Fei פ
Introduction
Ben-Shachar et al. (2023) developed this effect size measure for goodness-of-fit tests. They divide the found chi-square test statistic by the maximum possible chi-square value, and then take the square root from this.
The measure is therefor simply the square root from Johnston-Berry-Mielke E. Although Ben-Shachar et al. reference the article from Johnston et al (2006), in personal communication with them (pers. comm. 2024) they indicated that it was a case of independent discovery and only later found that similar attempts were made as theirs that were very close.
Taking the square root makes sense, since as the name implies the chi-square is a squared value, and also other effect sizes use the square root, making Fei more aligned with those.
Obtaining the Measure
(click below on program of interest to expand)
Formula
Fei (Ben-Shachar et al. ,2023, p. 6):.
\(פ=\sqrt{\frac{\chi^2}{n\times\left(\frac{1}{p_{E_{min}}} - 1\right)}}\)
In the above formula's \(p_{E_{min}}\) is the minimum expected proportion and \(n\) the sample size.
Interpretation
Unfortunately, no rule-of-thumb could be found for this measure. However, it can be converted to a Cohen w measure, for which there are rules-of-thumb, or convert this further to Cramer V. The formula for the conversion:
\(w = פ\times\sqrt{\frac{1}{p_{E_{min}}} - 1}\)
In this formula \(p_{E_{min}}\) is the minimum expected proportion.
Next step and Alternatives
Alternative effect sizes for a Goodness-of-Fit test could be:
If the test is significant a post-hoc analysis to pin-point which category is significantly different could be used. The post-hoc anlysis for a goodness-of-fit test is discussed here.
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