Biostatistics Study NotesFrom concepts and derivations to interpretation

CHAPTER 07 · TOPIC 03

Fisher’s z Transformation

Transform correlations for confidence intervals, tests against a specified population correlation, and comparisons between independent groups.

On this page
  1. Fisher? z Transformation
  2. Test whether ρ is equal to a specified value
  3. Confidence Interval for the Population Correlation
  4. Comparing Correlations from Two Independent Groups
  5. When This Two-Group Formula Does Not Apply

The sampling distribution of the sample correlation coefficient r is asymmetric, especially when |ρ| is large, and the variance also depends on ρ. The Fisher? z transformation maps r over the real numbers, making the distribution more normal and approximately constant in variance, making it possible to compare confidence intervals and correlation coefficients.

Fisher? z Transformation#

sample correlationzr=12ln⁡(1+r1−r)=arctanh⁡(r)z_r=\frac12\ln\left(\frac{1+r}{1-r}\right)=\operatorname{arctanh}(r)
population correlationzρ=12ln⁡(1+ρ1−ρ)z_\rho=\frac12\ln\left(\frac{1+\rho}{1-\rho}\right)
Inverse transformationr=tanh⁡(zr)=e2zr−1e2zr+1r=\tanh(z_r)=\frac{e^{2z_r}-1}{e^{2z_r}+1}
z(r)≈N(z(ρ),1n−3),SE[z(r)]≈1n−3z(r)\approx N\left(z(\rho),\frac1{n-3}\right),\qquad SE[z(r)]\approx\frac1{\sqrt{n-3}}

Test whether ρ is equal to a specified value#

hypothesisH0:ρ=ρ0,H1:ρ≠ρ0H_0:\rho=\rho_0,\qquad H_1:\rho\ne\rho_0
Z statisticZ=z(r)−z(ρ0)1/n−3Z=\frac{z(r)-z(\rho_0)}{1/\sqrt{n-3}}

The t-test on the previous page is an accurate standard method for testing ρ₀=0. Fisher Z is particularly useful for comparing nonzero specified values, confidence intervals, and independent correlations.

Confidence Interval for the Population Correlation#

lower limitLz=z(r)−z1−α/21n−3L_z=z(r)-z_{1-\alpha/2}\frac1{\sqrt{n-3}}
upper limitUz=z(r)+z1−α/21n−3U_z=z(r)+z_{1-\alpha/2}\frac1{\sqrt{n-3}}
Return to r scaleCIρ=[tanh⁡(Lz),tanh⁡(Uz)]CI_\rho=[\tanh(L_z),\tanh(U_z)]

Comparing Correlations from Two Independent Groups#

hypothesisH0:ρ1=ρ2,H1:ρ1≠ρ2H_0:\rho_1=\rho_2,\qquad H_1:\rho_1\ne\rho_2
Transform each groupz1=z(r1),z2=z(r2)z_1=z(r_1),\qquad z_2=z(r_2)
standard error of differenceSE(z1−z2)=1n1−3+1n2−3SE(z_1-z_2)=\sqrt{\frac1{n_1-3}+\frac1{n_2-3}}
Comparison ZZ=z(r1)−z(r2)1/(n1−3)+1/(n2−3)Z=\frac{z(r_1)-z(r_2)}{\sqrt{1/(n_1-3)+1/(n_2-3)}}
(z1−z2)±z1−α/21n1−3+1n2−3(z_1-z_2)\pm z_{1-\alpha/2}\sqrt{\frac1{n_1-3}+\frac1{n_2-3}}

When This Two-Group Formula Does Not Apply#

  • Correlation coefficients depend on each other, such as two correlations estimated from the same participants
  • Two correlations share one variable
  • The data involve clustering, repeated measures, or survey weights
  • The Pearson correlation assumption is significantly violated.
  • Applying the standard error formula directly to Spearman correlation