Statistical methods for estimating the protective effects of immune markers using test-negative designs.

While widely used to study vaccine effectiveness, test-negative designs (TNDs) also provide a platform for identifying and quantifying immunological correlates of protection against disease. A key component of such studies is the protection function, the mathematical relationship between the value of an immunological assay and the relative risk of disease. This function is often estimated using logistic regression, comparing the odds of disease at a given assay value to the odds at assay value zero. Here, we show through mathematical analysis and simulation experiments that logistic regression, while common, fundamentally constrains the functional forms of protection that can be inferred from data in TNDs, potentially leading to overly simplistic estimates of the protection function. To address this limitation, we adapt and analyze a scaled logit model, originally developed for case-control data, as a flexible alternative that allows for greater flexibility in estimating protection functions from TND data. We demonstrate that this approach improves accuracy across a range of biologically plausible protection functions, highlight conditions under which it may fail, and provide practical guidance for researchers to adopt it as a new standard for TND studies evaluating correlates of protection.