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Note that we do not compute a variance for the risk ratio in its original metric. Rather, we use the log risk ratio and its variance in the analysis to yield a summary effect, confidence limits, and so on, in log units. We then convert each of these values back to risk ratios using RiskRatio ¼ expðLogRiskRatioÞ; ð5:5Þ LLRiskRatio ¼ expðLLLogRiskRatioÞ; ð5:6Þ and ULRiskRatio ¼ expðULLogRiskRatioÞ ð5:7Þ where LL and UL represent the lower and upper limits, respectively. In the running example the risk ratio is RiskRatio ¼ 5=100 10=100 ¼ 0:5000: The log is LogRiskRatio ¼ ln ð0:5000Þ¼0:6932; with variance VLogRiskRatio ¼ 1 5 1 100 þ 1 10 1 100 ¼ 0:2800; and standard error SELogRiskRatio ¼ ffiffiffiffiffiffiffiffiffiffiffi 0:280 p ¼ 0:5292: Note 1. The log transformation is needed to maintain symmetry in the analysis. Assume that one study reports that the risk is twice as high in Group A while another reports that it is twice as high in Group B. Assuming equal weights, these studies should balance each other, with a combined effect showing equal risks (a risk ratio of 1.0). However, on the ratio scale these correspond to risk ratios of 0.50 and 2.00, which would yield a mean of 1.25. By working with log values we can avoid this problem. In log units the two estimates are 0.693 and þ0.693, which yield a mean of 0.00. We convert this back to a risk ratio of 1.00, which is the correct value for this data. Note 2. Although we defined the risk ratio in this example as RiskRatio ¼ 5=100 10=100 ¼ 0:5000 (which gives the risk ratio of dying) we could alternatively have focused on the risk of staying alive, given by RiskRatio ¼ 95=100 90=100 ¼ 1:0556: The ‘risk’ of staying alive is not the inverse of the risk of dying (that is, 1.056 is not the inverse of 0.50), and therefore this should be considered a different measure of effect size. Chapter 5: Effect Sizes Based on Binary Data (2 2 Tables) ODDS RATIO Where the risk ratio is the ratio of two risks, the odds ratio is the ratio of two odds. Here, the odds of death in the treated group would be 5/95, or 0.0526 (since probability of death in the treated group is 5/100 and the probability of life is 95/100), while the odds of death in the control group would be 10/90, or 0.1111. The ratio of the two odds would then be 0.0526/0.1111, or 0.4737. Many people find this effect size measure less intuitive than the risk ratio, but the odds ratio has statistical properties that often make it the best choice for a meta-analysis. When the risk of the event is low, the odds ratio will be similar to the risk ratio. For odds ratios, computations are carried out on a log scale (for the same reason as for risk ratios). We compute the log odds ratio, and the standard error of the log odds ratio, and will use these numbers to perform all steps in the meta-analysis. Only then will we convert the results back into the original metric. This is shown schematically in Figure 5.2. The computational formula for the odds ratio is OddsRatio ¼ AD BC : ð5:8Þ The log odds ratio is then LogOddsRatio ¼ lnð Þ OddsRatio ; ð5:9Þ with approximate variance VLogOddsRatio ¼ 1 A þ 1 B þ 1 C þ 1 D ð5:10Þ Study A 2×2 Table Odds ratio Log odds ratio Study B 2×2 Table Odds ratio Log odds ratio Study C 2×2 Table Odds ratio Log odds ratio Summary Odds ratio Summary Log odds ratio