Effective Clinical Practice

### Primer on Probability, Odds and Interpreting their Ratios

May/June 2000

All tables are attached.

Chance is measured using either probabilities (a ratio of occurrence to the whole) or odds (a ratio of occurrence to nonoccurrence). Consider measuring the chance of initiating breastfeeding among 1000 new mothers. If 600 ultimately breastfeed, the probability of breastfeeding is 600/1000 or .6 (often expressed as 60%) while the odds of breastfeeding is 600/400 or 1.5 (often expressed as 1.5 to 1). The characteristics of probabilities and odds are summarized below. (Table 1)

Probabilities and odds contain the same information and are equally valid measures of chance. In the case of infrequent events (i.e. probability < 0.1 or 10%) the distinction is not important (probability and odds have essentially the same value). As shown in the table below, however, probability and odds take on a very different values as the chance of an event increases. (Table 2)

Although probabilities are often reported in the medical literature, it is rare to see odds reported. On the other hand, ratios of both probabilities (i.e. Relative Risks or Risk Ratios) and odds (i.e. Odds Ratios) are frequently seen. And it is in these ratios of ratios were the distinction between probability and odds may be both important and ambiguous. When the chance of common events are being compared, calculated odds ratios and relative risk will substantially diverge in value. Let's return to the breastfeeding example. Imagine an randomized trial of a lactation support system. The probability of breastfeeding in the control group is 60% (or an odds of 1.5); in the intervention group it is 90% (or an odds of 9). As shown in the table below, the relative risk is 1.5 while the odds ratio is 6. (Table 3)

In general, odds ratios are more extreme (i.e. farther away from 1) than relative risks. Odds ratios that are greater than one exaggerate the increase in risk (i.e. OR > RR); odds ratios that are less than one exaggerate the decrease in risk (i.e. OR < RR). Practically speaking, the discrepancy between the two measures is only relevant when comparing relatively common events. Reader's should begin to worry about the distinction when baseline probabilities exceed 10 to 20%. And, as shown in the Table below, they might reasonably pursue a conversion when baseline probabilities are greater than 50%. (Table 4)

It is important to emphasize that relative risks and odds ratios are equally valid measures. But they are different. Readers are seeing more and more odds ratios in the medical literature, largely because of the increased use of logistic regression. Because most people are more familiar with probabilities than odds, odds ratios are often interpreted as relative risks. When events are common, this misinterpretation will substantially exaggerate the association being reported. If the goal is clarity, the probability (or absolute event rate) for each group is tough to beat.

Suggested Reading

1. Talfryn H, Davies O, Crombie IK, Tavakoli M. When can odds ratios mislead? BMJ. 1998;316:989-91.
2. Zhang J, Yu KF. What's the relative risk? A method of correcting the odds ratios in cohort studies of common outcomes. JAMA 1998;280:1690-1