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

No. The notation in the WWC handbook is poorly chosen and confusing, so
let me offer a slightly different notation to try and clear things up. This
correction applies to a study with m clusters, where cluster j has n_j
individuals. There are a total of N = sum(n_1,...,n_m) individuals, of
which N_t are assigned to the treatment condition and N_c are assigned to
the control condition. The average cluster size is then n_bar = N / m. With
this notation, the correction factor is

sqrt(1 - (2 * (n_bar - 1) * icc / (N - 2))) = sqrt(1 - (2 * (n_bar - 1) *
icc / (n_bar * m - 2)))

Thus, for a given study, you need to know the icc, the value of n_bar, and
the value of N (or the number of clusters m and the total sample size N,
from which you can calculate n_bar) in order to calculate the correction
factor.

On Thu, Nov 2, 2023 at 7:32?PM Yuhang Hu <yh342 at nau.edu> wrote:

Thanks so  much, James. This is how I got it set up. Do I have it right?

dat <- read.table(header=TRUE,text="
study  gi   vi  cluster n1  n2
1     .2   .05  T       25  23
1     .3   .08  T       18  11
2     1    .1   F       19  21
2     2    .2   F       12  36")

g_cluster <- \(gi, n1, n2, icc=.15){

  n <- mean(c(n1,n2), na.rm=TRUE)
  N <- sum(c(n1,n2), na.rm=TRUE)
  gi*sqrt( 1-((2*(n-1)*icc)/(N-2)) )
}
library(dplyr)
  group_by(dat , study) %>%
  mutate(gi= ifelse(cluster, g_cluster(gi,n1,n2),  gi))

On Thu, Nov 2, 2023 at 3:22?PM James Pustejovsky <jepusto at gmail.com>
wrote:

This correction applies to a single effect size estimate from a given
study. All of these values (n, N, n1, n2) are therefore specific to the
study and need to be recorded for each row in a meta-analysis database.

On Nov 2, 2023, at 5:14 PM, Yuhang Hu <yh342 at nau.edu> wrote:

?
Sure, I thought N is n1 + n2  which is unique to each row in the
dataset.

But it looks like I should compute N as n * m where "n" is (average of
n1, n2) for each row in the data but "m" is constant across all rows in the
dataset.

Thanks,
Yuhang

On Thu, Nov 2, 2023 at 2:16?PM James Pustejovsky <jepusto at gmail.com>
wrote:

Total sample size is the same thing as the average sample size per
cluster times the number of clusters. My previous message is just a
restatement of the formula to show how it is related to the number of
clusters.

On Thu, Nov 2, 2023 at 4:11?PM Yuhang Hu <yh342 at nau.edu> wrote:

Hi James,

If you look at Eq. number E.5.1 on p1 of this document: (
https://ies.ed.gov/ncee/wwc/Docs/referenceresources/WWC-41-Supplement-508_09212020.pdf)
they define the correction factor as: sqrt( 1-((2*(n-1)*icc)/(N-2)) )
where N is n1 + n2 (total sample size), and n as s the average number
of individuals per cluster.

Am I missing something? Or is the correction factor linked above from
WWC inaccurate?

Thank you,
Yuhang

On Thu, Nov 2, 2023 at 1:51?PM James Pustejovsky <jepusto at gmail.com>
wrote:

Responses inline below.

On Thu, Nov 2, 2023 at 3:30?PM Yuhang Hu <yh342 at nau.edu> wrote:

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?

N = n * m, where m is the number of clusters. So the correction

factor is
sqrt( 1-((2*(n-1)*icc)/(m * n - 2)) ~=  sqrt( 1- 2 * icc /m)

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.

No, the issue you've described here is pretty much unrelated to the
bias correction problem.

On Thu, Nov 2, 2023 at 8:41?AM James Pustejovsky <jepusto at 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 at 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 at 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 at 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 at r-project.org>
Cc: Yuhang Hu <yh342 at 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

R-sig-meta-analysis at r-project.org

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