Solved – Fisher R-to-Z transform for group correlation stats

correlationfisher-transformt-test

I have independent correlation coefficient measures for each subject. I would like to test whether the correlation coefficient of the group is significantly different from 0. My understanding is that the best way to do this would be to use a t-test with an r-value per subject.

It's been recommended to me that I first perform a Fisher's transformation on the r-values. I'm wondering why this is necessary. My understanding is that the Fisher's transform is used because the r's are not normally distributed.

Therefore, it seems that the transform makes sense if one is just comparing a single r-value to 0 (i.e. in lieu of testing against a t-distribution with the test statistic $t=\frac{r*\sqrt{n−2}}{\sqrt{1−r^2}}$). However, in my t-test, I am comparing the sample to the sampling distribution (which I think can be assumed normal even if the underlying distribution is not). If this is the case, does it still make sense to employ the transformation before performing the t-test?

Best Answer

You are right: it's not necessary to perform Fisher's transform. Do the t-test. It uses an exact null distribution, whereas comparing Fisher z-transform to a normal distribution would be an approximation.

Trying to do both the z-transform and the transformation to t-distribution would be complete nonsense. The formula for a t-statistic that you give is only for Pearson correlation coefficients, not for z-statistics.

Related Solutions

Solved – How to test whether average of ten independent correlations is different from zero

Treat r (or Z) as any effect size and calculate its mean using a standard formula:

$\bar{r} = \Sigma r_iw_i$/$\Sigma w_i$

where $r_i$ is the $i$th value of r and $w_i$ is a weighting factor which is $1/s^2(r_i)$

The standard error of $\bar{r}$ is given by the square root of $1/\Sigma w_i$ and you can use this to construct confidence intervals around $\bar{r}$ and check whether they include 0 or not.

Fisher’s Z-Transform – When to Use and Why

For questions like these I would just run a simulation and see if the $p$-values behave as I expect them to. The $p$-value is the probability of randomly drawing a sample that deviates at least as much from the null-hypothesis as the data you observed if the null-hypothesis is true. So if we had many such samples, and one of them had a $p$-value of .04 then we would expect 4% of those samples to have a value less than .04. The same is true for all other possible $p$-values.

Below is a simulation in Stata. The graphs check whether the $p$-values measure what they are supposed to measure, that is, they shows how much the proportion of samples with $p$-values less than the nominal $p$-value deviates from the nominal $p$-value. As you can see that test is somewhat problematic with such small number of observations. Whether or not it is too problematic for your research is your judgement call.

clear all
set more off

program define sim, rclass
    tempname z se
    foreach i of numlist 5/10 20(10)50 {
        drop _all
        set obs `i'
        gen x = rnormal()
        gen y = rnormal()
        corr x y 
        scalar `z'  = atanh(r(rho))
        scalar `se' = 1/sqrt(r(N)-3)
        return scalar p`i' = 2*normal(-abs(`z'/`se'))
    }
end

simulate p5 =r(p5)  p6 =r(p6)  p7  =r(p7)     ///
         p8 =r(p8)  p9 =r(p9)  p10 =r(p10)    ///
         p20=r(p20) p30=r(p30) p40 =r(p40)    ///
         p50=r(p50), reps(200000) nodots: sim 

simpplot p5 p6 p7 p8 p9 p10, name(small, replace) ///
    scheme(s2color) ylabel(,angle(horizontal))

enter image description here

simpplot p20 p30 p40 p50 , name(less_small, replace) ///
    scheme(s2color) ylabel(,angle(horizontal))

enter image description here

Best Answer

Related Solutions

Solved – How to test whether average of ten independent correlations is different from zero

Fisher’s Z-Transform – When to Use and Why

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