REVIEW ARTICLE

The Relative Merits of Risk Ratios and Odds Ratios

Peter Cummings, MD, MPH

W
hen a study outcome is rare in all strata used for an analysis, the odds ratio es-
timate of causal effects will approximate the risk ratio; therefore, odds ratios from
most case-control studies can be interpreted as risk ratios. However, if a study
outcome is common, the odds ratio will be further from 1 than the risk ratio.
There is debate regarding the merits of risk ratios compared with odds ratios for the analysis of
trials and cohort and cross-sectional studies with common outcomes. Odds ratios are conve-
niently symmetrical with regard to the outcome definition; the odds ratio for outcome Y is the in-
verse of the odds ratio for the outcome not Y. Risk ratios lack this symmetry, so it may be neces-
sary to present 1 risk ratio for outcome Y and another for outcome not Y. Risk ratios, but not odds
ratios, have a mathematical property called collapsibility; this means that the size of the risk ratio
will not change if adjustment is made for a variable that is not a confounder. Because of collapsibil-
ity, the risk ratio, assuming no confounding, has a useful interpretation as the ratio change in av-
erage risk due to exposure among the exposed. Because odds ratios are not collapsible, they usu-
ally lack any interpretation either as the change in average odds or the average change in odds (the
average odds ratio). Arch Pediatr Adolesc Med. 2009;163(5):438-445

For more than 20 years, there has been de- risk) of the outcome is A/(A⫹B) among
bate about the relative merits of risk ratios those exposed (Table 1) and C/(C ⫹D)
compared with odds ratios as estimates of among those not exposed; the correspond-
causal associations between an exposure ing odds are A/B and C/D, respectively. The
(such as smoking or medication for high risk ratio is therefore [A/(A ⫹ B)]/[C/
blood pressure) and a binary outcome (such (C ⫹ D)] and the odds ratio is (A/B)/(C/
as death vs life). In this article, I discuss how D). In the literature about these ratios,
these measures differ and review argu- there are 2 areas where there is general
ments for each. To limit my discussion, I agreement: if the outcome is rare, the odds
will ignore other useful measures of asso- ratio will approximate the risk ratio; and
ciation, such as the rate ratio, hazard ratio, in most case-control studies, odds ratios
risk difference, and rate difference. will approximate risk ratios.

AGREEMENTS REGARDING RISK Approximation of Risk Ratios by Odds

RATIOS AND ODDS RATIOS Ratios When Outcomes Are Rare

In a clinical trial or cohort study in which If the study outcome is rare among those
all subjects are followed up for the same exposed (Table 1), then A will be small
time, the cumulative incidence (average relative to B, so the risk (A/[A⫹B]) will
be close to the odds (A/B). Similarly, if the
Author Affiliations: Department of Epidemiology, School of Public Health and outcome is rare among those not ex-
Community Medicine, and Harborview Injury Prevention and Research Center, posed, then C will be small relative to D
University of Washington, Seattle. and [C/(C⫹ D)] will be close to (C/D). If

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 Table 1. Data From a Hypothetical Clinical Trial .50 .25 .10 .01
or Cohort Study a 10.0

Outcome 5.0

Exposed Yes No Risk of Outcome Odds of Outcome

Yes A B A/(A ⫹ B) A/B

Odds Ratio, Log Scale

No C D C/(C⫹ D) C/D

a A, B, C, and D are counts according to exposure and outcome. Formulae 1.0

for risk ratios and odds ratios (e, exposed; ne, not exposed; R, individual
risks for outcome = 1): Count A=sum of risks for outcome=1 if 0.5
exposed = sume(Re). Count B=sum of risks for outcome=0 if
exposed = sume(1 − Re). Count C =sum of risks for outcome=1 if not
exposed = sumne(Rne). Count D =sum of risks for outcome=0 if not
exposed = sumne(1 − Rne). Ratio change in cumulative incidence
risk = [A/(A ⫹B)] ÷ [C/(C ⫹D)]=sume (Re)/(A⫹B)÷sumne (Rne)/(C ⫹ D) = sume
(Re)/(A ⫹ B) ÷ sume(Rne)/(A ⫹B)=ratio change in average risk due to exposure
0.1
for the exposed. Average ratio change in risk due to exposure for the
exposed = average risk ratio=sume(Re /Rne)/(A⫹B). Ratio change in
0.1 0.5 1.0 5.0 10.0
cumulative incidence odds=(A/B)÷ (C/D)=sume(Re)/sume(1− Re)÷ sumne(Rne)/
Risk Ratio, Log Scale
sumne(1 − Rne) = sume (Re)/sume(1− Re)÷sume(Rne)/sume(1− Rne). Ratio change
in average odds due to exposure for the exposed=sume [Re /(1− Re)]/(A ⫹ B)
÷ sume [Rne /(1 −Rne)]/(A ⫹ B). Average ratio change in odds due to exposure Figure 1. Relationship of the odds ratio to the risk ratio according to 4 levels
for the exposed = average odds ratio=sume([Re /(1− Re)]÷ [Rne /(1− Rne)]). of outcome risk (cumulative incidence) for unexposed subjects: .01, .10, .25,
and .50.

Table 2. Hypothetical Data for a Trial of Drug X

10.0
Outcome, No.

Treatment Died Survived Risk of Death Odds of Death 5.0

Drug X 25 75 25/(25 ⫹75)=.25 25/75 = 0.33

Placebo 50 50 50/(50 ⫹50)=.50 50/50 = 1.00
Odds Ratio, Log Scale

the outcome is rare in both exposed and unexposed per- 1.0

sons, the odds ratio ([A/B]/[C/D]) will approximate the
risk ratio ([A/(A ⫹B)]/[C/(C⫹D)]). 0.5
However, when the study outcome is common and the
risk ratio is not close to 1, the odds ratio will be further
from 1 compared with the risk ratio. If the risk ratio is
greater than 1, the odds ratio will be greater still, and if
the risk ratio is smaller than 1, the odds ratio will be even 0.1
.01 .10 .25 .50
smaller. A hypothetical randomized trial of drug X is pre-
sented in Table 2; the risk ratio for death among pa- 0.1 0.5 1.0 5.0 10.0
Risk Ratio, Log Scale
tients given X, compared with those given placebo, is .25/
.50=0.5. The corresponding odds ratio is 0.33/1=0.33,
which is further from 1. Figure 2. Relationship of the odds ratio to the risk ratio according to 4 levels
of outcome risk (cumulative incidence) for exposed subjects: .01, .10, .25,
Figure 1 shows the relationship between odds and and .50.
risk ratios according to 4 levels of risk among unex-
posed persons; Figure 2 uses 4 risk levels for exposed sure subgroups formed by levels of the confounding vari-
persons; and Figure 3 shows 4 levels of average risk for able.1 For example, in hypothetical data for a cohort study
both unexposed and exposed subjects. When the out- of traffic crashes (Table 3), death was uncommon among
come risk is .01 or less, odds ratios and risk ratios agree those wearing a seat belt (risk=150/5500=.027) and among
well for risk ratio values ranging from 0.1 to 10 in all 3 those not wearing a seat belt (risk=300/5500=.055). The
figures. When cumulative incidence is .10, the odds ra- crude (unadjusted) odds ratio for death among belted oc-
tio is within 10% of the risk ratio for risk ratios ranging cupants compared with unbelted occupants is 0.49, which
from 0.1 to 1.8 in Figure 1, from 0.55 to 10 in Figure 2, is close to the risk ratio of 0.50. However, 375 of the 450
and from 0.4 to 2.5 in Figure 3. When cumulative inci- deaths (83%) were in high-speed crashes, in which 25%
dence is .25 or greater, the odds ratio differs notably from of those wearing seat belts and 50% of those not wearing
most of the risk ratios in all 3 figures. seat belts died. Using logistic regression to adjust for speed,
Even if the outcome is uncommon among all persons the adjusted odds ratio is 0.36; this does not approximate
in a study, the odds ratio may not approximate the risk the adjusted risk ratio of 0.5 well. Estimates such as those
ratio well if adjustment is made for potential confound- in Figures 1, 2, and 3 should be used with caution be-
ing variables and the outcome is not rare in some expo- cause they fail to account for the possibility that out-

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 10.0 Table 3. Hypothetical Cohort Study of Seat Belt Use and
Death in a Traffic Crash

5.0
Vehicle Outcome, No.
Crash Seat Belt Risk Odds
Speed Used Died Survived Risk Ratio Odds Ratio
Odds Ratio, Log Scale

Low Yes 25 4975 .005 0.005

0.50 0.50
No 50 4950 .010 0.010
1.0 High Yes 125 375 .250 0.333
0.50 0.33
No 250 250 .500 1.000
Total 450 10 550 .041 0.043
0.5

Ease of Interpretation of Risk Ratios by Clinicians

0.1
Some argue that risk ratios should be preferred because
.01 .10 .25 .50 they are more easily understood by clinicians.6,7 How-
0.1 0.5 1.0 5.0 10.0 ever, if odds ratios were otherwise superior, a better so-
Risk Ratio, Log Scale lution might be to use odds ratios and remedy any defi-
ciency in clinician education.
Figure 3. Relationship of the odds ratio to the risk ratio according to 4 levels
of outcome risk (cumulative incidence) for an average exposed and Symmetry of Odds Ratios Regarding
unexposed subject: .01, .10, .25, and .50. Estimates assume the number of Outcome Definitions
exposed subjects is equal to the number unexposed.

Some authors prefer odds ratios because they are sym-

metrical with regard to the outcome definition. Imagine
comes might be common in some exposure subgroups that a hypothetical trial of drug X and outcome Y (Y =death
contribute a notable portion of the outcomes. in Table 2). There is symmetry for both the odds and risk
ratios with regard to the definition of the exposure: both
Approximation of Risk Ratios by Odds Ratios ratio estimates for treatment with X compared with no
in Most Case-Control Studies X are the inverse of the ratio estimates for no X com-
pared with treatment with X.
Case-control studies are typically (but not always) used However, if we change the definition of the outcome
when outcomes are rare in the population from which from the occurrence of Y to no occurrence of Y, only the
study subjects are sampled. Outcome risks and odds of- odds ratio is symmetrical. The odds ratio for Y among
ten cannot be estimated directly from case-control data, those treated with X compared with those who did not
because the sampling proportions of cases and controls get X is (A/B)/(C/D) = (25/75)/(50/50) = (1/3)/(1) = 0.33
may be unknown. However, the odds ratio for the out- (Table 2). The odds ratio for no occurrence of Y among
come, (A/B)/(C/D) in Table 1, can be rewritten as (A/C)/ those treated with drug X compared with those who did
(B/D), which is the odds of exposure among the se- not get X is (B/A)/(D/C)=(75/25)/(50/50)=3/1=3. These
lected cases (persons with the outcome) divided by the odds ratios are simply the reciprocal of each other. The
odds of exposure among the selected controls (persons corresponding risk ratios are [A/(A⫹B)]/[C/(C⫹D)]=
without the outcome). This ratio can be estimated from (25/100)/(50/100) = 0.5 and [B/(A ⫹ B)]/[D/(C ⫹ D)] =
case-control data and it will approximate the risk ratio (75/100)/(50/100)=1.5; these risk ratios are not recip-
in the population from which the cases and controls were rocal. The symmetry property of the odds ratio is attractive
sampled when outcomes are rare in that population. This because 1 odds ratio can summarize the association of X
insight was described in 19512 and contributed to the use- with Y, and the choice between outcome Y and outcome
fulness of case-control studies. not Y is unimportant.8-12
For completeness, I note that there are case-control If outcome events are rare, the odds ratio and the risk
designs in which controls are sampled at the time each ratio for rare outcomes will be similar. The odds ratio for
case outcome occurs. In this design, the odds ratio will no event will be the inverse of the odds ratio for the event.
estimate the incidence rate ratio even when outcomes are The risk ratio for no event will necessarily be close to 1
common; no rare outcome assumption is needed.3-5 (Figure 4) and therefore of little interest. Thus, when
outcome events are rare, the symmetry issue is typically
ARGUMENTS REGARDING RISK RATIOS not important.
AND ODDS RATIOS
Constancy of Odds Ratios for Common Outcomes
There is debate regarding the merits of risk ratios com-
pared with odds ratios for the analysis of controlled trials, Some authors prefer odds ratios because they believe a
cohort studies, and cross-sectional studies with com- constant (homogenous) odds ratio may be more plau-
mon outcomes. I will discuss 4 arguments that have been sible than a constant risk ratio when outcomes are com-
used to advocate for one ratio or the other. mon. Risks range from 0 to 1. Risk ratios greater than 1

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 have an upper limit constrained by the risk when not
exposed. For example, if the risk when not exposed is 10.0
.5, the risk ratio when exposed cannot exceed 2:
.5⫻2=1. In a population with an average risk ratio of 2 5.0
for outcome Y among those exposed to X, assuming
that the risk for Y if not exposed to X varies from .1 to

Risk Ratio for Not Y, Log Scale

.9, the average risk ratio must be less than 2 for those
with risks greater than .5 when not exposed. Because
the average risk ratio for the entire population is 2, the 1.0
average risk ratio must be more than 2 for those with .01

risks less than .5 when not exposed. Therefore, a risk 0.5

ratio of 2 cannot be constant (homogeneous) for all
individuals in a population if risk when not exposed is
sometimes greater than .5. More generally, if the aver-
age risk ratio is greater than 1 in a population, the indi-
vidual risk ratios cannot be constant (homogeneous) 0.1
.90 .50 .25 .10
for all persons if any of them have risks when not
exposed that exceed 1/average risk ratio. 0.1 0.5 1.0 5.0 10.0
Odds range from 0 to infinity. Odds ratios greater Risk Ratio for Y, Log Scale
than 1 have no upper limit, regardless of the outcome
odds for persons not exposed. If we multiply any unex- Figure 4. Relationship of the risk ratio for outcome Y to the risk ratio for
posed outcome odds by an exposure odds ratio greater outcome not Y, according to 5 levels of outcome risk (cumulative incidence)
for unexposed subjects: .01, .10, .25, .50, and .90.
than 1 and convert the resulting odds when exposed to
a risk, that risk will fall between 0 and 1.8,11 Thus, it is
always hypothetically possible for an odds ratio to be tios, each an estimate of the change in average risk within
constant for all individuals in a population.12-14 a particular subgroup.

Rebuttals to Odds Ratio Constancy Argument Lack of Collapsibility−A Barrier to Estimating Odds Ra-
tios That Can Be Interpreted as Averages. It has been
Possibility of Constancy for Risk Ratios Less Than 1. known for years that risk ratios are collapsible while odds
For both risk and odds, the lower limit is 0. For any level ratios are not,1,11,16-18 but this is not mentioned in much
of risk or odds under no exposure, multiplication by a of the literature about these ratios. Some readers may find
risk or odds ratio less than 1 will produce a risk or odds this topic unfamiliar, technical, and even counterintui-
given exposure that is possible: 0 to 1 for risks and 0 to tive. To simplify my discussion, I will assume that there
infinity for odds. Thus, a constant risk or odds ratio is is no bias due to confounding. Had exposure not oc-
possible for ratios less than 1. If the risk ratio compar- curred, the average risk for the outcome would have been
ing exposed persons with those not exposed is greater the same in the exposed group as in the unexposed group.
than 1, the ratio can be inverted to be less than 1 by com- Collapsibility means that in the absence of confound-
paring persons not exposed with those exposed. There- ing, a weighted average of stratum-specific ratios (eg, using
fore, a constant risk ratio less than 1 is hypothetically pos- Mantel-Haenszel methods19) will equal the ratio from a
sible. This argument has been used to rebut the criticism single 2⫻ 2 table of the pooled (collapsed) counts from
of the risk ratio in the previous argument.15 the stratum-specific tables. This means that a crude (un-
adjusted) ratio will not change if we adjust for a variable
Constancy of Risk or Odds Ratios Not Necessary for that is not a confounder.18,20
Inference. It is not necessary that the risk or odds ratio I created hypothetical data for 2 randomized con-
be constant. We can use a ratio to make causal infer- trolled trials (Table 4 and Table 5). In both trials, the
ences or decisions about public and personal health if risk for the outcome, aside from exposure to treatment,
the ratio has an interpretation as an average effect: falls into just 2 categories. There is no confounding, as
either (1) the ratio change in average risk or odds or (2) the average risk without exposure is the same in each trial
the average ratio change in risk or odds. The ratio arm. Individual risks without exposure would not be
change in average risk (or odds) is not the same as the known in an actual trial; if they were knowable, we would
average ratio change in risk (or odds) (Table 1). not need a control group. The key difference between the
An estimate of the change in average risk is useful, tables is that the risk ratio is constant for all persons in
though it may not estimate the change expected for ev- Table 4, whereas the odds ratio is constant in Table 5.
ery individual or even for any particular individual. For In Table 4, the risk ratio due to exposure is 3.00 for
example, if research suggests that wearing a seat belt in all exposed persons, regardless of outcome risk (.05 or
a crash reduces the average risk of death by 50%, people .25) if not exposed. In a real trial, we could observe only
could use this information to make a decision about seat the data in the last column, where the risk ratio is 3.00
belt use, although this estimate might not apply to them for males, females, and all persons. The risk ratio of 3.00
in a given crash. If we had evidence that the average ra- for all persons has 3 interpretations1: (1) the ratio change
tio change in risk varied with levels of another factor, say in risk due to exposure for the exposed group, (2) the
age, the best choice might be to estimate several risk ra- ratio change in the average risk due to exposure among

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 Table 4. Hypothetical Randomized Trial of Treatment X and Outcome Y a

Low-Risk Subjects High-Risk Subjects All Subjects

Y No Y Y No Y Y No Y
Males
Treatment X, No.
Yes 15 85 225 75 240 160
No 5 95 75 225 80 320
Risk ratio 3.00 3.00 3.00
Odds ratio 3.35 9.00 6.00
Females
Treatment X, No.
Yes 45 255 75 25 120 280
No 15 285 25 75 40 360
Risk ratio 3.00 3.00 3.00
Odds ratio 3.35 9.00 3.86
All Persons
Treatment X, No.
Yes 60 340 300 100 360 440
No 20 380 100 300 120 680
Risk ratio 3.00 3.00 3.00
Odds ratio 3.35 9.00 4.64

a The risk for outcome Y, aside from exposure, is either low (.05) or high (.25). In this table, risk ratios are the same for all subjects. Ratio change in average
risk due to exposure among the exposed=([.05 ⫻(15 ⫹85 ⫹ 45 ⫹ 255)⫻ 3.00]⫹ [.25⫻ (225⫹ 75⫹ 75⫹ 25)⫻ 3.00]) ÷ ([.05⫻ (15⫹ 85⫹ 45⫹ 255)]⫹[.25⫻
(225 ⫹ 75 ⫹ 75 ⫹25)])=3.00. Average ratio change in risk for all exposed individuals (average risk ratio) = ([3.00 ⫻ (15⫹ 85)]⫹ [3.00⫻ (225⫹ 75)]⫹ [3.00 ⫻
(45 ⫹ 255)] ⫹ [3.00 ⫻ (75⫹ 25)])÷(15⫹ 85⫹ 225⫹75 ⫹ 45 ⫹255 ⫹ 75⫹ 25) = 3.00. Ratio change in average odds due to exposure among the
exposed = ([(.05/.95) ⫻(15 ⫹85 ⫹45 ⫹255)⫻ 3.35] ⫹[(.25/.75)⫻ (225⫹ 75⫹ 75 ⫹ 25)⫻ 9.00]) ÷ ([(.05/.95) ⫻ (15⫹ 85⫹ 45⫹ 255)]⫹
[(.25/.75) ⫻ (225 ⫹ 75 ⫹75⫹ 25)])=8.23. Average ratio change in odds for all exposed individuals (average odds ratio) = ([3.35 ⫻ (15⫹ 85)]⫹ [9.00⫻ (225⫹75)]
⫹ [3.35 ⫻(45 ⫹255)] ⫹[9.00⫻ (75⫹25)])÷(15 ⫹85 ⫹ 225⫹75 ⫹ 45⫹ 255⫹ 75⫹ 25) = 6.18.

Table 5. Hypothetical Randomized Trial of Treatment X2 and Outcome Y2 a

Low-Risk Subjects High-Risk Subjects All Subjects

Y2 No Y2 Y2 No Y2 Y2 No Y2
Males
Treatment X2, No.
Yes 51 51 288 18 339 69
No 20 80 240 60 260 140
Risk ratio 2.50 1.18 1.28
Odds ratio 4.00 4.00 2.65
Females
Treatment X2, No.
Yes 153 153 96 6 249 159
No 60 240 80 20 140 260
Risk ratio 2.50 1.18 1.74
Odds ratio 4.00 4.00 2.91
All Persons
Treatment X2, No.
Yes 204 204 384 24 588 228
No 80 320 320 80 400 400
Risk ratio 2.50 1.18 1.44
Odds ratio 4.00 4.00 2.58

a The risk for outcome Y2, aside from exposure, is either low (.2) or high (.8). In this table, the odds ratio is the same for all subjects. Ratio change in average
risk due to exposure among the exposed=([.2 ⫻(51 ⫹51 ⫹153 ⫹ 153)⫻ 2.50]⫹ [.8⫻ (288⫹ 18 ⫹ 96⫹ 6)⫻ 1.18]) ÷ ([.2⫻ (51⫹ 51⫹ 153⫹ 153)]⫹ [.8⫻
(288 ⫹ 18 ⫹ 96 ⫹6)])=1.44. Average ratio change in risk for all exposed individuals (average risk ratio) = ([2.50 ⫻ (51⫹ 51)]⫹ [1.18⫻ (288⫹ 18)]⫹ [2.50⫻
(153 ⫹ 153)] ⫹ [1.18 ⫻(96 ⫹6)])÷(51 ⫹ 51⫹ 288⫹18 ⫹ 153⫹153⫹ 96⫹ 6) = 1.84. Ratio change in average odds due to exposure among the
exposed = ([(.2/.8) ⫻ (51⫹ 51 ⫹153⫹ 153) ⫻4.00]⫹[(.8/.2)⫻(288⫹ 18⫹ 96⫹ 6)⫻ 4.00]) ÷ [(.2/.8)⫻ (51⫹ 51⫹ 153⫹ 153)⫹ (.8/.2)⫻
(288 ⫹ 18 ⫹96 ⫹ 6)] =4.00. Average ratio change in odds for all exposed individuals (average odds ratio) = ([4.00 ⫻ (51⫹ 51)]⫹ [4.00⫻ (288⫹ 18)]⫹ [4.00 ⫻
(153 ⫹ 153)] ⫹ [4.00⫻ (96⫹ 6)])÷(51 ⫹ 51⫹ 288⫹ 18 ⫹153⫹ 153⫹ 96 ⫹ 6) = 4.00.

the exposed, and (3) the average ratio change in risk for The risk ratios in Table 4 are collapsible. Any weighted
all exposed individuals (ie, the average risk ratio) average of the constant risk ratio of 3.00 should be 3.00;
(Table 1). in Table 4, all 5 collapsed tables do indeed yield risk ra-

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 tios of 3.00. Collapsibility means that if we adjust the risk will be closer to 1 than the ratio for the change in aver-
ratio of 3.00 for all persons for a variable that is not a age odds.1,17,21 If we had data from a large trial in which
confounder, sex in this example, the adjusted risk ratio randomization balanced all measured variables (thereby
will still be 3.00. This is true in Table 4. removing confounding by these factors), estimating the
The risk ratios for males and females are also collaps- ratio change in average odds due to treatment would re-
ible in Table 5. Sex is not a confounder in Table 5, and a quire adjustment for variables related to outcome risk;
Mantel-Haenszel–weighted average of the risk ratios for if variation in outcome risk under no exposure per-
males (1.28) and females (1.74) is equal to the risk ratio sisted within the adjustment covariate patterns, the ad-
of 1.44 from the collapsed table of counts for all per- justed odds ratio would still be closer to 1 than the de-
sons. If we adjust the risk ratio of 1.44 for sex, the result sired estimate.
is still 1.44. The odds ratio does not estimate the average change
Individual risks when not exposed would not be known in odds (the average odds ratio) among exposed indi-
in real data. But the 2⫻ 2 table of counts for all persons viduals either, except under implausible restrictions.1 This
could be obtained in a study, and from those counts, the means that estimating a constant (homogenous) odds ra-
ratio change in average risk (1.44) in Table 5 can be es- tio that applies to all exposed individuals, as proposed
timated. Calculations in Table 5 show that the ratio change in the argument that a constant odds ratio is more plau-
in average risk is indeed 1.44. However, unlike the ex- sible, will usually be impossible, even if a constant odds
ample in Table 4, the risk ratio of 1.44 in Table 5 is not ratio actually exists.
the average ratio change in risk (the average risk ratio) Useful discussions of this topic, with examples, can
for all individuals, which is 1.84 (calculations in Table 5). be found in publications by Greenland1 and Newman.11
The average risk ratio could be estimated in a study only Greenland suggests that odds ratios should not be used
if the risk ratio was the same for all exposed persons1; for inference unless they approximate risk ratios. New-
this unlikely scenario was shown in Table 4. man acknowledges the collapsibility problem and ar-
Odds ratios are not collapsible. The odds ratio is con- gues that the exposure-outcome association cannot be
stant (4.00) for all exposed persons in Table 5. However, summarized by a single odds ratio.11 He suggests report-
when we collapse across the categories of risk, the odds ra- ing 2 odds ratios. The first odds ratio is for the effect of
tios from each of the 3 collapsed tables are not equal to a exposure on the entire exposed group; this corresponds
weighted average of 4.00; all are closer to 1. Furthermore, to the ratio change in incidence odds (Table 1), which
the odds ratio of 2.58 for all persons is not a weighted av- was 4.64 in Table 4 and 2.58 in Table 5. The second odds
erage of the odds ratios of 2.65 for men and 2.91 for women, ratio is a summary across whatever stratum-specific odds
as 2.58 is closer to 1 than either stratum-specific estimate. ratios are available; this corresponds to a Mantel-Haenszel–
Adjusting the odds ratio of 2.58 for sex, using Mantel- adjusted odds ratio of 5.0 for Table 4 and 2.79 for Table 5.
Haenszel methods, produces an odds ratio of 2.79, though However, all of these odds ratios lack any interpretation
sex is not a confounder. If we could adjust for more vari- as an average.
ables, such as age, the adjusted odds ratio would tend to
move away from 2.58 and closer to 4.00.1 SUMMARY OF AGREEMENTS
The odds ratio of 2.58 in Table 5 is an unbiased esti- AND ARGUMENTS
mate of the ratio change in odds due to exposure for the
exposed group. However, it is not the ratio change in the Odds ratios approximate risk ratios when outcomes are
average odds due to exposure among the exposed, which rare in all noteworthy strata used for an analysis. When
is 4.00; nor is it the average ratio change in odds for all outcomes are rare, all 4 arguments can be ignored. This
exposed individuals (the average odds ratio), which is is most useful in case-control studies, in which odds ra-
also 4.00. The odds ratio will estimate the average change tios can be interpreted as risk ratios; whether the esti-
in odds (the average odds ratio) among exposed indi- mates are called odds or risk ratios is a matter of style.
viduals only when all individual odds ratios are equal and When event outcomes are common, odds ratios will
all individual outcome risks without exposure are equal1; not approximate risk ratios. Odds ratios are conve-
this implausible scenario is shown in Table 5, where col- niently symmetrical regarding the outcome, and a con-
lapsed counts for low- (or high-) risk subjects only pro- stant odds ratio may be more plausible than a constant
duce a 2⫻2 table with an odds ratios of 4.00. risk ratio, but estimating a constant odds ratio will usu-
Similarly, the odds ratio of 4.64 for all persons in ally be impossible. Even estimating the ratio change in
Table 4 is an unbiased estimate of the ratio change in odds average odds may involve insurmountable practical
due to exposure for the exposed group, but 4.64 is not difficulties.
the ratio change in average odds due to exposure, which Because risk ratios are not symmetrical, analysts using
is 8.23 (bottom of Table 4); and 4.64 is not the average risk ratios may wish to present 2 risk ratios when the out-
ratio change in odds (the average odds ratio), which is come is common: one for outcome Y and another for out-
6.18. If we use Mantel-Haenszel methods to adjust the come not Y. Because odds ratios are not collapsible, those
odds ratio of 4.64 for sex, a variable that is not a con- who report odds ratios could say explicitly that the es-
founder, the adjusted odds ratio is 5.00. timated odds ratio will be closer to 1 than the ratio change
In summary, the risk ratio has a useful interpretation in average odds. Also, because odds ratios are some-
as the ratio change in average risk due to exposure among times misinterpreted as risk ratios,22-25 studies that re-
the exposed.1 The odds ratio lacks any interpretation as port odds ratios when outcomes are common could state
an average. The odds ratio estimated from observed data that the estimates do not approximate risk ratios.

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 I have tried to give sufficient information to allow read- I have focused only on odds and risk ratios. How-
ers to choose between odds ratios or risk ratios. For my- ever, for some studies with binary outcomes, other mea-
self, I prefer risk ratios because they can be interpreted sures of association may be preferred, including the rate
as a ratio change in average risk. ratio, hazard ratio, risk difference, and rate difference.

HOW TO ESTIMATE RISK RATIOS Accepted for Publication: November 2, 2008.

Correspondence: Peter Cummings, MD, MPH, Depart-
Methods for estimating crude and adjusted risk ratios are ment of Epidemiology, University of Washington, 250
not widely described in textbooks, so I will briefly list Grandview Dr, Bishop, CA 93514 (peterc@u.washington
some with citations. Reviews are available.26,27 .edu).
When outcomes are rare, odds ratios will approxi- Financial Disclosure: None reported.
mate risk ratios, so Mantel-Haenszel methods for odds Funding/Support: This work was supported in part by
ratios or logistic regression can be used to estimate risk grant R49/CE000197-04 from the Centers for Disease
ratios.11,28 When outcomes are rare or common, risk ra- Control and Prevention.
tios can be estimated using several methods:
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What silent wonder is waked in the boy by

blowing bubbles from soap and water with
a pipe.
—Ralph Waldo Emerson

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