Improved Risk Ratio Approximation by Complementary Log-Log Models: A Comparison with Logistic Models

TL;DR

This paper compares complementary log-log models to logistic models, demonstrating that the former provides a more accurate approximation of risk ratios, especially for common outcomes, through theoretical and simulation analyses.

Contribution

It introduces the complementary log ratio as a better approximation to risk ratios and compares it with odds ratios within a unified link function framework.

Findings

Complementary log-log models yield smaller approximation bias than logistic models.

Simulation results confirm theoretical superiority of complementary log-log models.

Complementary log-log models are easy to implement in standard statistical software.

Abstract

Odds ratios obtained from logistic models fail to approximate risk ratios with common outcomes, leading to potential misinterpretations about exposure effects by practitioners. This article investigates the complementary log-log models as a practical alternative to produce risk ratio approximation. We demonstrate that the corresponding effect measure of complementary log-log models, called the complementary log ratio in this article, consistently provides a closer approximation to risk ratios than odds ratios. To compare the approximation accuracy, we adopt the one-parameter Aranda-Ordaz family of link functions, which includes both the logit and complementary log-log link functions as special cases. Within this unified framework, we implement a theoretical comparison of approximation accuracy between the complementary log ratio and the odds ratio, showing that the former always produces smaller approximation bias. Simulation studies further reinforce our theoretical findings. Given that the complementary log-log model is easily implemented in standard statistical software such as R and SAS, we encourage more frequent use of this model as a simple and effective alternative to logistic models when the goal is to approximate risk ratios more accurately.