A two-sample pseudo-observation-based regression approach for the relative treatment effect

TL;DR

The paper tackles the problem of assessing covariate-modified treatment effects in two-sample settings with potential censoring by introducing a distribution-free regression framework for the relative treatment effect $ heta = P(T_1 > T_2)$. It constructs two-sample jackknife pseudo-observations and fits a generalized estimating equation to model $ heta$ via a monotone link $oldsymbol eta$, with extensions to right-censored data through Kaplan-Meier estimators and a horizon $ au$. The authors prove a central limit theorem for $oldsymboleta$ in the uncensored case and establish censored-case results, complemented by bootstrap-based hypothesis tests that show competitive power to Cox-based tests, even under correct model specification. The method yields interpretable, patient-specific probabilities of benefit and supports flexible two-sample comparisons, including situations with historical controls, demonstrated through applications to SUCCESS-A DFS data. Overall, this work provides a robust, distribution-free alternative to Cox regression for evaluating treatment effects across covariate subgroups and offers practical tools for inference and subgroup discovery in survival analyses.

Abstract

The relative treatment effect is an effect measure for the order of two sample-specific outcome variables. It has the interpretation of a probability and also a connection to the area under the ROC curve. In the literature it has been considered for both ordinal or right-censored time-to-event outcomes. For both cases, the present paper introduces a distribution-free regression model that relates the relative treatment effect to a linear combination of covariates. To fit the model, we develop a pseudo-observation-based procedure yielding consistent and asymptotically normal coefficient estimates. In addition, we propose bootstrap-based hypothesis tests to infer the effects of the covariates on the relative treatment effect. A simulation study compares the novel method to Cox regression, demonstrating that the proposed hypothesis tests have high power and keep up with the z-test of the Cox model even in scenarios where the latter is specified correctly. The new methods are used to re-analyze data from the SUCCESS-A trial for progression-free survival of breast cancer patients.

Figures (6)

• Figure 1: Hazard rates of the Weibull$(2,1)$ and Weibull$(3,1)$ distributions. At the crossing point ($t = 2/3 \cdot \lambda_{2i}^3 \lambda_{1i}^{-2} = 2/3 \cdot 1^3 \cdot 1^{-2}=0.667$), the Weibull survival functions with shape parameters 2 and 3 assume the values $0.641$ and $0.744$, respectively.
• Figure 2: Results from the simulation study. The figure presents boxplots of the estimates of $\beta_{j1}$, $j=1,2$, for different covariate space dimensions $p\in \{2,4\}$ and unequal (I) and equal (II) Weibull shape parameters.
• Figure 3: Boxplots of type-I error rates (upper panels) and power (lower panels) of the tests for $H_0^{(j)}:\beta_{j1}=0$ vs. $H_a^{(j)}: \beta_{j1}\neq 0$, for $j=1$ (left panels) and $j=2$ (right panels), across all settings. The nominal significance level $\alpha=5\%$ is displayed as a horizontal line in the upper two panels. The labels on the x-axes correspond to the four tests $\varphi_{j,emp}$, $\varphi_{j,IQR}$, $\varphi_{j,MAD}$ and $\varphi_{j,quantile}$.
• Figure 4: Analysis of the SUCCESS-A study data. (A) Estimated coefficients $\hat{\boldsymbol\beta}_1 + \hat{\boldsymbol\beta}_2$ with 95% confidence intervals, obtained using the empirical standard deviation method. Positive values indicate a higher estimated probability of $T_1 > T_2$, corresponding to a benefit of the intervention compared to the reference categories of the covariates, whereas negative values indicate a benefit of the control treatment. Age and BMI are measured in years and $kg/m^2$, respectively. The reference categories are ER$-$, G1, HER2$-$, ductal, pN+, pre-menopausal, PR$-$ and pT1. The estimated value of the intercept $\beta_0$ is 0.1560 [$-$0.0006, 0.3125]. (B) Estimated group-by-covariate interactions ($\log(\hat{\text{HR}})$) with corresponding 95% confidence intervals, as obtained from the Cox proportional hazards model. The intervention serves as the reference group. Accordingly, negative estimates indicate an advantage of the control group compared to the reference categories of the covariates, whereas positive estimates indicate an advantage of the intervention. (C) Estimated probabilities $\hat{P}(T_1 > T_2 \mid \boldsymbol Z = \boldsymbol z)$ for all patients, grouped by estimated benefit in the intervention and control groups (using a probability threshold of 0.5). (D) Estimated probabilities $\hat{P}(T_1 > T_2 \mid \boldsymbol Z = \boldsymbol z)$ stratified by molecular tumor subtype, highlighting the heterogeneity of relative treatment effects across subgroups.
• Figure S1: Analysis of the SUCCESS-A study data. The figure presents the estimated coefficients $\hat{\boldsymbol\beta}_1 + \hat{\boldsymbol\beta}_2$ with 95% confidence intervals, obtained using the IQR, MAD, and empirical quantile methods. Positive values indicate a higher estimated probability of $T_1 > T_2$, corresponding to a benefit of the intervention compared to the reference categories of the covariates, whereas negative values indicate a benefit of the control treatment compared to the reference categories of the covariates. The estimated values of the intercept $\beta_0$ are 0.1560 [$-$0.0027, 0.3139] for the IQR method, 0.1560 [$-$0.0026, 0.3145] for the MAD method, and 0.1560 [$-$0.0014, 0.3168] for the empirical quantile method.

Theorems & Definitions (6)

• Remark 1
• Example 1: Identity link function
• Remark 2
• Theorem 1
• Remark 3
• Remark 4