[R-meta] Correcting Hedges' g vs. Log response ratio in nested studies

[Yuhang Hu]

Thank you, James and Wolfang,

James,

Regarding your first message, it looks like the correction factor for SMD
is: sqrt( 1-((2*(n-1)*icc)/(N-2)) ) where n is the average cluster size for
each comparison in a study, and N is the sum of the two groups' sample
sizes. So, I wonder how the number of clusters is impacting the correction
factor for SMD as you indicated?

Regarding my initial question, my hunch was that for SMD, the SMD estimate
and its sampling variance are (non-linearly) related to one another.
Therefore, correcting the sampling variance for a design issue will
necessitate correcting the SDM estimate as well.

On the other hand, the LRR estimate and its sampling variance are not as
much related to one another. Therefore, correcting the sampling variance
for a design issue will not necessitate correcting the LRR estimate as well.


vi_g <- function(gi, n1i, n2i) { ((n1i + n2i)/(n1i * n2i)) + ((gi^2)/(2 *
(n1i+n2i))) }

vi_lrr <- function(m1,m2,n1,n2,sd1,sd2) { (sd1 / sqrt(n1))^2 / m1^2 + (sd2
/ sqrt(n2))^2 / m2^2 }

curve(vi_g(x, 30,30), 0, 1, xlab="g", ylab = "sampling variance")

curve(vi_lrr(x, 1,30,30,.1,.1), 0, 1, xlab="LRR", ylab = "sampling
variance")


On Thu, Nov 2, 2023 at 8:41 AM James Pustejovsky <jepusto using gmail.com> wrote:

> One other thought on this question, for the extra-nerdy.
>
> The formulas for the Hedges' g SMD estimator involve what statisticians
> would call "second-order" bias corrections, meaning corrections arising
> from having a limited sample size. In contrast, the usual estimator of the
> LRR is just a "plug-in" estimator that works for large sample sizes but can
> have small biases with limited sample sizes. Lajeunesse (2015;
> https://doi.org/10.1890/14-2402.1) provides formulas for the second-order
> bias correction of the LRR estimator with independent samples. These bias
> correction formulas actually *would* need to be different if you
> have clustered observations. So, the two effect size metrics are maybe more
> similar than it initially seemed:
> - Both metrics have plug-in estimators that are not really affected by the
> dependence structure of the sample, but whose variance estimators do need
> to take into account the dependence structure
> - Both metrics have second-order corrected estimators, the exact form for
> which does need to take into account the dependence structure.
>
> James
>
> On Thu, Nov 2, 2023 at 8:14 AM James Pustejovsky <jepusto using gmail.com>
> wrote:
>
>>
>> Wolfgang is correct. The WWC correction factor arises because the sample
>> variance is not quite unbiased as an estimator for the total population
>> variance in a design with clusters of dependent observations, which leads
>> to a small bias in the SMD.
>>
>> The thing is, though, this correction factor is usually negligible. Say
>> you’ve got a clustered design with n = 21 kids per cluster and 20 clusters,
>> and an ICC of 0.2. Then the correction factor is going to be about 0.99 and
>> so will make very little difference for the effect size estimate. It only
>> starts to matter if you’re looking at studies with very few clusters and
>> non-trivial ICCs.
>>
>> James
>>
>> > On Nov 2, 2023, at 3:04 AM, Viechtbauer, Wolfgang (NP) via
>> R-sig-meta-analysis <r-sig-meta-analysis using r-project.org> wrote:
>> > Dear Yuhang,
>> >
>> > I haven't looked deeply into this, but an immediate thought I have is
>> that for SMDs, you divide by some measure of variability within the groups.
>> If that measure of variability is affected by your study design, then this
>> will also affect the SMD value. On the other hand, this doesn't have any
>> impact on LRRs since they are only the (log-transformed) ratio of the means.
>> >
>> > Best,
>> > Wolfgang
>> >
>> >> -----Original Message-----
>> >> From: R-sig-meta-analysis <r-sig-meta-analysis-bounces using r-project.org>
>> On Behalf
>> >> Of Yuhang Hu via R-sig-meta-analysis
>> >> Sent: Thursday, November 2, 2023 05:42
>> >> To: R meta <r-sig-meta-analysis using r-project.org>
>> >> Cc: Yuhang Hu <yh342 using nau.edu>
>> >> Subject: [R-meta] Correcting Hedges' g vs. Log response ratio in
>> nested studies
>> >>
>> >> Hello All,
>> >>
>> >> I know that when correcting Hedges' g (i.e., bias-corrected SMD, aka
>> "g")
>> >> in nested studies, we have to **BOTH** adjust our initial "g" and its
>> >> sampling variance "vi_g"
>> >> (
>> https://ies.ed.gov/ncee/wwc/Docs/referenceresources/WWC-41-Supplement-
>> >> 508_09212020.pdf).
>> >>
>> >> But when correcting Log Response Ratios (LRR) in nested studies, we
>> have to
>> >> **ONLY** adjust its initial sampling variance "vi_LRR" but not "LRR"
>> itself
>> >> (
>> https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2021-October/003486.html
>> ).
>> >>
>> >> I wonder why the two methods of correction differ for Hedge's g and
>> LRR?
>> >>
>> >> Thanks,
>> >> Yuhang
>> >
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>

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