Odds ratio

Imagine there is a rare disease, afflicting, say, only one in many thousands of adults in a country. Imagine we suspect that being exposed to something (say, having had a particular sort of injury in childhood) makes one more likely to develop that disease in adulthood. The most informative thing to compute would be the risk ratio, RR. To do this in the ideal case, for all the adults in the population we would need to know whether they (a) had the exposure to the injury as children and (b) whether they developed the disease as adults. From this we would extract the following information: the total number of people exposed to the childhood injury, ${\displaystyle N_{E},}$ out of which ${\displaystyle D_{E}}$ developed the disease and ${\displaystyle H_{E}}$ stayed healthy; and the total number of people not exposed, ${\displaystyle N_{N},}$ out of which ${\displaystyle D_{N}}$ developed the disease and ${\displaystyle H_{N}}$ stayed healthy. Since ${\displaystyle N_{E}=D_{E}+H_{E}}$ and similarly for the ${\displaystyle N_{N}}$ numbers, we only have four independent numbers, which we can organize in a table:

${\displaystyle {\begin{array}{|r|cc|}\hline &{\text{ Diseased }}&{\text{ Healthy }}\\\hline {\text{ Exposed }}&{D_{E}}&{H_{E}}\\{\text{ Not exposed }}&{D_{N}}&{H_{N}}\\\hline \end{array}}}$

To avoid possible confusion, we emphasize that all these numbers refer to the entire population, and not to some sample of it.

Now the risk of developing the disease given exposure is ${\displaystyle D_{E}/N_{E}}$ (where ${\displaystyle N_{E}=D_{E}+H_{E}}$), and of developing the disease given non-exposure is ${\displaystyle D_{N}/N_{N}.}$ The risk ratio, RR, is just the ratio of the two,

${\displaystyle RR={\frac {D_{E}/N_{E}}{D_{N}/N_{N}}}\,,}$

which can be rewritten as ${\displaystyle RR={\frac {D_{E}N_{N}}{D_{N}N_{E}}}={\frac {D_{E}/D_{N}}{N_{E}/N_{N}}}.}$

In contrast, the odds of disease if exposed ${\displaystyle D_{E}/H_{E}\,,}$ versus the odds of being diseased if not exposed ${\displaystyle D_{N}/H_{N}\,.}$ The odds ratio, OR, is the ratio of the two,

${\displaystyle OR={\frac {D_{E}/H_{E}}{D_{N}/H_{N}}}\,,}$ which can be rewritten as ${\displaystyle OR={\frac {D_{E}H_{N}}{D_{N}H_{E}}}={\frac {D_{E}/D_{N}}{H_{E}/H_{N}}}.}$

We may already note that if the disease is rare, then OR=RR. Indeed, for a rare disease, we will have ${\displaystyle D_{E}\ll H_{E},}$ and so ${\displaystyle D_{E}+H_{E}\approx H_{E};}$ but then ${\displaystyle D_{E}/(D_{E}+H_{E})\approx D_{E}/H_{E},}$ in other words, for the exposed population, the risk of developing the disease is approximately equal to the odds. Analogous reasoning shows that the risk is approximately equal to the odds for the non-exposed population as well; but then the ratio of the risks, which is RR, is approximately equal to the ratio of the odds, which is OR. Or, we could just notice that the rare disease assumption says that ${\displaystyle N_{E}\approx H_{E}}$ and ${\displaystyle N_{N}\approx H_{N},}$ from which it follows that ${\displaystyle N_{E}/N_{N}\approx H_{E}/H_{N},}$ in other words that the denominators in the final expressions for the RR and the OR are approximately the same. The numerators are exactly the same, and so, again, we conclude that OR ≈ RR. Returning to our hypothetical study, the problem we often face is that we may not have the data to estimate these four numbers. For example, we may not have the population-wide data on who did or did not have the childhood injury.

Often we may overcome this problem by employing random sampling of the population: namely, if neither the disease nor the exposure to the injury are too rare in our population, then we can pick (say) a hundred people at random, and find out these four numbers in that sample; assuming the sample is representative enough of the population, then the RR computed for this sample will be a good estimate for the RR for the whole population.

However, some diseases may be so rare that, in all likelihood, even a large random sample may not contain even a single diseased individual (or it may contain some, but too few to be statistically significant). This would make it impossible to compute the RR. But, we may nevertheless be able to estimate the OR, provided that, unlike the disease, the exposure to the childhood injury is not too rare. Of course, because the disease is rare, this is then also our estimate for the RR.

Looking at the final expression for the OR: the fraction in the numerator, ${\displaystyle D_{E}/D_{N},}$ we can estimate by collecting all the known cases of the disease (presumably there must be some, or else we likely wouldn't be doing the study in the first place), and seeing how many of the diseased people had the exposure, and how many did not. And the fraction in the denominator, ${\displaystyle H_{E}/H_{N},}$ is the odds that a healthy individual in the population was exposed to the childhood injury. Now note that this latter odds can indeed be estimated by random sampling of the population—provided, as we said, that the prevalence of the exposure to the childhood injury is not too small, so that a random sample of a manageable size would be likely to contain a fair number of individuals who have had the exposure. So here the disease is very rare, but the factor thought to contribute to it is not quite so rare; such situations are quite common in practice.

Thus we can estimate the OR, and then, invoking the rare disease assumption again, we say that this is also a good approximation of the RR. Incidentally, the scenario described above is a paradigmatic example of a case-control study.[2]

The same story could be told without ever mentioning the OR, like so: as soon as we have that ${\displaystyle N_{E}\approx H_{E}}$ and ${\displaystyle N_{N}\approx H_{N},}$ then we have that ${\displaystyle N_{E}/N_{N}\approx H_{E}/H_{N}.}$ Thus if, by random sampling, we manage to estimate ${\displaystyle H_{E}/H_{N},}$ then, by rare disease assumption, that will be a good estimate of ${\displaystyle N_{E}/N_{N},}$ which is all we need (besides ${\displaystyle D_{E}/D_{N},}$ which we presumably already know by studying the few cases of the disease) to compute the RR. However, it is standard in the literature to explicitly report the OR and then claim that the RR is approximately equal to it.

The odds ratio is the ratio of the odds of an event occurring in one group to the odds of it occurring in another group. The term is also used to refer to sample-based estimates of this ratio. These groups might be men and women, an experimental group and a control group, or any other dichotomous classification. If the probabilities of the event in each of the groups are p₁ (first group) and p₂ (second group), then the odds ratio is:

${\displaystyle {p_{1}/(1-p_{1}) \over p_{2}/(1-p_{2})}={p_{1}/q_{1} \over p_{2}/q_{2}}={\frac {\;p_{1}q_{2}\;}{\;p_{2}q_{1}\;}},}$

where q_x = 1 − p_x. An odds ratio of 1 indicates that the condition or event under study is equally likely to occur in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely to occur in the first group. And an odds ratio less than 1 indicates that the condition or event is less likely to occur in the first group. The odds ratio must be nonnegative if it is defined. It is undefined if p₂q₁ equals zero, i.e., if p₂ equals zero or q₁ equals zero.

The odds ratio can also be defined in terms of the joint probability distribution of two binary random variables. The joint distribution of binary random variables X and Y can be written

${\displaystyle {\begin{array}{c|cc}&Y=1&Y=0\\\hline X=1&p_{11}&p_{10}\\X=0&p_{01}&p_{00}\end{array}}}$

where p₁₁, p₁₀, p₀₁ and p₀₀ are non-negative "cell probabilities" that sum to one. The odds for Y within the two subpopulations defined by X = 1 and X = 0 are defined in terms of the conditional probabilities given X, i.e., P(Y|X):

${\displaystyle {\begin{array}{c|cc}&Y=1&Y=0\\\hline X=1&{\frac {p_{11}}{p_{11}+p_{10}}}&{\frac {p_{10}}{p_{11}+p_{10}}}\\X=0&{\frac {p_{01}}{p_{01}+p_{00}}}&{\frac {p_{00}}{p_{01}+p_{00}}}\end{array}}}$

Thus the odds ratio is

${\displaystyle {{\dfrac {p_{11}/(p_{11}+p_{10})}{p_{10}/(p_{11}+p_{10})}}{\bigg /}{\dfrac {p_{01}/(p_{01}+p_{00})}{p_{00}/(p_{01}+p_{00})}}}={\dfrac {p_{11}p_{00}}{p_{10}p_{01}}}}$

The simple expression on the right, above, is easy to remember as the product of the probabilities of the "concordant cells" (X = Y) divided by the product of the probabilities of the "discordant cells" (X ≠ Y). However note that in some applications the labeling of categories as zero and one is arbitrary, so there is nothing special about concordant versus discordant values in these applications.

If we had calculated the odds ratio based on the conditional probabilities given Y,

${\displaystyle {\begin{array}{c|cc}&Y=1&Y=0\\\hline X=1&{\frac {p_{11}}{p_{11}+p_{01}}}&{\frac {p_{10}}{p_{10}+p_{00}}}\\X=0&{\frac {p_{01}}{p_{11}+p_{01}}}&{\frac {p_{00}}{p_{10}+p_{00}}}\end{array}}}$

we would have obtained the same result

${\displaystyle {{\dfrac {p_{11}/(p_{11}+p_{01})}{p_{01}/(p_{11}+p_{01})}}{\bigg /}{\dfrac {p_{10}/(p_{10}+p_{00})}{p_{00}/(p_{10}+p_{00})}}}={\dfrac {p_{11}p_{00}}{p_{10}p_{01}}}.}$

Other measures of effect size for binary data such as the relative risk do not have this symmetry property.

If X and Y are independent, their joint probabilities can be expressed in terms of their marginal probabilities p_x =  P(X = 1) and p_y =  P(Y = 1), as follows

${\displaystyle {\begin{array}{c|cc}&Y=1&Y=0\\\hline X=1&p_{x}p_{y}&p_{x}(1-p_{y})\\X=0&(1-p_{x})p_{y}&(1-p_{x})(1-p_{y})\end{array}}}$

In this case, the odds ratio equals one, and conversely the odds ratio can only equal one if the joint probabilities can be factored in this way. Thus the odds ratio equals one if and only if X and Y are independent.

The odds ratio is a function of the cell probabilities, and conversely, the cell probabilities can be recovered given knowledge of the odds ratio and the marginal probabilities P(X = 1) = p₁₁ + p₁₀ and P(Y = 1) = p₁₁ + p₀₁. If the odds ratio R differs from 1, then

${\displaystyle p_{11}={\frac {1+(p_{1\cdot }+p_{\cdot 1})(R-1)-S}{2(R-1)}}}$

where p_1• = p₁₁ + p₁₀,  p_•1 = p₁₁ + p₀₁, and

${\displaystyle S={\sqrt {(1+(p_{1\cdot }+p_{\cdot 1})(R-1))^{2}+4R(1-R)p_{1\cdot }p_{\cdot 1}}}.}$

In the case where R = 1, we have independence, so p₁₁ = p_1•p_•1.

Once we have p₁₁, the other three cell probabilities can easily be recovered from the marginal probabilities.

Odds ratios have often been confused with relative risk in medical literature. For non-statisticians, the odds ratio is a difficult concept to comprehend, and it gives a more impressive figure for the effect.[8] However, most authors consider that the relative risk is readily understood.[9] In one study, members of a national disease foundation were actually 3.5 times more likely than nonmembers to have heard of a common treatment for that disease – but the odds ratio was 24 and the paper stated that members were ‘more than 20-fold more likely to have heard of’ the treatment.[10] A study of papers published in two journals reported that 26% of the articles that used an odds ratio interpreted it as a risk ratio.[11]

This may reflect the simple process of uncomprehending authors choosing the most impressive-looking and publishable figure.[9] But its use may in some cases be deliberately deceptive.[12] It has been suggested that the odds ratio should only be presented as a measure of effect size when the risk ratio cannot be estimated directly.[8]

The sample odds ratio n₁₁n₀₀ / n₁₀n₀₁ is easy to calculate, and for moderate and large samples performs well as an estimator of the population odds ratio. When one or more of the cells in the contingency table can have a small value, the sample odds ratio can be biased and exhibit high variance.

A number of alternative estimators of the odds ratio have been proposed to address limitations of the sample odds ratio. One alternative estimator is the conditional maximum likelihood estimator, which conditions on the row and column margins when forming the likelihood to maximize (as in Fisher's exact test).[13] Another alternative estimator is the Mantel–Haenszel estimator.