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In general conversation the terms ‘risk’ and ‘odds’ are used interchangeably (as are the terms ‘chance’, ‘probability’ and ‘likelihood’) as if they describe the same quantity. In statistics, however, risk and odds have particular meanings and are calculated in different ways. When the difference between them is ignored, the results of a systematic review may be misinterpreted.
Risk is the concept more familiar to patients and health professionals. Risk describes the probability with which a health outcome (usually an adverse event) will occur. In research, risk is commonly expressed as a decimal number between 0 and 1, although it is occasionally converted into a percentage. In ‘Summary of findings’ tables in Cochrane reviews, it is often expressed as a number of individuals per 1000 (see Chapter 11, Section 11.5). It is simple to grasp the relationship between a risk and the likely occurrence of events: in a sample of 100 people the number of events observed will on average be the risk multiplied by 100. For example, when the risk is 0.1, about 10 people out of every 100 will have the event; when the risk is 0.5, about 50 people out of every 100 will have the event. In a sample of 1000 people, these numbers are 100 and 500 respectively.
Odds is a concept that is more familiar to gamblers. The odds is the ratio of the probability that a particular event will occur to the probability that it will not occur, and can be any number between zero and infinity. In gambling, the odds describes the ratio of the size of the potential winnings to the gambling stake; in health care it is the ratio of the number of people with the event to the number without. It is commonly expressed as a ratio of two integers. For example, an odds of 0.01 is often written as 1:100, odds of 0.33 as 1:3, and odds of 3 as 3:1. Odds can be converted to risks, and risks to odds, using the formulae:
;
The interpretation of an odds is more complicated than for a risk. The simplest way to ensure that the interpretation is correct is to first convert the odds into a risk. For example, when the odds are 1:10, or 0.1, one person will have the event for every 10 who do not, and, using the formula, the risk of the event is 0.1/(1+0.1) = 0.091. In a sample of 100, about 9 individuals will have the event and 91 will not. When the odds is equal to 1, one person will have the event for every one who does not, so in a sample of 100, 100 × 1/(1+1) = 50 will have the event and 50 will not.
The difference between odds and risk is small when the event is rare (as illustrated in the first example above where a risk of 0.091 was seen to be similar to an odds of 0.1). When events are common, as is often the case in clinical trials, the differences between odds and risks are large. For example, a risk of 0.5 is equivalent to an odds of 1; and a risk of 0.95 is equivalent to odds of 19.
Measures of effect for clinical trials with dichotomous outcomes involve comparing either risks or odds from two intervention groups. To compare them we can look at their ratio (risk ratio or odds ratio) or their difference in risk (risk difference).