d_to_r: r and Coefficient of Determination (R2) from d

In MOTE: Effect Size and Confidence Interval Calculator

r and Coefficient of Determination (R2) from d

Description

**Note on function and output names:** This effect size translation is now implemented with the snake_case function name 'd_to_r()' to follow modern R style guidelines. The original dotted version 'd.to.r()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'r', 'rlow', 'rhigh', 'R2', 'R2low', 'R2high', 'se', 'n', 'dfm', 'dfe', 't', 'F', 'p', 'estimate', 'estimateR2', 'statistic') and newer snake_case aliases (e.g., 'r_lower_limit', 'r_upper_limit', 'r2_value', 'r2_lower_limit', 'r2_upper_limit', 'se_value', 'sample_size', 'degrees_freedom_model', 'degrees_freedom_error', 't_value', 'f_value', 'p_value', 'estimate_r', 'estimate_r2'). New code should prefer 'd_to_r()' and the snake_case output names, but existing code using the older names will continue to work.

Usage

d_to_r(d, n1, n2, a = 0.05)

d.to.r(d, n1, n2, a = 0.05)

Arguments

d	Effect size statistic.
n1	Sample size for group one.
n2	Sample size for group two.
a	Significance level.

Details

Calculates r from d and then translates r to r2 to calculate the non-central confidence interval for r2 using the F distribution.

The correlation coefficient (r) is calculated by dividing Cohen's d by the square root of the total sample size squared, divided by the product of the sample sizes of group one and group two.

r = \frac{d}{\sqrt{d^2 + \frac{(n_1 + n_2)^2}{n_1 n_2}}}

Learn more on our example page.

Value

Provides the effect size (correlation coefficient) with associated confidence intervals, the t-statistic, F-statistic, and other estimates appropriate for d to r translation. Note this CI is not based on the traditional r-to-z transformation but rather non-central F using the ci.R function from MBESS.

r

Correlation coefficient.

rlow

Lower level confidence interval for r.

rhigh

Upper level confidence interval for r.

R2

Coefficient of determination.

R2low

Lower level confidence interval of R2.

R2high

Upper level confidence interval of R2.

se

Standard error.

n

Sample size.

dfm

Degrees of freedom of mean.

dfe

Degrees of freedom error.

t

t-statistic.

F

F-statistic.

p

p-value.

estimate

The r statistic and confidence interval in APA style for markdown printing.

estimateR2

The R^2 statistic and confidence interval in APA style for markdown printing.

statistic

The t-statistic in APA style for markdown printing.

Examples

# The following example is derived from the "indt_data"
# dataset, included in the MOTE library.

# A forensic psychologist conducted a study to examine whether
# being hypnotized during recall affects how well a witness
# can remember facts about an event. Eight participants
# watched a short film of a mock robbery, after which
# each participant was questioned about what he or she had
# seen. The four participants in the experimental group
# were questioned while they were hypnotized. The four
# participants in the control group received the same
# questioning without hypnosis.

# Contrary to the hypothesized result, the group that underwent
# hypnosis were significantly less accurate while reporting
# facts than the control group with a large effect size, t(6) = -2.66,
# p = .038, d_s = -1.88.

     d_to_r(d = -1.88, n1 = 4, n2 = 4, a = .05)

MOTE documentation built on Dec. 15, 2025, 9:06 a.m.