, the strong convergence for the estimator of non parametric regression model are established.