Signature
lnOR = 1.814 * d; OR = exp(lnOR)
| Inputs | Definition | Unit |
|---|---|---|
d | Cohen's d, or a pooled standardised mean difference such as Hedges' g, with higher values favouring the intervention | standard deviation units |
lnOR | Natural logarithm of the odds ratio for responding, intervention against comparator | log odds |
|---|---|---|
OR | Odds ratio for responding implied by the standardised difference | ratio |
Function
Cohen's d standardisation and re-expression
Maps a difference between two group means on a continuous outcome to standard deviation units, so that trials which measured the same construct on different instruments can be compared and pooled, and maps a standardised difference back to a quantity a decision model can use, here a log odds ratio for responding. A confidence interval for the standardised difference is the large-sample interval HE-FM-AN-001 applied with its standard error, and a comparator response probability combined with an odds ratio gives the intervention probability by HE-FM-NMA-004; neither is restated here. Hedges' small-sample correction and its standard error belong on the Hedges' g page.
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Implementations
Excel
Log odds ratio and odds ratio from a standardised difference
With the standardised difference in a cell named SMD, the first formula returns the log odds ratio in a cell named LogOR and the second returns the odds ratio. Replacing 1.814 with PI()/SQRT(3) gives the unrounded factor.
=1.814*SMD; =EXP(LogOR)
Assumptions
Logistic distribution with equal spread in both groups
The conversion assumes that the underlying continuous outcome follows a logistic distribution with the same standard deviation in both groups. The assumption will not hold exactly, so the Cochrane Handbook treats the result as an approximation.
Comparator response needed for a probability
The odds ratio alone does not give a probability. A comparator proportion of responders is needed, and the intervention probability then follows from HE-FM-NMA-004.
Worked examples
Article standardised effect of 0.54
The article rounds Hedges' g to 0.54. The log odds ratio is about 0.9796, shown as 0.980 in the article, and the odds ratio is about 2.6633; the article exponentiates the rounded 0.980 and reports 2.664.
d = 0.54; lnOR = 0.9796; OR = 2.6633
Lower confidence limit of the article's estimate
Computed here for illustration: carried through the same conversion, the article's lower 95% limit of 0.021 gives a log odds ratio of about 0.0381 and an odds ratio of about 1.0388, close to no effect.
d = 0.021; lnOR = 0.0381; OR = 1.0388
Upper confidence limit of the article's estimate
The upper limit of 1.051 gives a log odds ratio of about 1.9065 and an odds ratio of about 6.7296, computed here for illustration from the article's interval.
d = 1.051; lnOR = 1.9065; OR = 6.7296
No difference in means gives an odds ratio of 1
A standardised difference of 0 converts to a log odds ratio of 0 and an odds ratio of 1, a limiting case.
d = 0; lnOR = 0; OR = 1
Common errors
Exponentiating the standardised difference without the factor
Taking the exponential of 0.54 directly gives about 1.716 instead of the odds ratio of about 2.663, computed here for illustration, so the response probability and the QALY gain are understated.
Reading the converted odds ratio as a relative risk
With a comparator response of 0.30, an odds ratio of about 2.66 gives an intervention response of about 0.533, a relative risk of about 1.78, computed here for illustration. Multiplying 0.30 by the odds ratio would give about 0.80 instead.
Sources
Re-expressing standardised mean differences as log odds ratios
Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10: Analysing data and undertaking meta-analyses. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 10.6, which states that standardised mean differences can be re-expressed as log odds ratios by multiplying by pi over the square root of 3, 1.814.
Conversion from an odds ratio to an effect size
Chinn S. A simple method for converting an odds ratio to effect size for use in meta-analysis. Statistics in Medicine. 2000;19(22):3127-3131. Abstract: a log odds ratio can be converted to an effect size by dividing by 1.81.
Standardised mean difference to odds ratio and risk in interpretation
Schünemann HJ, Vist GE, Higgins JPT, Santesso N, Deeks JJ, Glasziou P, Akl EA, Guyatt GH. Chapter 15: Interpreting results and drawing conclusions. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 15.5.3.3, which re-expresses a standardised mean difference as an odds ratio under the assumption of a logistic distribution with equal standard deviations, and warns that the result is an approximation.
Canonical Identity
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