I write a script in R to generate simulated data for meta-analysis for teaching purposes. Given the "population" correlation coefficient, it generates "individual studies" with various sample sizes and correlation coefficients. In order to generate these data correctly, I need to calculate the SE of the coefficient. The problem is that I found two different approaches to it. The first one uses the formula SE=sqrt ((1-r^2)/(n-2)) which results in a symmetrical confidence interval. The second is based on the Fisher's r-to-z transformation and uses the formula SE=1/sqrt(n-3). This approach results in asymmetical CIs. Which formula should I use?
Hi Igor Kupriyanov. First, note that Fisher's r-to-z transformation is the inverse hyperbolic tangent, or atanh() in many programing languages. As you note, if we let Y = atanh(r), SE(Y)=1/sqrt(n-3). The standard (approximate) approach is to compute the CI for Y, and then back-transform the limits to get the CI for r.
- Lower limit for r = tanh(lower limit for Y)
- Upper limit for r = tanh(upper limit for Y)
For more detail, you could take a look at Cox (2008) in the Stata Journal. HTH.
Article Speaking Stata: Correlation with Confidence, or Fisher's z revisited
The only requirement for a confidence interval is that the coverage probability (of the procedure) is close to the nominal value. As long as approximately 95% of the confidence intevals calculated with procedure include the "true" value, the procedure is ok and the intervals are 95% confidence intervals.
Practically, confidence intervals can be (and often are) constructed by inverting a corresponding hypothesis test: the 95% confidence inetrval is the set of hypothesis values that can not be rejected with 95% confidence (=5% significance). This results in correct coverage.
For Pearson's r, there exists a test based on the approximate normal distribution of sqrt(r² / (1-r²)) with a standard error of sqrt(n-2). Inverting this test leads to the symmetric CI.
This approximation is bad for r close to 0 or 1 and for small samples. In these cases there is a better approximation based on Fisher's z-transformation, which accordingly gives non-symmetric CIs, particularily when r is close to 0 or 1.
I would prefer the CI calculated via Fisher's z transformation.
Loftus, G.R. & Loftus, E.F. (1988). Essence of Statistics (2nd Edition). New York: McGraw Hill.
You haven't said what the data are like (e.g., are they discrete, skewed) which you are simulating, but given you are creating presumably lots of simulated samples, you could think about bootstrapping methods. Also, since the typical CI is asymmetric I assume you are not wanting a single SE for it. Of the two approaches you mention, like Bruce Weaver and Jochen Wilhelm , I think most folks prefer the asymmetric intervals for Pearson's correlation (I assume that is what you are referring to). If you go to the code for cor.test(), you will see the bits that Bruce Weaver suggests (I have removed some stuff just to make it clearer, to get code type getAnywhere(cor.test.default)):
...
t < sqrt(df) * r/sqrt(1 - r^2)
z
Daniel Wright , Jochen Wilhelm , Bruce Weaver , thank you! So I'll use the atanh () function. What I try to get is a data frame with two variables: n (the "sample size" of each study) and r (the Pearson's sample correlation coefficient). So I wrote the following script. I generate the sample r values from the normal distribution with the mean = z transformed population r and SD = SE (which is calculated as 1/sqrt(n-3)). Is it correct to do so?
require (randomNames) # The package that generates random "names of the studies"
st.number
Looking at histograms of bootstrapped values for Pearson's r from smallish simulated data sets. If the absolute value of r is say 0.4 or 0.5 or greater, the histogram is notably skewed and the confidence interval is notably asymmetric. (Percentile or BCa). I don't know if that observation is helpful.
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