[R-meta] between-study variance

Thu Jan 19 19:32:41 CET 2023

Dear Wolfgang and James,

Thank you both!

I had not thought about the possibility that the variance could be shifting from between-study to within-study so I definitely appreciate the resources and a course of action for me to evaluate.

Best,
Karen

Wolfgang's points are spot-on. I would add, the second point about accounting for the correlation between effect size estimates is worth digging into a bit. The assumption you make about correlation here (or lack of correlation) can definitely influence the variance component estimates, shifting variance from the between-study level to the within-study level or vice versa. I looked at this a little bit in the supplementary materials (https://osf.io/nyv4u<https://urldefense.com/v3/__https://osf.io/nyv4u__;!!PvXuogZ4sRB2p-tU!FBOeJleDwMuTkpbLYEV-KxsxaASgnqeqiP8z2nnk-x0fRMvEh4bUfEmTLJ76z3eZvsUFxwW3cHDZWsZKLYmtBXE$>) of my paper with Tipton on expanding working models for RVE meta-analysis (https://www.jepusto.com/publication/rve-meta-analysis-expanding-the-range/<https://urldefense.com/v3/__https://www.jepusto.com/publication/rve-meta-analysis-expanding-the-range/__;!!PvXuogZ4sRB2p-tU!FBOeJleDwMuTkpbLYEV-KxsxaASgnqeqiP8z2nnk-x0fRMvEh4bUfEmTLJ76z3eZvsUFxwW3cHDZWsZKJDN_FeM$>). See Section S4.3 especially.

James
On Thu, Jan 19, 2023 at 9:51 AM Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl<mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>> wrote:
Dear Karen,

In principle, variance components can be equal to 0. That is not automatically an indication that there is something wrong.

A few observations:

1) You are including "factor(Cell)-1" as a moderator, which apparently is a factor with 10 levels (hence, the QM-statistic has 10 dfs). If this moderator varies (mostly) at the study level, then with 27 studies it could be the case that this moderator 'sucks up' the study level heterogeneity. What happens when you leave out this moderator?

2) I see that the input to V is called 'invn'. This appears to be a vector with the inverse sample sizes (e.g., 1/0.000509684 = 1962). This is not quite right for computing the sampling variance of correlation coefficients. More importantly, this does not capture the covariance between multiple correlation coefficients coming from the same sample. See help(rcalc) for a function that will construct the var-cov matrix for dependent correlation coefficients.

As for simplified dataset: The copy of dput() you put below got cut off.

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org<mailto:r-sig-meta-analysis-bounces using r-project.org>] On
>Behalf Of Karen Alexander
>Sent: Thursday, 19 January, 2023 16:33
>To: r-sig-meta-analysis using r-project.org<mailto:r-sig-meta-analysis using r-project.org>
>Subject: Re: [R-meta] between-study variance
>
>Hello,
>
>I am a PhD candidate at the University of Nebraska-Lincoln in the Quantitative,
>Qualitative, and Psychometric program and am running a meta-analysis with complex
>data using the framework from Wilson, et al., (2016) (Fitting meta-analytic
>structural equation models with complex datasets) https://doi-<https://urldefense.com/v3/__https://doi-__;!!PvXuogZ4sRB2p-tU!FBOeJleDwMuTkpbLYEV-KxsxaASgnqeqiP8z2nnk-x0fRMvEh4bUfEmTLJ76z3eZvsUFxwW3cHDZWsZKpHBJQoU$>
?
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Karen M. Alexander Glueckert, M. Ed.

Ph. D. Candidate

University of Nebraska-Lincoln

________________________________
From: James Pustejovsky <jepusto using gmail.com>
Sent: Thursday, January 19, 2023 10:29 AM
To: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
Cc: r-sig-meta-analysis using r-project.org <r-sig-meta-analysis using r-project.org>; Karen Alexander <kalexander using huskers.unl.edu>
Subject: Re: [R-meta] between-study variance

Non-NU Email
________________________________
Wolfgang's points are spot-on. I would add, the second point about accounting for the correlation between effect size estimates is worth digging into a bit. The assumption you make about correlation here (or lack of correlation) can definitely influence the variance component estimates, shifting variance from the between-study level to the within-study level or vice versa. I looked at this a little bit in the supplementary materials (https://osf.io/nyv4u<https://urldefense.com/v3/__https://osf.io/nyv4u__;!!PvXuogZ4sRB2p-tU!FBOeJleDwMuTkpbLYEV-KxsxaASgnqeqiP8z2nnk-x0fRMvEh4bUfEmTLJ76z3eZvsUFxwW3cHDZWsZKLYmtBXE$>) of my paper with Tipton on expanding working models for RVE meta-analysis (https://www.jepusto.com/publication/rve-meta-analysis-expanding-the-range/<https://urldefense.com/v3/__https://www.jepusto.com/publication/rve-meta-analysis-expanding-the-range/__;!!PvXuogZ4sRB2p-tU!FBOeJleDwMuTkpbLYEV-KxsxaASgnqeqiP8z2nnk-x0fRMvEh4bUfEmTLJ76z3eZvsUFxwW3cHDZWsZKJDN_FeM$>). See Section S4.3 especially.

James

On Thu, Jan 19, 2023 at 9:51 AM Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl<mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>> wrote:
Dear Karen,

In principle, variance components can be equal to 0. That is not automatically an indication that there is something wrong.

A few observations:

1) You are including "factor(Cell)-1" as a moderator, which apparently is a factor with 10 levels (hence, the QM-statistic has 10 dfs). If this moderator varies (mostly) at the study level, then with 27 studies it could be the case that this moderator 'sucks up' the study level heterogeneity. What happens when you leave out this moderator?

2) I see that the input to V is called 'invn'. This appears to be a vector with the inverse sample sizes (e.g., 1/0.000509684 = 1962). This is not quite right for computing the sampling variance of correlation coefficients. More importantly, this does not capture the covariance between multiple correlation coefficients coming from the same sample. See help(rcalc) for a function that will construct the var-cov matrix for dependent correlation coefficients.

As for simplified dataset: The copy of dput() you put below got cut off.

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org<mailto:r-sig-meta-analysis-bounces using r-project.org>] On
>Behalf Of Karen Alexander
>Sent: Thursday, 19 January, 2023 16:33
>To: r-sig-meta-analysis using r-project.org<mailto:r-sig-meta-analysis using r-project.org>
>Subject: Re: [R-meta] between-study variance
>
>Hello,
>
>I am a PhD candidate at the University of Nebraska-Lincoln in the Quantitative,
>Qualitative, and Psychometric program and am running a meta-analysis with complex
>data using the framework from Wilson, et al., (2016) (Fitting meta-analytic
>structural equation models with complex datasets) https://doi-<https://urldefense.com/v3/__https://doi-__;!!PvXuogZ4sRB2p-tU!FBOeJleDwMuTkpbLYEV-KxsxaASgnqeqiP8z2nnk-x0fRMvEh4bUfEmTLJ76z3eZvsUFxwW3cHDZWsZKpHBJQoU$>
>org.libproxy.unl.edu/10.1002/jrsm.1199<http://org.libproxy.unl.edu/10.1002/jrsm.1199>
>
>Specifically, when running the following code to get the unadjusted pooled
>correlation matrix without the intercept
>
>step1 <- rma.mv<https://urldefense.com/v3/__http://rma.mv__;!!PvXuogZ4sRB2p-tU!FBOeJleDwMuTkpbLYEV-KxsxaASgnqeqiP8z2nnk-x0fRMvEh4bUfEmTLJ76z3eZvsUFxwW3cHDZWsZKRDKpvuc$>(yi = r, V = invn,
>                data = data1,
>                random = list(~ 1|studyID, ~ 1|ESID),
>                method = "ML", mods = ~ factor(Cell)-1)
>summary(step1)
>
>I get the following output which indicates that the between-study variance is 0.
>
>
>Multivariate Meta-Analysis Model (k = 376; method: ML)
>
>   logLik   Deviance        AIC        BIC       AICc
>  87.6057  1749.7597  -151.2114  -104.0563  -150.3519
>
>Variance Components:
>
>            estim    sqrt  nlvls  fixed   factor
>sigma^2.1  0.0000  0.0000     28     no  studyID
>sigma^2.2  0.0339  0.1841    376     no     ESID
>
>Test for Residual Heterogeneity:
>QE(df = 366) = 12265.8732, p-val < .0001
>
>Test of Moderators (coefficients 1:10):
>QM(df = 10) = 110.8337, p-val < .0001
>
>Model Results:
>
>---
>Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
>1) I find that hard to believe. So, I made a subset of my data and ran the model
>with the subset, and while all other values changed, the between-study variance
>was still 0.
>
>2) So, now I am questioning either my code and/or my dataset.
>
>Because of the complexity of the data, the dataset is rather wieldy. I have a
>simplified version below, but it may need to be sent as an attachment, which I am
>happy to do as well.
>
>structure(list(studyID = c(10000L, 10000L, 10000L, 10000L, 10000L,
>10000L, 10000L, 10000L, 10000L, 10000L, 10000L, 10000L, 10000L,
>10000L, 10000L, 10000L, 10000L, 10000L, 10000L, 13000L, 13000L,
>13000L, 13000L, 13000L, 13000L, 13000L, 13000L, 13000L, 13000L,
>13000L, 13000L, 13000L, 13000L, 13000L, 13000L, 13000L, 13000L,
>13000L, 13000L, 13000L, 13000L, 13000L, 13000L, 13000L, 13000L,
>14000L, 14000L, 14000L, 14000L, 14000L, 14000L, 15000L, 15000L,
>15000L, 15000L, 15000L, 15000L, 15000L, 15000L, 15000L, 15000L,
>15000L, 15000L, 15000L, 15000L, 15000L, 15000L, 16000L, 17000L,
>17000L, 17000L, 17000L, 17000L, 17000L, 17000L, 17000L, 18000L,
>18000L, 19000L, 20000L, 20000L, 20000L, 20000L, 20000L, 20000L,
>20000L, 20000L, 20000L, 21000L, 21000L, 21000L, 21000L, 21000L,
>21000L, 21000L, 21000L, 21000L, 21000L, 21000L, 23000L, 23000L,
>23000L, 23000L, 23000L, 23000L, 24000L, 24000L, 24000L, 24000L,
>24000L, 24000L, 24000L, 24000L, 24000L, 24000L, 24000L, 24000L,
>24000L, 24000L, 24000L, 24000L, 26000L, 26000L, 26000L, 26000L,
>26000L, 26000L, 26000L, 26000L, 28000L, 28000L, 29000L, 29000L,
>29000L, 29000L, 29000L, 30000L, 30000L, 30000L, 30000L, 30000L,
>30000L, 30000L, 30000L, 30000L, 32000L, 32000L, 32000L, 32000L,
>32000L, 32000L, 32000L, 32000L, 32000L, 32000L, 32000L, 32000L,
>32000L, 32000L, 32000L, 32000L, 33000L, 33000L, 33000L, 33000L,
>33000L, 33000L, 33000L, 33000L, 33000L, 33000L, 33000L, 33000L,
>35000L, 35000L, 36000L, 36000L), ESID = c(10001L, 10002L, 10003L,
>10004L, 10005L, 10006L, 10007L, 10008L, 10009L, 10010L, 10011L,
>10012L, 10013L, 10014L, 10015L, 10016L, 10017L, 10018L, 10019L,
>13001L, 13002L, 13003L, 13004L, 13005L, 13006L, 13007L, 13008L,
>13009L, 13010L, 13011L, 13012L, 13013L, 13014L, 13015L, 13016L,
>13017L, 13018L, 13019L, 13020L, 13021L, 13022L, 13023L, 13024L,
>13025L, 13026L, 14001L, 14002L, 14003L, 14004L, 14005L, 14006L,
>15001L, 15002L, 15003L, 15004L, 15005L, 15006L, 15007L, 15008L,
>15009L, 15010L, 15011L, 15012L, 15013L, 15014L, 15015L, 15016L,
>16001L, 17001L, 17002L, 17003L, 17004L, 17005L, 17006L, 17007L,
>17008L, 18001L, 18002L, 19001L, 20001L, 20002L, 20003L, 20004L,
>20005L, 20006L, 20007L, 20008L, 20009L, 21001L, 21002L, 21003L,
>21004L, 21005L, 21006L, 21007L, 21008L, 21009L, 21010L, 21011L,
>23001L, 23002L, 23003L, 23004L, 23005L, 23006L, 24001L, 24002L,
>24003L, 24004L, 24005L, 24006L, 24007L, 24008L, 24009L, 24010L,
>24011L, 24012L, 24013L, 24014L, 24015L, 24016L, 26001L, 26002L,
>26003L, 26004L, 26005L, 26006L, 26007L, 26008L, 28001L, 28002L,
>29001L, 29002L, 29003L, 29004L, 29005L, 30001L, 30002L, 30003L,
>30004L, 30005L, 30006L, 30007L, 30008L, 30009L, 32001L, 32002L,
>32003L, 32004L, 32005L, 32006L, 32007L, 32008L, 32009L, 32010L,
>32011L, 32012L, 32013L, 32014L, 32015L, 32016L, 33001L, 33002L,
>33003L, 33004L, 33005L, 33006L, 33007L, 33008L, 33009L, 33010L,
>33011L, 33012L, 35001L, 35002L, 36001L, 36002L), Cell = c(4L,
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>"r12", "r12", "r12", "r12", "r12", "r12", "r12", "r12", "r12",
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>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 1L,
>0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 1L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L,
>0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L,
>0L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L), Cell5 = c(0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L
>), Cell6 = c(0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L), Cell7 = c(0L, 0L, 1L, 1L, 0L, 0L, 1L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 1L, 1L,
>1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>1L, 1L, 1L, 1L, 1L, 0L, 1L, 0L, 0L, 1L, 0L, 1L, 1L, 1L, 1L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 0L, 0L,
>0L, 0L, 0L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>1L, 0L, 1L, 0L, 1L, 0L, 1L, 1L, 1L, 0L, 0L, 0L, 1L, 1L, 1L, 1L,
>1L, 1L, 0L, 1L, 1L, 0L, 1L, 0L, 1L, 1L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 1L, 1L, 0L, 0L, 0L, 1L, 1L, 0L, 1L, 1L, 1L, 1L, 0L, 0L, 0L,
>1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 0L, 0L, 0L, 1L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
>1L, 1L, 1L, 1L, 1L, 0L, 1L, 1L, 1L), Cell8 = c(0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L), Cell9 = c(0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 1L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L
>), Cell10 = c(0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 1L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
>0L, 0L, 0L, 0L, 0L), r = c(0.01, 0.27, 0.19, 0.19, 0.29, 0.24,
>0.19, 0.25, 0.22, 0.24, 0.31, 0.24, 0.3, -0.17, -0.25, -0.31,
>0.09, 0.01, -0.01, 0.03, 0.05, 0.09, 0.07, 0.02, 0, 0.07, 0.04,
>0.02, 0.04, 0.04, 0.04, 0.01, 0.15, 0.16, 0.15, 0.15, 0.17, 0,
>0.02, -0.02, 0.01, 0.05, 0.08, 0.06, 0.06, 0.1, 0.02, 0.07, 0.07,
>0, 0.06, -0.03, -0.06, 0.17, 0.27, 0.06, 0.08, 0.1, 0.08, 0.26,
>0.02, 0.49, 0.41, 0.25, 0.21, 0.31, 0.36, 0.354, 0.1, 0.15, 0.46,
>0.22, 0.01, 0.23, 0.09, -0.27, -0.048, 0.2, 0.15, 0.04, 0, -0.4,
>0.02, 0.03, 0.08, 0.18, 0.26, 0.24, 0.02, -0.17, -0.04, -0.13,
>-0.07, -0.15, 0.11, 0.13, 0.16, 0.42, 0.57, 0.093, 0.633, 0.264,
>0.302, 0.055, 0.252, 0.01, 0.08, -0.01, -0.03, 0.08, 0.06, 0.1,
>0.01, 0.08, 0.14, 0.44, 0.18, 0.02, 0.08, 0.44, 0.54, 0.09, -0.07,
>0.09, 0.49, 0.11, 0.04, 0.01, 0.06, -0.52727, 0.0686, -0.01,
>-0.09, -0.01, 0.5, 0.78, -0.04, -0.07, 0.47, -0.04, -0.03, 0.55,
>-0.06, -0.04, 0.53, 0.09, 0.03, -0.07, 0.14, 0.11, 0.08, -0.03,
>0.12, -0.25, 0.07, 0.09, -0.18, -0.1, 0.16, -0.09, 0.04, 0.06,
>-0.003, 0.055, 0.025, -0.039, -0.076, 0.01, -0.055, -0.002, -0.111,
>-0.102, -0.132, -0.05, 0.04, 0.07, 0.06), invn = c(0.000509684,
>0.000509684, 0.000509684, 0.000509684, 0.000509684, 0.000509684,
>0.000509684, 0.000509684, 0.000509684, 0.000509684, 0.000509684,
>0.000509684, 0.000509684, 0.000509684, 0.000509684, 0.000509684,
>0.000509684, 0.000509684, 0.000509684, 0.000615764, 0.000661813,
>0.000666223, 0.000710732, 0.000615764, 0.000661813, 0.000666223,
>0.000710732, 0.000661813, 0.000666223, 0.000710732, 0.000666223,
>0.000710732, 0.000710732, 0.000615764, 0.000661813, 0.000666223,
>0.000710732, 0.000615764, 0.000661813, 0.000666223, 0.000710732,
>0.000615764, 0.000661813, 0.000666223, 0.000710732, 0.000931966,
>0.000931966, 0.000931966, 0.000931966, 0.000931966, 0.000931966,
>0.000693481, 0.000693481, 0.000693481, 0.000693481, 0.000693481,
>0.000693481, 0.000693481, 0.000693481, 0.000693481, 0.000693481,
>0.000693481, 0.000693481, 0.000693481, 0.000693481, 0.000693481,
>0.000693481, 0.001061571, 0.001824818, 0.001824818, 0.001824818,
>0.001824818, 0.001824818, 0.001824818, 0.001824818, 0.001824818,
>0.017241379, 0.017241379, 0.005, 0.000714796, 0.000714796, 0.000714796,
>0.000714796, 0.000714796, 0.000714796, 0.000714796, 0.000714796,
>0.000714796, 0.005524862, 0.005524862, 0.005524862, 0.005524862,
>0.005524862, 0.005524862, 0.005524862, 0.005524862, 0.005524862,
>0.005524862, 0.005524862, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05,
>0.000305623, 0.000305623, 0.000305623, 0.000305623, 0.000305623,
>0.000305623, 0.000305623, 0.000305623, 0.000305623, 0.000305623,
>0.000305623, 0.000305623, 0.000305623, 0.000305623, 0.000305623,
>0.000305623, 0.005681818, 0.005681818, 0.005681818, 0.005681818,
>0.005681818, 0.005681818, 0.005681818, 0.005681818, 0.041666667,
>0.041666667, 0.006535948, 0.006535948, 0.006535948, 0.006535948,
>0.006535948, 0.003816794, 0.003816794, 0.003816794, 0.003816794,
>0.003816794, 0.003816794, 0.003816794, 0.003816794, 0.003816794,
>0.003546099, 0.003546099, 0.003546099, 0.003546099, 0.003584229,
>0.003584229, 0.003584229, 0.003584229, 0.003225806, 0.003225806,
>0.003225806, 0.003225806, 0.003225806, 0.003225806, 0.003225806,
>0.003225806, 0.005405405, 0.005405405, 0.005405405, 0.005405405,
>0.007751938, 0.007751938, 0.007751938, 0.007751938, 0.005780347,
>0.005780347, 0.005780347, 0.005780347, 0.001305483, 0.001305483,
>0.004484305, 0.008130081)), class = "data.frame
>
>Any assistance that you can provide would be greatly appreciated. And, if in the
>end there really is no between-study variance, how interesting!
>Thank you in advance for your help!
>
>Karen M. Alexander Glueckert, M. Ed.
>Ph. D. Candidate
>
>University of Nebraska-Lincoln

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