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

Background In a population, when a disease is causing a symptom, the overall symptom incidence can be determined by proportions diseased, baseline symptom incidence, and risk ratios of developing the symptom due to the disease. There are various measures of association, including risk ratios. How risk ratios are linked to other measures of association, such as correlation coefficients and chi-squared statistics, has not been explicitly discussed. This study aims to demonstrate their connection via equations and simulations, assuming one disease causes symptoms. Methods The equations for correlation coefficients and chi-square statistics were rewritten using epidemiological measures: proportions diseased, baseline symptom incidence, and risk ratios. Simulations were conducted to test the accuracy of the equations. The baseline symptom incidence and the proportions diseased were assumed to be 0.05, 0.1, 0.2, 0.4, or 0.8. The risk ratios were assumed to be 0.5, 1, 2, 5, 10, and 25. Another disease that correlates with this disease was created (correlation = 0, 0.3, or 0.7). For each combination of symptom incidence, proportions diseased, risk ratios, and between-disease correlations, 10,000 subjects were simulated. The correlation coefficients and chi-squared statistics were approximated with epidemiologic measures and their interaction terms. R-squared was used to assess the importance of the epidemiologic measures. Results In the simulations, the overall symptom incidence, correlation coefficients, and chi-squared statistics between the disease and symptoms could be fully explained by the epidemiologic measures in the equations (R-squared = 1). When approximating correlation coefficients and chi-squared statistics with individual measures or their interaction terms, the importance of these measures depended on whether the at-risk incidence reached 1 or not. The numbers in the four cells in the contingency table predicted correlation coefficients, or chi-squared statistics, with different R-squared. Conclusion To our knowledge, this is the first study to translate the three epidemiologic measures (risk ratios, baseline symptom incidence, and proportions diseased) into correlation coefficients and chi-squared statistics. However, chi-squared statistics also depend on sample sizes. This study also provides a platform for developing teaching cases for students to investigate the causal relationship between diseases and symptoms or exposure and outcomes.