[R-meta] Standardized response mean

[Reza Norouzian]

To add quickly to Wolfgang's comment, if you're lucky, there might be
some studies in which a paired-samples t-test (t.pair) for one or more
groups with some sample size (n) has been reported. Other than being
able to directly compute your (biased) SMCC for those groups using:

SMCC = t.pair / sqrt(n)

you can also get the denominator (sd_c) of your SMCC effect size using:

sd_c = sqrt((n*(mean_post - mean_pre)^2) / t.pair^2)

and then, obtain the r_pre_post by plugging that obtained sd_c in:

r_pre_post = (sd_pre^2 + sd_post^2 - sd_c^2) / (2*sd_pre*sd_post)

This r_pre_post may help you impute the missing r_pre_post for other
groups that have not reported a t.pair in that same study, as well as
giving you some sense as to what a defensible "guesstimate" for use in
other similar studies, where no direct info. about r_pre_post is
available, might look like.

Such manually computed SMCC effect sizes are systematically biased, so
make sure to apply a correction factor to them:

cfactor <- function(df) exp(lgamma(df/2)-log(sqrt(df/2)) - lgamma((df-1)/2))

unbiased_SMCC = SMCC * cfactor(n - 1)

You can also manually compute their unbiased sampling variances (unbiased_vi):

unbiased_vi = 1/n_c + (1 - ( (n-3) / ((n-1)*cfactor(n-1)^2) )) * unbiased_SMCC^2

Best,
Reza

On Thu, Sep 9, 2021 at 3:16 AM Viechtbauer, Wolfgang (SP)
<wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>
> Hi Hazel,
>
> If you have the SD of the change scores, then you do not need the correlation. See:
>
> "Also, for the raw mean change ("MC") or the standardized mean change using change score standardization ("SMCC"), if sd1i, sd2i, and ri are unknown, but the standard deviation of the change scores is directly reported, then one can set sd1i to that value and both sd2i and ri to 0."
>
> The equation you posted [1] was essentially:
>
> 1/n1 + 1/n2 + d^2 / (2*(n1+n2))
>
> which is the sampling variance for a standardized mean difference (for comparing two independent groups). This is not the right equation for measures such as SMCC or SMCR. As a sidenote: SMD, SMCC, SMCR, MD, MC, and dozens of other measures are "effect sizes", so there is not a generic equation for the variance of an "effect size".
>
> [1] Note that this is a plain-text mailing list, so such inline images get scrubbed from the archives -- see here: https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2021-September/003213.html -- which makes the inclusion of inline images problematic for those going through the archives for answers (I could see the image before my email client converted your message to plain text, at which point also the image was lost -- see below).
>
> Best,
> Wolfgang
>
> >-----Original Message-----
> >From: Hazel Wellington [mailto:hazel.wellington13 using gmail.com]
> >Sent: Thursday, 09 September, 2021 4:39
> >To: Viechtbauer, Wolfgang (SP)
> >Cc: r-sig-meta-analysis using r-project.org
> >Subject: Re: [R-meta] Standardized response mean
> >
> >Hi Wolfgang,
> >
> >Thanks so much for your reply. I am actually wondering if there is another way to
> >compute ‘standardized mean change using change score standardization’ or SMCC as
> >the approach given in your article requires correlation between the measurements.
> >The correlations are missing in the studies that I am working on.
> >
> >Also, as standardized response mean (SRM) is a kind of effect size, will it be OK
> >to calculate its standard error using the attached formula where ‘d’ will be
> >replaced by SRM?
> >
> >Any thoughts?
> >
> >Please let me know.
> >
> >Thanks again,
> >
> >Hazel
> >
> >On Wed, Aug 25, 2021 at 5:23 AM Viechtbauer, Wolfgang (SP)
> ><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
> >Dear Hazel,
> >
> >Based on some online searching, the 'standardized response mean' appears to be the
> >mean change in some dependent variable divided by the SD of the change scores.
> >This is the same as the 'standardized mean change using change score
> >standardization', which is measure="SMCC" in escalc(). See here:
> >
> >https://wviechtb.github.io/metafor/reference/escalc.html#arguments
> >
> >Best,
> >Wolfgang
> >
> >>-----Original Message-----
> >>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
> >>Behalf Of Hazel Wellington
> >>Sent: Wednesday, 25 August, 2021 11:13
> >>To: r-sig-meta-analysis using r-project.org
> >>Subject: [R-meta] Standardized response mean
> >>
> >>Hi everyone,
> >>Hope you are well. I have a query re: meta-analysis of responsiveness
> >>measures.
> >>
> >>My effect size is a standardized response mean and I am wondering how can I
> >>compute its variance or standard error.
> >> Also, should I use the same code in metafor for standardized response mean
> >>as we do for Cohen’s d?
> >>
> >>Please let me know.
> >>Thank you,
> >>Hazel
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