2 × 2 contingency table of Table 3.3. The odds ratio for the corn resolving in the plaster group compared to the scalpel group is OR = (32 × 74)/(20 × 63) = 1.88.
The OR = 1.88 and RR = 1.58 with these trial data are clearly different. However, when the probability of an event happening is rare, the odds and probabilities are close, because when a is much smaller than c then the risk a/(a + c) is approximately a/c. Further if b is much smaller than d then b/(b + d) is approximately b/d, then RR = (a/c)/(b/d). Thus, the OR approximates the RR when the successes are rare, say with a maximum incidence less than 10% of either pTest or pControl Sometimes the odds ratio is referred to as ‘the approximate relative risk’. The approximation is demonstrated in Table 3.6.
Table 3.6 Comparison of RR and OR for different baseline rates.
| p Test | p Control | RR | OR | RR and OR |
|---|---|---|---|---|
| 0.05 | 0.1 | 0.5 | 0.47 | Close |
| 0.1 | 0.2 | 0.5 | 0.44 | Close |
| 0.2 | 0.4 | 0.5 | 0.38 | Not close |
| 0.4 | 0.2 | 2 | 2.66 | Not close |
| 0.2 | 0.1 | 2 | 2.25 | Close |
| 0.1 | 0.05 | 2 | 2.11 | Close |
Why Should One Use the Odds Ratio?
The calculation for an OR may seem rather perverse, given that we can calculate the relative risk directly from the 2 × 2 table and the odds ratio is only an approximation of this. However, the OR appears quite often in the literature, so it is important to be aware of it. It has certain mathematical properties which render it attractive as an alternative to the RR as a summary measure. The OR features in logistic regression (see Chapter 10) and as a summary measure for case‐control studies (see Section 14.8). An example where the authors quote an odds ratio is given below.
One point about the OR that can be seen immediately from the formula is that the odds ratio for failure as opposed to the odds ratio for success in Table 3.3 is given by OR = bc/ad. Thus, the OR for failure is just the inverse of the OR for success.
Thus in the corn plaster trial, the odds ratio for the corn not healing or resolving at three months in the plaster group compared to the scalpel group is (20 × 63)/(32 × 74) = 0.532; which is the same as the reciprocal or inverse of the odds ratio for the corn resolving at three months in the plaster group compared to the scalpel group or 1/1.879 = 0.53. In contrast the relative risk ratio for the corn not healing or resolving at three months in the plaster group compared to the scalpel group (1 − 0.337)/(1 − 213) = 0.842, which is not the same as the inverse of the relative risk for the corn resolving at three months in the plaster compared to the scalpel group, which is 1/1.583 = 0.632. This symmetry of interpretation of the OR is one of the reasons for its continued use.
The Odd Ratios are Symmetrical but the Relative Risk Is Not
Consider the data in the 2 × 2 contingency table of Table 3.7 where the relative risk of being alive in the exposed compared to the not exposed group is relative risk (alive) = 0.96 /0 99 = 0.97; the reciprocal is 1/relative risk (alive) = 1/ 0.97 = 1.03. The relative risk (dead) = 0.04/0.01 = 4. Thus, note that the relative risk (dead) is not equal to 1 / relative risk (alive).
Table 3.7 Example of a two by two contingency table with a binary outcome (alive or dead) and two groups of subjects (exposed or not exposed).
| Outcome | Test treatment exposed | Control treatment not exposed |
|---|---|---|
| Alive | 0.96 | 0.99 |
| Dead | 0.04 | 0.01 |
| Total | 1.00 | 1.00 |
The odds ratio (alive) = (0.96/0.04) / (0.99/0.01) = 0.24; the reciprocal is: 1/odds ratio (alive) = 1/ 0.24 = 4.13. The odds ratio (dead) = (0.04/0.96)/(0.01/0.99) = 4.13; and hence the odds ratio (dead) is equal to 1/odds ratio (alive).
How Are Risks Compared?
To understand risks that are smaller than 1% (or 1 in 100) you may find it helpful to compare these risks to other risks in life. Some people use words like ‘high’ or ‘low’ to talk about risk. So Calman (1996), an expert in risk communication, has produced a ‘risk classification’ scale that looks at particular risks and suggests words that the public and health care professionals can use to describe them. An outline of the scale is given in Table 3.8.
Table 3.8 Risk of an individual dying (D) in one year or developing an adverse response (A)
(Source: Calman 1996).
| Term used | Risk range | Example | Risk estimate |
|---|---|---|---|
| High | >1:100 | (A) Transmission to susceptible household contacts of measles and chickenpox (A) Transmission of HIV from mother to child (Europe) | 1:1–1:2 1:6 |
| Moderate | 1:100–1:1000 | (D) Smoking 10 cigarettes per day (D) All natural causes, age 40 | 1:200 1:850 |
| Low | 1:1000–1:10 000 | (D) All kinds of violence (D) Influenza (D) Accident on road | 1: 300 1:5000 1:8000 |
| Very low |
1:10 000–1:100
|