#### 9.2.2.2 Measures of relative effect: the risk ratio and odds ratio

Measures of relative effect express the outcome in one group relative to that in the other. The risk ratio (or relative risk) is the ratio of the risk of an event in the two groups, whereas the odds ratio is the ratio of the odds of an event (see Box 9.2.a). For both measures a value of 1 indicates that the estimated effects are the same for both interventions.

Neither the risk ratio nor the odds ratio can be calculated for a study if there are no events in the control group. This is because, as can be seen from the formulae in Box 9.2.a, we would be trying to divide by zero. The odds ratio also cannot be calculated if everybody in the intervention group experiences an event. In these situations, and others where standard errors cannot be computed, it is customary to add ½ to each cell of the 2×2 table (RevMan automatically makes this correction when necessary). In the case where no events (or all events) are observed in both groups the study provides no information about relative probability of the event and is automatically omitted from the meta-analysis. This is entirely appropriate. Zeros arise particularly when the event of interest is rare – such events are often unintended adverse outcomes. For further discussion of choice of effect measures for such sparse data (often with lots of zeros) see Chapter 16 (Section 16.9).

Risk ratios describe the multiplication of the risk that occurs with use of the experimental intervention. For example, a risk ratio of 3 for a treatment implies that events with treatment are three times more likely than events without treatment. Alternatively we can say that treatment increases the risk of events by 100 × (RR – 1)% = 200%. Similarly a risk ratio of 0.25 is interpreted as the probability of an event with treatment being one-quarter of that without treatment. This may be expressed alternatively by saying that treatment decreases the risk of events by 100 × (1 –RR)% = 75%. This is known as the relative risk reduction (see also Chapter 12, Section 12.5.1). The interpretation of the clinical importance of a given risk ratio cannot be made without knowledge of the typical risk of events without treatment: a risk ratio of 0.75 could correspond to a clinically important reduction in events from 80% to 60%, or a small, less clinically important reduction from 4% to 3%.

The numerical value of the observed risk ratio must always be between 0 and 1/ CGR, where CGR (abbreviation of ‘control group risk’, sometimes referred to as the control event rate) is the observed risk of the event in the control group (expressed as a number between 0 and 1). This means that for common events large values of risk ratio are impossible. For example, when the observed risk of events in the control group is 0.66 (or 66%) then the observed risk ratio cannot exceed 1.5. This problem applies only for increases in risk, and causes problems only when the results are extrapolated to risks above those observed in the study.

Odds ratios, like odds, are more difficult to interpret (Sinclair 1994, Sackett 1996). Odds ratios describe the multiplication of the odds of the outcome that occur with use of the intervention. To understand what an odds ratio means in terms of changes in numbers of events it is simplest to first convert it into a risk ratio, and then interpret the risk ratio in the context of a typical control group risk, as outlined above. The formula for converting an odds ratio to a risk ratio is provided in Chapter 12 (Section 12.5.4.4). Sometimes it may be sensible to calculate the RR for more than one assumed control group risk.