The Upper Limits of Risk Ratios and Recommendations for Reporting Risk Ratios, Odds Ratios, and Rate Ratios

Abstract

BackgroundRelative measures, including risk ratios (RRs) and odds ratios (ORs), are reported in many epidemiological studies. RRs represent how many times a condition is likely to develop when exposed to a risk factor. The upper limit of RRs is the multiplicative inverse of the baseline incidence. Ignoring the upper limits of RRs can lead to reporting exaggerated relative effect sizes. ObjectivesThis study aims to demonstrate the importance of such upper limits for effect size reporting via equations, examples, and simulations and provide recommendations for the reporting of relative measures. MethodsEquations to calculate RRs and their 95% confidence intervals (CIs) were listed. We performed simulations with 10,000 simulated subjects and three population variables: proportions at risk (0.05, 0.1, 0.3, 0.5, and 0.8), baseline incidence (0.05, 0.1, 0.3, 0.5, and 0.8), and RRs (0.5, 1.0, 5.0, 10.0, and 25.0). Subjects were randomly assigned with a risk based on the set of proportions-at-risk values. A disease occurred based on the baseline incidence among those not at risk. The incidence of those at risk was the product of the baseline incidence and the RRs. The 95% CIs of RRs were calculated according to Altman. ResultsThe calculation of RR 95% CIs is not connected to the RR upper limits in equations. The RRs in the simulated populations at risk could reach the upper limits of RRs: multiplicative inverse of the baseline incidence. The upper limits to the derived RRs were around 1.25, 2, 3.3, 10, and 20, when the assumed baseline incidence rates were 0.8, 0.5, 0.3, 0.2, and 0.05, respectively. We demonstrated five scenarios in which the RR 95% CIs might exceed the upper limits. ConclusionsStatistical significance does not imply the RR 95% CIs not exceeding the upper limits of RRs. When reporting RRs or ORs, the RR upper limits should be assessed. The rate ratio is also subject to a similar upper limit. In the literature, ORs tend to overestimate effect sizes. It is recommended to correct ORs that aim to approximate RRs assuming outcomes are rare. A reporting guide for relative measures, RRs, ORs, and rate ratios, is provided. Researchers are recommended to report whether the 95% CIs of relative measures, RRs, ORs, and rate ratios, overlap with the range of upper limits and discuss whether the relative measure estimates may exceed the upper limits.

Translating Risk Ratios, Baseline Incidence, and Proportions Diseased to Correlations and Chi-Squared Statistics: Simulation Epidemiology

“…For both diseases and symptoms, a value of 1 represented the presence of diseases or symptoms, and a value of 0 represented their absence. The products of the incidence rates and risk ratios could not exceed 1, a maximum of 100% incidence rate among those with or without the disease [12]. The Pearson correlation coefficients between the disease and the other associated disease were 0, 0.3, or 0.7.…”