Clinical Review & Education
JAMA Guide to Statistics and Methods
Odds Ratios—Current Best Practice and Use
Edward C. Norton, PhD; Bryan E. Dowd, PhD; Matthew L. Maciejewski, PhD
Odds ratios frequently are used to present strength of association dard test is whether the parameter (log odds) equals 0, which cor-
between risk factors and outcomes in the clinical literature. Odds responds to a test of whether the odds ratio equals 1. Odds ratios
and odds ratios are related to the probability of a binary outcome typically are reported in a table with 95% CIs. If the 95% CI for an
(an outcome that is either present or absent, such as mortality). odds ratio does not include 1.0, then the odds ratio is considered to
The odds are the ratio of the probability that an outcome occurs to be statistically significant at the 5% level.
the probability that the outcome does not occur. For example, sup-
pose that the probability of mortality is 0.3 in a group of patients. What Are the Limitations of Odds Ratios?
This can be expressed as the odds of dying: 0.3/(1 − 0.3) = 0.43. Several caveats must be considered when reporting results with
When the probability is small, odds are virtually identical to the odds ratios. First, the interpretation of odds ratios is framed in
probability. For example, for a probability of 0.05, the odds are terms of odds, not in terms of probabilities. Odds ratios often are
0.05/(1 − 0.05) = 0.052. This similarity does not exist when the mistaken for relative risk ratios.2,3 Although for rare outcomes
value of a probability is large. odds ratios approximate relative risk ratios, when the outcomes
Probability and odds are different ways of expressing similar con- are not rare, odds ratios always overestimate relative risk ratios, a
cepts. For example, when randomly selecting a card from a deck, the problem that becomes more acute as the baseline prevalence of
probability of selecting a spade is 13/52 = 25%. The odds of select- the outcome exceeds 10%. Odds ratios cannot be calculated
ing a card with a spade are 25%/75% = 1:3. Clinicians usually are in- directly from relative risk ratios. For example, an odds ratio for
terested in knowing probabilities, whereas gamblers think in terms men of 2.0 could correspond to the situation in which the prob-
of odds. Odds are useful when wagering because they represent fair ability for some event is 1% for men and 0.5% for women. An odds
payouts. If one were to bet $1 on selecting a spade from a deck of ratio of 2.0 also could correspond to a probability of an event
cards, a payout of $3 is necessary to have an even chance of win- occurring 50% for men and 33% for women, or to a probability of
ning your money back. From the gambler’s perspective, a payout 80% for men and 67% for women.
smaller than $3 is unfavorable and greater than $3 is favorable. Second, and less well known, the magnitude of the odds ratio
Differences between 2 different groups having a binary out- from a logistic regression is scaled by an arbitrary factor (equal to
come such as mortality can be compared using odds ratios, the ra- the square root of the variance of the unexplained part of binary
tio of 2 odds. Differences also can be compared using probabilities outcome).4 This arbitrary scaling factor changes when more or bet-
by calculating the relative risk ratio, which is the ratio of 2 probabili- ter explanatory variables are added to the logistic regression model
ties. Odds ratios commonly are used to express strength of asso- because the added variables explain more of the total variation and
ciations from logistic regression to predict a binary outcome.1 reduce the unexplained variance. Therefore, adding more indepen-
dent explanatory variables to the model will increase the odds ratio
Why Report Odds Ratios From Logistic Regression? of the variable of interest (eg, treatment) due to dividing by a
Researchers often analyze a binary outcome using multivariable smaller scaling factor. In addition, the odds ratio also will change if
logistic regression. One potential limitation of logistic regression is the additional variables are not independent, but instead are corre-
that the results are not directly interpretable as either probabilities lated with the variable of interest; it is even possible for the odds
or relative risk ratios. However, the results from a logistic regression ratio to decrease if the correlation is strong enough to outweigh the
are converted easily into odds ratios because logistic regression change due to the scaling factor.
estimates a parameter, known as the log odds, which is the natural Consequently, there is no unique odds ratio to be estimated,
logarithm of the odds ratio. For example, if a log odds estimated by even from a single study. Different odds ratios from the same study
logistic regression is 0.4 then the odds ratio can be derived by cannot be compared when the statistical models that result in odds
exponentiating the log odds (exp(0.4) = 1.5). It is the odds ratio ratio estimates have different explanatory variables because each
that is usually reported in the medical literature. The odds ratio is model has a different arbitrary scaling factor.4-6 Nor can the magni-
always positive, although the estimated log odds can be positive or tude of the odds ratio from one study be compared with the mag-
negative (log odds of −0.2 equals odds ratio of 0.82 = exp(−0.2)). nitude of the odds ratio from another study, because different
The odds ratio for a risk factor contributing to a clinical out- samples and different model specifications will have different arbi-
come can be interpreted as whether someone with the risk factor trary scaling factors. A further implication is that the magnitudes of
is more or less likely than someone without that risk factor to expe- odds ratios of a given association in multiple studies cannot be syn-
rience the outcome of interest. Logistic regression modeling al- thesized in a meta-analysis.4
lows the estimates for a risk factor of interest to be adjusted for other
risk factors, such as age, smoking status, and diabetes. One nice fea- How Did the Authors Use Odds Ratios?
ture of the logistic function is that an odds ratio for one covariate is In a recent JAMA article, Tringale and colleagues7 studied industry
constant for all values of the other covariates. payments to physicians for consulting, ownership, royalties, and re-
Another nice feature of odds ratios from a logistic regression is search as well as whether payments differed by physician specialty
that it is easy to test the statistical strength of association. The stan- or sex. Industry payments were received by 50.8% of men across
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JAMA Guide to Statistics and Methods Clinical Review & Education
all specialties compared with 42.6% of women across all special- more likely to receive payments than women, even after control-
ties. Converting these probabilities to odds, the odds that men re- ling for confounders. The magnitude of the odds ratio, about 1.4,
ceive industry payments is 1.03 (0.51/0.49), and the odds that indicates the direction of the effect, but the magnitude of the num-
women receive industry payments is 0.74 = (0.43/0.57). ber itself is hard to interpret. The estimated odds ratio is 1.4 when
The odds ratio for men compared with women is the ratio of simultaneously accounting for specialty, spending region, sole pro-
the odds for men divided by the odds for women. In this case, the prietor status, sex, and the interaction between specialty and sex.
unadjusted odds ratio is 1.03/0.74 = 1.39. Therefore, the odds for A different odds ratio would be found if the model included a differ-
men receiving industry payments are about 1.4 as large (40% ent set of explanatory variables. The 1.4 estimated odds ratio
higher) compared with women. Note that the ratio of the odds is should not be compared with odds ratios estimated from other
different than the ratio of the probabilities because the probability data sets with the same set of explanatory variables, or to odds
is not close to 0. The unadjusted ratio of the probabilities for men ratios estimated from this same data set with a different set of
and women (Tringale et al7 report each probability, but not the explanatory variables.4
ratio), the relative risk ratio, is 1.19 (0.51/0.43).
Greater odds that men may receive industry payments may be What Caveats Should the Reader Consider?
explained by their disproportionate representation in specialties Odds ratios are one way, but not the only way, to present an asso-
more likely to receive industry payments. After controlling for spe- ciation when the main outcome is binary. Tringale et al7 also report
cialty (and other factors), the estimated odds ratio was reduced from absolute rate differences. The reader should understand odds ra-
1.39 to 1.28, with a 95% CI of 1.26 to 1.31, which did not include 1.0 tios in the context of other information, such as the underlying prob-
and, therefore, is statistically significant. The odds ratio probably de- ability. When the probabilities are small, odds ratios and relative risk
clined after adjusting for more variables because they were corre- ratios are nearly identical, but they can diverge widely for large prob-
lated with physicians’ sex. abilities. The magnitude of the odds ratio is hard to interpret be-
cause of the arbitrary scaling factor and cannot be compared with
How Should the Findings Be Interpreted? odds ratios from other studies. It is best to examine study results pre-
In exploring the association between physician sex and receiving sented in several ways to better understand the true meaning of
industry payments, Tringale and colleagues7 found that men are study findings.
ARTICLE INFORMATION Conflict of Interest Disclosures: All authors have catheterization. N Engl J Med. 1999;341(4):279-283.
Author Affiliations: Department of Health completed and submitted the ICMJE Form for doi:10.1056/NEJM199907223410411
Management and Policy, Department of Disclosure of Potential Conflicts of Interest . 3. Holcomb WL Jr, Chaiworapongsa T, Luke DA,
Economics, University of Michigan, Ann Arbor Dr Maciejewski reported receiving personal fees Burgdorf KD. An odd measure of risk: use and
(Norton); National Bureau of Economic Research, from the University of Alabama at Birmingham misuse of the odds ratio. Obstet Gynecol. 2001;98
Cambridge, Massachusetts (Norton); Division of for a workshop presentation; receiving grants from (4):685-688.
Health Policy and Management, School of Public NIDA and the Veterans Affairs; receiving a contract
from NCQA to Duke University for research; 4. Norton EC, Dowd BE. Log odds and the
Health, University of Minnesota, Minneapolis interpretation of logit models. Health Serv Res.
(Dowd); Center for Health Services Research in being supported by a research career scientist
award 10-391 from the Veterans Affairs Health 2018;53(2):859-878. doi:10.1111/1475-6773.12712
Primary Care, Durham Veterans Affairs Medical
Center, Durham, North Carolina (Maciejewski); Services Research and Development; and that his 5. Miettinen OS, Cook EF. Confounding: essence
Department of Population Health Sciences, spouse owns stock in Amgen. No other disclosures and detection. Am J Epidemiol. 1981;114(4):593-603.
Duke University School of Medicine, Durham, were reported. doi:10.1093/oxfordjournals.aje.a113225
North Carolina (Maciejewski); Division of General 6. Hauck WW, Neuhaus JM, Kalbfleisch JD,
Internal Medicine, Department of Medicine, Duke REFERENCES Anderson S. A consequence of omitted covariates
University School of Medicine, Durham, North 1. Meurer WJ, Tolles J. Logistic regression when estimating odds ratios. J Clin Epidemiol. 1991;
Carolina (Maciejewski). diagnostics: understanding how well a model 44(1):77-81. doi:10.1016/0895-4356(91)90203-L
Section Editors: Roger J. Lewis, MD, PhD, predicts outcomes. JAMA. 2017;317(10):1068-1069. 7. Tringale KR, Marshall D, Mackey TK, Connor M,
Department of Emergency Medicine, Harbor-UCLA doi:10.1001/jama.2016.20441 Murphy JD, Hattangadi-Gluth JA. Types and
Medical Center and David Geffen School of 2. Schwartz LM, Woloshin S, Welch HG. distribution of payments from industry to
Medicine at UCLA; and Edward H. Livingston, MD, Misunderstandings about the effects of race and physicians in 2015. JAMA. 2017;317(17):1774-1784.
Deputy Editor, JAMA. sex on physicians’ referrals for cardiac doi:10.1001/jama.2017.3091
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