Odds Ratio and Risk Ratio

Compares an exposed group against an unexposed one to express how strongly the exposure relates to the outcome. The risk ratio compares the rates of the outcome; the odds ratio compares the odds. A value of 1 means no association either way.

To investigate whether a factor is linked to a disease, an exposed group and an unexposed group are tabulated by whether or not the outcome occurred. From that two-by-two table, the strength of the association can be expressed in two different ways.

The first is the risk ratio: work out the proportion affected in each group and divide.

RR = \dfrac{\dfrac{a}{a + b}}{\dfrac{c}{c + d}}

The second is the odds ratio. Odds means the number affected divided by the number unaffected, and the odds ratio compares those.

OR = \dfrac{a / b}{c / d} = \dfrac{ad}{bc}

$a$ is exposed with the outcome, $b$ exposed without it
$c$ is unexposed with the outcome, $d$ unexposed without it

A value of 1 means no association either way. Above 1 the exposure goes with more of the outcome, below 1 with less.

Example

Take the defaults: among the exposed, 40 developed the outcome and 60 did not; among the unexposed, 20 did and 80 did not.

The risk when exposed is $40 \div 100$ , or 40%; unexposed it is $20 \div 100$ , or 20%. The risk ratio is $0.4 \div 0.2$ , exactly 2. Exposure doubles the chance of the outcome — a statement anyone can read.

The odds ratio is $(40 \times 80) \div (60 \times 20)$ , about 2.6667. The same data, yet a noticeably larger number.

Why the two diverge

Odds put the unaffected in the denominator, so as the outcome becomes more common that denominator shrinks and the ratio stretches. With 40% affected here, that stretching is pronounced.

Rare outcomes behave differently. When only a few percent are affected, the unaffected are nearly the whole group, and the odds ratio and risk ratio come out almost identical.

So reading an odds ratio as a multiple of risk is an approximation licensed only when the outcome is rare. Calling this example a 2.6667-fold increase in risk overstates the real doubling.

Which to use

A risk ratio can only be computed by following a group forward and counting outcomes. A case-control study, which selects affected and unaffected people first and then looks back at exposure, has no access to the underlying rate and therefore cannot produce a risk ratio. The odds ratio can still be calculated, which is why it is so prevalent in medical research.