Predicting Survival Probabilities with Semiparametric Transformation Models

Functional proportional hazards mixture cure model with applications in cancer mortality in NHANES and post ICU recovery

We develop a functional proportional hazards mixture cure model with scalar and functional covariates measured at the baseline. The mixture cure model, useful in studying populations with a cure fraction of a particular event of interest is extended to functional data. We employ the expectation-maximization algorithm and develop a semiparametric penalized spline-based approach to estimate the dynamic functional coefficients of the incidence and the latency part. The proposed method is computationally efficient and simultaneously incorporates smoothness in the estimated functional coefficients via roughness penalty. Simulation studies illustrate a satisfactory performance of the proposed method in accurately estimating the model parameters and the baseline survival function. Finally, the clinical potential of the model is demonstrated in two real data examples that incorporate rich high-dimensional biomedical signals as functional covariates measured at the baseline and constitute novel domains to apply cure survival models in contemporary medical situations. In particular, we analyze (i) minute-by-minute physical activity data from the National Health And Nutrition Examination Survey 2003-2006 to study the association between diurnal patterns of physical activity at baseline and all cancer mortality through 2019 while adjusting for other biological factors; (ii) the impact of daily functional measures of disease severity collected in the intensive care unit on post intensive care unit recovery and mortality event. Our findings provide novel epidemiological insights into the association between daily patterns of physical activity and cancer mortality. Software implementation and illustration of the proposed estimation method are provided in R.

Quantifying and Interpreting the Prediction Accuracy of Models for the Time of a Cardiovascular Event-Moving Beyond C Statistic: A Review

Importance

For personalized or stratified medicine, it is critical to establish a reliable and efficient prediction model for a clinical outcome of interest. The goal is to develop a parsimonious model with fewer predictors for broad future application without compromising predictability. A general approach is to construct various empirical models via individual patients' specific baseline characteristics/biomarkers and then evaluate their relative merits. When the outcome of interest is the timing of a cardiovascular event, a commonly used metric to assess the adequacy of the fitted models is based on C statistics. These measures quantify a model's ability to separate those who develop events earlier from those who develop them later or not at all (discrimination), but they do not measure how closely model estimates match observed outcomes (prediction accuracy). Metrics that provide clinically interpretable measures to quantify prediction accuracy are needed.

Observations

C statistics measure the concordance between the risk scores derived from the model and the observed event time observations. However, C statistics do not quantify the model prediction accuracy. The integrated Brier Score, which calculates the mean squared distance between the empirical cumulative event-free curve and its individual patient's counterparts, estimates the prediction accuracy, but it is not clinically intuitive. A simple alternative measure is the average distance between the observed and predicted event times over the entire study population. This metric directly quantifies the model prediction accuracy and has often been used to evaluate the goodness of fit of the assumed models in settings other than survival data. This time-scale measure is easier to interpret than the C statistics or the Brier score.

Conclusions and Relevance

This article enhances our understanding of the model selection/evaluation process with respect to prediction accuracy. A simple, intuitive measure for quantifying such accuracy beyond C statistics can improve the reliability and efficiency of the selected model for personalized and stratified medicine.

Dynamic semiparametric transformation models for recurrent event data with a terminal event

Recurrent event data with a terminal event commonly arise in many longitudinal follow-up studies. This article proposes a class of dynamic semiparametric transformation models for the marginal mean functions of the recurrent events with a terminal event, where some covariate effects may be time-varying. An estimation procedure is developed for the model parameters, and the asymptotic properties of the resulting estimators are established. In addition, relevant significance tests are suggested for examining whether or not covariate effects vary with time, and a model checking procedure is presented for assessing the adequacy of the proposed models. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a medical cost study of chronic heart failure patients is provided.

Rank-Based Greedy Model Averaging for High-Dimensional Survival Data

Model averaging is an effective way to enhance prediction accuracy. However, most previous works focus on low-dimensional settings with completely observed responses. To attain an accurate prediction for the risk effect of survival data with high-dimensional predictors, we propose a novel method: rank-based greedy (RG) model averaging. Specifically, adopting the transformation model with splitting predictors as working models, we doubly use the smooth concordance index function to derive the candidate predictions and optimal model weights. The final prediction is achieved by weighted averaging all the candidates. Our approach is flexible, computationally efficient, and robust against model misspecification, as it neither requires the correctness of a joint model nor involves the estimation of the transformation function. We further adopt the greedy algorithm for high dimensions. Theoretically, we derive an asymptotic error bound for the optimal weights under some mild conditions. In addition, the summation of weights assigned to the correct candidate submodels is proven to approach one in probability when there are correct models included among the candidate submodels. Extensive numerical studies are carried out using both simulated and real datasets to show the proposed approach's robust performance compared to the existing regularization approaches. Supplementary materials for this article are available online.

Semi-supervised approach to event time annotation using longitudinal electronic health records

Large clinical datasets derived from insurance claims and electronic health record (EHR) systems are valuable sources for precision medicine research. These datasets can be used to develop models for personalized prediction of risk or treatment response. Efficiently deriving prediction models using real world data, however, faces practical and methodological challenges. Precise information on important clinical outcomes such as time to cancer progression are not readily available in these databases. The true clinical event times typically cannot be approximated well based on simple extracts of billing or procedure codes. Whereas, annotating event times manually is time and resource prohibitive. In this paper, we propose a two-step semi-supervised multi-modal automated time annotation (MATA) method leveraging multi-dimensional longitudinal EHR encounter records. In step I, we employ a functional principal component analysis approach to estimate the underlying intensity functions based on observed point processes from the unlabeled patients. In step II, we fit a penalized proportional odds model to the event time outcomes with features derived in step I in the labeled data where the non-parametric baseline function is approximated using B-splines. Under regularity conditions, the resulting estimator of the feature effect vector is shown as root-n consistent. We demonstrate the superiority of our approach relative to existing approaches through simulations and a real data example on annotating lung cancer recurrence in an EHR cohort of lung cancer patients from Veteran Health Administration.

Testing for Differences in Survival When Treatment Effects Are Persistent, Decaying, or Delayed

A statistical test for the presence of treatment effects on survival will be based on a null hypothesis (absence of effects) and an alternative (presence of effects). The null is very simply expressed. The most common alternative, also simply expressed, is that of proportional hazards. For this situation, not only do we have a very powerful test in the log-rank test but also the outcome is readily interpreted. However, many modern treatments fall outside this relatively straightforward paradigm and, as such, have attracted attention from statisticians eager to do their best to avoid losing power as well as to maintain interpretability when the alternative hypothesis is less simple. Examples include trials where the treatment effect decays with time, immunotherapy trials where treatment effects may be slow to manifest themselves as well as the so-called crossing hazards problem. We review some of the solutions that have been proposed to deal with these issues. We pay particular attention to the integrated log-rank test and how it can be combined with the log-rank test itself to obtain powerful tests for these more complex situations.

Bayesian transformation models with partly interval-censored data

In many scientific fields, partly interval-censored data, which consist of exactly observed and interval-censored observations on the failure time of interest, appear frequently. However, methodological developments in the analysis of partly interval-censored data are relatively limited and have mainly focused on additive or proportional hazards models. The general linear transformation model provides a highly flexible modeling framework that includes several familiar survival models as special cases. Despite such nice features, the inference procedure for this class of models has not been developed for partly interval-censored data. We propose a fully Bayesian approach coped with efficient Markov chain Monte Carlo methods to fill this gap. A four-stage data augmentation procedure is introduced to tackle the challenges presented by the complex model and data structure. The proposed method is easy to implement and computationally attractive. The empirical performance of the proposed method is evaluated through two simulation studies, and the model is then applied to a dental health study.

Landmark Linear Transformation Model for Dynamic Prediction with Application to a Longitudinal Cohort Study of Chronic Disease

Dynamic prediction of the risk of a clinical event using longitudinally measured biomarkers or other prognostic information is important in clinical practice. We propose a new class of landmark survival models. The model takes the form of a linear transformation model, but allows all the model parameters to vary with the landmark time. This model includes many published landmark prediction models as special cases. We propose a unified local linear estimation framework to estimate time-varying model parameters. Simulation studies are conducted to evaluate the finite sample performance of the proposed method. We apply the methodology to a dataset from the African American Study of Kidney Disease and Hypertension and predict individual patient's risk of an adverse clinical event.

IMPROVING EFFICIENCY IN BIOMARKER INCREMENTAL VALUE EVALUATION UNDER TWO-PHASE DESIGNS

Cost-effective yet efficient designs are critical to the success of biomarker evaluation research. Two-phase sampling designs, under which expensive markers are only measured on a subsample of cases and non-cases within a prospective cohort, are useful in novel biomarker studies for preserving study samples and minimizing cost of biomarker assaying. Statistical methods for quantifying the predictiveness of biomarkers under two-phase studies have been proposed (Cai and Zheng, 2012; Liu, Cai and Zheng, 2012). These methods are based on a class of inverse probability weighted (IPW) estimators where weights are 'true' sampling weights that simply reflect the sampling strategy of the study. While simple to implement, existing IPW estimators are limited by lack of practicality and efficiency. In this manuscript, we investigate a variety of two-phase design options and provide statistical approaches aimed at improving the efficiency of simple IPW estimators by incorporating auxiliary information available for the entire cohort. We consider accuracy summary estimators that accommodate auxiliary information in the context of evaluating the incremental values of novel biomarkers over existing prediction tools. In addition, we evaluate the relative efficiency of a variety of sampling and estimation options under two-phase studies, shedding light on issues pertaining to both the design and analysis of biomarker validation studies. We apply our methods to the evaluation of a novel biomarker for liver cancer risk conducted with a two-phase nested case control design (Lok et al., 2010).

Estimating time-dependent ROC curves using data under prevalent sampling

Prevalent sampling is frequently a convenient and economical sampling technique for the collection of time-to-event data and thus is commonly used in studies of the natural history of a disease. However, it is biased by design because it tends to recruit individuals with longer survival times. This paper considers estimation of time-dependent receiver operating characteristic curves when data are collected under prevalent sampling. To correct the sampling bias, we develop both nonparametric and semiparametric estimators using extended risk sets and the inverse probability weighting techniques. The proposed estimators are consistent and converge to Gaussian processes, while substantial bias may arise if standard estimators for right-censored data are used. To illustrate our method, we analyze data from an ovarian cancer study and estimate receiver operating characteristic curves that assess the accuracy of the composite markers in distinguishing subjects who died within 3-5 years from subjects who remained alive. Copyright © 2016 John Wiley & Sons, Ltd.

Nonparametric Maximum Likelihood Estimators of Time-Dependent Accuracy Measures for Survival Outcome Under Two-Stage Sampling Designs

Large prospective cohort studies of rare chronic diseases require thoughtful planning of study designs, especially for biomarker studies when measurements are based on stored tissue or blood specimens. Two-phase designs, including nested case-control (Thomas, 1977) and case-cohort (Prentice, 1986) sampling designs, provide cost-effective strategies for conducting biomarker evaluation studies. Existing literature for biomarker assessment under two-phase designs largely focuses on simple inverse probability weighting (IPW) estimators (Cai and Zheng, 2011; Liu et al., 2012). Drawing on recent theoretical development on the maximum likelihood estimators for relative risk parameters in two-phase studies (Scheike and Martinussen, 2004; Zeng et al., 2006), we propose nonparametric maximum likelihood based estimators to evaluate the accuracy and predictiveness of a risk prediction biomarker under both types of two-phase designs. In addition, hybrid estimators that combine IPW estimators and maximum likelihood estimation procedure are proposed to improve efficiency and alleviate computational burden. We derive large sample properties of proposed estimators and evaluate their finite sample performance using numerical studies. We illustrate new procedures using a two-phase biomarker study aiming to evaluate the accuracy of a novel biomarker, des-γ-carboxy prothrombin, for early detection of hepatocellular carcinoma (Lok et al., 2010).

Bayesian analysis of transformation latent variable models with multivariate censored data

Transformation latent variable models are proposed in this study to analyze multivariate censored data. The proposed models generalize conventional linear transformation models to semiparametric transformation models that accommodate latent variables. The characteristics of the latent variables were assessed based on several correlated observed indicators through measurement equations. A Bayesian approach was developed with Bayesian P-splines technique and the Markov chain Monte Carlo algorithm to estimate the unknown parameters and transformation functions. Simulation shows that the performance of the proposed methodology is satisfactory. The proposed method was applied to analyze a cardiovascular disease data set.

Conditional transformation models for survivor function estimation

In survival analysis, the estimation of patient-specific survivor functions that are conditional on a set of patient characteristics is of special interest. In general, knowledge of the conditional survival probabilities of a patient at all relevant time points allows better assessment of the patient's risk than summary statistics, such as median survival time. Nevertheless, standard methods for analysing survival data seldom estimate the survivor function directly. Therefore, we propose the application of conditional transformation models (CTMs) for the estimation of the conditional distribution function of survival times given a set of patient characteristics. We used the inverse probability of censoring weighting approach to account for right-censored observations. Our proposed modelling approach allows the prediction of patient-specific survivor functions. In addition, CTMs constitute a flexible model class that is able to deal with proportional as well as non-proportional hazards. The well-known Cox model is included in the class of CTMs as a special case. We investigated the performance of CTMs in survival data analysis in a simulation that included proportional and non-proportional hazard settings and different scenarios of explanatory variables. Furthermore, we re-analysed the survival times of patients suffering from chronic myelogenous leukaemia and studied the impact of the proportional hazards assumption on previously published results.

Causal estimation using semiparametric transformation models under prevalent sampling

This article presents methods and inference for causal estimation in semiparametric transformation models for the prevalent survival data. Through the estimation of the transformation models and covariate distribution, we propose a few analytical procedures to estimate the causal survival function. As the data are observational, the unobserved potential outcome (survival time) may be associated with the treatment assignment, and therefore there may exist a systematic imbalance between the data observed from each treatment arm. Further, due to prevalent sampling, subjects are observed only if they have not experienced the failure event when data collection began, causing the prevalent sampling bias. We propose a unified approach, which simultaneously corrects the bias from the prevalent sampling and balances the systematic differences from the observational data. We illustrate in the simulation study that standard analysis without proper adjustment would result in biased causal inference. Large sample properties of the proposed estimation procedures are established by techniques of empirical processes and examined by simulation studies. The proposed methods are applied to the Surveillance, Epidemiology, and End Results (SEER) and Medicare-linked data for women diagnosed with breast cancer.

A Model-Free Machine Learning Method for Risk Classification and Survival Probability Prediction

Risk classification and survival probability prediction are two major goals in survival data analysis since they play an important role in patients' risk stratification, long-term diagnosis, and treatment selection. In this article, we propose a new model-free machine learning framework for risk classification and survival probability prediction based on weighted support vector machines. The new procedure does not require any specific parametric or semiparametric model assumption on data, and is therefore capable of capturing nonlinear covariate effects. We use numerous simulation examples to demonstrate finite sample performance of the proposed method under various settings. Applications to a glioma tumor data and a breast cancer gene expression survival data are shown to illustrate the new methodology in real data analysis.

On optimal treatment regimes selection for mean survival time

In clinical studies with time-to-event as a primary endpoint, one main interest is to find the best treatment strategy to maximize patients' mean survival time. Due to patient's heterogeneity in response to treatments, great efforts have been devoted to developing optimal treatment regimes by integrating individuals' clinical and genetic information. A main challenge arises in the selection of important variables that can help to build reliable and interpretable optimal treatment regimes as the dimension of predictors may be high. In this paper, we propose a robust loss-based estimation framework that can be easily coupled with shrinkage penalties for both estimation of optimal treatment regimes and variable selection. The asymptotic properties of the proposed estimators are studied. Moreover, a model-free estimator of restricted mean survival time under the derived optimal treatment regime is developed, and its asymptotic property is studied. Simulations are conducted to assess the empirical performance of the proposed method for parameter estimation, variable selection, and optimal treatment decision. An application to an AIDS clinical trial data set is given to illustrate the method.

Adopting nested case-control quota sampling designs for the evaluation of risk markers

Two-phase study methods, in which more detailed or more expensive exposure information is only collected on a sample of individuals with events and a small proportion of other individuals, are expected to play a critical role in biomarker validation research. One major limitation of standard two-phase designs is that they are most conveniently employed with study cohorts in which information on longitudinal follow-up and other potential matching variables is electronically recorded. However for many practical situations, at the sampling stage such information may not be readily available for every potential candidates. Study eligibility needs to be verified by reviewing information from medical charts one by one. In this manuscript, we study in depth a novel study design commonly undertaken in practice that involves sampling until quotas of eligible cases and controls are identified. We propose semiparametric methods to calculate risk distributions and a wide variety of prediction indices when outcomes are censored failure times and data are collected under the quota sampling design. Consistency and asymptotic normality of our estimators are established using empirical process theory. Simulation results indicate that the proposed procedures perform well in finite samples. Application is made to the evaluation of a new risk model for predicting the onset of cardiovascular disease.

Semiparametric inference on the absolute risk reduction and the restricted mean survival difference

For time-to-event data, when the hazards are non-proportional, in addition to the hazard ratio, the absolute risk reduction and the restricted mean survival difference can be used to describe the time-dependent treatment effect. The absolute risk reduction measures the direct impact of the treatment on event rate or survival, and the restricted mean survival difference provides a way to evaluate the cumulative treatment effect. However, in the literature, available methods are limited for flexibly estimating these measures and making inference on them. In this article, point estimates, pointwise confidence intervals and simultaneous confidence bands of the absolute risk reduction and the restricted mean survival difference are established under a semiparametric model that can be used in a sufficiently wide range of applications. These methods are motivated by and illustrated for data from the Women's Health Initiative estrogen plus progestin clinical trial.

Imputation for semiparametric transformation models with biased-sampling data

Widely recognized in many fields including economics, engineering, epidemiology, health sciences, technology and wildlife management, length-biased sampling generates biased and right-censored data but often provide the best information available for statistical inference. Different from traditional right-censored data, length-biased data have unique aspects resulting from their sampling procedures. We exploit these unique aspects and propose a general imputation-based estimation method for analyzing length-biased data under a class of flexible semiparametric transformation models. We present new computational algorithms that can jointly estimate the regression coefficients and the baseline function semiparametrically. The imputation-based method under the transformation model provides an unbiased estimator regardless whether the censoring is independent or not on the covariates. We establish large-sample properties using the empirical processes method. Simulation studies show that under small to moderate sample sizes, the proposed procedure has smaller mean square errors than two existing estimation procedures. Finally, we demonstrate the estimation procedure by a real data example.

Estimating Regression Parameters in an Extended Proportional Odds Model

The proportional odds model may serve as a useful alternative to the Cox proportional hazards model to study association between covariates and their survival functions in medical studies. In this article, we study an extended proportional odds model that incorporates the so-called "external" time-varying covariates. In the extended model, regression parameters have a direct interpretation of comparing survival functions, without specifying the baseline survival odds function. Semiparametric and maximum likelihood estimation procedures are proposed to estimate the extended model. Our methods are demonstrated by Monte-Carlo simulations, and applied to a landmark randomized clinical trial of a short course Nevirapine (NVP) for mother-to-child transmission (MTCT) of human immunodeficiency virus type-1 (HIV-1). Additional application includes analysis of the well-known Veterans Administration (VA) Lung Cancer Trial.

Estimation of the 2-sample hazard ratio function using a semiparametric model

The hazard ratio provides a natural target for assessing a treatment effect with survival data, with the Cox proportional hazards model providing a widely used special case. In general, the hazard ratio is a function of time and provides a visual display of the temporal pattern of the treatment effect. A variety of nonproportional hazards models have been proposed in the literature. However, available methods for flexibly estimating a possibly time-dependent hazard ratio are limited. Here, we investigate a semiparametric model that allows a wide range of time-varying hazard ratio shapes. Point estimates as well as pointwise confidence intervals and simultaneous confidence bands of the hazard ratio function are established under this model. The average hazard ratio function is also studied to assess the cumulative treatment effect. We illustrate corresponding inference procedures using coronary heart disease data from the Women's Health Initiative estrogen plus progestin clinical trial.

Linear transformation models for interval-censored data

In statistical analysis, when the value of a random variable is only known to be between two bounds, we say that this random variable is interval censored. This complicated censoring pattern is a common problem in research fields such as clinical trials or actuarial studies and raises challenges for statistical analysis. In this paper, we focus on regression analysis of case 2 interval-censored data. We first briefly review existing regression methods and an estimation approach under the class of linear transformation models developed by Zhang et al. We then propose a method for survival probability prediction via generalized estimating equations. We also consider a graphical model checking technique and a model selection tool. Some theoretical properties are established and the performance of our procedures is evaluated and illustrated by numerical studies including a real-life data analysis.

Checking the linear transformation model for clustered failure time observations

The linear transformation model is a semiparametric model which contains the Cox proportional hazards model and the proportional odds model as special cases. Cai et al. (Biometrika 87:867-878, 2000) have proposed an inference procedure for the linear transformation model with correlated censored observations. In this article, we develop formal and graphical model checking techniques for the linear transformation models based on cumulative sums of martingale-type residuals. The proposed method is illustrated with a clinical trial data.

A flexible semiparametric transformation model for survival data

I suggest an extension of the semiparametric transformation model that specifies a time-varying regression structure for the transformation, and thus allows time-varying structure in the data. Special cases include a stratified version of the usual semiparametric transformation model. The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline. Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given. The method is illustrated by a worked example and a small simulation study. A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggested.

Marginal regression of multivariate event times based on linear transformation models

Multivariate event time data are common in medical studies and have received much attention recently. In such data, each study subject may potentially experience several types of events or recurrences of the same type of event, or event times may be clustered. Marginal distributions are specified for the multivariate event times in multiple events and clustered events data, and for the gap times in recurrent events data, using the semiparametric linear transformation models while leaving the dependence structures for related events unspecified. We propose several estimating equations for simultaneous estimation of the regression parameters and the transformation function. It is shown that the resulting regression estimators are asymptotically normal, with variance-covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. Simulation studies show that the proposed approach is appropriate for practical use. An application to the well-known bladder cancer tumor recurrences data is also given to illustrate the methodology.