Ratio and Difference of Average Hazard with Survival Weight: New Measures to Quantify Survival Benefit of New Therapy

The hazard ratio (HR) has been the most popular measure to quantify the magnitude of treatment effect on time-to-event outcomes in clinical research. However, the HR estimated by Cox\u27s method has several drawbacks. One major issue is that there is no clear interpretation when the proportional hazards (PH) assumption does not hold, because it is affected by study-specific censoring time distribution in non-PH cases. Another major issue is that the lack of a group-specific absolute hazard value in each group obscures the clinical significance of the magnitude of the treatment effect. Given these, we propose average hazard with survival weight (AH-SW) as a summary metric of event time distribution and will use difference in AH-SW (DAH-SW) or ratio of AH-SW (RAH-SW) to quantify the treatment effect magnitude. The AH-SW we propose is a new digestible metric interpreted as a person-years event rate when random censoring would not exist. It is defined as the ratio of tau-year event rate and restricted mean survival time, which can be estimated non-parametrically. Numerical studies demonstrate that DAH-SW and RAH-SW offer almost identical power to Cox\u27s method under PH scenarios and can be more powerful for delayed-difference patterns that are often seen in immunotherapy trials. The proposed metrics (i.e., AH-SW, DAH-SW and RAH-SW) and the inferential methods for them offer a digestible interpretation that the conventional Cox\u27s method could not provide about the survival benefit of a new therapy. These metrics will increase the likelihood that results from clinical studies are correctly interpreted

The Optimal Confidence Region for a Random Parameter

Under a two-level hierarchical model, suppose that the distribution of the random parameter is known or can be estimated well. Data are generated via a fixed, but unobservable realization of this parameter. In this paper, we derive the smallest confidence region of the random parameter under a joint Bayesian/frequentist paradigm. On average this optimal region can be much smaller than the corresponding Bayesian highest posterior density region. The new estimation procedure is appealing when one deals with data generated under a highly parallel structure, for example, data from a trial with a large number of clinical centers involved or genome-wide gene-expession data for estimating individual gene- or center-specific parameters simultaneously. The new proposal is illustrated with a typical microarray data set and its performance is examined via a small simulation study

On assessing survival benefit of immunotherapy using long-term restricted mean survival time

The pattern of the difference between two survival curves we often observe in randomized clinical trials for evaluating immunotherapy is not proportional hazards; the treatment effect typically appears several months after the initiation of the treatment (i.e., delayed difference pattern). The commonly used logrank test and hazard ratio estimation approach will be suboptimal concerning testing and estimation for those trials. The long-term restricted mean survival time (LT-RMST) approach is a promising alternative for detecting the treatment effect that potentially appears later in the study. A challenge in employing the LT-RMST approach is that it must specify a lower end of the time window in addition to a truncation time point that the RMST requires. There are several investigations and suggestions regarding the choice of the truncation time point for the RMST. However, little has been investigated to address the choice of the lower end of the time window. In this paper, we propose a flexible LT-RMST-based test/estimation approach that does not require users to specify a lower end of the time window. Numerical studies demonstrated that the potential power loss by adopting this flexibility was minimal, compared to the standard LT-RMST approach using a prespecified lower end of the time window. The proposed method is flexible and can offer higher power than the RMST-based approach when the delayed treatment effect is expected. Also, it provides a robust estimate of the magnitude of the treatment effect and its confidence interval that corresponds to the test result

Evaluating Prediction Rules for t-Year Survivors With Censored Regression Models

Suppose that we are interested in establishing simple, but reliable rules for predicting future t-year survivors via censored regression models. In this article, we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate, sensitivity and specificity, and positive and negative predictive values. Specifically, under various working models we derive consistent estimators for the above measures via substitution and cross validation estimation procedures. Furthermore, we provide large sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified. Confidence intervals, for example, for the difference of the precision measures between two competing rules can then be constructed. All the proposals are illustrated with two real examples and their finite sample properties are evaluated via a simulation study

A versatile test for equality of two survival functions based on weighted differences of Kaplan-Meier curves

With censored event time observations, the logrank test is the most popular tool for testing the equality of two underlying survival distributions. Although this test is asymptotically distribution-free, it may not be powerful when the proportional hazards assumption is violated. Various other novel testing procedures have been proposed, which generally are derived by assuming a class of specific alternative hypotheses with respect to the hazard functions. The test considered by Pepe and Fleming (1989) is based on a linear combination of weighted differences of two Kaplan-Meier curves over time and is a natural tool to assess the difference of two survival functions directly. In this article, we take a similar approach, but choose weights which are proportional to the observed standardized difference of the estimated survival curves at each time point. The new proposal automatically makes weighting adjustments empirically. The new test statistic is aimed at a one-sided general alternative hypothesis, and is distributed with a short right tail under the null hypothesis, but with a heavy tail under the alternative. The results from extensive numerical studies demonstrate that the new procedure performs well under various general alternatives. The survival data from a recent cancer comparative study are utilized for illustrating the implementation of the process

Graphical Procedures for Evaluating Overall and Subject-Specific Incremental Values from New Predictors with Censored Event Time Data

Quantitative procedures for evaluating added values from new markers over a conventional risk scoring system for predicting event rates at specific time points have been extensively studied. However, a single summary statistic, for example, the area under the receiver operating characteristic curve or its derivatives, may not provide a clear picture about the relationship between the conventional and the new risk scoring systems. When there are no censored event time observations in the data, two simple scatterplots with individual conventional and new scores for “cases” and “controls” provide valuable information regarding the overall and the subject-specific level incremental values from the new markers. Unfortunately, in the presence of censoring, it is not clear how to construct such plots. In this paper, we propose a nonparametric estimation procedure for the distributions of the differences between two risk scores conditional on the conventional score. The resulting quantile curves of these differences over the subject-specific conventional score provide extra information about the overall added value from the new marker. They also help us to identify a subgroup of future subjects who need the new predictors, especially when there is no unified utility function available for cost-risk-benefit decision making. The procedure is illustrated with two data sets. The first is from a well-known Mayo Clinic PBC liver study. The second is from a recent breast cancer study on evaluating the added value from a gene score, which is relatively expensive to measure compared with the routinely used clinical biomarkers for predicting the patient's survival after surgery

Efficiency of Two Sample Tests via the t-Mean Survival Time for Analyzing Event Time Observations

In comparing two treatments with the event time observations, the hazard ratio (HR) estimate is routinely used to quantify the treatment difference. However, this model dependent estimate may be difficult to interpret clinically especially when the proportional hazards (PH) assumption is violated. An alternative estimation procedure for treatment efficacy based on the restricted means survival time or t-year mean survival time (t-MST) has been discussed extensively in the statistical and clinical literature. On the other hand, a statistical test 1 via the HR or its asymptotically equivalent counterpart, the logrank test, is asymptotically distribution-free. In this paper, we assess the relative efficiency of the hazard ratio and t-MST tests with respect to the statistical power using various PH and non-PH models under theoretical and practical settings. When the PH assumption is valid, the t-MST test performs almost as well as the HR test. For non-PH models, the t-MST test can substantially outperform its HR counter- part. On the other hand, the HR test can be powerful when the true difference of two survival functions is quite large at end of the study. Unfortunately, for this case, the HR estimate may not have a simple clinical interpretation for the treatment effect due to the violation of the PH assumption