Censored Outcomes Research Articles

Assessing heterogeneity in surrogacy using censored data.

Determining whether a surrogate marker can be used to replace a primary outcome in a clinical study is complex. While many statistical methods have been developed to formally evaluate a surrogate marker, they generally do not provide a way to examine heterogeneity in the utility of a surrogate marker. Similar to treatment effect heterogeneity, where the effect of a treatment varies based on a patient characteristic, heterogeneity in surrogacy means that the strength or utility of the surrogate marker varies based on a patient characteristic. The few methods that have been recently developed to examine such heterogeneity cannot accommodate censored data. Studies with a censored outcome are typically the studies that could most benefit from a surrogate because the follow-up time is often long. In this paper, we develop a robust nonparametric approach to assess heterogeneity in the utility of a surrogate marker with respect to a baseline variable in a censored time-to-event outcome setting. In addition, we propose and evaluate a testing procedure to formally test for heterogeneity at a single time point or across multiple time points simultaneously. Finite sample performance of our estimation and testing procedure are examined in a simulation study. We use our proposed method to investigate the complex relationship between change in fasting plasma glucose, diabetes, and sex hormones using data from the diabetes prevention program study.

Type I Tobit Bayesian Additive Regression Trees for censored outcome regression

Censoring occurs when an outcome is unobserved beyond some threshold value. Methods that do not account for censoring produce biased predictions of the unobserved outcome. This paper introduces Type I Tobit Bayesian Additive Regression Tree (TOBART-1) models for censored outcomes. Simulation results and real data applications demonstrate that TOBART-1 produces accurate predictions of censored outcomes. TOBART-1 provides posterior intervals for the conditional expectation and other quantities of interest. The error term distribution can have a large impact on the expectation of the censored outcome. Therefore, the error is flexibly modeled as a Dirichlet process mixture of normal distributions. An R package is available at https://github.com/EoghanONeill/TobitBART.

Causal effect estimation with censored outcome and covariate selection

We estimate the causal effect in the presence of censored outcome and high-dimensional covariates. To improve the efficiency of the estimation of average causal effect, we propose the censored outcome adaptive Lasso (COAL) to select covariates.

Discussion on "Instrumental variable estimation of the causal hazard ratio," by Linbo Wang, Eric Tchetgen Tchetgen, Torben Martinussen, Stijn Vansteelandt.

Wang, etal.'s estimator for causal hazard ratios for endogenous treatments makes an important addition to a researcher's toolkit for analyzing censored duration outcomes. Their method complements existing methods in the semiparametric treatment effects literature and suggests useful avenues for future research.

Two-Stage Least Squares with a Randomly Right-Censored Outcome

This note develops a simple two-stage least squares (2SLS) procedure to estimate the causal effect of some endogenous regressors on a randomly right censored outcome in the linear model. The proposal replaces the usual ordinary least squares regressions of the standard 2SLS by weighted least squares regressions. The weights correspond to the inverse probability of censoring. We show consistency and asymptotic normality of the estimator. The estimator exhibits good finite sample performances in simulations.

Model free feature screening for ultrahigh dimensional covariates with right censored outcomes

This paper is concerned with feature screening for the ultrahigh dimensional survival data. We propose a new feature screening procedure by extending the method of Zhu et al. via inverse probability censoring weighting. The proposed procedure enjoys two appealing merits. First, it does not need to specify any model assumption between the response and the covariates. Thus, it is robust to the model mis-specification. Second, our procedure is robust in the presence of outliers or extreme values since it only uses the rank of censored outcomes. We establish the sure screening property under some regular conditions. The simulations and analysis of the real data demonstrate that our procedure exhibits favorably in comparison with the existing competitors.

Approximate Bayesian Bootstrap procedures to estimate multilevel treatment effects in observational studies with application to type 2 diabetes treatment regimens.

Randomized clinical trials are considered as the gold standard for estimating causal effects. Nevertheless, in studies that are aimed at examining adverse effects of interventions, randomized trials are often impractical because of ethical and financial considerations. In observational studies, matching on the generalized propensity scores was proposed as a possible solution to estimate the treatment effects of multiple interventions. However, the derivation of point and interval estimates for these matching procedures can become complex with non-continuous or censored outcomes. We propose a novel Approximate Bayesian Bootstrap algorithm that results in statistically valid point and interval estimates of the treatment effects with categorical outcomes. The procedure relies on the estimated generalized propensity scores and multiply imputes the unobserved potential outcomes for each unit. In addition, we describe a corresponding interpretable sensitivity analysis to examine the unconfoundedness assumption. We apply this approach to examine the cardiovascular safety of common, real-world anti-diabetic treatment regimens for type 2 diabetes mellitus in a large observational database.

342-OR: Optimal Ages for Screening for T1D Risk in Children

Screening for pre-symptomatic T1D can significantly reduce the risk of DKA at diagnosis. Screening can also identify children who may benefit from interventions to delay/prevent progression from islet autoimmunity (IA) to diabetes. To help design a rational screening schedule, we analyzed data to determine the optimal ages to screen children for IA. Longitudinal data sets from birth cohort studies (DIPP, BABYDIAB, DiPiS, and DAISY) were harmonized for analyses. We analyzed the age of screening for multiple autoantibodies to insulin, GAD and IA-2 to maximize the sensitivity and positive predictive value (PPV) to predict development of T1D by age 15. Inverse probability censoring weighting was used to account for right censored outcomes. Cumulative sensitivity, dynamic specificity, positive and negative predictive values were used to describe the models. In the combined cohorts (n=6061), a one-time screening for multiple autoantibodies achieved a maximum sensitivity of 35% and a PPV of 70-80% when applied at age 3-5 yr. To increase the sensitivity above 50% and retain PPV >70%, two screenings should occur: at 2-3 yr and at 5-6 yr (Fig). Public health screening efforts to identify children progressing to T1D must make compromises based on feasibility, cost, optimal metrics. Our data represent a starting point for these considerations that should be customized based on the population’s underlying disease characteristics and public health infrastructure. Disclosure M. Ghalwash: None. V. Anand: None. W. Hagopian: Consultant; Self; Novo Nordisk Inc. M. Lundgren: None. M. Rewers: None. R. Veijola: None. A. Ziegler: None. J.L. Dunne: None.

Maximum likelihood estimation with missing outcomes: From simplicity to complexity.

Many clinical or prevention studies involve missing or censored outcomes. Maximum likelihood (ML) methods provide a conceptually straightforward approach to estimation when the outcome is partially missing. Methods of implementing ML methods range from the simple to the complex, depending on the type of data and the missing-data mechanism. Simple ML methods for ignorable missing-data mechanisms (when data are missing at random) include complete-case analysis, complete-case analysis with covariate adjustment, survival analysis with covariate adjustment, and analysis via propensity-to-be-missing scores. More complex ML methods for ignorable missing-data mechanisms include the analysis of longitudinal dropouts via a marginal model for continuous data or a conditional model for categorical data. A moderately complex ML method for categorical data with a saturated model and either ignorable or nonignorable missing-data mechanisms is a perfect fit analysis, an algebraic method involving closed-form estimates and variances. A complex and flexible ML method with categorical data and either ignorable or nonignorable missing-data mechanisms is the method of composite linear models, a matrix method requiring specialized software. Except for the method of composite linear models, which can involve challenging matrix specifications, the implementation of these ML methods ranges in difficulty from easy to moderate.

Quantile regression for survival data in modern cancer research: expanding statistical tools for precision medicine.

Quantile regression links the whole distribution of an outcome to the covariates of interest and has become an important alternative to commonly used regression models. However, the presence of censored data such as survival time, often the main endpoint in cancer studies, has hampered the use of quantile regression techniques because of the incompleteness of data. With the advent of the precision medicine era and availability of high throughput data, quantile regression with high-dimensional predictors has attracted much attention and provided added insight compared to traditional regression approaches. This paper provides a practical guide for using quantile regression for right censored outcome data with covariates of low- or high-dimensionality. We frame our discussion using a dataset from the Boston Lung Cancer Survivor Cohort, a hospital-based prospective cohort study, with the goals of broadening the scope of cancer research, maximizing the utility of collected data, and offering useful statistical alternatives. We use quantile regression to identify clinical and molecular predictors, for example CpG methylation sites, associated with high-risk lung cancer patients, for example those with short survival.

Adjustments of multi-sample [formula omitted]-statistics to right censored data and confounding covariates

U-statistics that can be used for comparing distribution of outcomes in two groups are considered. Adjustments to the classical U-statistics are proposed for overcoming potential biases arising from right-censoring of the outcomes and presence of confounding covariates. These newly proposed U-statistics are appropriate when, in addition to right censored outcomes, some fixed covariates are observed and deemed as confounders in an observational study. The summands of U-statistics are re-weighted and normalized based on a combination of inverse probability of censoring weights and propensity score based weights. Censoring times may depend on the group membership, confounders or some potentially observed time-dependent covariates, which may result in censoring mechanisms of varying degrees of complexity. In total, four censoring mechanisms are considered for the two-group comparison. Simulation results are used to illustrate the impact of right-censoring and confounding covariates on the performance of the newly proposed U-statistics under different censoring mechanisms. It is also demonstrated that large sample inferences for the adjusted U-statistics are valid using jackknife variance estimator. Comparisons of more than two groups are also considered from certain ways of pairwise two-group comparisons. The procedure is applied to analyze two real data sets for comparing two or more groups of event times. R codes of our procedure are available under supplementary material.

K-Sample comparisons using propensity analysis.

In this paper, we investigate K-group comparisons on survival endpoints for observational studies. In clinical databases for observational studies, treatment for patients are chosen with probabilities varying depending on their baseline characteristics. This often results in noncomparable treatment groups because of imbalance in baseline characteristics of patients among treatment groups. In order to overcome this issue, we conduct propensity analysis and match the subjects with similar propensity scores across treatment groups or compare weighted group means (or weighted survival curves for censored outcome variables) using the inverse probability weighting (IPW). To this end, multinomial logistic regression has been a popular propensity analysis method to estimate the weights. We propose to use decision tree method as an alternative propensity analysis due to its simplicity and robustness. We also propose IPW rank statistics, called Dunnett-type test and ANOVA-type test, to compare 3 or more treatment groups on survival endpoints. Using simulations, we evaluate the finite sample performance of the weighted rank statistics combined with these propensity analysis methods. We demonstrate these methods with a real data example. The IPW method also allows us for unbiased estimation of population parameters of each treatment group. In this paper, we limit our discussions to survival outcomes, but all the methods can be easily modified for any type of outcomes, such as binary or continuousvariables.

Semiparametric linear transformation models: Effect measures, estimators, and applications.

Semiparametric linear transformation models form a versatile class of regression models with the Cox proportional hazards model being the most well-known member. These models are well studied for right censored outcomes and are typically used in survival analysis. We consider transformation models as a tool for situations with uncensored continuous outcomes where linear regression is not appropriate. We introduce the probabilistic index as a uniform effect measure for the class of transformation models. We discuss and compare three estimators using a working Cox regression model: the partial likelihood estimator, an estimator based on binary generalized linear models and one based on probabilistic index model estimating equations. The latter has a superior performance in terms of bias and variance when the working model is misspecified. For the purpose of illustration, we analyze data that were collected at an urban alcohol and drug detoxification unit.

Substantial Bias in the Tobit Estimator: Making a Case for Alternatives

Censored outcome data are commonly encountered in criminology. Criminologists sometimes use the tobit model to address these censored data. While tobit models make more realistic demands of censored outcome data than ordinary least squares (OLS) regression, they require the researcher to make strong distributional assumptions. When these assumptions are not met, as is often the case in criminological data, tobit models yield biased and inconsistent estimates. We seek to demonstrate this substantial bias in simulation analyses and present easily applied alternative methods. The tobit model and semiparametric alternatives for censored outcome data are applied with simulated data under varying conditions. These simulations are followed with an empirical example using sentencing data. The bias from tobit can be corrected through application of semiparametric alternatives. Criminologists should begin their analyses of censored outcome data with the least restrictive of the available models (CLAD) before progressing to more efficient, but potentially biased, estimators.

Partial Least Squares Method for Treatment Effect in Observational Studies with Censored Outcomes

To estimate the true treatment effect on a censored outcome in observational studies, potential confounding effect and complex heterogeneity in the treatment assignment have to be properly adjusted. In this article, we demonstrate that the partial least squares method could be a valuable tool in this regard. It is showed that the partial least squares method not only can adequately account for the heterogeneity in treatment assignment, but also be robust to treatment assignment model misspecifications. Numerical results show that the partial least squares estimator is more efficient and robust. A real data set is analyzed to illustrate the proposed method.

Semiparametric Bayesian analysis of high-dimensional censored outcome data

ABSTRACTThe Surveillance, Epidemiology and End Results (SEER) cancer database contains survival data for US individuals diagnosed with cancer. Semiparametric Bayesian methods are computationally expensive to fit for such large data-sets. This paper develops a cost-effective Markov chain Monte Carlo strategy for censored outcomes to fit a semiparametric bayesian analysis of SEER data of New Mexico. We use an accelerated failure time model, with Dirichlet process random effects for inter-subject variation, and intrinsic conditionally autoregressive random effects for spatial correlations. The results offer insights into differences in breast cancer mortality rates between ethnic groups, tumor grade and spatial effect of counties.

Estimation in the Semiparametric Accelerated Failure Time Model With Missing Covariates: Improving Efficiency Through Augmentation

ABSTRACTThis article considers linear regression with missing covariates and a right censored outcome. We first consider a general two-phase outcome sampling design, where full covariate information is only ascertained for subjects in phase two and sampling occurs under an independent Bernoulli sampling scheme with known subject-specific sampling probabilities that depend on phase one information (e.g., survival time, failure status and covariates). The semiparametric information bound is derived for estimating the regression parameter in this setting. We also introduce a more practical class of augmented estimators that is shown to improve asymptotic efficiency over simple but inefficient inverse probability of sampling weighted estimators. Estimation for known sampling weights and extensions to the case of estimated sampling weights are both considered. The allowance for estimated sampling weights permits covariates to be missing at random according to a monotone but unknown mechanism. The asymptotic properties of the augmented estimators are derived and simulation results demonstrate substantial efficiency improvements over simpler inverse probability of sampling weighted estimators in the indicated settings. With suitable modification, the proposed methodology can also be used to improve augmented estimators previously used for missing covariates in a Cox regression model. Supplementary materials for this article are available online.

Estimating overall exposure effects for the clustered and censored outcome using random effect Tobit regression models.

The random effect Tobit model is a regression model that accommodates both left- and/or right-censoring and within-cluster dependence of the outcome variable. Regression coefficients of random effect Tobit models have conditional interpretations on a constructed latent dependent variable and do not provide inference of overall exposure effects on the original outcome scale. Marginalized random effects model (MREM) permits likelihood-based estimation of marginal mean parameters for the clustered data. For random effect Tobit models, we extend the MREM to marginalize over both the random effects and the normal space and boundary components of the censored response to estimate overall exposure effects at population level. We also extend the 'Average Predicted Value' method to estimate the model-predicted marginal means for each person under different exposure status in a designated reference group by integrating over the random effects and then use the calculated difference to assess the overall exposure effect. The maximum likelihood estimation is proposed utilizing a quasi-Newton optimization algorithm with Gauss-Hermite quadrature to approximate the integration of the random effects. We use these methods to carefully analyze two real datasets. Copyright © 2016 John Wiley & Sons, Ltd.

Doubly robust survival trees.

Estimating a patient's mortality risk is important in making treatment decisions. Survival trees are a useful tool and employ recursive partitioning to separate patients into different risk groups. Existing 'loss based' recursive partitioning procedures that would be used in the absence of censoring have previously been extended to the setting of right censored outcomes using inverse probability censoring weighted estimators of loss functions. In this paper, we propose new 'doubly robust' extensions of these loss estimators motivated by semiparametric efficiency theory for missing data that better utilize available data. Simulations and a data analysis demonstrate strong performance of the doubly robust survival trees compared with previously used methods. Copyright © 2016 John Wiley & Sons, Ltd.

Collective versus Individual Effects in Survival Analysis of Multiple Failures

AbstractIn survival analysis, we sometimes encounter data with multiple censored outcomes. Under certain scenarios, partial or even all covariates have ‘similar’ relative risks on the multiple outcomes in the Cox regression analysis. The similarity in covariate effects can be quantified using the proportionality of regression coefficients. Identifying the proportionality structure, or equivalently whether covariates have individual or collective effects, may have important scientific implications. In addition, it can lead to a smaller set of unknown parameters, which in turn results in more accurate estimation. In this article, we develop a novel approach for identifying the proportionality structure. Simulation shows the satisfactory performance of the proposed approach and its advantage over estimation under no assumed structure. We analyse three datasets to demonstrate the practical application of the proposed approach.