Explaining Odds Ratios

3.6 Explanation

3.6.1 The basics

In some regression models (examples here are logistic and Poisson regression), instead of a beta coefficient that can be interpreted as-is (unit of predictor change results in coefficient amount of response change), you are given log odds (DWin 2011). The log odds must be converted to odds ratios, or their Poisson similar cousin relative risk/relative odds. You can present both the original log odds estimate plus the odds ratio / relative risk (Perraillon, n.d.) in your results table.

UCLA’s Statistics Center has an overview (“FAQ: How Do I Interpret Odds Ratios in Logistic Regression?” n.d.) for logistic regression that goes from probabilities to odds to log odds to odds ratios. Helpfully, they show examples for a variety of types of predictor variables (none, categorical, numeric, interactions).

3.6.2 More technical

3.6.2.1 Questions and data types

Example problem structures and types of data you need.

• The whole analysis page (“Poisson Regression | R Data Analysis Examples” n.d.), specifically look at the secion called “Poisson regression” about halfway down the page and the second bullet point for interpreting coefficients specifically.

• The whole analysis page (“Logit Regression | R Data Analysis Examples” n.d.), section where it will talk about coefficients.

• (Hector 2015) has a good overview of conversion on pp 130-134 for binary/logistic and 137-138 for Poisson data.

• Box 13.2 in Quinn and Keough also has a good work-through (logistic regression, but as long as both use the default log link the interpretation is similar).

3.6.2.2 Key assumptions

Did you run a generalized linear model (GLM Monica (2013)) with a log link or logit link? Then yes.

3.6.2.3 Key distinctions among related methods

Odds ratio for logistic vs relative risk for Poisson (DWin 2011). There is a good summary chapter about odds ratios vs log odds vs other related confusing terms.

3.6.2.4 Implementations and controversies

Very helpful to plot your data (Cameron 2021) so you are interpreting in the correct direction, as for any type of regression.

3.6.3 Most technical

• Full explanation of the equations used to calculate and differences among odds ratios, probability, relative risk (Perraillon and Lindrooth, n.d.)

3.7 Examples “in the wild”

Citations and what is useful in the paper.

References

Cameron, Allan. 2021. “Answer to "How Can I Create a Ggplot with a Regression Line Based on the Predicted Values of a Glm?".” Stack Overflow. November 2, 2021. https://stackoverflow.com/a/69811892.

DWin. 2011. “Answer to "How to Interpret Coefficients in a Poisson Regression?".” Cross Validated. May 21, 2011. https://stats.stackexchange.com/a/11097.

“FAQ: How Do I Interpret Odds Ratios in Logistic Regression?” n.d. Accessed May 20, 2022. https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faq-how-do-i-interpret-odds-ratios-in-logistic-regression/.

Hector, Andy. 2015. “9: Generalized Linear Models for Data with Non-Normal Distributions.” In New Statistics with R: An Introduction for Biologists, 1st ed., 121–40. Oxford (GB): Oxford university press.

“Logit Regression | R Data Analysis Examples.” n.d. Accessed May 20, 2022. https://stats.oarc.ucla.edu/r/dae/logit-regression/.

Monica, gung-Reinstate. 2013. “Answer to "How Does the Generalized Linear Model Generalize the General Linear Model?".” Cross Validated. June 27, 2013. https://stats.stackexchange.com/a/62692.

Perraillon, Marcelo Coca. n.d. “Interpreting Model Estimates: Marginal Effects.”

Perraillon, Marcelo Coca, and Richard C Lindrooth. n.d. “Chapter 6: Marginal Effects.” In Health Services Research and Program Evaluation.

“Poisson Regression | R Data Analysis Examples.” n.d. Accessed July 25, 2025. https://stats.oarc.ucla.edu/r/dae/poisson-regression/.