{"id":475,"date":"2023-04-04T13:07:14","date_gmt":"2023-04-04T13:07:14","guid":{"rendered":"https:\/\/www.editage.com\/blog\/?p=475"},"modified":"2026-08-05T17:29:19","modified_gmt":"2026-08-05T11:59:19","slug":"risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers","status":"publish","type":"post","link":"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/","title":{"rendered":"Risk Ratios, Odds Ratios, and Hazard Ratios: Definition, Calculation, Examples"},"content":{"rendered":"\r\n<p class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"><strong>Key Takeaways:<\/strong><\/p>\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3 print:block print:space-y-1\" dir=\"ltr\">\r\n<li class=\"font-claude-response-body whitespace-normal break-words pl-2\"><strong>Risk and odds are not the same thing, and the gap matters.<\/strong> Risk is events divided by everyone at risk (bounded 0\u20131); odds is events divided by non-events (0 to infinity). They converge only when the outcome is uncommon \u2014 roughly under 10% prevalence \u2014 and diverge sharply as the outcome becomes more frequent.<\/li>\r\n<li class=\"font-claude-response-body whitespace-normal break-words pl-2\"><strong>Each ratio answers a different question.<\/strong> Risk ratios compare cumulative probabilities between groups, odds ratios compare odds, and hazard ratios compare the instantaneous event rate at any moment, making them the right choice for time-to-event data with variable follow-up or censoring.<\/li>\r\n<li class=\"font-claude-response-body whitespace-normal break-words pl-2\"><strong>Study design dictates the measure.<\/strong> Cohort studies and RCTs support RR (and HR for survival data); case-control studies can only yield OR, since outcome-based sampling makes true risk unknowable; cross-sectional studies call for a prevalence ratio.<\/li>\r\n<li class=\"font-claude-response-body whitespace-normal break-words pl-2\"><strong>Reading an odds ratio as a risk ratio is a documented and consequential error.<\/strong> It appears in roughly a quarter of studies in some fields and systematically overstates effect size when the outcome is common. Where that risk exists, use Poisson or log-binomial regression to estimate RR directly, and always pair relative measures with absolute ones (ARR, NNT\/NNH) so readers can judge clinical significance.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>In biomedical research and clinical literature, the terms risk, odds, and rate appear constantly \u00a0yet they are frequently misunderstood or used interchangeably. In reality, each describes a distinct mathematical relationship, and choosing the wrong measure can lead to misleading conclusions. This guide explains the foundational concepts, walks through step-by-step calculations with numerical examples, compares all major effect measures, and provides practical guidance on which measure to use for different study designs.<\/p>\r\n\r\n\r\n\r\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Foundational_Concepts\" >Foundational Concepts<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Probability_Risk_and_Odds\" >Probability, Risk, and Odds<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Key_Distinction_Risk_vs_Odds\" >Key Distinction: Risk vs. Odds<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Rate_vs_Risk\" >Rate vs. Risk<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Risk_Factors\" >Risk Factors<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#The_Three_Core_Ratio_Measures\" >The Three Core Ratio Measures<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Risk_Ratio_Relative_Risk\" >Risk Ratio (Relative Risk)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Odds_Ratio\" >Odds Ratio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Hazard_Ratio\" >Hazard Ratio<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Step-by-Step_Calculations_A_Worked_Example\" >Step-by-Step Calculations: A Worked Example<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#The_2%C3%972_Contingency_Table\" >The 2\u00d72 Contingency Table<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Absolute_Effect_Measures_Risk_Difference_ARR_RRR_NNT_and_NNH\" >Absolute Effect Measures: Risk Difference, ARR, RRR, NNT, and NNH<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Risk_Difference_Absolute_Risk_Reduction\" >Risk Difference (Absolute Risk Reduction)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Relative_Risk_Reduction\" >Relative Risk Reduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Number_Needed_to_Treat_NNT\" >Number Needed to Treat (NNT)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Number_Needed_to_Harm_NNH\" >Number Needed to Harm (NNH)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Survival_Ratio\" >Survival Ratio<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Comprehensive_Comparison_of_Effect_Measures\" >Comprehensive Comparison of Effect Measures<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Which_Measure_to_Use_Study_Design_Matters\" >Which Measure to Use: Study Design Matters<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Cohort_Studies_and_Randomised_Controlled_Trials\" >Cohort Studies and Randomised Controlled Trials<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Case-Control_Studies\" >Case-Control Studies<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Cross-sectional_Studies\" >Cross-sectional Studies<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Quick_Reference_Study_Design_and_Appropriate_Measures\" >Quick Reference: Study Design and Appropriate Measures<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#The_OR-as-RR_Error_A_Common_Pitfall\" >The OR-as-RR Error: A Common Pitfall<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Controlling_for_Confounders_Regression_Models\" >Controlling for Confounders: Regression Models<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Logistic_Regression_and_Adjusted_Odds_Ratios\" >Logistic Regression and Adjusted Odds Ratios<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Cox_Proportional_Hazards_Model_and_Adjusted_Hazard_Ratios\" >Cox Proportional Hazards Model and Adjusted Hazard Ratios<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Poisson_Regression_for_Risk_Ratios\" >Poisson Regression for Risk Ratios<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Best_Practices_for_Reporting_Statistical_Measures\" >Best Practices for Reporting Statistical Measures<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Always_Accompany_Relative_Measures_with_Absolute_Measures\" >Always Accompany Relative Measures with Absolute Measures<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Match_the_Measure_to_the_Study_Design\" >Match the Measure to the Study Design<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Transparency_About_Biases\" >Transparency About Biases<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Interpreting_Values_Near_the_Null\" >Interpreting Values Near the Null<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Real-World_Example_Smoking_and_Lung_Cancer_Mortality\" >Real-World Example: Smoking and Lung Cancer Mortality<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Glossary_of_Key_Terms\" >Glossary of Key Terms<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-36\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Frequently_Asked_Questions\" >Frequently Asked Questions<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-37\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Can_a_hazard_ratio_and_a_risk_ratio_for_the_same_study_ever_give_opposite_conclusions\" >Can a hazard ratio and a risk ratio for the same study ever give opposite conclusions?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-38\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#Is_it_possible_to_convert_an_odds_ratio_to_a_risk_ratio_after_the_fact\" >Is it possible to convert an odds ratio to a risk ratio after the fact?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-39\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#What_does_a_confidence_interval_that_crosses_10_mean_for_a_ratio_measure\" >What does a confidence interval that crosses 1.0 mean for a ratio measure?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#When_is_it_appropriate_to_pool_odds_ratios_in_a_meta-analysis_versus_risk_ratios\" >When is it appropriate to pool odds ratios in a meta-analysis versus risk ratios?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#How_does_loss_to_follow-up_affect_the_calculation_of_these_measures\" >How does loss to follow-up affect the calculation of these measures?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/#What_is_the_difference_between_a_rate_ratio_and_a_risk_ratio\" >What is the difference between a rate ratio and a risk ratio?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Foundational_Concepts\"><\/span><a id=\"_Toc231545808\"><\/a>Foundational Concepts<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>Before diving into ratios, it is essential to understand the building blocks.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Probability_Risk_and_Odds\"><\/span>Probability, Risk, and Odds<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>These three terms are related but mathematically distinct:<\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li>Probability is the fraction of times an outcome is expected to occur across many trials. It ranges from 0 to 1 (0% to 100%).<\/li>\r\n<li>Risk is the probability of an adverse outcome in a defined population over a specified time period. Risk is dimensionless and confined to values between 0 and 1. Example: if 30 out of 100 patients develop an infection, the risk is 0.30 or 30%.<\/li>\r\n<li>Odds express the ratio of the probability of an event occurring to the probability of it not occurring. If the probability of an event is p, then: Odds = p \/ (1 \u2212 p). Using the example above: Odds = 0.30 \/ 0.70 = 0.43. Odds have no upper bound and can take any non-negative value.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Key_Distinction_Risk_vs_Odds\"><\/span>Key Distinction: Risk vs. Odds<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>When disease prevalence is low (under approximately 10%), risk and odds are numerically similar and odds ratios closely approximate risk ratios. As prevalence rises, the two diverge \u2014 and the odds ratio will always exaggerate the effect size compared to the risk ratio. This is one of the most important practical points in biostatistics.<\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-table\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Feature<\/strong><\/td>\r\n<td><strong>Risk (Probability)<\/strong><\/td>\r\n<td><strong>Odds<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Definition<\/strong><\/td>\r\n<td>Events \/ Total people at risk<\/td>\r\n<td>Events \/ Non-events<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Range<\/strong><\/td>\r\n<td>0 to 1 (0% to 100%)<\/td>\r\n<td>0 to infinity<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Example (30 of 100 infected)<\/strong><\/td>\r\n<td>30\/100 = 0.30<\/td>\r\n<td>30\/70 = 0.43<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>When they are similar<\/strong><\/td>\r\n<td>\u2014<\/td>\r\n<td>When disease prevalence &lt; 10%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Rate_vs_Risk\"><\/span>Rate vs. Risk<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>A rate expresses change over time. For example, the incidence rate of a disease equals the number of new cases divided by the total person-time at risk. Unlike risk, a rate has a time dimension and can exceed 1.0. A risk is a cumulative probability over a fixed follow-up period, whereas a rate is an instantaneous measure expressed per unit time (e.g., per 100 person-years).<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Risk_Factors\"><\/span>Risk Factors<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>A risk factor is any variable that increases the probability of developing a disease or experiencing an adverse outcome. Examples include smoking (lung cancer), obesity (type 2 diabetes), and radiation exposure (certain cancers). Identifying and quantifying risk factors is the cornerstone of epidemiology and preventive medicine.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"The_Three_Core_Ratio_Measures\"><\/span><a id=\"_Toc231545809\"><\/a>The Three Core Ratio Measures<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Risk_Ratio_Relative_Risk\"><\/span>Risk Ratio (Relative Risk)<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>The risk ratio (RR), also called relative risk, compares the probability of an outcome in an exposed group to the probability in an unexposed (control) group.<\/p>\r\n\r\n\r\n\r\n<h4>Formula:\u00a0<\/h4>\r\n\r\n\r\n\r\n<p><strong>RR = Risk in exposed group \/ Risk in unexposed group<\/strong><\/p>\r\n\r\n\r\n\r\n<h4>Interpretation:<\/h4>\r\n\r\n\r\n\r\n<ul>\r\n<li>RR = 1: No association; both groups have equal risk.<\/li>\r\n<li>RR &gt; 1: The exposure is associated with higher risk (harmful).<\/li>\r\n<li>RR &lt; 1: The exposure is associated with lower risk (protective).<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Odds_Ratio\"><\/span>Odds Ratio<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>The odds ratio (OR) compares the odds of an outcome in an exposed group to the odds in an unexposed group.<\/p>\r\n\r\n\r\n\r\n<h4>Formula:\u00a0<\/h4>\r\n\r\n\r\n\r\n<p><strong>OR = Odds in exposed group \/ Odds in unexposed group\u00a0 =\u00a0 (a\/b) \/ (c\/d)\u00a0 =\u00a0 ad \/ bc<\/strong><\/p>\r\n\r\n\r\n\r\n<p>where a = exposed with outcome, b = exposed without outcome, c = unexposed with outcome, d = unexposed without outcome.<\/p>\r\n\r\n\r\n\r\n<h4>Interpretation:<\/h4>\r\n\r\n\r\n\r\n<ul>\r\n<li>OR = 1: No association.<\/li>\r\n<li>OR &gt; 1: Positive association between exposure and outcome.<\/li>\r\n<li>OR &lt; 1: Negative (protective) association.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Hazard_Ratio\"><\/span>Hazard Ratio<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>The hazard ratio (HR) is used in time-to-event (survival) analyses. Unlike RR or OR \u2014 which are cumulative over a study period \u2014 the HR compares the instantaneous rate of events between groups at any given moment in time. It is derived from survival analysis techniques such as the Cox proportional hazards model.<\/p>\r\n\r\n\r\n\r\n<h4>Formula:\u00a0<\/h4>\r\n\r\n\r\n\r\n<p><strong>HR = Hazard rate in intervention group \/ Hazard rate in control group<\/strong><\/p>\r\n\r\n\r\n\r\n<h4>Interpretation:<\/h4>\r\n\r\n\r\n\r\n<ul>\r\n<li>HR = 1: Both groups experience events at the same rate at any given time.<\/li>\r\n<li>HR &gt; 1: The intervention group experiences events at a higher rate at any point in time.<\/li>\r\n<li>HR &lt; 1: The intervention group experiences events at a lower rate at any point in time (protective effect).<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>A key feature of the HR: two trials with the same final RR can have very different HRs if one group experienced earlier events. Survival analysis captures this temporal nuance.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Step-by-Step_Calculations_A_Worked_Example\"><\/span><a id=\"_Toc231545810\"><\/a>Step-by-Step Calculations: A Worked Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>The following example uses a hypothetical influenza vaccine trial to show how RR and OR are calculated from a 2\u00d72 contingency table.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"The_2%C3%972_Contingency_Table\"><\/span>The 2\u00d72 Contingency Table<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>Suppose 48 vaccinated and 40 unvaccinated individuals are followed to see who becomes infected:<\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-table\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\u00a0<\/td>\r\n<td><strong>Infected<\/strong><\/td>\r\n<td><strong>Not Infected<\/strong><\/td>\r\n<td><strong>Total<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Vaccinated (exposed)<\/strong><\/td>\r\n<td>a = 16<\/td>\r\n<td>b = 32<\/td>\r\n<td>48<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Not vaccinated (unexposed)<\/strong><\/td>\r\n<td>c = 30<\/td>\r\n<td>d = 10<\/td>\r\n<td>40<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Total<\/strong><\/td>\r\n<td>46<\/td>\r\n<td>42<\/td>\r\n<td>88<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\r\n\r\n\r\n<h4>Calculating Risk Ratio<\/h4>\r\n\r\n\r\n\r\n<ul>\r\n<li>Risk in vaccinated group = a \/ (a + b) = 16 \/ 48 = 0.33 (33%)<\/li>\r\n<li>Risk in unvaccinated group = c \/ (c + d) = 30 \/ 40 = 0.75 (75%)<\/li>\r\n<li>RR = 0.33 \/ 0.75 = 0.44<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h4>Interpretation:<\/h4>\r\n\r\n\r\n\r\n<p>Vaccinated individuals are 0.44 times as likely to become infected compared to unvaccinated individuals \u2014 the vaccine reduces the risk of infection by about 56%.<\/p>\r\n\r\n\r\n\r\n<h4>Calculating Odds Ratio<\/h4>\r\n\r\n\r\n\r\n<ul>\r\n<li>Odds of infection (vaccinated) = a \/ b = 16 \/ 32 = 0.50<\/li>\r\n<li>Odds of infection (unvaccinated) = c \/ d = 30 \/ 10 = 3.0<\/li>\r\n<li>OR = 0.50 \/ 3.0 = 0.17\u00a0 (equivalently: ad \/ bc = 16\u00d710 \/ 32\u00d730 = 160\/960 = 0.17)<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h4>Interpretation:<\/h4>\r\n\r\n\r\n\r\n<p>The odds of infection among vaccinated individuals are 0.17 times the odds among unvaccinated individuals.<\/p>\r\n\r\n\r\n\r\n<h4>Why RR and OR Differ in This Example<\/h4>\r\n\r\n\r\n\r\n<p>The infection rate among unvaccinated individuals is 75%, far above the 10% threshold. This is why the OR (0.17) is considerably smaller than the RR (0.44). When the outcome is common, OR exaggerates the effect compared to RR. The relationship between OR and RR is:<\/p>\r\n\r\n\r\n\r\n<p><strong>OR = RR \u00d7 [(1 \u2212 risk in unexposed) \/ (1 \u2212 risk in exposed)]<\/strong><\/p>\r\n\r\n\r\n\r\n<p>When both risks are small (&lt; 10%), this multiplier approaches 1 and OR \u2248 RR.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Absolute_Effect_Measures_Risk_Difference_ARR_RRR_NNT_and_NNH\"><\/span><a id=\"_Toc231545811\"><\/a>Absolute Effect Measures: Risk Difference, ARR, RRR, NNT, and NNH<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>Ratio measures (RR, OR, HR) describe relative effects. Absolute measures describe the actual magnitude of benefit or harm \u2014 and are often more relevant for clinical decision-making.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Risk_Difference_Absolute_Risk_Reduction\"><\/span>Risk Difference (Absolute Risk Reduction)<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p><strong>Formula:\u00a0 ARR = Risk in control group \u2212 Risk in treatment group<\/strong><\/p>\r\n\r\n\r\n\r\n<p>Using the vaccine example: ARR = 0.75 \u2212 0.33 = 0.42 (42 percentage points). This means that for every 100 people vaccinated, 42 infections are prevented.<\/p>\r\n\r\n\r\n\r\n<p>A large RR can accompany a trivially small ARR. For example, an RR of 2.0 (doubled risk) sounds alarming, but if the baseline risk is 0.1%, doubling it to 0.2% is clinically insignificant. This is why absolute measures should always accompany relative ones.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Relative_Risk_Reduction\"><\/span>Relative Risk Reduction<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p><strong>Formula:\u00a0 RRR = 1 \u2212 RR\u00a0 (or equivalently, ARR \/ Risk in control group)<\/strong><\/p>\r\n\r\n\r\n\r\n<p>Using the vaccine example: RRR = 1 \u2212 0.44 = 0.56 (56%). The vaccine reduces relative risk by 56%.<\/p>\r\n\r\n\r\n\r\n<p>Caution: RRR reported without ARR can be misleading. A drug that reduces risk from 0.2% to 0.1% has an RRR of 50% \u2014 impressive-sounding \u2014 but an ARR of only 0.1%, meaning 1,000 people must be treated to prevent one event.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Number_Needed_to_Treat_NNT\"><\/span>Number Needed to Treat (NNT)<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p><strong>Formula:\u00a0 NNT = 1 \/ ARR<\/strong><\/p>\r\n\r\n\r\n\r\n<p>NNT answers: how many patients must receive the treatment for one additional patient to benefit? Lower NNT = more effective treatment.<\/p>\r\n\r\n\r\n\r\n<p>Using the vaccine example: NNT = 1 \/ 0.42 = 2.4 (approximately 2\u20133 people must be vaccinated to prevent one infection).<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Number_Needed_to_Harm_NNH\"><\/span>Number Needed to Harm (NNH)<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p><strong>Formula:\u00a0 NNH = 1 \/ Absolute Risk Increase<\/strong><\/p>\r\n\r\n\r\n\r\n<p>NNH answers: how many patients must be exposed to a risk factor or treatment for one additional patient to be harmed? Higher NNH = safer intervention.<\/p>\r\n\r\n\r\n\r\n<p>NNT and NNH together provide a balanced picture of benefit versus risk, and are widely used in evidence-based medicine to communicate clinical significance.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Survival_Ratio\"><\/span>Survival Ratio<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>The survival ratio is the complement of the risk ratio. It compares the probability of not experiencing the outcome between groups:<\/p>\r\n\r\n\r\n\r\n<p><strong>Survival Ratio = (1 \u2212 Risk in exposed) \/ (1 \u2212 Risk in unexposed)<\/strong><\/p>\r\n\r\n\r\n\r\n<p>In the vaccine example: (1 \u2212 0.33) \/ (1 \u2212 0.75) = 0.67 \/ 0.25 = 2.68. Unvaccinated individuals are 2.68 times as likely to survive infection-free, compared to vaccinated individuals but wait, the direction flips: vaccinated individuals are 2.68 times as likely to remain infection-free. This framing can be intuitive when communicating protective effects.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Comprehensive_Comparison_of_Effect_Measures\"><\/span><a id=\"_Toc231545812\"><\/a>Comprehensive Comparison of Effect Measures<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<figure class=\"wp-block-table\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\u00a0<\/td>\r\n<td><strong>Risk Ratio (RR)<\/strong><\/td>\r\n<td><strong>Odds Ratio (OR)<\/strong><\/td>\r\n<td><strong>Hazard Ratio (HR)<\/strong><\/td>\r\n<td><strong>Risk Difference (ARR)<\/strong><\/td>\r\n<td><strong>NNT \/ NNH<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Also called<\/td>\r\n<td>Relative risk<\/td>\r\n<td>\u2014<\/td>\r\n<td>\u2014<\/td>\r\n<td>Absolute risk reduction<\/td>\r\n<td>\u2014<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Calculation<\/td>\r\n<td>Risk_exp \/ Risk_ctrl<\/td>\r\n<td>(a\/b) \/ (c\/d)<\/td>\r\n<td>Survival analysis (Cox model)<\/td>\r\n<td>Risk_ctrl \u2212 Risk_exp<\/td>\r\n<td>1 \/ ARR<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Null value<\/td>\r\n<td>1<\/td>\r\n<td>1<\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<td>\u221e (no effect)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Range<\/td>\r\n<td>0 to \u221e<\/td>\r\n<td>0 to \u221e<\/td>\r\n<td>0 to \u221e<\/td>\r\n<td>\u22121 to +1<\/td>\r\n<td>Any positive number<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Type of effect<\/td>\r\n<td>Relative<\/td>\r\n<td>Relative<\/td>\r\n<td>Relative (time-varying)<\/td>\r\n<td>Absolute<\/td>\r\n<td>Absolute<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Best study design<\/td>\r\n<td>Cohort \/ RCT<\/td>\r\n<td>Case-control, logistic regression<\/td>\r\n<td>RCT, cohort (time-to-event)<\/td>\r\n<td>Cohort \/ RCT<\/td>\r\n<td>Cohort \/ RCT<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Time dimension<\/td>\r\n<td>No<\/td>\r\n<td>No<\/td>\r\n<td>Yes<\/td>\r\n<td>No<\/td>\r\n<td>No<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Accounts for censoring<\/td>\r\n<td>No<\/td>\r\n<td>No<\/td>\r\n<td>Yes<\/td>\r\n<td>No<\/td>\r\n<td>No<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>When OR \u2248 RR<\/td>\r\n<td>\u2014<\/td>\r\n<td>When outcome prevalence &lt; 10%<\/td>\r\n<td>\u2014<\/td>\r\n<td>\u2014<\/td>\r\n<td>\u2014<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Which_Measure_to_Use_Study_Design_Matters\"><\/span><a id=\"_Toc231545813\"><\/a>Which Measure to Use: Study Design Matters<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>The choice of effect measure is not arbitrary \u2014 it is constrained by <a href=\"https:\/\/www.editage.com\/insights\/qualitative-quantitative-or-mixed-methods-a-quick-guide-to-choose-the-right-design-for-your-research\">study design<\/a>.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Cohort_Studies_and_Randomised_Controlled_Trials\"><\/span>Cohort Studies and Randomised Controlled Trials<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>In cohort studies and <a href=\"https:\/\/www.editage.com\/insights\/a-young-researchers-guide-to-a-clinical-trial\">clinical trials<\/a>, participants are not selected based on outcome status. Both RR and OR can be calculated. RR is generally preferred because it is more directly interpretable.<\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li>Use RR when the outcome is binary and follow-up time is equal or short.<\/li>\r\n<li>Use HR when follow-up times vary across participants, or when censoring occurs (i.e., survival analysis is needed).<\/li>\r\n<li>OR from logistic regression is valid but should be clearly labelled, especially when outcome prevalence is above 10%.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Case-Control_Studies\"><\/span>Case-Control Studies<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>In case-control studies, participants are selected based on their outcome status (cases vs. controls). Because the sampling is outcome-based, true population prevalence cannot be estimated. So RR cannot be directly calculated.<\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li>OR is the appropriate and only valid measure of association in standard case-control studies.<\/li>\r\n<li>The OR from a case-control study can approximate the RR only when the disease is rare in the source population (&lt; 10%).<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Cross-sectional_Studies\"><\/span>Cross-sectional Studies<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p><a href=\"https:\/\/www.editage.com\/blog\/cross-sectional-study-definition-examples-and-tips-for-survey-research-design-and-reporting\/\">Cross-sectional studies<\/a> measure exposure and outcome at the same point in time. In this design, prevalence ratio (PR), which is computed identically to RR but using prevalence rather than incidence, is the preferred measure. OR can be reported but is less intuitive.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Quick_Reference_Study_Design_and_Appropriate_Measures\"><\/span>Quick Reference: Study Design and Appropriate Measures<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<figure class=\"wp-block-table\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Study design<\/strong><\/td>\r\n<td><strong>RR valid?<\/strong><\/td>\r\n<td><strong>OR valid?<\/strong><\/td>\r\n<td><strong>HR valid?<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Randomised controlled trial<\/strong><\/td>\r\n<td>Yes<\/td>\r\n<td>Yes (with caution if outcome common)<\/td>\r\n<td>Yes (if time-to-event)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cohort study<\/strong><\/td>\r\n<td>Yes<\/td>\r\n<td>Yes (with caution if outcome common)<\/td>\r\n<td>Yes (if time-to-event)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Case-control study<\/strong><\/td>\r\n<td>No (usually)<\/td>\r\n<td>Yes \u2014 preferred<\/td>\r\n<td>No<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cross-sectional study<\/strong><\/td>\r\n<td>Prevalence ratio<\/td>\r\n<td>Yes (less preferred)<\/td>\r\n<td>No<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"The_OR-as-RR_Error_A_Common_Pitfall\"><\/span><a id=\"_Toc231545814\"><\/a>The OR-as-RR Error: A Common Pitfall<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>One of the most documented errors in clinical literature is treating the odds ratio as if it were a risk ratio. Published research has found this misinterpretation in approximately 23\u201326% of studies in fields such as obstetrics and gynaecology and obesity research.<\/p>\r\n\r\n\r\n\r\n<p>The consequences:<\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li>When OR &gt; 1 and the outcome is common, interpreting OR as RR overestimates the strength of the association.<\/li>\r\n<li>When OR &lt; 1, interpreting OR as RR exaggerates the protective effect.<\/li>\r\n<li>The error grows as the event rate increases and as the OR moves further from the null (further from 1.0).<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>A useful rule of thumb: if the outcome prevalence in unexposed individuals is under 10%, OR and RR will differ by less than 20%, and the error is unlikely to change clinical conclusions. Above 10%, always report both or use the appropriate measure.<\/p>\r\n\r\n\r\n\r\n<p>Alternatives when OR is not appropriate but logistic regression has been used: researchers can use Poisson regression with robust variance, log-binomial regression, or modified Poisson regression to estimate RR directly from multivariable models.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Controlling_for_Confounders_Regression_Models\"><\/span><a id=\"_Toc231545815\"><\/a>Controlling for Confounders: Regression Models<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>In observational studies, simple 2\u00d72 table calculations do not account for confounding variables. Multivariable models are needed to estimate adjusted effect measures.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Logistic_Regression_and_Adjusted_Odds_Ratios\"><\/span>Logistic Regression and Adjusted Odds Ratios<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>Logistic regression is the standard method for obtaining adjusted odds ratios while controlling for multiple covariates simultaneously. The exponentiated coefficient (exp(\u03b2)) from logistic regression equals the adjusted OR for a one-unit change in the predictor, holding all other variables constant.<\/p>\r\n\r\n\r\n\r\n<p>Logistic regression is appropriate when:<\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li>The outcome is binary.<\/li>\r\n<li>The study is a case-control design.<\/li>\r\n<li>The outcome is rare (&lt; 10%) in a cohort study and OR is an adequate approximation of RR.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Cox_Proportional_Hazards_Model_and_Adjusted_Hazard_Ratios\"><\/span>Cox Proportional Hazards Model and Adjusted Hazard Ratios<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>The Cox model estimates the adjusted HR while accounting for time-to-event data and censoring. It assumes proportional hazards, meaning the ratio of hazards between groups remains constant over time. When this assumption is violated, alternative models (e.g., time-varying covariates, restricted mean survival time) are needed.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Poisson_Regression_for_Risk_Ratios\"><\/span>Poisson Regression for Risk Ratios<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>When the goal is to estimate adjusted RR from a cohort study with a common outcome, Poisson regression with robust (sandwich) standard errors is a well-established alternative to logistic regression. This approach avoids the OR-as-RR error and produces directly interpretable relative risk estimates.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Best_Practices_for_Reporting_Statistical_Measures\"><\/span><a id=\"_Toc231545816\"><\/a>Best Practices for Reporting Statistical Measures<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Always_Accompany_Relative_Measures_with_Absolute_Measures\"><\/span>Always Accompany Relative Measures with Absolute Measures<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>Reporting a relative measure alone (RR, OR, HR) is insufficient for clinical interpretation. Always include:<\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li>The baseline risk (or control group event rate).<\/li>\r\n<li>The absolute risk reduction or increase (ARR \/ ARI).<\/li>\r\n<li>NNT or NNH where relevant.<\/li>\r\n<li>95% confidence intervals for all estimates.<\/li>\r\n<li>The p-value (though confidence intervals are more informative).<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Match_the_Measure_to_the_Study_Design\"><\/span>Match the Measure to the Study Design<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>Do not report RR from a case-control study. Do not interpret OR as RR when the outcome is common. Use HR when follow-up time is variable or censoring has occurred.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Transparency_About_Biases\"><\/span>Transparency About Biases<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>Clearly report:<\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li>Inclusion and exclusion criteria to define the population.<\/li>\r\n<li>Data sources and their limitations.<\/li>\r\n<li>Potential <a href=\"https:\/\/www.editage.com\/insights\/7-tips-to-avoid-biases-in-biomedical-data-collection\">biases<\/a>: selection bias, measurement bias, confounding, and loss to follow-up.<\/li>\r\n<li>Whether estimates are crude (unadjusted) or adjusted, and for which covariates.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Interpreting_Values_Near_the_Null\"><\/span>Interpreting Values Near the Null<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>A ratio of 1.0 (or a risk difference of 0) means no association. Always consider whether a <a href=\"https:\/\/www.editage.com\/blog\/what-is-confidence-intervals-and-why-is-it-important\/\">confidence interval<\/a> crossing 1.0 (for ratios) or 0 (for differences) renders the finding statistically non-significant. Statistical significance does not equal clinical significance, and vice versa.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Real-World_Example_Smoking_and_Lung_Cancer_Mortality\"><\/span><a id=\"_Toc231545817\"><\/a>Real-World Example: Smoking and Lung Cancer Mortality<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<p>Smoking is among the most studied risk factors in epidemiology. Data from large US prospective cohort studies illustrate how different effect measures tell different parts of the story:<\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-table\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Outcome<\/strong><\/td>\r\n<td><strong>Risk in smokers<\/strong><\/td>\r\n<td><strong>Risk in non-smokers<\/strong><\/td>\r\n<td><strong>Risk Ratio<\/strong><\/td>\r\n<td><strong>Absolute Risk Increase<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Lung cancer mortality (men)<\/td>\r\n<td>~1.4%<\/td>\r\n<td>~0.07%<\/td>\r\n<td>~21\u00d7<\/td>\r\n<td>~1.3 percentage points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Heart disease mortality<\/td>\r\n<td>~6.5%<\/td>\r\n<td>~3.5%<\/td>\r\n<td>~2\u00d7<\/td>\r\n<td>~3 percentage points<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\r\n\r\n\r\n<p>This illustrates that the risk ratio for lung cancer (approximately 21) is far higher than for heart disease (approximately 2). Yet in absolute terms, smoking causes far more deaths from heart disease simply because heart disease is more common. Both relative and absolute measures are essential for a complete picture.<\/p>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Glossary_of_Key_Terms\"><\/span><a id=\"_Toc231545818\"><\/a>Glossary of Key Terms<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<figure class=\"wp-block-table\">\r\n<table>\r\n<thead>\r\n<tr>\r\n<td><strong>Term<\/strong><\/td>\r\n<td><strong>Definition<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Absolute Risk Reduction (ARR)<\/td>\r\n<td>The arithmetic difference in risk between two groups (control risk minus treatment risk). Indicates how much a treatment reduces the absolute probability of an outcome.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Case-control study<\/td>\r\n<td>A study design in which participants are selected based on their outcome status (cases vs. controls). Exposure history is then compared between groups. Only OR \u2014 not RR \u2014 can be directly calculated.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Censoring<\/td>\r\n<td>In survival analysis, censoring occurs when a participant leaves the study before experiencing the outcome, or the study ends. Censored individuals contribute follow-up time but not an event to the analysis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cohort study<\/td>\r\n<td>A study design in which participants are enrolled based on exposure status (exposed vs. unexposed) and followed over time to observe outcomes. Both RR and OR can be calculated.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Confounding<\/td>\r\n<td>A distortion of the true association between exposure and outcome, caused by a third variable related to both. Adjusted analyses (e.g., logistic regression, Cox model) are used to control for confounders.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cox proportional hazards model<\/td>\r\n<td>A regression model used in survival analysis to estimate adjusted hazard ratios while accounting for time-to-event data and censoring.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Hazard rate<\/td>\r\n<td>The instantaneous probability of an event occurring at a specific point in time, given that the individual has survived until that time.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Hazard Ratio (HR)<\/td>\r\n<td>The ratio of the hazard rate in the intervention group to the hazard rate in the control group at any given time. Used in survival analyses with time-to-event outcomes.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Incidence rate<\/td>\r\n<td>The number of new cases of a disease per unit of person-time at risk. Has a time dimension (e.g., per 100 person-years).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Logistic regression<\/td>\r\n<td>A multivariable statistical model for binary outcomes that yields adjusted odds ratios. The standard method for estimating ORs while controlling for covariates.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Null value<\/td>\r\n<td>The value of an effect measure that indicates no association. For ratio measures (RR, OR, HR), the null value is 1. For difference measures (ARR, risk difference), it is 0.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Number Needed to Harm (NNH)<\/td>\r\n<td>The number of individuals who must be exposed to a risk factor or intervention for one additional person to experience harm. Calculated as 1 \/ Absolute Risk Increase.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Number Needed to Treat (NNT)<\/td>\r\n<td>The number of patients who must receive a treatment for one additional patient to benefit. Calculated as 1 \/ ARR. A smaller NNT indicates greater treatment efficacy.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Odds<\/td>\r\n<td>The ratio of the probability of an event occurring to the probability of it not occurring: p \/ (1 \u2212 p).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Odds Ratio (OR)<\/td>\r\n<td>The ratio of the odds of an outcome in an exposed group to the odds in an unexposed group. The preferred measure in case-control studies.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Person-time at risk<\/td>\r\n<td>The accumulated time that all individuals in a study contribute while being at risk of the outcome. Used to calculate incidence rates.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Poisson regression<\/td>\r\n<td>A regression model that can estimate risk ratios directly from cohort data, particularly useful when the outcome is common and logistic regression would yield ORs that overestimate RR.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Prevalence ratio (PR)<\/td>\r\n<td>The ratio of the prevalence of an outcome in exposed vs. unexposed groups. Conceptually equivalent to RR but used in cross-sectional studies.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Probability<\/td>\r\n<td>The expected frequency of an event across many trials. Ranges from 0 to 1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Proportional hazards assumption<\/td>\r\n<td>The assumption in the Cox model that the ratio of hazards between groups remains constant over time.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Relative Risk Reduction (RRR)<\/td>\r\n<td>The proportional reduction in risk in the treatment group compared to the control group: 1 \u2212 RR, or ARR \/ control risk.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Risk<\/td>\r\n<td>The probability of an adverse outcome in a defined population over a specific time period. Ranges from 0 to 1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Risk Difference<\/td>\r\n<td>Synonymous with Absolute Risk Reduction. The arithmetic difference in risk between two groups.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Risk factor<\/td>\r\n<td>Any variable associated with an increased probability of developing a disease or adverse outcome.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Risk Ratio (RR)<\/td>\r\n<td>The ratio of the risk of an outcome in the exposed group to the risk in the unexposed group. Also called relative risk. Preferred in cohort studies and RCTs.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Survival analysis<\/td>\r\n<td>A set of statistical methods for analysing time-to-event data. Accounts for censoring and variable follow-up times. Produces hazard ratios.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Survival ratio<\/td>\r\n<td>The ratio of the probability of not experiencing the outcome in one group versus another: (1 \u2212 Risk in exposed) \/ (1 \u2212 Risk in unexposed).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\r\n\r\n\r\n<h2><span class=\"ez-toc-section\" id=\"Frequently_Asked_Questions\"><\/span><a id=\"_Toc231545819\"><\/a>Frequently Asked Questions<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Can_a_hazard_ratio_and_a_risk_ratio_for_the_same_study_ever_give_opposite_conclusions\"><\/span>Can a hazard ratio and a risk ratio for the same study ever give opposite conclusions?<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>In theory, yes, though it is unusual. If events in the treatment group are concentrated early in the study (front-loaded), the HR can be greater than 1 (apparent harm) while the final RR is less than 1 (apparent benefit) because many treated patients who die early no longer appear in later risk windows. This is why examining Kaplan-Meier survival curves and testing the proportional hazards assumption is important. An HR alone, without the accompanying survival curves, can give an incomplete picture.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"Is_it_possible_to_convert_an_odds_ratio_to_a_risk_ratio_after_the_fact\"><\/span>Is it possible to convert an odds ratio to a risk ratio after the fact?<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>Yes, if you know the baseline risk (the event rate in the control or unexposed group). The formula is: RR = OR \/ [(1 \u2212 baseline risk) + (baseline risk \u00d7 OR)]. This conversion is increasingly used in meta-analyses and systematic reviews to standardise effect measures across studies with different designs. When the baseline risk is very low, OR and RR will already be close and conversion has little effect.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"What_does_a_confidence_interval_that_crosses_10_mean_for_a_ratio_measure\"><\/span>What does a confidence interval that crosses 1.0 mean for a ratio measure?<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>A 95% confidence interval (CI) that includes 1.0 for a ratio measure (RR, OR, HR) means the result is not statistically significant at the conventional \u03b1 = 0.05 threshold; the data are compatible with there being no association. However, the width of the CI is equally important: a narrow CI crossing 1.0 suggests a small, precisely estimated null effect, while a very wide CI crossing 1.0 suggests the study was underpowered and cannot rule out a clinically meaningful effect in either direction.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"When_is_it_appropriate_to_pool_odds_ratios_in_a_meta-analysis_versus_risk_ratios\"><\/span>When is it appropriate to pool odds ratios in a meta-analysis versus risk ratios?<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>OR is generally preferred for pooling in meta-analyses because it has better statistical properties (it is symmetric and less affected by variation in baseline risk across studies). However, if the outcome is common and the studies included mostly cohort designs, pooling RRs or converting ORs to RRs before pooling may give more clinically interpretable results. Some guidelines, such as those from the Cochrane Collaboration, recommend reporting both the pooled OR and an illustrative ARR at a representative baseline risk.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"How_does_loss_to_follow-up_affect_the_calculation_of_these_measures\"><\/span>How does loss to follow-up affect the calculation of these measures?<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>Loss to follow-up (attrition) can bias estimates of RR, OR, and HR if the reasons for dropout are related to the outcome or the exposure, a form of informational censoring. For RR and OR in studies without censoring, lost participants are simply excluded from the denominator, which is acceptable only if missingness is completely at random (MCAR). In survival analyses, informative censoring violates the assumption that censoring is independent of the hazard, potentially biasing the HR. Sensitivity analyses, multiple imputation, and inverse probability weighting are common strategies for addressing attrition bias.<\/p>\r\n\r\n\r\n\r\n<h3><span class=\"ez-toc-section\" id=\"What_is_the_difference_between_a_rate_ratio_and_a_risk_ratio\"><\/span>What is the difference between a rate ratio and a risk ratio?<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n\r\n\r\n\r\n<p>A risk ratio compares cumulative risks (proportions) over a defined period; it is dimensionless and ranges from 0 to infinity. A rate ratio compares incidence rates, which are expressed per unit of person-time and can exceed 1.0. They are numerically similar when follow-up times are short and uniform, but diverge when follow-up is long or when person-time denominators are used instead of head counts. Rate ratios are appropriate when person-time data are available; risk ratios when only proportions are reported.<\/p>\r\n\r\n\r\n\r\n<p><em>Would you like guidance from an expert statistician on how to define your study variables and conduct your analysis? Check out Editage\u2019s\u00a0<a href=\"https:\/\/www.editage.com\/services\/publishing-services-packs\/statistical-analysis\"><strong>Statistical Analysis &amp; Review Services<\/strong><\/a>!<\/em><\/p>\r\n","protected":false},"excerpt":{"rendered":"In biomedical research and literature, the terms risks, rates, and odds are used very frequently. In medicine (particularly epidemiology), risks, rates, and odds are statistical measures calculated to understand the likelihood of an event occurring (e.g., an infectious disease or response to a treatment in a population or group of individuals). These calculations are important for researchers to determine the relative effectiveness of different treatments or interventions, identify potential risk factors or predictors of an outcome, and make informed decisions about clinical practice or public health policy. Calculating rates of disease can help identify risk factors and guide public health interventions.","protected":false},"author":2,"featured_media":479,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_ayudawp_aiss_exclude":false,"_ayudawp_aiss_summary":"When disease prevalence is low (under approximately 10%), risk and odds are numerically similar and odds ratios closely approximate risk ratios. When is it appropriate to pool odds ratios in a meta-analysis versus risk ratios?. Rate ratios are appropriate when person-time data are available; risk ratios when only proportions are reported.","_ayudawp_aiss_summary_provider":"manual","_ayudawp_aiss_summary_hash":"9e276d950169346266b4ab38f1e3460430a4418a"},"categories":[14],"tags":[23,24],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Risk Ratios, Odds Ratios, and Hazard Ratios: Definition, Calculation, Examples<\/title>\n<meta name=\"description\" content=\"Learn what are risk ratios, odds ratios, hazard ratios, how to calculate them, differences between them, how to use them in meta-analyses.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Risk Ratios, Odds Ratios, and Hazard Ratios: Definition, Calculation, Examples\" \/>\n<meta property=\"og:description\" content=\"Learn what are risk ratios, odds ratios, hazard ratios, how to calculate them, differences between them, how to use them in meta-analyses.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.editage.com\/blog\/risk-ratios-odds-ratios-and-hazard-ratios-for-biomedical-researchers\/\" \/>\n<meta property=\"og:site_name\" content=\"Educational Articles For Researchers, Students And Authors - 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