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

[Yuhang Hu]

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 using 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 using 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 using gmail.com>
>> wrote:
>>
>>> Responses inline below.
>>>
>>> On Thu, Nov 2, 2023 at 3:30 PM Yuhang Hu <yh342 using 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 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
>>>>>> >
>>>>>> > _______________________________________________
>>>>>> > R-sig-meta-analysis mailing list @
>>>>>> R-sig-meta-analysis using r-project.org
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>>>>>>
>>>>>

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