Odds Ratios—Current Best Practice and Use

Odds Ratios—Current B est 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

between risk factors and outcome s in the clinical literature. Odds

and odds ratios are related to the probability of a binary outcome

(an outcome that is either present or absent, such as mor tality).

The odds are the ratio of the probability that an outcome occurs to

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.

This can be expressed as the odds of dying: 0.3/(1 − 0.3) = 0.43.

When the probability is small, odds are virtually identical to the

probability. For example, for a probability of 0.05, the odds are

0.05/(1 − 0.05) = 0.052. This similarity does not exist when the

value of a probability is large.

Probability and odds are different ways of expressing similar con-

cepts. Forexample,when randomly selecting a card from a deck, the

probability of selecting a spade is 13/52 = 25%. The odds of select-

ing a card with a spade are 25%/75% = 1:3. Clinicians usually are in-

terested in knowing probabilities, whereas gamblers think in terms

of odds. Odds are useful when wageringbecause they represent fair

payouts. If one were to bet $1 on selecting a spade from a deck of

cards, a payout of $3 is necessar y to have an even chance of win-

ning your money back. From the gambler’s perspective, a payout

smaller than $3 is unfavorable and greater than $3 is favorable.

Differences between 2 different groups having a binary out-

come such as mortality can be compared using odds ratios, the ra-

tio of 2 odds. Differences also can be compared using probabilities

by calculating the relative risk ratio, which is the ratio of 2 probabili-

ties. Odds ratios commonly are used to express strength of asso-

ciations from logistic regression to predict a binary outcome.

Why Report Odds Ratios From Logistic Regression?

Researchers often analyze a binar y outcome using multivariable

logistic regression. One potential limitation of logistic regression is

that the results are not directly interpretable as either probabilities

or relative risk ratios. However, the results from a logistic regression

are converted easily into odds ratios because logistic regression

estimates a parameter, known as the log odds, which is the natural

logarithm of the odds ratio. For example, if a log odds estimated b y

logistic regression is 0.4 then the odds ratio can be derived by

exponentiating the log odds (exp(0.4) = 1.5). It is the odds ratio

that is usually reported in the medical literature. The odds ratio is

always positive, although the estimated log odds can be positive or

negative (log odds of −0 .2 equals odds ratio of 0.82 = exp(−0 .2)).

The odds ratio for a risk factor contributing to a clinical out-

come can be interpreted as whether someone with the risk factor

is more or less likely than someone without that risk factor to expe-

rience the outcome of intere st. Logistic regression modeling al-

lows the estimates for arisk factor of interest to be adjusted for other

risk factors, such as age, smoking status, and diabetes. One nice fea-

ture of the logistic function is that an odds ratio for one covariate is

constant for all values of the other covariates.

Another nice feature of odds ratios from a logistic regression is

that it is easy to test the statistical strength of association. Thestan-

dard test is whether the parameter (log odds) equals 0, which cor-

responds to a test of whether the odds ratio equals 1. Odds ratios

typically are reported in a table with 95% CIs. If the 95% CI for an

odds ratio does not include 1.0, then the odds ratio is considered to

be statistically significant at the 5% level.

What Are the Limitations of Odds Ratios?

Several caveats must be considered when reporting results with

odds ratios. First, the interpre tation of odds ratios is framed in

terms of odds, not in terms of probabilities. Odds ratios often are

mistaken for relative risk ratios.

2,3

Although for rare outcomes

odds ratios approximate relative risk ratios, when the outcomes

are not rare, odds ratios always overestimate relative risk ratios, a

problem that becomes more acute as the baseline prevalence of

the outcome exceeds 10%. Odds ratios cannot be calculated

directly from relative risk ratios. For example, an odds ratio for

men of 2.0 could correspond to the situation in which the prob-

ability for some event is 1% for men and 0.5 % for women. An odds

ratio of 2.0 also could correspond to a probability of an event

occurring 50% for men and 33% for women, or to a probability of

80% for men and 6 7% for women.

Second, and less well known, the magnitude of the odds ratio

from a logistic regression is scaled by an arbitrary factor (equal to

the square root of the variance of the unexplained par t of binary

outcome).

This arbitrary scaling factor changes when more or bet-

ter explanatory variables are added to the logistic regression model

because the added v ariables explain more of the total v ariation and

reduce the unexplained variance . Therefore, adding more indepen-

dent explanatory variables to the model will increase the odds ratio

of the variable of interest (eg, treatment) due to dividing by a

smaller scaling factor. In addition, the odds ratio also will change if

the additional variables are not independent, but instead are corre-

lated with the variable of interest; it is even possible for the odds

ratio to decrease if the correlation is strong enough to outweigh the

change due to the scaling f actor.

Consequently, there is no unique odds ratio to be estimated,

even from a single study. Different odds ratios from the same study

cannot be compared when the statistical models that result in odds

ratio estimates have different explanatory variables because each

model has a different arbitrary scaling factor.

4-6

Nor can the magni-

tude of the odds ratio from one study be compared with the mag-

nitude of the odds ratio from another study, because different

samples and different model specifications will have different arbi-

trary scaling factors. A further implication is that the magnitudes of

odds ratios of a given association in multiple studies cannot be syn-

thesized in a meta-analysis.

How Did the Authors Use Odds Ratios?

In a recent JAMA article, Tringale and colleagues

studied industry

payments to physicians for consulting, ownership, royalties, and re-

search as well as whether payments differed by physician specialty

or sex. Industry payments were received by 50.8% of men across

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all specialties compared with 42.6% of women across all special-

ties. Converting these probabilities to odds, the odds that men re-

ceive industry payments is 1.03 (0.51/0.49), and the odds that

women receive industry payments is 0.74 = (0.43/0.57).

The odds ratio for men compared with women is the ratio of

the odds for men divided by the odds for women. In this case, the

unadjusted odds ratio is 1.03/0.74 = 1.39. Therefore, the odds for

men receiving industry payments are about 1.4 as large (40%

higher) compared with women. Note that the ratio of the odds is

different than the ratio of the probabilities because the probability

is not close to 0. The unadjusted ratio of the probabilities for men

and women (Tringale et al

report each probability, but not the

ratio), the relative risk ratio , is 1.19 (0.5 1/ 0.43).

Greater odds that men may receive industry payments may be

explained by their dispropor tionate representation in specialties

more likely to receive industry payments. After controlling for spe-

cialty (and other factors), theestimated odds ratio was reduced from

1.39 to 1.28, with a 95% CI of 1.26 to 1.31, which did not include 1.0

and, therefore, is statistically significant. The odds ratio probably de-

clined after adjusting for more variables because they were corre-

lated with physicians’ sex.

How Should the Findings Be Interpreted?

In exploring the association be tween physician sex and receiving

industry payments, Tringale and colleagues

found that men are

more likely to receive payments than women, even after control-

ling for confounders. The magnitude of the odds ratio, about 1.4,

indicates the direction of the effect, but the magnitude of the num-

ber itself is hard to interpret. The estimated odds ratio is 1.4 when

simultaneously accounting for specialty, spending region, sole pro-

prietor status, sex, and the interaction between specialty and sex.

A different odds ratio would be found if the model included a differ-

ent set of explanatory variables. The 1.4 estimated odds ratio

should not be compared with odds ratios estimated from other

data se ts with the same set of explanatory variables, or to odds

ratios estimated from this same data set with a different se t of

explanatory v ariables.

What Caveats Should the Reader Consider?

Odds ratios are one way, but not the only way, to present an asso-

ciation when the main outcome is binary. Tringale et al

also report

absolute rate differences. The reader should understand odds ra-

tios in the contextof other information, such as the underlying prob-

ability. When the probabilities are small, odds ratios and relative risk

ratios are nearly identical, but they can diverge widely for large prob-

abilities. The magnitude of the odds ratio is hard to interpret be-

cause of the arbitrary scaling factor and cannot be compared with

odds ratios from other studies. Itis best to examine study results pre-

sented in several ways to better understand the true meaning of

study findings.

ARTICLE INFORMATION

Author Affiliations: Department of Health

Management and Policy, Department of

Economics, University of Michigan, Ann Arbor

(Norton); National Bureau of Economic Research,

Cambridge, Massachusetts (Norton); Division of

Health Policy and Management, School of Public

Health, University of Minnesota, Minneapolis

(Dowd); Center for Health Services Research in

Primary Care, Durham Veterans Affairs Medical

Center, Durham, North Carolina (Maciejewski);

Department of Population Health Sciences,

Duke University School of Medicine, Durham,

North Carolina (Maciejewski); Division of General

Internal Medicine, Department of Medicine, Duke

University School of Medicine, Durham, North

Carolina (Maciejewski).

Section Editors: Roger J. Lewis, MD, PhD,

Department of Emergency Medicine, Harbor-UCLA

Medical Center and David Geffen School of

Medicine at UCLA; and Edward H. Livingston, MD,

Deputy Editor, JAMA.

Conflict of Interest Disclosures: All authors have

completed and submitted the ICMJE Form for

Disclosure of Potential Conflicts of Interest .

Dr Maciejewski repor ted receiving personal fees

from the University of Alabama at Birmingham

for a workshop presentation; receiving grants from

NIDA and the Veterans Affairs; receiving a contract

from NCQA to Duke University for research;

being supported by a research career scientist

award 10-391 from the Veterans Affairs Health

Services Research and Development; and that his

spouse owns stock in Amgen. No other disclosures

were reported.

REFERENCES

1. Meurer WJ, Tolles J. Logistic regression

diagnostics: understanding how well a model

predicts outcomes. JAMA. 2017;317(10):1068-1069.

doi:10.1001/jama.2016.20441

2. Schwartz LM, Woloshin S, Welch HG.

Misunderstandings about the effects of race and

sex on physicians’ referrals for cardiac

catheterization. N Engl J Med. 1999;341(4):279-283.

doi:10.1056/NEJM199907223410411

3. Holcomb WL Jr, Chaiworapongsa T, Luke DA,

Burgdorf KD. An odd measure of risk: use and

misuse of the odds ratio. Obstet Gynecol. 2001;98

(4):685-688.

4. Norton EC, Dowd BE. Log odds and the

interpretation of logit models. Health Serv Res.

2018;53(2):859-878. doi:10.1111/1475-6773.12712

5. Miettinen OS, Cook EF. Confounding: essence

and detection. Am J Epidemiol. 1981;114(4):593-603.

doi:10.1093/oxfordjournals.aje.a113225

6. Hauck WW, Neuhaus JM, Kalbfleisch JD,

Anderson S. A consequence of omitted covariates

when estimating odds ratios. J Clin Epidemiol. 1991;

44(1):77-81. doi:10.1016/0895-4356(91)90203-L

7. Tringale KR, Marshall D, Mackey TK, Connor M,

Murphy JD, Hattangadi-Gluth JA. Types and

distribution of payments from industry to

physicians in 2015. JAMA. 2017;317(17):1774-1784.

doi:10.1001/jama.2017.3091

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