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(Risk) odds ratio
The odds ratio (OR) is a ratio of “odds”, which are transformations of risks or probabilities.
odds = p/(1-p), where p = probability
The OR addresses the question “how many times greater is the odds of disease for exposed persons
than for unexposed persons?” Since odds have a different scale of measurement than risk, the
answer to this question can sometimes differ from the answer to the corresponding question about
risk. Often, however, we are concerned with rare diseases, for which risk and odds are very close
and CIR’s and OR’s (and IDR’s) are very close. Since the OR can be defined in terms of odds of
disease among exposed or odds of exposure among cases, there are two mathematical formulations:
Odds in “exposed”
Odds ra ti o = ————————————
Odds in “unexposed”
The odds is simply an algebraic transformation of probability, so any probability (which must, of
course, be less than 1.0) can be expressed as “odds”. The probability that something may happen,
especially something bad, is often referred to as a “risk”. Odds derived from a risk are termed,
appropriately, risk odds, so that a ratio of two risk odds is a risk odds ratio, or ROR.
(Exposure) odds ratio
A prevalence is commonly referred to as an estimate of probability (e.g., of exposure). A justification
for this usage is that if we were to select an individual at random from the group, the probability that
that individual would have a certain characteristic is estimated by the prevalence in the group. Odds
that correspond to the probability of exposure are called “exposure odds”, so their ratio is an
exposure odds ratio, or EOR. Although conceptually distinct, for a two-by-two table these two
odds ratios are algebraically identical, as we shall see. Thus, our ability to estimate an (exposure)
odds ratio in a situation where we do not know disease incidence is a powerful tool for examining
associations involving disease incidence even where we do not have incidence data, as was first
presented in a classic paper by Jerome Cornfield (see the chapter on Analytic Study Designs for
elaboration).
Risk odds in “exposed” odds1 CI1 / (1-CI1)
ORr = Risk odds ratio = —————————————— = ——— = ———————
Risk odds in “unexposed” odds0 CI0 / (1-CI0)
Exposure odds in “cases” odds1
ORe = Exposure odds ratio = ——————————————— = ————
Exposure in “noncases” odds0
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www.epidemiolog.net, © Victor J. Schoenbach 2000 7. Relating risk factors to health - 172
rev. 10/9/2000, 5/8/2001, 2/19/2002, 4/26/2002