[R-meta] Plotting the correlation among true/random effects across categories

-----Original Message-----
From: Yuhang Hu [mailto:yh342 at nau.edu]
Sent: Friday, 16 June, 2023 17:06
To: Viechtbauer, Wolfgang (NP)
Cc: R Special Interest Group for Meta-Analysis
Subject: Re: [R-meta] Plotting the correlation among true/random effects across
categories

Hi Wolfgang,

Thank you very much for your response. I wanted to ask two questions?regarding
your responses.

1-- If instead of struct="UN", I use struct="GEN" (below), then what does the
correlation reported between AL and PD represent?

My understanding is that it represents the correlation between the 'differences'
(not between the means of AL and PD) between AL and PD across the trials, right?

model <-?rma.mv(yi~ outcome, vi, data = ?dat.berkey1998,
? ? ? ? ? ? ? ? random = ~ outcome | trial, struct = "GEN")

2-- You mentioned that the estimated variation by the model (say tau2) accounts
for an additional piece (i.e., E(var(u_i|y_i))) in random effects (ui) that is
not present in the BLUPs.

My?question is then, is the variation, or correlation obtained by the model
preferred over (and more reliable?than) that obtained by hand-calculating them
from the BLUPs?

Many thanks,
Yuhang

On Thu, Jun 1, 2023 at 8:18?PM Viechtbauer, Wolfgang (NP)
<wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
You get the estimated variation in the underlying true means of each category
(around the fixed effects, which correspond to the expected value of the
underlying true means of each category) and the correlation of that variation
between the two categories. I have now added the model to the write-up of this
example here:

https://www.metafor-project.org/doku.php/analyses:berkey1998

so you can see exactly what the code (given on that page) corresponds to model-
wise.

Best,
Wolfgang

-----Original Message-----
From: Yuhang Hu [mailto:yh342 at nau.edu]
Sent: Thursday, 04 May, 2023 23:39
To: Viechtbauer, Wolfgang (NP)
Cc: R Special Interest Group for Meta-Analysis
Subject: Re: [R-meta] Plotting the correlation among true/random effects across
categories

Thank you very much, Wolfgang. Regarding my second (unclear) follow-up, I might
have a conceptual misunderstanding that I hope to clear up.

When we use a "categorical" moderator (with struct="UN") to the left of | (e.g.,
~outcome | trial_id), conceptually it means that in each unique trial_id, we fit
"effect_size ~ outcome+0", and obtain the Mean for each outcome category (say

two

categories). Then in the?rma.mv() output, we get the variation in means of each
category as well as the correlation between the means of the two categories
across all unique trial_ids.

Is my conceptual understanding correct?

Thanks,
Yuhang

On Wed, May 3, 2023 at 1:59?PM Viechtbauer, Wolfgang (NP)
<wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
Please see below for responses.

Best,
Wolfgang

-----Original Message-----
From: Yuhang Hu [mailto:yh342 at nau.edu]
Sent: Wednesday, 03 May, 2023 22:48
To: Viechtbauer, Wolfgang (NP)
Cc: R Special Interest Group for Meta-Analysis
Subject: Re: [R-meta] Plotting the correlation among true/random effects across
categories

Thank you very much, Wolfgang. Two quick follow-ups:

1) To convert these estimated random deviations to true effects, I should add

the

fixed effect estimates (assuming I use 'outcome + 0' in model formula) for each
category of outcome to its relevant column, right?

AL = paired[,1] + model$b[1,1]
PD = paired[,2] + model$b[2,1]
plot(PD~AL, pch=21, bg="gray", cex=1.5, lwd=1.2)

Correct.

2) When using categorical variables (with "UN") to the left of |, I think we

drop

the intercept in the random-effects design matrix, so what is actually allowed

to

vary across the trials given that eac trial has only one instance of AL and PD

in

it?

Not entirely sure what you mean by this question. The help file explains what

the

different structures do:

https://wviechtb.github.io/metafor/reference/rma.mv.html#specifying-random-
effects

Thank you for your time.
Yuhang

On Wed, May 3, 2023 at 12:51?AM Viechtbauer, Wolfgang (NP)
<wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
You can extract the BLUPs of the random effects and create a scatterplot based

on

them:

re <- ranef(model)
re

paired <- do.call(rbind, split(re[[1]]$intrcpt, dat.berkey1998$trial))
paired

plot(paired, pch=21, bg="gray", cex=1.5, lwd=1.2)

And before somebody asks why cor(paired) does not yield 0.7752 (or why the

values

in var(paired) do not match up with the variances as estimated from the model),
see for example:

https://stats.stackexchange.com/q/69882/1934

Or let me give a more technical explanation based on the standard RE model:

dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)
res <- rma(yi, vi, data=dat)
res$tau2
var(ranef(res)$pred)

You will notice that the latter is smaller than tau^2. By the law of total
variance:

tau^2 = var(u_i) = E(var(u_i|y_i)) + var(E(u_i|y_i)).

The conditional means of the random effects (which is what ranef() provides
estimates of) are E(u_i|y_i) and hence their variance is only part of the total
variance. Therefore, the estimate of tau^2 and the estimated variance of the
BLUPs of the random effects will not match up.

In more complex models, this then also affects things like the correlation
between the BLUPs of the random effects.

Best,
Wolfgang

-----Original Message-----
From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces at r-project.org]

On

Behalf Of Yuhang Hu via R-sig-meta-analysis
Sent: Wednesday, 03 May, 2023 1:21
To: R meta
Cc: Yuhang Hu
Subject: [R-meta] Plotting the correlation among true/random effects across
categories

Hello Colleagues,

I was wondering if there is a way to scatterplot the correlation between
the categories of variable "outcome" (AL and PD) which is estimated to be
rho = .7752 in my model below?

model <- rma.mv(yi~ outcome, vi, data =? dat.berkey1998,
? ? ? ? ? ? ? ? random = ~ outcome | trial, struct = "UN")

? ? rho.AL? rho.PD? ? AL? PD
AL? ? ? ?1? ? ? ? ? ? ?-? ?5
PD? 0.7752? ? ? ?1? ? no? ?-

Thanks,
Yuhang