Comparative analysis of two rates

TLDR

The chi‑square function for odds‑ratio analysis, introduced by Cornfield and extended by Gart, is based on the efficient score and embodies optimality properties. This study investigates comparative analysis of rates (RD, RR, OR) and proposes improved chi‑square methods that avoid log transformation and restrict variance estimation. The authors develop chi‑square formulations grounded in the efficient score for RD and RR, analogous to the existing OR framework, to provide parameter‑constrained interval estimation and hypothesis testing. Simulations demonstrate that the proposed chi‑square procedures outperform traditional practices, revealing the inadequacy of current methods for RD and RR.

Abstract

In this paper, we examine comparative analysis of rates with a view to each of the usual comparative parameters-rate difference (RD), rate ratio (RR) and odds ratio (OR)-and with particular reference to first principles. For RD and RR we show the prevailing statistical practices to be rather poor. We stress the need for restricted estimation of variance in the chi-square function underlying interval estimation (and also point estimation and hypothesis testing). For RR analysis we propose a chi-square formulation analogous to that for RD and, thus, one which obviates the present practice of log transformation and its associated use of Taylor series approximation of the variance. As for OR analysis, we emphasize that the chi-square function, introduced by Cornfield for unstratified data, and extended by Gart to the case of stratified analysis, is based on the efficient score and thus embodies its optimality properties. We provide simulation results to evince the better performance of the proposed (parameter-constrained) procedures over the traditional ones.