Learning Statistics with R - A tutorial for psychology students and other beginners, 2018a

13.8.2 Cohen’s d from a Student t test
The majority of discussions of Cohen’s d focus on a situation that is analogous to Student’s independent
samples t test, and it’s in this context that the story becomes messier, since there are several
different versions of d that you might want to use in this situation, and you can use the method argument
to the cohensD() function to pick the one you want. To understand why there are multiple versions of d,
it helps to take the time to write down a formula that corresponds to the true population effect size δ.
It’s pretty straightforward,
δ “ μ 1 ´ μ 2
σ
where, as usual, μ 1 and μ 2 are the population means corresponding to group 1 and group 2 respectively,
and σ is the standard deviation (the same for both populations). The obvious way to estimate δ is to
do exactly the same thing that we did in the t-test itself: use the sample means as the top line, and a
pooled standard deviation estimate for the bottom line:
d “ ¯X 1 ´ ¯X 2
ˆσ p
where ˆσ p is the exact same pooled standard deviation measure that appears in the t-test. This is the most
commonly used version of Cohen’s d when applied to the outcome of a Student t-test ,and is sometimes
referred to as Hedges’ g statistic (Hedges, 1981). It corresponds to method = "pooled" in the cohensD()
function, and it’s the default.
However, there are other possibilities, which I’ll briefly describe. Firstly, you may have reason to
want to use only one of the two groups as the basis for calculating the standard deviation. This approach
(often called Glass’ Δ) only makes most sense when you have good reason to treat one of the two groups
as a purer reflection of “natural variation” than the other. This can happen if, for instance, one of the
two groups is a control group. If that’s what you want, then use method = "x.sd" or method = "y.sd"
when using cohensD(). Secondly, recall that in the usual calculation of the pooled standard deviation we
divide by N ´ 2 to correct for the bias in the sample variance; in one version of Cohen’s d this correction
is omitted. Instead, we divide by N. This version (method = "raw") makes sense primarily when you’re
trying to calculate the effect size in the sample; rather than estimating an effect size in the population.
Finally, there is a version based on Hedges and Olkin (1985), who point out there is a small bias in the
usual (pooled) estimation for Cohen’s d. Thus they introduce a small correction (method = "corrected"),
by multiplying the usual value of d by pN ´ 3q{pN ´ 2.25q.
In any case, ignoring all those variations that you could make use of if you wanted, let’s have a look
at how to calculate the default version. In particular, suppose we look at the data from Dr Harpo’s
class (the harpo data frame). The command that we want to use is very similar to the relevant t.test()
command, but also specifies a method
> cohensD( formula = grade ~ tutor, # outcome ~ group
+ data = harpo, # data frame
+ method = "pooled" # which version to calculate?
+ )
[1] 0.7395614
This is the version of Cohen’s d that gets reported by the independentSamplesTTest() function whenever
it runs a Student t-test.
13.8.3 Cohen’s d from a Welch test
Suppose the situation you’re in is more like the Welch test: you still have two independent samples,
- 414 -