Back-Transform a Log-Scale Effect and Its Interval Together

Identify what was logged

A log-scale estimate may represent a log risk ratio, log odds ratio, log rate ratio or a difference in mean logarithms. These are not interchangeable merely because exponentiation produces a positive number. Record the original estimand and the logarithm base before transforming anything.

Cochrane’s effect-measure guidance discusses ratio measures on logarithmic scales. The transformation is useful for estimation and intervals, but it does not turn an odds ratio into a risk ratio or a geometric-mean ratio into an arithmetic-mean difference.

Transform endpoints, not an isolated standard error

For a natural-log ratio estimate L with a compatible standard error, form the interval on the log scale under the stated method, then exponentiate the estimate and both interval endpoints. Exponentiation is monotone, so endpoint ordering is preserved.

Do not exponentiate the standard error and call the result the standard error of the ratio. A ratio-scale uncertainty calculation needs an appropriate transformation method; simply applying exp to every number in an output row gives different quantities with different meanings.

Check a hypothetical result

Suppose a log ratio is ln(1.5), approximately 0.4055, and a fictional log-scale interval is [ln(1.1), ln(2.0)]. Exponentiating gives a ratio estimate of 1.5 with interval [1.1, 2.0]. The interval is asymmetric around 1.5 on the ratio scale, which is expected.

If the contrast is reversed, the estimate becomes 1/1.5 = 0.6667 and the interval becomes [1/2.0, 1/1.1], approximately [0.5, 0.9091]. Reciprocating endpoints without reversing their order would produce an invalid lower-to-upper presentation.

Check the narrative as well as the numbers

Interpret the ratio using the correct event, comparator and follow-up. A ratio of 1.5 means a 50% higher value of the specified relative measure, not a 50-percentage-point increase in probability. Absolute risks require an appropriate baseline risk and compatible assumptions.

Keep a column identifying the analysis scale and a separate column for the reporting scale. Preserve sufficient precision internally and round only the displayed result. The final table should let readers trace the transformed estimate and interval back to their original model output without mistaking log units, ratio units and standard-error units for the same object.

Source verification: 6 November 2026. Examples are hypothetical.