[R-meta] d_z and d_av effect sizes in metafor

[Viechtbauer, Wolfgang (NP)]

The formula for the bias-correction factor is the same:

gamma(mi/2)/(sqrt(mi/2)*gamma((mi-1)/2))

but for SMD = n1i+n2i-2, for SMCC, mi = ni-1, and for SMCRP, mi = 2*(ni-1) / (1 + ri^2). As for references, see the documentation of escalc():

https://wviechtb.github.io/metafor/reference/escalc.html#-a-measures-for-quantitative-variables-2

Best,
Wolfgang

> -----Original Message-----
> From: Zhouhan Jin <zjin65 using uwo.ca>
> Sent: Monday, April 7, 2025 20:40
> To: R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-
> project.org>; Viechtbauer, Wolfgang (NP)
> <wolfgang.viechtbauer using maastrichtuniversity.nl>
> Subject: RE: d_z and d_av effect sizes in metafor
>
> Thank you, Wolfgang, for the clarification! Is it possible to share the formula
> for bias correction for SMCRP and how it is different from the correction factor
> used in SMCC or SMD? Any references for the difference in correction factors are
> highly appreciated?
>
> Best wishes,
>
> Zhouhan Jin
> On Apr 7, 2025 at 07:18 -0400, Viechtbauer, Wolfgang (NP)
> <mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>, wrote:
>
> Dear Zhouhan,
>
> If r=0.5 *and* sd_pre = sd_post, *then* the SMCC and SMCR are identical:
>
> n <- 30
> r <- 0.5 # r set to 0.5 for demonstration
> mean_pre <- 0
> mean_post <- 0.5
> sd_pre <- 0.75
> sd_post <- 0.75
>
> escalc(measure = "SMCC", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n)
>
> escalc(measure = "SMCR", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n)
>
> However, r=0.5 is not sufficient for this equivalence to hold.
>
> Even then, SMCRP will be slightly different, since it uses a different bias-
> adjustment factor:
>
> escalc(measure = "SMCRP", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n)
>
> We can switch off the bias correction and then get equivalence for this measure
> as well:
>
> escalc(measure = "SMCC", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n, correct=FALSE)
>
> escalc(measure = "SMCR", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n, correct=FALSE)
>
> escalc(measure = "SMCRP", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n, correct=FALSE)
>
> The sampling variance will still be different for SMCRP, since it uses more
> information compared to SMCR.
>
> Best,
> Wolfgang
>
> -----Original Message-----
> From: R-sig-meta-analysis <mailto:r-sig-meta-analysis-bounces using r-project.org> On
> Behalf
> Of Zhouhan Jin via R-sig-meta-analysis
> Sent: Saturday, April 5, 2025 21:18
> To: R Special Interest Group for Meta-Analysis <mailto:r-sig-meta-analysis using r-
> project.org>
> Cc: Zhouhan Jin <mailto:zjin65 using uwo.ca>
> Subject: [R-meta] d_z and d_av effect sizes in metafor
>
> Hello Wolfgang and Rmeta Community,
>
> I always assumed that under pre-post correlation = 0.5, metafor::escal()'s
> measure = "SMCC" and "SMCRP" referred to in the article below as "d_z" and
> "d_av" respectively, are equivalent. But that doesn't seem to be the case (see R
> code below).
>
> Questions: 1- Can they be equivalent under any condition?
> 2- If yes, will their sampling variances then also be equal?
> 3- If no, are there any other two SMC variants that will be
> equivalent in both effect size and sampling variances under r=0.5?
>
> ARTICLE:
> https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2013.0086
> 3/full
>
> # R code:
> library(metafor)
> n <- 30
> r <- 0.5 # r set to 0.5 for demonstration
> mean_pre <- 0
> mean_post <- 0.5
> sd_pre <- 1
> sd_post <- .5
>
> esc_dz <- escalc(measure = "SMCC", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n)
>
> esc_dav <- escalc(measure = "SMCRP", m1i = mean_post, m2i = mean_pre,
> sd1i = sd_post, sd2i = sd_pre, ri = r, ni = n)
>
> Best wishes,
>
> Zhouhan Jin