# [R-meta] Converting smd to r-coefficients

Tarun Khanna kh@nn@ @end|ng |rom hert|e-@choo|@org
Thu Oct 8 16:18:51 CEST 2020

```Let me also add that i use the r to Z transformation:

zi = 0.5 * log((1+r_coefficient) / (1-r_coefficient))

vi   = 1/(`total_sample_size` - 3)

From your paper it seems that I should use instead used equation 17 for the transformation of the point biserial coefficients by including the factor a as defined in the paper. Is that correct?

Thank you!

Tarun

Tarun Khanna
Research Associate

Hertie School

Friedrichstraße 180
10117 Berlin ∙ Germany
khanna using hertie-school.org ∙ www.hertie-school.org<http://www.hertie-school.org/>

________________________________
From: Tarun Khanna
Sent: 08 October 2020 16:08:20
To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
Subject: Re: Converting smd to r-coefficients

Dear Wolfgang,

I transformed the regression coefficients from primary studies to r_coefficients using the following formulae:

r_coefficient =
ifelse(significance_test == "t-test", ((`test_statistic`)^2/((`test_statistic`)^2 + `total_sample_size`))^(1/2),
ifelse((significance_test == "z-test" | significance_test == "X^2 Test, (`test_statistic`^2/`total_sample_size`)^(1/2)

In most of the underlying studies, y is a continuous variable and x is a discontinuous variable.

I transformed the mean differences from primary studies to smd using the following:

smd =
ifelse(`statistical_technique` == "Difference of means" | `statistical_technique` == "ANOVA" , `diff_mean`/`pooled_sd`, NA),
ifelse(((`statistical_technique` == "Difference of means" | `statistical_technique` == "ANOVA") & (significance_test == "F-test" | significance_test == "H-test") , (`test_statistic`/`control_sample_size`/`treatment_sample_size`*(`treatment_sample_size` + `control_sample_size`))^(1/2)

I then transform the smd to r_coefficients

r_coefficient = ifelse(`statistical_technique`  %in% c("Difference of means", "ANOVA"), smd/(smd^2+4)^(1/2), r_coefficient)

I read the paper that you referenced and it seems that you are right in that the point-biserial correlation must first be transformed into the biserial correlation coefficient. I am not really sure how to implement the fomula (8)  in the paper to do this though.

Best
Tarun

Tarun Khanna

Research Associate

Hertie School

Friedrichstraße 180

10117 Berlin ∙ Germany
khanna using hertie-school.org ∙ www.hertie-school.org<http://www.hertie-school.org/>

________________________________
From: Viechtbauer, Wolfgang (SP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
Sent: 07 October 2020 21:50:39
To: Tarun Khanna; r-sig-meta-analysis using r-project.org
Subject: RE: Converting smd to r-coefficients

Dear Tarun,

For the first set of studies: How did you actually calculate a correlation coefficient? Did the authors that conducted these regression analyses also report correlation coefficients or did you use some kind of conversion equation? If so, which? Also, are the two variables for which you computed a correlation coefficient continuous variables or was one of the two actually a dichotomous variable that corresponds to the grouping variable based on which the SMD values for the second set of studies were computed?

And how did you transform the standardized mean differences into correlations? The usual equation that one can find in textbooks for this purpose yields a point-biserial correlation coefficient. That is not comparable to a correlation coefficient based on two continuous variables. If one can reasonably assume that the variable that creates the two groups is actually a dichotomized version of some latent continuous variable, then one can convert a SMD value also into a biserial correlation coefficient. The latter will be larger, which might explain why you find that negative coefficient in your meta-regression model.

The following article discusses the transformation of SMD values to biserial correlations:

Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research Synthesis Methods, 8(2), 161-180.

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org]
>On Behalf Of Tarun Khanna
>Sent: Wednesday, 07 October, 2020 21:06
>To: r-sig-meta-analysis using r-project.org
>Subject: [R-meta] Converting smd to r-coefficients
>
>Dear all,
>
>In the meta regression that I am conducting, there were two types of primary
>studies. One that employed regression analysis and others that employed a
>difference of means test. I calculated an r coefficient for the first set of
>studies and standardised mean differences for the second set and then
>converted these into r coefficients. I conducted the final analysis on r
>coefficients. In the meta regression models I added “statistical technique”
>as a moderator variable and the coefficient of the variable signifying mean
>differences studies is significantly negative.
>
>I'm not sure if this difference being captured in the regression is on
>account of difference in the underlying studies or whether conversion of smd
>to r coefficients might have somehow biased estimates from the second set of
>studies downwards. Is this this possible? What can I do to check?
>
>Best regards
>Tarun
>
>Tarun Khanna
>Research Associate
>
>Hertie School
>
>Friedrichstraße 180
>10117 Berlin ∙ Germany
>khanna using hertie-school.org ∙ www.hertie-school.org<http://www.hertie-<http://www.hertie-school.org<http://www.hertie->
>school.org/>

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