[R-meta] Calculating variances and z transformation for tetrachoric, biserial correlations?

As James mentioned, just use:

Var(z) = Var(r) / (1 - r^2)^2

to compute the sampling variance of a Fisher's r-to-z transformed
coefficient. So, for example:

dat1 <- escalc(measure="COR", ri=0.42, ni=23, add.measure=TRUE)
dat2 <- escalc(measure="RBIS", m1i=2.5, m2i=2.0, sd1i=1.1, sd2i=0.9,
n1i=20, n2i=20, add.measure=TRUE)
dat3 <- escalc(measure="RTET", ai=10, bi=4, ci=6, di=12, add.measure=TRUE)
dat <- rbind(dat1, dat2, dat3)
dat

dat$vi <- dat$vi / (1 - dat$yi^2)^2
dat$yi <- transf.rtoz(dat$yi)
dat

You can also do this with the polyserial coefficient.

Note that for standard correlations, this results in using to 1/(n-1) for
the variance (e.g., 1/22 in this example). To use the slightly more
accurate 1/(n-3):

dat$vi[1] <- 1/(23-3)
dat

To compare:

escalc(measure="ZCOR", ri=0.42, ni=23, add.measure=TRUE)

Best,
Wolfgang

-----Original Message-----
From: Mark White [mailto:markhwhiteii at gmail.com]
Sent: Monday, July 03, 2017 00:02
To: Viechtbauer Wolfgang (SP)
Cc: r-sig-meta-analysis at r-project.org
Subject: Re: [R-meta] Calculating variances and z transformation for
tetrachoric, biserial correlations?

Thanks for your prompt and detailed responses!

All of the effect sizes I culled that were from 2x2 tables, Ms and SDs, or
t- and F-statistics were artificially dichotomized (either both or one
variable, respectively). So they are, in fact, coming from a truly
continuous distribution, so I believe that they can all be compared to one
another.

So it seems like:

1. The 217 "regular" correlations can be converted from r to z, and then I
can use the 1/(N-3) variance for that.

2. The 10 effect sizes where only one variable was dichotomized can be
converted to d (via Ms and SDs, or ts and Fs), which can then be converted
to r_{eg} to z, via James's 2014 paper. I can also use his calculations
for the variance of z from r_{eg}.

(I would be doing this instead of `metafor::escalc`, because even though I
could directly convert r_{bis} to z using the normal Fisher's r to z
transformation, there is no way to go from var(RBIS) to var(Z), and using
1/(N-3) is not appropriate).

3. The issue is the 12 effect sizes from 2x2 contingency tables since even
though I could convert directly from r_{tet} to z using Fisher's
transformation, there is no way to go from var(RTET) to var(Z), and using
1/(N-3) is not appropriate. I suppose I could go from an odds ratio to d
to r_{eg} to z, using James's 2014 paper?

4. The other issue is, even though I could get the r_{poly} to z, I could
not get the var(r_{poly}) to var(z), and again using 1/(N-3) is not
appropriate.

How much would it harm the meta-analysis if 217 of my 240 effect sizes had
the correct estimation of 1/(N-3), but the other 23 effects?transformed
from r_{bis}, r_{poly}, r_{tet}?to z and then their variances estimated
incorrectly using 1/(N-3)? It seems like, although I can get comparable
effect sizes now, I cannot transform their variances appropriately.

Thanks,
Mark

On Sun, Jul 2, 2017 at 4:30 PM, Viechtbauer Wolfgang (SP)
<wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
Let me address the computations first (that's the easy part).

Tetrachoric correlation: For tetrachoric correlations, escalc() computes
the MLE (requires an iterative routine -- optim() is used for that). The
sampling variance is estimated based on the inverse of the Hessian
evaluated at the MLE. There is no closed form solution for that.

Biserial correlation (from *t*- or *F*-statistic): You can use a trick
here if you still want to use escalc(). If you know t (or t = sqrt(F)),
then just use escalc(measure="RBIS", m1i=t*sqrt(2)/sqrt(n), m2i=0, sd1i=1,
sd2i=1, n1i=n, n2i=n), where n is the size of the groups (not the total
sample size). For example, using the example from Jacobs & Viechtbauer
(2017):

escalc(measure="RBIS", m1i=1.68*sqrt(2)/sqrt(10), m2i=0, sd1i=1, sd2i=1,
n1i=10, n2i=10)

yields yi = 0.4614 and vi = 0.0570, exactly as in the example. You used
equation (13) to compute the sampling variances, which is the approximate
equation. escalc() uses the 'exact' one (equation 12). That way, you are
also consistent with what you get for the case of "Biserial correlation
(from *M *and *SD*)".

Biserial correlation (from *M *and *SD*): As mentioned above, escalc()
uses equation (12) from Jacobs & Viechtbauer (2017) to compute/estimate
the sampling variance.

Square-root of eta-squared: You cannot use the large-sample variance of a
regular correlation coefficient for this. The right thing to do is to
compute a polyserial correlation coefficient here (the extension of the
biserial to more than two groups). You can do this using the polycor
package. Technically, the polyserial() function from that package requires
you to input the raw data, which you don't have. If you have the means and
SDs, you can just simulate raw data with exactly those means and SDs and
use that as input to polyserial(). The means and SDs are sufficient
statistics here, so you should always get the same result regardless of
what specific values are simulated. Here is an example:

x1 <- scale(rnorm(10)) * 2.4 + 10.4
x2 <- scale(rnorm(10)) * 2.8 + 11.2
x3 <- scale(rnorm(10)) * 2.1 + 11.5

x <- c(x1, x2, x3)
y <- rep(1:3,each=10)

polyserial(x, y, ML=TRUE, std.err=T, control=list(reltol=1e12))

If you run this over and over, you will (should) always get the same
polyserial correlation coefficient of 0.2127. The standard error is
~0.195, but it changes very slightly from run to run due to minor
numerical differences in the optimization routine. Note that I increased
the convergence tolerance a bit to avoid that those numerical issues also
affect the estimate itself. But these minor differences are essentially
inconsequential anyway.

If you do not have the means and SDs, then well, don't know what to do off
the top of my head. But again, don't treat the converted value as if it
was a correlation coefficient. It is not.

Now for your question what/how to combine:

The various coefficients (Pearson product-moment correlation coefficients,
biserial correlations, polyserial correlations, tetrachoric correlations)
are directly comparable, at least in principle (assuming that the
underlying assumptions hold -- e.g., bivariate normality for the
observed/latent variables). I just saw that James also posted an answer
and he raises an important issue about the theoretical comparability of
the various coefficients, esp. when they arise from different sampling
designs. I very much agree that this needs to be considered. You could
take a pragmatic / empirical approach though by coding the type of
coefficient / design from which the coefficient arose and examine
empirically whether there are any systematic differences (i.e., via a
meta-regression analysis) between the types.

As James also points out, you can use Fisher's r-to-z transformation on
all of these coefficients, but to be absolutely clear: Only for Pearson
product-moment correlation coefficients is the variance then approximately
1/(n-3). I have seen many cases where people converted all kinds of
statistics to 'correlations', then applied Fisher's r-to-z transformation,
and then used 1/(n-3) as the variance, which is just flat out wrong in
most cases. Various books on meta-analysis even make such faulty
suggestions.

Also, Fisher's r-to-z transformation will *only* be a variance stabilizing
transformation for Pearson product-moment correlation coefficients (e.g.,
the actual variance stabilizing transformation for biserial correlation
coefficients is given by equation 17 in Jacobs & Viechtbauer, 2017 -- and
even that is just an approximation, since it is based on Soper's
approximate formula). If you apply Fisher's r-to-z transformation to other
types of coefficients, you have to use the right sampling variance (see
James' mail). Also note: You cannot mix different transformations (i.e.,
use Fisher's r-to-z transformation for all).

Whether applying Fisher's r-to-z transformation to other coefficients
(other than 'regular' correlation coefficients) is actually advantageous
is debatable. Again, you do not get the nice variance stabilizing
properties here (the transformation may still have some normalizing
properties). If I remember correctly, James examined this in his 2014
paper, at least for biserial correlations (James, please correct me if I
misremember).

Best,
Wolfgang