Conversion between Standardized Mean Difference and Odds Ratio (OR)

Scenario 1: Convert standardized mean difference to odds ratio (OR).
Standardized Mean Difference (d) Standard Error (SE)
Conversion method

Confidence Level
Result OR
Confidence Interval Lower Limit Upper Limit

Standard Error of d can be estimated here.

Assumptions:

  • Hasselblad and Hedges Method: Requires that the means of the two groups follow a logistic distribution, and variances are homogeneous[1].
  • Cox and Snell Method: Assumes that the means of the two groups follow a normal distribution, and variances are homogeneous[2].

Formula: $$log(OR)=d*(conversion factor)$$ $$SE(log(OR))=SE(d)*(conversion factor)$$

Scenario 2: Convert odds ratio (OR) to standardized mean difference.
OR Standard Error (SE)
Conversion method

Confidence Level
Result d
Confidence Interval Lower Limit Upper Limit

Standard Error of OR can be estimated here.

Assumptions:

  • Hasselblad and Hedges Method: Requires that the means of the two groups follow a logistic distribution, and variances are homogeneous[1].
  • Cox and Snell Method: Assumes that the means of the two groups follow a normal distribution, and variances are homogeneous[2].

Formula: $$d=log(OR)/(conversion factor) $$ $$SE(d)=SE(log(OR))/(conversion factor) $$

Reference:

[1] Hasselblad V, Hedges LV. Meta-analysis of screening and diagnostic tests, Psychol Bull, 1995, vol. 117 (pg. 167-78)
[2] Cox DR, Snell EJ. , Analysis of Binary Data, 1989 London Chapman and Hall
[3] Bruno R da Costa, Anne WS Rutjes, et. al., Methods to convert continuous outcomes into odds ratios of treatment response and numbers needed to treat: meta-epidemiological study, International Journal of Epidemiology, Volume 41, Issue 5, October 2012, Pages 1445–1459,