# Variance of Cohen's $d$ for within subjects designs

Based on this paper by Lakens 2013, I understand there are two recommended ways for calculating Cohen's $d$ for within subjects measures.

1. Cohen's $d_{rm}$ (which assumes the correlation between measures is known) and can be calculated as: $$\text{Cohen's }d_{rm}=( M_{diff}/\sqrt{(SD_1^2+SD_2^2-2*r*SD_1*SD_2})*\sqrt{2(1-r)}$$ Where $M_{diff}$ is the difference in means, $SD_1$ and $SD_2$ are the standard deviations of these means and $r$ is the correlation between measures.

the variance of Cohen's $d_{rm}$ can be calculated using the following: $$V(d_{rm})=(1/n+d_{rm}^2/2n)2(1-r)$$ where $n$ is the sample size. Again this assumes the correlation is known.

2. Cohen's $d_{av}$ (which ignores any correlation between measures and used the average of the standard deviations). $$\text{Cohen's }d_{av}=M_{diff}/((SD_1+SD_2)/2)$$ My question is how to calculate the variance of Cohen's $d_{av}$ $(V(d_{av}))$?

Can it be calculate in a similar manner to the calculation for  independent groups - simply by substituting $d$ for $d_{av}$? $$V(d_{av}) = (n_1+n_2/n_1*n_2)+(d_{av}^2/2(n_1+n_2))$$ if so some elements of this equation appear unclear as the sample size is the same for both observations.

Calculating variance of Cohen's d for repeated measures designs? - ResearchGate (accessed Jan 13, 2017).

• I have made some more edits subsequent to Gung's major formatting effort. Can you check the equations carefully as I am still not sure they are all correct. – mdewey Jan 13 '17 at 14:08
• Thank-you for editing these equations ! This is my first post on here so I am still unsure how to properly format !! These equations look to be correct now - thank-you! – RebeccaHirst Jan 13 '17 at 16:00
• Yes, once you ignore the correlations, you can just plug-in the estimator of Cohen's $\delta$. BTW, see here slightly different formula for its variance. – V. Vancak Jan 13 '17 at 16:36
• @V.V thanks for your response! Does it not matter that n1 and n2 are the same sample (as this is within subjects)? – RebeccaHirst Jan 13 '17 at 16:49
• It makes the formula slightly more elegant, nothing more. It is equivalent to a so-called "balanced design" Cohen's $\delta$. Anyway, it seems like the main issue should be the omitting of the autocorrelation that you have to be concerned about. – V. Vancak Jan 13 '17 at 17:50

There is no know derivation of the variance of $d_{av}$. In fact, this is not how one should compute the $d$ value for a within-subjects design (neither with $d_{rm}$ or the $d_{av}$).

There are two approaches for computing a $d$ value for a within-subjects design. The first uses change score standardization and is given by $$d_c = \frac{M_d}{SD_d},$$ where $M_d$ is the mean change and $SD_d$ is the SD of the change scores (which is equal to $SD_d = \sqrt{SD_1^2 + SD_2^2 - 2 \times r \times SD_1SD_2}$). The large-sample variance of $d_c$ is $$Var[d_c] = \frac{1}{n} + \frac{d_c^2}{2 \times n}.$$

The second approach uses raw score standardization. Here, we standardize based on the SD of either the pre- or the post-test scores (typically, the SD of the pre-test scores is used). So, we compute $$d_r = \frac{M_d}{SD_1}.$$ The large-sample variance of $d_r$ is $$Var[d_r] = \frac{2(1-r)}{n} + \frac{d_r^2}{2 \times n}.$$

We do not take the average of $SD_1$ and $SD_2$ in the second approach (and if anything, we would compute $\sqrt{(SD_1^2 + SD_2^2)/2}$ but again we also do not do that). If you would do this, then it becomes nearly impossible to derive the variance, because $SD_1^2 + SD_2^2$ does not follow a (scaled) $\chi^2$ distribution, since the pre- and post-test scores are not independent. If you look at the answer given here (which is about the independent samples case), you will see that the derivation of the variance involves the non-central t-distribution, which is also involved in deriving the two variance equations above. However, to get that non-central t, you need (the square-root of) a (scaled) $\chi^2$ distribution in the denominator.

• My reason for choosing dav is that it does not require knowledge of r. The problem with drm, dc and dr, is that knowledge of r is required. I am conducting a meta-analysis in which r is not typically reported for each study. I can either try to estimate r based on the scores available in the literature, or use dav - which does not require r. My aim was to compute both and perform a sensitivity analysis to observe the differences, however I cannot do this without knowing how to calculate variance for dav- are you suggesting that this is not possible? Thanks for your help – RebeccaHirst Jan 14 '17 at 12:15
• Again, I do not know of any derivation of the variance of $d_{av}$, and I don't think there is. But it would involve $r$, just like the variance for $d_r$. So, there is no way around knowing or 'guestimating' the correlation here, regardless of what kind of d value you compute. – Wolfgang Jan 14 '17 at 12:49
• There are at least 5 different ways in actual practice that people compute a d statistic for a within-subjects design. Curious for any comments you have on that (which you could leave here or at my blog, whichever is more appropriate). – Jake Westfall Jan 28 '17 at 3:55
• @JakeWestfall This was too long for a comment here, so I commented at your blog. – Wolfgang Jan 28 '17 at 12:17
• Dear Wolfang, $(1)$ could you please provide a reference for the large-sample variance of $d_c$? $(2)$ is the variance of bias-corrected $d_c$ (Hedge's g) the following: $Var(d_c) \times (Hedge's~ correction factor)^2$ – rnorouzian Apr 25 at 15:43

Structural equation modeling (SEM) is your friend in this case. We need the sample means and the covariance matrix as inputs. We may label the parameters (m1, m2, sd1, and sd2) and define $d_{av}$=(m2-m1)/((sd1+sd2)/2). SEM automatically calculates $Var(d_{av})$ by taken the correlation between the pre- and post-test scores into account.

The followings are the R code using the lavaan package. You may refer to Chapter 3 in Cheung (2015) for the details.

library("lavaan")

## Sample covariance matrix on pre- and post-test scores
lower <- '10
8 12'
( Cov <- getCov(lower, diag=TRUE, names=c("x_pre","x_post")) )

## Means of the pre- and post-tests
Mean <- c(10, 13)

## Sample size
N <- 50

model1 <- '# Label the sds with sd1 and sd2
eta_pre =~ sd1*x_pre
eta_post =~ sd2*x_post
# Fix the error variances at 0
x_pre ~~ 0*x_pre
x_post ~~ 0*x_post
# Label the means with m1 and m2
x_pre ~ m1*1
x_post ~ m2*1
# Define d_av
d_av := (m2-m1)/((sd1+sd2)/2)'

## Fit the model
fit1 <- cfa(model1, sample.cov=Cov, sample.mean=Mean,
sample.nobs=N, std.lv=TRUE,
sample.cov.rescale=FALSE)

## Display d_av and its SE
parameterEstimates(fit1)[12, -c(1,2,3)]

## Output
label   est    se   z pvalue ci.lower ci.upper
12  d_av 0.905 0.131 6.9      0    0.648    1.163


Reference

Cheung, M. W.-L. (2015). Meta-analysis: A structural equation modeling approach. Chichester, West Sussex: John Wiley & Sons, Inc.

• Nice approach, Mike! – Wolfgang Jan 28 '17 at 12:19
• Thanks. It is based on maximum likelihood estimation. Thus, it is likely biased in small samples. – Mike Cheung Jan 28 '17 at 12:49
• Yeah, I know. Tried it out and compared it against the bias-corrected version of $d_r$ in my answer. Would be interesting to see which has lower MSE. – Wolfgang Jan 28 '17 at 13:30