Converting an odds ratio to a range of plausible relative risks for better communication of research findings
BMJ 2014; 348 doi: https://doi.org/10.1136/bmj.f7450 (Published 24 January 2014) Cite this as: BMJ 2014;348:f7450
All rapid responses
Rapid responses are electronic comments to the editor. They enable our users to debate issues raised in articles published on bmj.com. A rapid response is first posted online. If you need the URL (web address) of an individual response, simply click on the response headline and copy the URL from the browser window. A proportion of responses will, after editing, be published online and in the print journal as letters, which are indexed in PubMed. Rapid responses are not indexed in PubMed and they are not journal articles. The BMJ reserves the right to remove responses which are being wilfully misrepresented as published articles or when it is brought to our attention that a response spreads misinformation.
From March 2022, the word limit for rapid responses will be 600 words not including references and author details. We will no longer post responses that exceed this limit.
The word limit for letters selected from posted responses remains 300 words.
Grant notes that the odds ratio will be further from 1 compared with the risk ratio if outcomes are common and expresses concern that authors sometimes misinterpret odds ratios as if they were risk ratios.1 He urges authors to convert adjusted odds ratios for a binary exposure to an adjusted risk ratio using a simple formula: Adjusted risk ratio = (adjusted odds ratio)/[(1 – risk0)+(risk0 x adjusted odds ratio)], where risk0 is the risk for the outcome among those not exposed. Unfortunately this formula will almost always produce biased estimates of the adjusted risk ratio.
Grant’s formula1 for converting an adjusted odds ratio to an adjusted risk ratio was described by Holland in 1989; see his equation 2.9.2 But Holland had second-thoughts, and in 1991 Greenland and Holland showed that this formula can produce biased estimates.3 The same formula was suggested anew by Zhang and Yu in 1998.4 Since then many authors have pointed out that this formula will produce adjusted risk ratios that are biased.5-9
The formula suggested by Grant1 and previous authors2,4 produces adjusted risk ratios that are biased away from 1. The formula will produce an unbiased risk ratio only when the adjusted odds ratio is also the average odds ratio. The adjusted odds ratio will equal the average odds ratio if all subjects have the same outcome risk when not exposed and all subjects experience the same change in outcome odds when exposed. This equality is unlikely as variation in outcome risk when not exposed is ubiquitous. Further details about the properties of odds ratios, risk ratios, and risk differences as effect estimates can be found in discussions by Greenland10 and myself.8
Adjusted risk ratios can be estimated for randomized trials or cohort studies, without using odds ratios from logistic regression.6 We can use Mantel-Haenszel methods,11 logistic regression with marginal standardization,6,7,9,12-15 generalized linear regression with a log link and binomial distribution,5,6,16-21 Poisson regression with a robust or GEE variance estimator,6,22 p724-730,23,24 and other25,26 methods. These methods can also estimate adjusted risk differences. Two articles describe how these approaches can be implemented in Stata software.27,28 For case-control studies, if the outcome is rare in the population, then odds ratios from logistic regression can be interpreted as risk ratios. When a common outcome is studied with the case-control design, other techniques can be used to estimate risk ratios.6
Sincerely yours,
Peter Cummings
Professor Emeritus of Epidemiology, University of Washington, Seattle WA
Address for correspondence: Peter Cummings, 250 Grandview Drive, Bishop CA 93514, USA
peterc@uw.edu
1. Grant RL. Converting an odds ratio to a range of plausible relative risks for better communication of research findings. BMJ. 2014;348:f7450.
2. Holland PW. A note on the covariance of the Mantel-Haenszel log-odds-ratio estimator and the sample marginal rates. Biometrics. 1989;45(3):1009-1016.
3. Greenland S, Holland P. Estimating standardized risk differences from odds ratios. Biometrics. 1991;47(1):319-322.
4. Zhang J, Yu KF. What's the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA. 1998;280(19):1690-1691.
5. McNutt L-A, Wu C, Xue X, Hafner JP. Estimating the relative risk in cohort studies and clinical trials of common outcomes. Am J Epidemiol. 2003;157(10):940-943.
6. Greenland S. Model-based estimation of relative risks and other epidemiologic measures in studies of common outcomes and in case-control studies. Am J Epidemiol. 2004;160(4):301-305.
7. Localio AR, Margolis DJ, Berlin JA. Relative risks and confidence intervals were easily computed indirectly from multivariable logistic regression. J Clin Epidemiol. 2007;60(9):874-882.
8. Cummings P. The relative merits of risk ratios and odds ratios. Arch Pediatr Adolesc Med. 2009;163(5):438-445.
9. Kleinman LC, Norton EC. What's the Risk? A simple approach for estimating adjusted risk measures from nonlinear models including logistic regression. Health Serv Res. 2009;44(1):288-302.
10. Greenland S. Interpretation and choice of effect measures in epidemiologic analyses. Am J Epidemiol. 1987;125(5):761-768.
11. Greenland S, Rothman KJ. Introduction to stratified analysis. In: Rothman KJ, Greenland S, Lash TL, eds. Modern Epidemiology. 3rd ed. Philadelphia: Lippincott Williams & Wilkins; 2008:258-282.
12. Lane PW, Nelder JA. Analysis of covariance and standardization as instances of prediction. Biometrics. 1982;38(3):613-621.
13. Flanders WD, Rhodes PH. Large sample confidence intervals for regression standardized risks, risk ratios, and risk differences. J Chronic Dis. 1987;40(7):697-704.
14. Greenland S. Estimating standardized parameters from generalized linear models. Stat Med. 1991;10(7):1069-1074.
15. Greenland S. Introduction to regression models. In: Rothman KJ, Greenland S, Lash TL, eds. Modern Epidemiology. 3rd ed. Philadelphia: Lippincott Williams & Wilkins; 2008:381-417.
16. Wacholder S. Binomial regression in GLIM: estimating risk ratios and risk differences. Am J Epidemiol. 1986;123(1):174-184.
17. Skov T, Deddens J, Petersen MR, Endahl L. Prevalence proportion ratios: estimation and hypothesis testing. Int J Epidemiol. 1998;27(1):91-95.
18. Robbins AS, Chao SY, Fonseca VP. What's the relative risk? A method to directly estimate risk ratios in cohort studies of common outcomes. Ann Epidemiol. 2002;12(7):452-454.
19. Blizzard L, Hosmer DW. Parameter estimation and goodness-of-fit in log binomial regression. Biom J. 2006;48(1):5-22.
20. Lumley T, Kronmal R, Ma S. Relative risk regression in medical research: models, contrasts, estimators, and algorithms. UW Biostatistics Working Paper Series. 2006(Paper 293). http://www.bepress.com/uwbiostat/paper293. Accessed September 9, 2008.
21. Deddens JA, Petersen MR. Approaches for estimating prevalence ratios. Occup Environ Med. 2008;65(7):501-506.
22. Wooldridge JM. Econometric Analysis of Cross Section and Panel Data. 2nd ed. Cambridge, MA: The MIT Press; 2010.
23. Zou G. A modified Poisson regression approach to prospective studies with binary data. Am J Epidemiol. 2004;159(7):702-706.
24. Carter RE, Lipsitz SR, Tilley BC. Quasi-likelihood estimation for relative risk regression models. Biostatistics. 2005;6(1):39-44.
25. Savu A, Liu Q, Yasui Y. Estimation of relative risk and prevalence ratio. Stat Med. 2010;29(22):2269-2281.
26. Chu H, Cole SR. Estimation of risk ratios in cohort studies with common outcomes: a Bayesian approach. Epidemiology. 2010;21(6):855-862.
27. Cummings P. Methods for estimating adjusted risk ratios. Stata J. 2009;9(2):175-196.
28. Cummings P. Estimating adjusted risk ratios for matched and unmatched data: an update. Stata J. 2011;11(2):290-298.
Competing interests: No competing interests
I agree heartily with Dr Grant that odds ratios are easily and frequently misinterpreted and I support his recommendations about the use of absolute and relative risks to facilitate comprehension by readers. Unfortunately, I think there are a number of small errors in the example that is included in this article.
Under the heading "Box: Definitions of odds and risks", the total for control is stated as 'c+d' but it should be 'b+d'.
Under the heading "An applied example", the allocation of 'a', 'b', 'c' and 'd' are not consistent with the definitions given earlier. For example, 'b' is meant to be 'Outcome present for control', but the label 'b' has been allocated to "Number of participants who did not quit at four weeks after the hospital based intervention", which equates to 'Outcome absent for treatment: c'. Even if the reader accepts the new labelling system in this example, the totals are not labelled correctly. The 'Total for the hospital based intervention' should be 'a+b', although the total of 260 is correct.
Fortunately, the relative risk and odds ratios at the end of the example are calculated correctly.
Is there are scope for these small errors to be corrected? I am also a journal editor and I would like to refer authors to this publication when encouraging them to report relative risks instead of odds ratios. Apart from the small errors, Dr Grant explains beautifully the many reasons for preferring relative risks to odds ratios.
Competing interests: No competing interests
Re: Converting an odds ratio to a range of plausible relative risks for better communication of research findings
Thank you to my two respondents.
I agree with Cummings that simply plugging the adjusted OR into the formula is not wise, hence ideally the use marginal effects calculations to take all covariates, interactions and non-linearities in the model into account. Thankfully, the implementation in Stata software is particularly simple for those analysing data without familiarity with the numerical integration that takes place behind the scenes for this.
Regarding the typos pointed out in the box, I do regret this - originally I wanted the a,b,c,d symbols to be in a familiar 2x2 table but this was not possible under house style and was rewritten by the typesetter. I checked and corrected some of it but obviously missed this - and I accept all responsibility for that. I will ask for it to be amended.
Competing interests: No competing interests