Discrepancy in `t_to_d`

Describe the bug
I was doing a comparison between the results of emmeans::eff_size() and effectsize::t_to_d(), and come up with very different results

To Reproduce

require(effectsize)
#> Loading required package: effectsize
require(emmeans)
#> Loading required package: emmeans

warp.lm = lm(breaks ~ tension, data = warpbreaks)
emm = emmeans(warp.lm, "tension")

### Using emmeans::eff_size
eff_size(emm, sigma = sigma(warp.lm), edf = df.residual(warp.lm))

#>  contrast effect.size    SE df lower.CL upper.CL
#>  L - M          0.842 0.344 51    0.152     1.53
#>  L - H          1.239 0.355 51    0.526     1.95
#>  M - H          0.397 0.336 51   -0.276     1.07
#> 
#> sigma used for effect sizes: 11.88 
#> Confidence level used: 0.95

### Using t_to_d
t = summary(pairs(emm)) $ t.ratio
t_to_d(t, df = df.residual(warp.lm))

#>    d |        95% CI
#> --------------------
#> 0.71 | [ 0.14, 1.27]
#> 1.04 | [ 0.45, 1.62]
#> 0.33 | [-0.22, 0.89]

Note that the estimated effect sizes are somewhat smaller than yielded by eff_size().

My own equivalent function

• Since d = delta / sigma (where delta is the difference of means), and t = delta / (sigma * sqrt (2 / n)), we have that d = sqrt(2 / n) * t.
• Meanwhile, df = k * (n - 1) where k is the number of groups, so that n = 1 + df / k.
• Finally, to get k, the number of t ratios is k * (k - 1) / 2, so k = (1 + sqrt(1 + 8 * L)) / 2, where L is the number of t ratios.

Hence, my own version of t_to_d(), implementing these steps backwards:

### My own equivalent (just the point estimates)
rvl_t2d = function(t, df) {
    k = (1 + sqrt(1 + 8 * length(t))) / 2
    n = 1 + df / k
    sqrt(2 / n) * t
}

rvl_t2d(t, df.residual(warp.lm))

#> [1] 0.8417098 1.2391839 0.3974741

These results agree with the emmeans::eff_size() results.

Additional comment

My rvl_t2d() function depends on our having a balanced experiment with independent samples. Anyn such function that depends only on t ratios and df would require the same assumption (or something equally restrictive). The function emmeans::eff_size() does not require such a simplifying assumption, as it actually estimates each pairwise difference and divides it by the estimate of sigma. That's one reason I am quite sure that its results are correct. But I would think that effectsize::t_to_d() should yield the same results when those restrictive assumptions are satisfied, as they are in this illustration. I am not well-versed in the effect-size literature so I cannot comment with any authority on the formulas given in the documentation for t_to_d().

My system

R.Version()
#> $platform
#> [1] "x86_64-w64-mingw32"
#> 
#> $arch
#> [1] "x86_64"
#> 
#> $os
#> [1] "mingw32"
#> 
#> $system
#> [1] "x86_64, mingw32"
#> 
#> $status
#> [1] ""
#> 
#> $major
#> [1] "4"
#> 
#> $minor
#> [1] "0.3"
#> 
#> $year
#> [1] "2020"
#> 
#> $month
#> [1] "10"
#> 
#> $day
#> [1] "10"
#> 
#> $`svn rev`
#> [1] "79318"
#> 
#> $language
#> [1] "R"
#> 
#> $version.string
#> [1] "R version 4.0.3 (2020-10-10)"
#> 
#> $nickname
#> [1] "Bunny-Wunnies Freak Out"

packageVersion("effectsize")
#> [1] '0.4.0'

packageVersion("emmeans")
#> [1] '1.5.2.10003'

^{Created on 2020-11-28 by the reprex package (v0.3.0)}

Start at the t_to_d() entry point and run the supplied warpbreaks, emmeans, and effectsize reproduction. Compare its point estimates and confidence intervals with emmeans::eff_size() and the reported balanced-design calculation; done means the discrepancy is explained and the expected behavior is clearly established.