# [R-meta] Combining studies with different within-subject designs

Pablo Grassi p@b|o@gr@@@| @end|ng |rom c|n@un|-tueb|ngen@de
Fri Feb 18 15:22:07 CET 2022

```Dear Wolfgang,

Thanks for your detailed answer. To clarify, all cases involve the /same
group of participants/ and are thus paired data ("within-subject designs"):

Case 1 = pre-Treament and post_Treatment both in day 1 and
pre-Control and post-Control both in day 2, in the same participants
(order of T and C counterbalanced)

Case 2 = post-Treatment in day 1 vs post-Control in day 2, in
the same participants (order of T and C counterbalanced).

Case 3 = pre-Treatment and post-Treatment in day 1.

So, if I understand you correctly, your suggestion to calculate SMDs
using "raw score standardization" would be for each case:

For Case 1:

SMD1 = ((MpostT - MpreT) - (MpostC-MpreC)) / SD_pre_pooled

with SD_pre_pooled = sqrt(SDpreT^2 + SDpreC^2 +
2*r1*SDpreT*SDpreC) / 2

and r1 = correlation between preT and preC values (as they
T and C are the same participants)

SE1 =  2*(1-r2)/n + SMD^2 /(2n)

n = n of participants

and r2 = correlation between individual change score
differences (postT-preT) and (postC-preC)

Then, case 2 is not so clear for me (as I believe you assumed treatment
and control are different group of participants):

SMD2 = (MpostT-MpostC) / SD_???

Which SD do you suggest to use?

SD_diff =  sqrt(SDpostT^2 + SDpostC^2 - 2*r*SDpostT*SDpostC) or

SD_pooled = sqrt(SDpostT^2 + SDpostC^2 + 2*r*SDpostT*SDpostC) / 2

or just one of the SD (SDt or SDc)?

And SE2 = ???

And case 3, you suggest:

SMD3 = (MpostT-MpreT) / SD_pre

And SE3 (same as case SE1) = 2*(1-r2)/n + SMD^2 /(2n)

Is this correct? (I am also a bit unsure about the SE formulas)

And unfortunately yes, most of the authors do not report any of the
required SDs. Paired t-tests and related values (e.g. often reported
effect sizes as cohens d_z) would be of no help then, as they use the SD
of the differences/change scores (and would only be of use using a
change score standardization).

Alternative: Wouldn't it be possible to check if the variance
post-treatment differs to post-control in those studies were the raw
data is available and then based on this information, leave or drop the
change score standardization?

In any case thanks for your suggestions! I'll definitely add the design
as a moderator in the meta-analysis.

Best,

Pablo

On 18.02.22 11:10, Viechtbauer, Wolfgang (SP) wrote:
> Dear Pablo,
>
> I am confused why you call case 2 a 'within-study design'. That looks like a two-group post-test only design to me.
>
> In any case, for all three cases, you want to use 'raw score standardization' to make the various effect size measures comparable, at least under some restrictive assumptions, such as that there is no inherent time effect or time by group interaction.
>
> So, for case 2, you compute standard SMDs, which use the SD at a single timepoint (i.e. at post). This is in essence 'raw score standardization'.
>
> Ananlogously, for case 3, you also use 'raw score standardization', so measure "SMCR" in escalc() lingo.
>
> And finally, for case 1, you again want to standardize based on the SD of a single timepoint, not the SD of the change scores. See:
>
> https://www.metafor-project.org/doku.php/analyses:morris2008
>
> for a discussion of how to do this.
>
> If authors do not report the required raw score SDs, then you will have to get creative getting these (e.g., contacting authors, back-calculating them based on the SD of the change scores and other information provided, guestimating them).
>
> I would also code the type of design that was used in each study (and hence the type of effect size measure that was computed) to examine whether there are still systematic differences between these groups, even if the type of standardization was the same across measures.
>
> Best,
> Wolfgang
>
>> -----Original Message-----
>> From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>> Behalf Of Pablo Grassi
>> Sent: Wednesday, 16 February, 2022 16:14
>> To:r-sig-meta-analysis using r-project.org
>> Subject: [R-meta] Combining studies with different within-subject designs
>>
>> Dear all,
>>
>> Maybe some of you can help me out with the following design-conundrum. I
>> am currently performing a series of meta-analysis investigating the
>> effect on an intervention in closely related outcomes. Most of the
>> reviewed studies are within-subject designs (one group of participants),
>> for which the standardized mean differences (SMD; Hedge's g) are
>> differences of change scores divided by the SD of the difference (with
>> change score standardization), as follows:
>>
>>      SMD = M_diff / SDdiff   (for simplicity in this E-mail without the
>> bias-correction factor)
>>
>> Unfortunately, there is a huge variability in the control measurements
>> used. Roughtly following the nomenclature from Morris 2008, I have the
>> following different design-cases:
>>
>>      Case 1) Within-subject desing, pre-post-control (WS_PPC):
>>
>>                 M_diff_ws_pcc = (post_Treatment - pre_Treatment) -
>> (post_Control - pre_Control)
>>
>> However, some others had no baseline pre-intervention measurement (i.e.
>> only report post-intervention measurements), i.e. Post-test only with
>> control design (WS_POWC)
>>
>>      Case 2) M_diff_ws_powc = (post_Treatment - post_Control)
>>
>> And some few others just have a post-pre measurement but no control
>> (i.e. only report a change score, single-group pre-post, SGPP), so that:
>>
>>      Case 3) M_diff_ws_sgpp = (post_Treatment - pre_Treatment)
>>
>> Then, while the M_diff if Case 1 and 2 is measuring +/- the same effect,
>> their SDdiffs (and thus SMDs) are not really comparable as they reflect
>> different things.
>>
>> Thus:
>>
>>   * What would be the "standard" approach to include the within-subject
>>     design studies (Case 1, 2, 3) in the same meta-analysis, if this is
>>     possible at all? (Please consider that most of the publications
>>     report ONLY the SD of the change scores and NOT the SD of the pre
>>     and post conditions separately or in case 2 of the post-interventions).
>>
>> Best,
>>
>> Pablo

--
Dr. Pablo R. Grassi
'Vision and Cognition Lab' Centre for Integrative Neurosciences, University of Tuebingen, Germany.
Otfried-Müller-Str. 25, 72076 Tuebingen, Germany.
+49 7071 29 89103
--

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