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### 9.2.7Expressing intervention effects on log scales

The values of ratio intervention effects (such as the odds ratio, risk ratio, rate ratio and hazard ratio) usually undergo log transformations before being analysed, and they may occasionally be referred to in terms of their log transformed values. Typically the natural log transformation (log base e, written ‘ln’) is used.

Ratio summary statistics all have the common feature that the lowest value that they can take is 0, that the value 1 corresponds with no intervention effect, and the highest value that an odds ratio can ever take is infinity. This number scale is not symmetric. For example, whilst an odds ratio of 0.5 (a halving) and an OR of 2 (a doubling) are opposites such that they should average to no effect, the average of 0.5 and 2 is not an OR of 1 but an OR of 1.25. The log transformation makes the scale symmetric: the log of 0 is minus infinity, the log of 1 is zero, and the log of infinity is infinity. In the example, the log of the OR of 0.5 is –0.69 and the log of the OR of 2 is 0.69. The average of –0.69 and 0.69 is 0 which is the log transformed value of an OR of 1, correctly implying no average intervention effect.

Graphical displays for meta-analysis performed on ratio scales usually use a log scale. This has the effect of making the confidence intervals appear symmetric, for the same reasons.