The distribution of Cohen's d

These functions use some conversion to and from the t distribution to provide the Cohen's d distribution. There are four versions that act similar to the standard distribution functions (the d., p., q., and r. functions, and their longer aliases .Cohensd), three convenience functions (pdExtreme, pdMild, and pdInterval), a function to compute the confidence interval for a Cohen's d estimate cohensdCI, and a function to compute the sample size required to obtain a confidence interval around a Cohen's d estimate with a specified accuracy (pwr.cohensdCI and its alias pwr.confIntd).

dd(x, df, populationD = 0)
pd(q, df, populationD = 0, lower.tail = TRUE)
qd(p, df, populationD = 0, lower.tail = TRUE)
rd(n, df, populationD = 0)

dCohensd(x, df, populationD = 0)
pCohensd(q, df, populationD = 0, lower.tail = TRUE)
qCohensd(p, df, populationD = 0, lower.tail = TRUE)
rCohensd(n, df, populationD = 0)

pdExtreme(d, n, populationD=0)
pdMild(d, n, populationD=0)
pdInterval(ds, n, populationD=0)

cohensdCI(d, n, conf.level = .95, plot=FALSE, silent=TRUE)
confIntD(d, n, conf.level = .95, plot=FALSE, silent=TRUE)

pwr.cohensdCI(d, w = 0.1, conf.level = 0.95,
              extensive = FALSE, silent = TRUE)
pwr.confIntd(d, w = 0.1, conf.level = 0.95,
             extensive = FALSE, silent = TRUE)

Arguments

x, q, d	Vector of quantiles, or, in other words, the value(s) of Cohen's d.
ds	A vector with two Cohen's d values.
p	Vector of probabilites (p-values).
df	Degrees of freedom.
n	Desired number of Cohen's d values for rCohensd and rd, and the number of participants/datapoints for pdExtreme, pdMild, pdInterval, and cohensdCI.
populationD	The value of Cohen's d in the population; this determines the center of the Cohen's d distribution. I suppose this is the noncentrality parameter.
lower.tail	logical; if TRUE (default), probabilities are the likelihood of finding a Cohen's d smaller than the specified value; otherwise, the likelihood of finding a Cohen's d larger than the specified value.
conf.level	The level of confidence of the confidence interval.
plot	Whether to show a plot of the sampling distribution of Cohen's d and the confidence interval. This can only be used if specifying one value for d, n, and conf.level.
w	The desired 'half-width' or margin of error of the confidence interval.
extensive	Whether to only return the required sample size, or more extensive results.
silent	Whether to provide FALSE or suppress (TRUE) warnings. This is useful because function 'qt', which is used under the hood (see qt for more information), warns that 'full precision may not have been achieved' when the density of the distribution is very close to zero. This is normally no cause for concern, because with sample sizes this big, small deviations have little impact.

Details

The functions use convert.d.to.t and convert.t.to.d to provide the Cohen's d distribution.

More details about cohensdCI and pwr.cohensdCI are provided in Peters & Crutzen (2017).

Value

dCohensd (or dd) gives the density, pCohensd (or pd) gives the distribution function, qCohensd (or qd) gives the quantile function, and rCohensd (or rd) generates random deviates.

pdExtreme returns the probability (or probabilities) of finding a Cohen's d equal to or more extreme than the specified value(s).

pdMild returns the probability (or probabilities) of finding a Cohen's d equal to or less extreme than the specified value(s).

pdInterval returns the probability of finding a Cohen's d that lies in between the two specified values of Cohen's d.

cohensdCI provides the confidence interval(s) for a given Cohen's d value.

pwr.cohensdCI provides the sample size required to obtain a confidence interval for Cohen's d with a desired width.

References

Peters, G. J. Y. & Crutzen, R. (2017) Knowing exactly how effective an intervention, treatment, or manipulation is and ensuring that a study replicates: accuracy in parameter estimation as a partial solution to the replication crisis. http://dx.doi.org/

Maxwell, S. E., Kelley, K., & Rausch, J. R. (2008). Sample size planning for statistical power and accuracy in parameter estimation. Annual Review of Psychology, 59, 537-63. https://doi.org/10.1146/annurev.psych.59.103006.093735

Cumming, G. (2013). The New Statistics: Why and How. Psychological Science, (November). https://doi.org/10.1177/0956797613504966

See also

convert.d.to.t, convert.t.to.d, dt, pt, qt, rt

Examples

### Confidence interval for Cohen's d of .5
### from a sample of 200 participants, also
### showing this visually: this clearly shows
### how wildly our Cohen's d value can vary
### from sample to sample.
cohensdCI(.5, n=200, plot=TRUE);
#>                     lo        hi
#> d=0.5, n=200 0.2249466 0.7859536

### How many participants would we need if we
### would want a more accurate estimate, say
### with a maximum confidence interval width
### of .2?
pwr.cohensdCI(.5, w=.1);
#> [1] 1564

### Show that 'sampling distribution':
cohensdCI(.5,
          n=pwr.cohensdCI(.5, w=.1),
          plot=TRUE);
#>                      lo        hi
#> d=0.5, n=1564 0.4000818 0.6013028

### Generate 10 random Cohen's d values
rCohensd(10, 20, populationD = .5);
#>  [1]  1.00915778  0.09919953  0.29601339  0.61056978  0.08845328  0.12103648
#>  [7]  1.39755050  0.26858049 -0.20755801  0.69554566

### Probability of findings a Cohen's d smaller than
### .5 if it's 0 in the population (i.e. under the
### null hypothesis)
pCohensd(.5, 64);
#> [1] 0.9783415

### Probability of findings a Cohen's d larger than
### .5 if it's 0 in the population (i.e. under the
### null hypothesis)
1 - pCohensd(.5, 64);
#> [1] 0.02165847

### Probability of findings a Cohen's d more extreme
### than .5 if it's 0 in the population (i.e. under
### the null hypothesis)
pdExtreme(.5, 64);
#> [1] 0.04654743

### Probability of findings a Cohen's d more extreme
### than .5 if it's 0.2 in the population.
pdExtreme(.5, 64, populationD = .2);
#> [1] 0.2336917