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

James Pustejovsky jepusto at gmail.com
Sun Jul 2 23:01:12 CEST 2017

```Mark,

I do not know the specifics of how metafor makes these calculations, but I
will offer the following:

1. The variance of the statistic (whether r or z) depends on the
sampling design from which it was collected. So for instance, the variance
of the tetrachoric correlation will typically differ from the variance of
the Pearson correlation, and both will differ from the variance of the
point biserial correlation. This is as should be expected because each of
these statistics uses progressively more detailed information about the
underlying distribution.
2. If you have the variance of the r estimates, you can convert them
using Fisher's z transformation and then calculate the sampling variance of
z using the delta method. If V^r is the variance of the r estimate, then
the variance of z(r) is V^z = V^r / [(1 - r^2)^2].
3. Application of the z transformation to the point biserial correlation
is complicated by the possibility that r_{pbs} estimates can be larger than
1 in absolute value. See the reference below for more detailed discussion
and one possible solution.
4. In the reference below, I argued that it is important to consider
whether it makes theoretical sense to combine evidence across different
sampling designs. For example, I don't think it is sensible to
combine biserial correlation estimates based on data from between-groups
randomized experiments with Pearson correlation estimates from purely
observational studies. This is because the two parameters represent
different types of relationships: the former being a causal relationship
arising from manipulating one of the variables, the other being a purely
descriptive/correlational relationship. Similarly, tetrachoric correlations
can arise from either an experimental design, in which one variable is
manipulated, and the other, dichotomous variable observed as the outcome,
or from an observational design, in which both variables are
dichotomizations of a continuous latent construct.

Cheers,
James

Pustejovsky, J. E. (2014). Converting from d to r to z when the design uses
extreme groups, dichotomization, or experimental control. *Psychological
Methods*, *19*(1), 92–112. DOI: 10.1037/a0033788

On Sun, Jul 2, 2017 at 1:05 PM, Mark White <markhwhiteii at gmail.com> wrote:

> Hello,
>
> I have converted a number of summary statistics (contingency tables, *t-*
> and
> *F*-statistics,* M*s and *SD*s) to tetrachoric and biserial correlations.
> The other effect sizes that I directly observed were raw correlations. I
> have my model all set up to run, but I am unsure as to what to do about
> these effect sizes. I see two options:
>
> 1. Submit raw, tetrachoric, and biserial correlations and their variances
> to analyses directly (what I have now).
>
> 2. Do Fisher's r-to-z transformation and *then *submit those to analyses.
> The problem here is: How do I convert tetrachoric and biserial correlations
> to Fisher's z? And if I do that, can I just use N to calculate the
> variance? Or, do I have to also convert the variances of tetrachoric and
> biserial correlations?
>
> In either case, I am not sure how `metafor::escalc` calculates variances
> for tetrachoric (`RTET`) and biserial (`RBIS`) correlations. I tried
> looking through the code for `metafor::escalc` on GitHub, but could not
> figure out the calculations.
>
> I have included a table describing my effect sizes and how I calculated
> them/their variances below.
>
> What do you all think would be the best way to handle these data?
>
> *Effect size*
>
> *k*
>
> *Effect size calculation*
>
> *Variance calculation*
>
> Raw correlation
>
> 217
>
> Directly observed
>
> Typical large-samples estimation (see Hedges, 1989, Equation 5), using
> `metafor::escalc`
>
> Tetrachoric correlation
>
> 12
>
> From 2 x 2 contingency tables, using `metafor::escalc`
>
> From 2 x 2 contingency tables, using `metafor::escalc`
>
>
> *Unsure what the formula is*
>
> Biserial correlation (from *t*- or *F*-statistic)
>
> 8
>
> From *t*- or *F*-statistic to point-biserial correlation (using
> `compute.es::tes`
> and `compute.es::fes`) to biserial correlation (self-written function
> based
> on Jacobs & Viechtbauer, 2016, assuming *n*s equal across conditions)
>
> From *n*, using self-written function based on Soper's method (Jacobs &
> Viechtbauer, 2016, Equation 13, assuming *n*s equal across conditions)
>
> Biserial correlation (from *M *and *SD*)
>
> 2
>
> From means and standard deviations directly, using `metafor::escalc`
>
> From means and standard deviations directly, using `metafor::escalc`
>
>
> *Unsure what the formula is*
>
> Square-root of eta-squared
>
> 1
>
> F-statistic to Cohen’s *f*  (Cohen, 1988) to eta-squared to square-root of
> eta-squared as an approximation of a raw correlation coefficient (Lakens,
> 2013), using self-written function
>
>
>
> *This was a one-way ANOVA with three means (low, medium, high prejudice).*
>
> Typical large-samples estimation (see Hedges, 1989, Equation 5), using
> `metafor::escalc`
>
> Best,
> Mark
>
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>
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