Fisher Transformations

Instructions

• This tool does multiple Fisher Transformations, r-to-Z or Z-to-r, where
  • r is Pearson's Correlation Coefficient
  • Z is a variance stabilizing transformation of the sampling distribution of r.

• To use the tool, enter one value per line, ending each line by pressing the Enter key.

What is computed?

• Z is calculated by applying an inverse hyperbolic tangent function to r. Conversely, r is calculated by applying a hyperbolic tangent function to Z.
• Unlike similar tools which transform one value at a time, our tool applies Fisher Transformations to an entire set of values, saving users time and effort.
• WARNING: Contrary to popular opinions, Fisher Transformations should not be applied arbitrarily to (x, y) paired data, but only when both (x, y) random variables are approximately normally distributed; i.e., both variables must describe bell-shaped curves (Garcia, 2016; 2017). If the variables are bivariate normal, the sampling distribution of Z is approximately normally distributed with mean as described by the r-to-Z transformation (Anderson, 2007).
  Ignoring the requisite of bivariate normality can introduce non-trivial errors and induce researchers to draw misleading conclusions (Zimmerman, Zumbo, & Williams, 2003). This is particularly true when Fisher Transformations are used to find confidence intervals for r, or as workarounds for averaging correlation coefficients. Before applying Fisher Transformations, be sure that you understand

  • Bivariate Normality
  • Normal Distribution Tests
  • Quantile-Quantile Plots
  • r and Z Sampling Distributions
  • why correlation coefficients are not additive

Who can use it?

• Data miners, statisticians, or anyone that need to apply Fisher Transformations.

References

• Anderson, C. J. Correlation. Edpsy 580 Course. Department of Education Psychology, University of Illinois at Urbana-Champaign.
• Garcia, E. (2012). The Self-Weighting Model. Communications in Statistics - Theory and Methods, 41:8,1421-1427.
• Garcia, E. (2015a). The Self-Weighting Model Tutorial: Part 1.
• Garcia, E. (2015b). The Self-Weighting Model Tutorial: Part 2.
• Garcia, E. (2016). A Tutorial on Standard Errors.
• Garcia, E. (2017). On the Nonadditivity of Correlation Coefficients Part 2: Fisher Transformations.
• Zimmerman, D. W., Zumbo, B. D., and Williams, R. H. (2003). Bias in estimation and hypothesis testing of correlation. Psicologica 24:133-158.

Feedback

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