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

[Mark White]

Ah, that is brilliant! Thank you for the reproducible example, as well. I
wasn't sure if he meant the delta method could be applied to all flavors of
the r, or just the "regular" r we usually use by just doing `cor()`. So we
can apply it to all flavors (RTET, RBIS, POLY, etc.)

And forgive my ignorance, but where could I reference the delta method?
Looking around, it seems to be a very general rule, but is there somewhere
that mentions using the delta method with regards to transforming variances
of effect sizes particularly? It seems like it is just taking the numerator
of var(r) for the large sample approximation and also putting it back in
the denominator?

On Sun, Jul 2, 2017 at 5:27 PM, Viechtbauer Wolfgang (SP) <
wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:

> 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
>

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