principal axis factoring

[maxheld83]

Job van Exel recently mentioned after a talk in Bremen, Germany, that principal axis extraction would be a new, convincing way to extract factors for Q. He mentioned a talk by some other researcher about this, will try to find more info.

[maxheld83]

Principal Axis Factoring (PAF), was suggested as an extraction method for q methodology by Noori Akhtar-Danesh at the q conference in 2007:

• PAF is similar to PCA with the only difference that in the correlation matrix “ones” in the positive diagonal are substituted with the estimates of communalities.
• For each Q-sort the communality is the proportion of the variance that is shared (or explained) with the other Q-sorts.
• PAF is generally considered best for exploring the underlying factors for theoretical purposes. (Source: Presentation by Dr. Akhtar-Danesh from 2007, via email from Job van Exel in 2014)

[maxheld83]

there seems to be a problem when number of sorts > number of items (as would often be the case):

Second, in Q studies we usually have more Q-sorts than the number of statements where PAF crashes because of singularity in the data matrix. (Source: Presentation by Dr. Akhtar-Danesh from 2007, via email from Job van Exel in 2014)

[maxheld83]

good news – psych seems to offer something (like?) PAF with fa.

Will:

• [x] check whether this does what Noori (above)
• [ ] try and shoehorn fa into qmethod as an alternative to principal

[maxheld83]

Noori just confirmed that PAF in psych is, in fact, principal axis factoring:

Noori: would you mind having a look at the psych documentation (starting page 104) and give us some indication as to whether this facility would be what you have in mind for an appropriate PAF?

Yes, I looked at it and it seems to be PAF.

[maxheld83]

some more resources to implement this:

• http://www.statmethods.net/advstats/factor.html
• http://cran.r-project.org/web/packages/nFactors/index.html
• http://cran.r-project.org/web/packages/FactoMineR/index.html
• psych also appears to do parallel analysis by fa.parallel

[maxheld83]

I've done some more reading on this, and I am -- for now -- more in favor of using PCA, mostly because I am not sure if/how the latent variable assumption of PAF (or other factor analyses) make sense for Q. Bottom line: I'm not going to work on this anytime soon.

[bobbraswell]

wrt using PAF: the problem is most modern PAF implementations estimate the diagonals using multiple regression, which fails with many Q data sets for the reasons you mentioned. It would be easy for me to implement a robust PAF that didn't fail with Q data. This would make a reduction averaging .4 in the size of each associated eigenvalue (when around 50 statements and 25 sorts) compared to PCA, just from the size of the diagonal element. This gets spread into loadings, the diagonal making up a larger proportion of each loading the more factors are extracted. Whether one sees this as PCA being a bad estimate of FA or the other way around depends on which theoretical model we are working with. It might be as well to offer a choice for those who feel strongly one way or the other, though the centroid routine may fill that niche for many.

[maxheld83]

duplicated here https://github.com/QWrks/pensieve/issues/251