Hello,
Background: I am doing a meta-analysis using Cohen's d. I am calculating the effect size of having a higher BMI on social cognition abilities. Some studies have a high BMI group and a control "normal weight" control group, and the effect size calculation is pretty straightforward. For studies that used a single group design and reported a simple bivariate r-stats between BMI and my "outcome" measure (social cognition), I used an r-to-d conversion using the following formula:
d = 2r/ √(1-r2)
[Friedman, H. (1968). "Magnitude of experimental effect and a table for its rapid estimation," Psychological Bulletin, 70(4): 245-251.]
My question. Firstly, I'm hoping I used the r-to-d equation appropriately. Secondly, in order to complete my meta-analysis, I need to calculate a standard error (SE) of the effect size to enter into the meta-analysis software. Most standard error equations assume the study design uses two groups, rather than a single group study design. I just know r-values, sample size, and means/SDs of BMI & the outcome (social cognition) variables. Does anyone know how to calculate the standard error in this situation?
Thanks so much!!
Hi Melani,
maybe this stack-post is of interest for you.
First, it shows how to compute variance from r for d (which you can probably use to calculate SD and SE from). and second also opens the question, whether transforming r to d makes any sense :)
https://stats.stackexchange.com/questions/308724/interpretation-when-converting-correlation-of-continuous-data-to-cohens-d
Best René
Thanks for sending, René. I did see that citation inArticle Converting among effect sizes
but I was not what the variance of r was in my sample. I also an not sure the best way to convert variance to SE, but I appreciate the resource.
Well... in this case: my statistics course from University tought me.
The variance of a correlations is:
V = ((1-r^2)^2) / (n-1)
and that one should use, however, Fisher-z transformed correlation coefficients for calculating stuff, to make the meta-analyses less affected by outliers.
With:
z= .5 ln( (1+r)/(1-r) )
V= 1/(n-3) - for this you only need sample size (because z values)
SE =sqrt(V)
(I guess one can calculate the meta-analytics stuff with this, and then backtransform; from my weak memory for this course, one can use the z-variance to calculate r-confidence intervals and backtransform those as well. Not sure whether back-transforming variance itself would make sense. Probably not. So maybe this is only a partial answer.)
r = (exp(2z-1) / exp(2z+1))
Unfortunately, I have no exact citation/source in my script, but only general ones:
Borenstein (2010). Introduction to meta-analysis.
Sedlmeier & Renkewitz (2007). Chapter 22. (which is a german book, if you can read german... Statistik für Psychologen).
Schmidt, F. L. (1992). What do data really mean?
Hope this helps.
Best
René
Ps: If you want to go for the r to d conversion and then calculate the variance of d, this seems not really possible, at first sight, since this requires to know the sample sizes of the two samples. As you mention. However, I guess, there could be a work around (which I am not aware of). The standard formula would be:
V= (n_1 + n_2) / n_1*n_2 +(d^2)(2*n_1 + n_2)
Since there is only one sample size, a (probably questionable) approximation could be to use half of it (but see the criticism about interpreting the transformed d in the above blog post, i.e. d interpretion as if there would be a median split of sub-groups seems not permitted or is at least questionable).
V (n / (n/2)^2) + (d^2)/2n
no guarantee :)
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