Hazard rate ratio and prospective epidemiological studies

Abstract

Analysis of prospective follow-up data usually includes a Cox regression model. When a hazard rate ratio, obtained as the exponential of an estimated regression coefficient from the Cox model, is greater than 1.0, it consistently exceeds relative risk, and is exceeded by the odds ratio. The divergence of these distinct epidemiologic measures increases with the product of three factors: (1) the length of follow-up, (2) the average rate of the end point occurence over the follow-up period, and (3) the magnitude of risk, either above or below 1. Cornfield's rare disease assumption is basically the product of the first two of these factors. However, risks in excess of 2.5 have a powerful effect on the divergence of these measures, and this point has received less emphasis. Conversely, and as seen frequently in applications, relative risk, hazard rate ratio, and odds ratio numerically approximate one another with shorter follow-up, rarer end points, and risks closer to 1. Although the hazard rate ratio is not always distinguished from relative risk, it is commonly close to, and is always between, relative risk and the odds ratio. Consistent and accurate terminology would have us use hazard rate ratio with Cox regression and odds ratio with logistic regression. The term “relative risk” seems to be a default choice, regardless of the model being used. However, when relative risk is the object of the model chosen, as in a Poisson regression approximation of two binomial proportions or an equivalent weighted least squares, then for us, relative risk is the accurate terminology.