The Fisher Z transformation is a formula we can use to transform Pearson’s correlation coefficient (r) into a value (zr) that can be used to calculate a confidence interval for Pearson’s correlation coefficient.
The formula is as follows:
zr = ln((1+r) / (1-r)) / 2
For example, if the Pearson correlation coefficient between two variables is found to be r = 0.55, then we would calculate zr to be:
- zr = ln((1+r) / (1-r)) / 2
- zr = ln((1+.55) / (1-.55)) / 2
- zr = 0.618
This is important because it allows us to calculate a confidence interval for a Pearson correlation coefficient.
Without performing this Fisher Z transformation, we would be unable to calculate a reliable confidence interval for the Pearson correlation coefficient.
The following example shows how to calculate a confidence interval for a Pearson correlation coefficient in practice.
Example: Calculating a Confidence Interval for Correlation Coefficient
Suppose we want to estimate the correlation coefficient between height and weight of residents in a certain county. We select a random sample of 60 residents and find the following information:
- Sample size n = 60
- Correlation coefficient between height and weight r = 0.56
Here is how to find a 95% confidence interval for the population correlation coefficient:
Step 1: Perform Fisher transformation.
Let zr = ln((1+r) / (1-r)) / 2 = ln((1+.56) / (1-.56)) / 2 = 0.6328
Step 2: Find log upper and lower bounds.
Let L = zr – (z1-α/2 /√) = .6328 – (1.96 /√ ) = .373
Let U = zr + (z1-α/2 /√) = .6328 + (1.96 /√ ) = .892
Step 3: Find confidence interval.
Confidence interval = [(e2L-1)/(e2L+1), (e2U-1)/(e2U+1)]
Confidence interval = [(e2(.373)-1)/(e2(.373)+1), (e2(.892)-1)/(e2(.892)+1)] = [.3568, .7126]
Note: You can also find this confidence interval by using the Confidence Interval for a Correlation Coefficient Calculator.
This interval gives us a range of values that is likely to contain the true population Pearson correlation coefficient between weight and height with a high level of confidence.
Note the importance of the Fisher Z transformation: It was the first step we had to perform before we could actually calculate the confidence interval.