Testing similarity of competing risks models by comparing transition probabilities

TL;DR

This work addresses the problem of assessing whether two patient groups exhibit similar event dynamics under competing risks, by shifting focus from transition intensities to transition probabilities. It defines a maximum-deviation metric $d_\infty$ between transition probability trajectories and develops a constrained parametric bootstrap procedure that yields asymptotically valid tests under both administrative and random censoring. Through theory and extensive simulations, the authors show that the proposed transition-probability–based test maintains the nominal level and generally achieves higher power than existing intensity-based approaches. The method is demonstrated on prostate cancer surgery pathways, providing a practical tool to quantify similarity and justify pooling data across cohorts when clinically acceptable differences are small.

Abstract

Assessing whether two patient populations exhibit comparable event dynamics is essential for evaluating treatment equivalence, pooling data across cohorts, or comparing clinical pathways across hospitals or strategies. We introduce a statistical framework for formally testing the similarity of competing risks models based on transition probabilities, which represent the cumulative risk of each event over time. Our method defines a maximum-type distance between the transition probability matrices of two multistate processes and employs a novel constrained parametric bootstrap test to evaluate similarity under both administrative and random right censoring. We theoretically establish the asymptotic validity and consistency of the bootstrap test. Through extensive simulation studies, we show that our method reliably controls the type I error and achieves higher statistical power than existing intensity-based approaches. Applying the framework to routine clinical data of prostate cancer patients treated with radical prostatectomy, we identify the smallest similarity threshold at which patients with and without prior in-house fusion biopsy exhibit comparable readmission dynamics. The proposed method provides a robust and interpretable tool for quantifying similarity in event history models.

Figures (4)

• Figure 1: Empirical rejection probabilities of the similarity test based on transition probabilities for the scenarios in Table \ref{['tab:transition_intensities']} ($n_1=n_2=200$). The different lines represent the power for different censoring mechanisms.
• Figure 2: Empirical rejection probabilities of the similarity test based on transition probabilities under alternative $2$ (left) and alternative $3$ (right), in dependence of the group sample sizes. The lines represent the power for different censoring mechanisms.
• Figure 3: Empirical rejection probabilities of the similarity test based on transition probabilities and the test based on transition intensities under the scenarios from Table \ref{['tab:transition_intensities']}. The sample size is $(n_1,n_2) = (200,200)$ and four censoring scenarios are considered.
• Figure 4: Estimates of the transition probabilities from competing risks models 1 (black) and 2 (grey) in the application example. Illustrated for each transition are the non-parametric Aalen Johansen estimates (solid lines) with 95% confidence intervals (CI) as well as the parametric model fit assuming exponential distribution (dot-dashed lines).

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Theorem 1
• proof