Restricted mean survival time as a function of restriction time

Semiparametric Additive Modeling of the Restricted Mean Survival Time

Analysis of the restricted mean survival time (RMST) has become increasingly common in biomedical studies during the last decade as a means of estimating treatment or covariate effects on survival. Advantages of RMST over the hazard ratio (HR) include increased interpretability and lack of reliance on the often tenuous proportional hazards assumption. Some authors have argued that RMST regression should generally be the frontline analysis as opposed to methods based on counting process increments. However, in order for the use of the RMST to be more mainstream, it is necessary to broaden the range of data structures to which pertinent methods can be applied. In this report, we address this issue from two angles. First, most of existing methodological development for directly modeling RMST has focused on multiplicative models. An additive model may be preferred due to goodness of fit and/or parameter interpretation. Second, many settings encountered nowadays feature high-dimensional categorical (nuisance) covariates, for which parameter estimation is best avoided. Motivated by these considerations, we propose stratified additive models for direct RMST analysis. The proposed methods feature additive covariate effects. Moreover, nuisance factors can be factored out of the estimation, akin to stratification in Cox regression, such that focus can be appropriately awarded to the parameters of chief interest. Large-sample properties of the proposed estimators are derived, and a simulation study is performed to assess finite-sample performance. In addition, we provide techniques for evaluating a fitted model with respect to risk discrimination and predictive accuracy. The proposed methods are then applied to liver transplant data to estimate the effects of donor characteristics on posttransplant survival time.

Time-varying effect in older patients with early-stage breast cancer: a model considering the competing risks based on a time scale

Background

Patients with early-stage breast cancer may have a higher risk of dying from other diseases, making a competing risks model more appropriate. Considering subdistribution hazard ratio, which is used often, limited to model assumptions and clinical interpretation, we aimed to quantify the effects of prognostic factors by an absolute indicator, the difference in restricted mean time lost (RMTL), which is more intuitive. Additionally, prognostic factors of breast cancer may have dynamic effects (time-varying effects) in long-term follow-up. However, existing competing risks regression models only provide a static view of covariate effects, leading to a distorted assessment of the prognostic factor.

Methods

To address this issue, we proposed a dynamic effect RMTL regression that can explore the between-group cumulative difference in mean life lost over a period of time and obtain the real-time effect by the speed of accumulation, as well as personalized predictions on a time scale.

Results

A simulation validated the accuracy of the coefficient estimates in the proposed regression. Applying this model to an older early-stage breast cancer cohort, it was found that 1) the protective effects of positive estrogen receptor and chemotherapy decreased over time; 2) the protective effect of breast-conserving surgery increased over time; and 3) the deleterious effects of stage T2, stage N2, and histologic grade II cancer increased over time. Moreover, from the view of prediction, the mean C-index in external validation reached 0.78.

Conclusion

Dynamic effect RMTL regression can analyze both dynamic cumulative effects and real-time effects of covariates, providing a more comprehensive prognosis and better prediction when competing risks exist.

Semiparametric estimation of restricted mean survival time as a function of restriction time

The restricted mean survival time (RMST) is an appealing measurement in clinical or epidemiological studies with censored survival outcome and receives a lot of attention in the past decades. It provides a useful alternative to the Cox model for evaluating the covariate effect on survival time. The covariate effect on RMST usually varies with the restriction time. However, existing methods cannot address this problem properly. In this article, we propose a semiparametric framework that directly models RMST as a function of the restriction time. Our proposed model adopts a widely-used proportional form, enabling the estimation of RMST predictions across an interval using a unified model. Furthermore, the covariate effect for multiple restriction time points can be derived simultaneously. We develop estimators based on estimating equations theories and establish the asymptotic properties of the proposed estimators. The finite sample properties of the estimators are evaluated through extensive simulation studies. We further illustrate the application of our proposed method through the analysis of two real data examples. Supplementary Material are available online.

Analysis of dynamic restricted mean survival time based on pseudo-observations

In clinical follow-up studies with a time-to-event end point, the difference in the restricted mean survival time (RMST) is a suitable substitute for the hazard ratio (HR). However, the RMST only measures the survival of patients over a period of time from the baseline and cannot reflect changes in life expectancy over time. Based on the RMST, we study the conditional restricted mean survival time (cRMST) by estimating life expectancy in the future according to the time that patients have survived, reflecting the dynamic survival status of patients during follow-up. In this paper, we introduce the estimation method of cRMST based on pseudo-observations, the statistical inference concerning the difference between two cRMSTs (cRMSTd), and the establishment of the robust dynamic prediction model using the landmark method. Simulation studies are conducted to evaluate the statistical properties of these methods. The results indicate that the estimation of the cRMST is accurate, and the dynamic RMST model has high accuracy in coefficient estimation and good predictive performance. In addition, an example of patients with chronic kidney disease who received renal transplantations is employed to illustrate that the dynamic RMST model can predict patients' expected survival times from any prediction time, considering the time-dependent covariates and time-varying effects of covariates.

Restricted mean survival time to estimate an intervention effect in a cluster randomized trial

For time-to-event outcomes, the difference in restricted mean survival time is a measure of the intervention effect, an alternative to the hazard ratio, corresponding to the expected survival duration gain due to the intervention up to a predefined time t*. We extended two existing approaches of restricted mean survival time estimation for independent data to clustered data in the framework of cluster randomized trials: one based on the direct integration of Kaplan-Meier curves and the other based on pseudo-values regression. Then, we conducted a simulation study to assess and compare the statistical performance of the proposed methods, varying the number and size of clusters, the degree of clustering, and the magnitude of the intervention effect under proportional and non-proportional hazards assumption. We found that the extended methods well estimated the variance and controlled the type I error if there was a sufficient number of clusters (≥ 50) under both proportional and non-proportional hazards assumption. For cluster randomized trials with a limited number of clusters (< 50), a permutation test for pseudo-values regression was implemented and corrected the type I error. We also provided a procedure to estimate permutation-based confidence intervals which produced adequate coverage. All the extended methods performed similarly, but the pseudo-values regression offered the possibility to adjust for covariates. Finally, we illustrated each considered method with a cluster randomized trial evaluating the effectiveness of an asthma-control education program.

A Dynamic Prediction Model Supporting Individual Life Expectancy Prediction Based on Longitudinal Time-Dependent Covariates

In the field of clinical chronic diseases, common prediction results (such as survival rate) and effect size hazard ratio (HR) are relative indicators, resulting in more abstract information. However, clinicians and patients are more interested in simple and intuitive concepts of (survival) time, such as how long a patient may live or how much longer a patient in a treatment group will live. In addition, due to the long follow-up time, resulting in generation of longitudinal time-dependent covariate information, patients are interested in how long they will survive at each follow-up visit. In this study, based on a time scale indicator-restricted mean survival time (RMST)-we proposed a dynamic RMST prediction model by considering longitudinal time-dependent covariates and utilizing joint model techniques. The model can describe the change trajectory of longitudinal time-dependent covariates and predict the average survival times of patients at different time points (such as follow-up visits). Simulation studies through Monte Carlo cross-validation showed that the dynamic RMST prediction model was superior to the static RMST model. In addition, the dynamic RMST prediction model was applied to a primary biliary cirrhosis (PBC) population to dynamically predict the average survival times of the patients, and the average C-index of the internal validation of the model reached 0.81, which was better than that of the static RMST regression. Therefore, the proposed dynamic RMST prediction model has better performance in prediction and can provide a scientific basis for clinicians and patients to make clinical decisions.

Nonparametric inference of general while-alive estimands for recurrent events

Measuring the treatment effect on recurrent events like hospitalization in the presence of death has long challenged statisticians and clinicians alike. Traditional inference on the cumulative frequency unjustly penalizes survivorship as longer survivors also tend to experience more adverse events. Expanding a recently suggested idea of the "while-alive" event rate, we consider a general class of such estimands that adjust for the length of survival without losing causal interpretation. Given a user-specified loss function that allows for arbitrary weighting, we define as estimand the average loss experienced per unit time alive within a target period and use the ratio of this loss rate to measure the effect size. Scaling the loss rate by the width of the corresponding time window gives us an alternative, and sometimes more photogenic, way of showing the data. To make inferences, we construct a nonparametric estimator for the loss rate through the cumulative loss and the restricted mean survival time and derive its influence function in closed form for variance estimation and testing. As simulations and analysis of real data from a heart failure trial both show, the while-alive approach corrects for the false attenuation of treatment effect due to patients living longer under treatment, with increased statistical power as a result. The proposed methods are implemented in the R-package WA, which is publicly available from the Comprehensive R Archive Network (CRAN).

Restricted mean survival time regression model with time-dependent covariates

In clinical or epidemiological follow‐up studies, methods based on time scale indicators such as the restricted mean survival time (RMST) have been developed to some extent. Compared with traditional hazard rate indicator system methods, the RMST is easier to interpret and does not require the proportional hazard assumption. To date, regression models based on the RMST are indirect or direct models of the RMST and baseline covariates. However, time‐dependent covariates are becoming increasingly common in follow‐up studies. Based on the inverse probability of censoring weighting (IPCW) method, we developed a regression model of the RMST and time‐dependent covariates. Through Monte Carlo simulation, we verified the estimation performance of the regression parameters of the proposed model. Compared with the time‐dependent Cox model and the fixed (baseline) covariate RMST model, the time‐dependent RMST model has a better prediction ability. Finally, an example of heart transplantation was used to verify the above conclusions.

Clustered restricted mean survival time regression

For multicenter randomized trials or multilevel observational studies, the Cox regression model has long been the primary approach to study the effects of covariates on time-to-event outcomes. A critical assumption of the Cox model is the proportionality of the hazard functions for modeled covariates, violations of which can result in ambiguous interpretations of the hazard ratio estimates. To address this issue, the restricted mean survival time (RMST), defined as the mean survival time up to a fixed time in a target population, has been recommended as a model-free target parameter. In this article, we generalize the RMST regression model to clustered data by directly modeling the RMST as a continuous function of restriction times with covariates while properly accounting for within-cluster correlations to achieve valid inference. The proposed method estimates regression coefficients via weighted generalized estimating equations, coupled with a cluster-robust sandwich variance estimator to achieve asymptotically valid inference with a sufficient number of clusters. In small-sample scenarios where a limited number of clusters are available, however, the proposed sandwich variance estimator can exhibit negative bias in capturing the variability of regression coefficient estimates. To overcome this limitation, we further propose and examine bias-corrected sandwich variance estimators to reduce the negative bias of the cluster-robust sandwich variance estimator. We study the finite-sample operating characteristics of proposed methods through simulations and reanalyze two multicenter randomized trials.

Deep Neural Networks For Predicting Restricted Mean Survival Times

 

Restricted mean survival time (RMST) is a useful summary measurement of the time-to-event data, and it has attracted great attention for its straightforward clinical interpretation. In this article, I propose a deep neural network model that directly relates the RMST to its baseline covariates for simultaneous prediction of RSMT at multiple times. Each subject's survival time is transformed into a series of jackknife pseudo observations and then used as quantitative response variables in a deep neural network model. By using the pseudo values, a complex survival analysis is reduced to a standard regression problem, which greatly simplifies the neural network construction. By jointly modelling RMST at multiple times, the neural network model gains prediction accuracy by information sharing across times. The proposed network model was evaluated by extensive simulation studies and was further illustrated on three real datasets. In real data analyses, I also used methods to open the blackbox by identifying subject-specific predictors and their importance in contributing to the risk prediction.

AVAILABILITY AND IMPLEMENTATION

The source code is freely available at http://github.com/lilizhaoUM/DnnRMST.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.