The use of odds was introduced in Statistics review 8 [3]. The odds of an individual exposed to a risk factor developing a disease is the ratio of the number exposed who develop the disease to the number exposed who do not develop the disease. For the data given in Table 1, the estimated odds of developing ARDS if the C allele is present are 11/208 = 0.053.

The odds ratio (OR) is the ratio of the odds of the disease in the group exposed to the factor, to the odds of the disease in the unexposed group. For the data given in Table 1, the OR is estimated by the following:

This value is similar to that obtained for the RR for these data. Generally, when the risk of the disease in the unexposed is low, the OR approximates to the risk ratio. This applies in the ARDS study, where the estimate of the risk for ARDS for those with the C allele absent was 1/183 = 0.005. Therefore, again, the OR implies that patients with the C allele present are approximately nine times as likely to develop ARDS as those with genotype TT. In general, using the notation given in Table 2, the OR can be expressed as follows:

An approximate 95% confidence interval for the true population OR can be calculated in a similar manner to that for the RR, but the SE of ln OR is approximated by

For the data given in Table 1, ln OR = 2.26 and the SE of ln OR is given by the following:

Therefore, the 95% confidence interval for the population ln OR is given by

2.26 - 1.96 × 1.049 to 2.26 + 1.96 × 1.049 (i.e. 0.204 to 4.316)

Again, we need to antilog (e^{x}) these lower and upper limits in order to obtain the 95% confidence interval for the OR. The 95% confidence interval for the population RR is given by the following:

e^{0.204} to e^{4.316} (i.e. 1.23 to 74.89)

Therefore the population OR is likely to be between 1.23 and 74.89 – a similar confidence interval to that obtained for the risk ratio. Again, the fact that the interval does not contain 1 indicates that there is a significant difference between the genotype groups.

The OR has several advantages. Risk cannot be estimated directly from a case–control study, in which patients are selected because they have a particular disease and are compared with a control group who do not, and therefore RRs are not calculated for this type of study. However, the OR can be used to give an indication of the RR, particularly when the incidence of the disease is low. This often applies in case–control studies because such studies are particularly useful for rare diseases.

The OR is a symmetric ratio in that the OR for the disease given the risk factor is the same as the OR for the risk factor given the disease. ORs also form part of the output when carrying out logistic regression, an important statistical modelling technique in which the effects of one or more factors on a binary outcome variable (e.g. survival/death) can be examined simultaneously. Logistic regression will be covered in a future review.

In the case of both the risk ratio and the OR, the reciprocal of the ratio has a direct interpretation. In the example given in Table 1, the risk ratio of 9.19 measures the increased risk of those with the C allele having ARDS. The reciprocal of this (1/9.19 = 0.11) is also a risk ratio but measures the reduced risk of those without the C allele having ARDS. The reciprocal of the odds ratio – 1/9.63 = 0.10 – is interpreted similarly.

Both the RR and the OR can also be used in the context of clinical trials to assess the success of the treatment relative to the control.