Converting Peto’s odds ratio to risk difference

Article type
Authors
Tam W, Ling Tang J
Abstract
Background: Peto’s odds ratio is often used in meta-analyses to combine studies of binary outcomes. However, the odds ratio needs to be converted to an absolute effect such as the risk difference to inform decision making. In such conversions, Peto’s odds ratio is normally taken as an ordinary odds ratio in such conversions. Objectives: To examine the bias in the risk difference derived from the conventional method and to develop a new method for the conversion. Methods: We examined the difference between Peto’s odds ratio and the ordinary odds ratio in various hypothetical trials, developed a new formula for deriving the risk difference from Peto’s odds ratio, compared with the true risk difference with those derived from the new and conventional methods. Results: Peto’s odds ratio and ordinary odds ratio could differ substantively when the event rate in the control group is very small or very large and/or the odds ratio is relatively large. A quadratic equation of the risk in the treatment group (Pt) was derived (Figure 1) where Pc is the risk in the control group and ORp is the Peto’s odds ratio. Figure 2 shows the risk difference (RDp) derived from different combinations of Pc and ORp using the proposed formula. The RDp from the new formula is almost identical to the true risk difference, whereas the risk difference derived from the conventional method is always smaller than the truth and the difference can be quite large (Table 1). We also showed in a real meta-analysis, the risk difference derived from the two methods is as large as 72%. Conclusions: The risk difference derived by taking Peto’s odds ratio as an ordinary odds ratio can be considerably biased when the event rate in the control group is relatively small or large and/or the odds ratio is relatively large. The new formula should be used for deriving the risk difference in such situations.