Convert Between the Standardized Mean Difference and the Odds Ratio

Invertible conversions between a two-group standardized mean difference (Cohen's d) and an odds ratio, by the logistic-distribution method of Hasselblad and Hedges (1995): a continuous outcome split at a threshold under logistic errors implies $$d = \log(\mathrm{OR}) \cdot \frac{\sqrt{3}}{\pi}, \qquad \mathrm{OR} = \exp\!\bigl(d \cdot \pi / \sqrt{3}\bigr).$$ These conversions let binary-outcome studies enter a synthesis on the standardized mean difference scale, or mean-difference studies enter one on the odds ratio scale (Borenstein, Hedges, Higgins, & Rothstein, 2009, Chapter 7).

Usage

convert_d_or(d)

convert_or_d(or)

Arguments

d

The standardized mean difference.

or

The odds ratio, a single positive number.

Value

A data.frame (class dmar_tbl) with a single row: term odds_ratio (for convert_d_or) or smd (for convert_or_d) and its value.

References

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Wiley.

Hasselblad, V., & Hedges, L. V. (1995). Meta-analysis of screening and diagnostic tests. Psychological Bulletin, 117(1), 167–178. doi:10.1037/0033-2909.117.1.167

See also

convert_d_r / convert_r_d for the correlation leg of the same triangle.

Other parameterization conversions: convert_F_chisq(), convert_R2, convert_Z_r(), convert_cor_cov(), convert_d_r(), convert_r_Z(), convert_t_smd, convert_z_normal()

Author

Ken Kelley kkelley@nd.edu

Examples

# d = 0.5 corresponds to an odds ratio of about 2.48.
convert_d_or(d = 0.5)
#>  term       value
#>  odds_ratio 2.48 

# And back, exactly.
convert_or_d(or = convert_d_or(d = 0.5)$value)
#>  term value
#>  smd  0.5  

# The null maps to the null: d = 0 is an odds ratio of 1.
convert_d_or(d = 0)
#>  term       value
#>  odds_ratio 1