Computes Cohen's h, the effect size for the difference between two
proportions on the arcsine (variance-stabilizing) scale,
$$h = \varphi_1 - \varphi_2, \qquad \varphi_i = 2\,\arcsin\!\sqrt{p_i}.$$
The arcsine transform spaces proportions so that a given h carries the
same detectability wherever the proportions sit, which a raw difference
\(p_1 - p_2\) does not: a shift from .01 to .05 is easier to detect than one
from .41 to .45, and h reflects that while the raw difference does not.
Cohen's h is the proportion analogue of the standardized mean
difference (smd): the effect size on which power and sample size
planning for a difference between two proportions is conventionally based.
Value
A 1-row data.frame with columns term and value;
term is "cohen_h" and value is the signed effect size.
Details
Cohen's h is a population quantity: supplied with population
proportions it returns the population value, supplied with sample proportions
it returns the corresponding sample value. It is signed, positive when
p1 exceeds p2; its magnitude abs() is the size of the
effect irrespective of direction.
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum. (See Chapter 6.)
See also
smd for the standardized mean difference,
cohen_f, ci_proportion
Author
Ken Kelley kkelley@nd.edu
Examples
# A shift from .40 to .55.
cohen_h(p1 = 0.55, p2 = 0.40)
#> term value
#> cohen_h 0.302
# The same raw difference near the floor is a larger h, since a difference is
# easier to detect where the proportions are small.
cohen_h(p1 = 0.20, p2 = 0.05)
#> term value
#> cohen_h 0.476
# Its magnitude is the size irrespective of direction.
abs(cohen_h(p1 = 0.40, p2 = 0.55)$value)
#> [1] 0.3015253