Convert Between the Standardized Mean Difference and the Correlation
Source:R/convert_d_r.R
convert_d_r.RdInvertible conversions between a two-group standardized mean difference
(Cohen's d) and the (point-biserial) correlation between the
outcome and group membership. convert_d_r() maps d to
r; convert_r_d() maps r back to d. These are
the standard conversions used to bring effect sizes reported in different
metrics onto a common scale, for example when synthesizing a literature in
which some studies report mean differences and others report correlations
(Borenstein, Hedges, Higgins, & Rothstein, 2009, Chapter 7).
Value
A data.frame (class dmar_tbl) with a single row:
term r (for convert_d_r) or smd (for
convert_r_d) and its value.
Details
With \(a = (n_1 + n_2)^2/(n_1 n_2)\) (equal to 4 for equal groups), the
two directions are
$$r = \frac{d}{\sqrt{d^2 + a}}, \qquad
d = \frac{\sqrt{a}\, r}{\sqrt{1 - r^2}},$$
exact inverses of one another for a given \(a\). The same n_1 and
n_2 must be supplied to both directions for the round trip to be
exact.
References
Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Wiley.
See also
convert_d_or / convert_or_d for the
odds ratio leg of the same triangle; smd and
ci_r for estimating the quantities being converted.
Other parameterization conversions:
convert_F_chisq(),
convert_R2,
convert_Z_r(),
convert_cor_cov(),
convert_d_or(),
convert_r_Z(),
convert_t_smd,
convert_z_normal()
Author
Ken Kelley kkelley@nd.edu
Examples
# Equal groups: d = 0.5 corresponds to r about .243.
convert_d_r(d = 0.5)
#> term value
#> r 0.243
# And back, exactly.
convert_r_d(r = convert_d_r(d = 0.5)$value)
#> term value
#> smd 0.5
# Unequal groups change the conversion factor.
convert_d_r(d = 0.5, n_1 = 20, n_2 = 80)
#> term value
#> r 0.196