Bartlett Test in PCA Output: Exact Statistic and Degrees of Freedom in JMP

[#1]

Hi everyone,

I am trying to better understand the Bartlett test reported in the Principal Components output in JMP, specifically the one shown in the “Eigenvalues” table with Chi-square, DF, and p-values for each component.

From the documentation, I understand that this test is used to assess whether the remaining eigenvalues are equal (i.e., a test of homogeneity after extracting the first k principal components). However, I would like to clarify the exact form of the test used in JMP:

1. What is the exact test statistic used for this Bartlett test?
   Is it based on a likelihood ratio involving the remaining eigenvalues (for example, using logarithms of ratios relative to their mean)?
2. In the correction term of the statistic, is the number of remaining components (m = p − k) used, or the total number of variables (p)?
3. The reported degrees of freedom are sometimes non-integers.
   Does JMP apply an additional correction or approximation to the degrees of freedom?
4. Is this implementation directly based on Bartlett (1937, 1954), or are there JMP-specific modifications?

I would appreciate any clarification or references regarding the exact formulation used.

Thank you!

[#2]

If the documentation and references there in are not sufficient, then you might reach out to JMP Technical Support (support@jmp.com) for further details. Keep in mind that some aspects of the implementation are proprietary and, therefore, confidential.