# A Spline‐Based Approach to Smoothly Constrain Hazard Ratios With a View to Apply Treatment Effect Waning

## TL;DR

This paper introduces a new method using splines to model how treatment effects decrease over time in survival analysis.

## Contribution

A novel spline-based approach is proposed to smoothly constrain hazard ratios, allowing treatment effect waning in survival analysis.

## Key findings

- Bias in survival time estimates was small unless constraints contradicted true waning patterns.
- Including a knot at the 95th percentile minimized biases during extrapolation.
- Later placement of constraints slowed HR waning, leading to less conservative assumptions.

## Abstract

To describe and assess, via simulation, a constraint‐based spline approach to implement smooth hazard ratio (HR) waning in time‐to‐event analyses.

A common consideration when extrapolating survival functions to evaluate the long‐term performance of a novel intervention is scenarios where the beneficial effect of an intervention eventually disappears (treatment effect waning). One approach to relaxing the proportional hazards assumption for a treatment effect is to model it as a function of the timescale, with a spline function offering a flexible approach. We consider the constraint of coefficients of spline variables to 0 during estimation, leading to log‐treatment effects that are constrained to 0 (HR = 1) from a given time‐point: enforcing treatment efficacy waning. An example is reported. Datasets were simulated under a variety of scenarios and analyzed with treatment effect waning assumptions under various modeling choices. Bias in mean survival time difference, given fully observed waning or fully censored waning, was assessed and constrained HR estimates were visualized.

Given full waning, biases were small unless constraints directly contradicted truths. When waning was extrapolated, akin to real‐life practice, biases over observed periods were minimized through the inclusion of a knot at the 95th percentile. The rate at which the HR waned slowed as the upper boundary knot/constraint was placed later, inducing less conservative treatment effect waning assumptions.

An alternative approach to modeling smooth treatment efficacy waning is demonstrated, enabling HR conditioning and marginal RMST calculation in a single framework, along with applications of the method beyond this use.