6.4 Odds and odds ratios

What are odds?

The odds of an event are the ratio of how likely the event is to occur and how likely it is to not occur. If \(p\) (probability) tells use how likely an event is to occur then \(1 - p\) tells us how likely it is not to occur and the odds are the ratio of these two, \(p/(1-p)\). For example, suppose an event has a probability of \(75\%\). The odds of that event are \(75\% / 25\% = 3:1\), read as “3 to 1”. As another example, suppose you attend a meeting of five people, including yourself. You each write your name on a piece of paper for a randomly drawn door prize. Your chance, or probability, of winning the prize is 1/5 (0.20). Your odds of winning, however, are 1 to 4 (1:4) (0.25). There is one piece of paper with your name and four without, so you have one chance of winning and four chances to lose.

There is a one-to-one relationship between odds and probability. They are two ways of expressing the same thing, but on different scales. The odds are greater (less) than 1 if and only if the probability is greater (less) than 0.5, and the odds are exactly 1 if and only if the probability is 0.5.

What is an odds ratio (OR)?

An odds ratio (OR) compares the odds of an event between two different groups. For example, suppose the probability of disease is 0.35 for men and 0.25 for women. The OR is, therefore, \([0.35 / (1 - 0.35)]\) \(\div\) \([0.25 / (1 - 0.25)]\) \(=\) \(0.54\) \(\div\) \(0.33\) \(=\) \(1.62\).

Interpreting an OR

An OR > 1 (< 1) implies a positive (negative) association between a continuous predictor and the outcome. An OR of 1 implies no association. For an OR \(>\) 1, 100% \(\times\) (OR – 1) represents the % greater odds for \(X = x + 1\) compared to \(X = x\) for a continuous predictor, or for a level compared to the reference level for a categorical predictor. For an OR < 1, 100% \(\times\) (1 – OR) represents the % lower odds. Also, if you reverse the order of the groups being compared, the OR will be inverted.

For example:

  • OR = 1.45 implies that the first group has 45% greater odds of the outcome than the second group, or 1.45 times the odds of the second group.
  • OR = 0.81 implies that the first group has 19% lower odds of the outcome than the second group, or 0.81 times the odds of the second group.
  • If the OR comparing group A to group B is 0.667, then the OR comparing group B to group A is 1/0.667 = 1.50.

The “times the odds” style of interpreting an OR may be more clear than the “% greater odds” style when OR \(\ge\) 2. For example, OR = 2.50 could be interpreted as the first group having “150% greater odds than” or “2.5 times the odds of” the second group. Both interpretations are correct, but the latter may be more clear.