Pitman-Morgan test
Variances (spread) · Two paired samples · reference distribution: t(n−2)
When to use it
Compare the variances of two paired measurements, such as the precision of two instruments applied to the same samples.
Null hypothesis
The two variances are equal: σ₁² = σ₂².
Assumptions
- Pairs independent of each other
- Bivariate normality
Test statistic
r = \operatorname{corr}(x - y,\; x + y), \qquad t = \dfrac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}How to report it
r(D, S) = .48, t(18) = 2.32, p = .032
In R and Python
R
cor.test(x - y, x + y)Python
stats.pearsonr(x - y, x + y)In Python, stats is scipy.stats and np is numpy.
Variants and alternatives
- F for two variances, with independent samples
Where it sits in the catalog
Variances (spread). They compare how spread out the data are, not the center. Useful on their own, as in quality control, and to check the equal-variance assumption of other tests.
Two paired samples. Two measurements linked in pairs: the same individual before and after, or matched pairs, such as twins. The test uses the difference within each pair.
In the decision tree
- What do you want to do? Compare groups, or one group with a reference value
- What kind of response did you measure? The spread, not the center
- How many groups or measurements? Two measurements on the same individuals