Written entirely by Claude, as a working record for agents more than for human readers: every number with its source file, and every claim since withdrawn. Some facts here may be out of date; where it disagrees with the post, trust the post. For the readable version, see the companion.

Factor-analysis methods reference

Parallel analysis, Tucker congruence, Procrustes rotation, PAF versus PCA and the rotation criteria -- short reference entries for the methods the geometry pages use.

Short entries the geometry pages can link to. The project's implementation is qwen35/analyse_fa_qwen35.py, adapted from sweep100/analyse_fa.py with the mathematics -- "principal axis factoring with SMC starts, Horn's parallel analysis, varimax x2 + direct oblimin rotation, Tucker congruence, and the verify() self-test" -- copied verbatim, so the two runs are directly comparable.

One point of context before the entries. Applying factor analysis to a set of weight deltas is, as far as this project could establish, new. qwen35/paper_notes.md section 3.8 states it as the contribution: Persona Cartography does factor analysis, but on questionnaire responses, and Open Character Training does no weight analysis at all. Section 4 item 8 is careful about the strength of that claim: "I found no claim in either paper that this has been done, but absence in two papers is not a literature search." Every method below is therefore being borrowed from psychometrics into a setting its originators never considered, and the borrowing is the thing that needs defending -- see Weight-space geometry of the 134-adapter zoo.

Principal axis factoring, and why not PCA

Principal axis factoring (PAF) and principal component analysis (PCA) answer different questions and are routinely confused. PCA decomposes the total variance of the observed variables: components are exact linear combinations of the data, and the first component is whatever direction has the most variance. PAF decomposes only the common variance: it puts communality estimates on the diagonal of the correlation matrix in place of ones -- classically the squared multiple correlations (SMC) of each variable with all the others -- and iterates to convergence, so the factors model what the variables share and the variable-specific and error variance is left out.

For personality this distinction is the whole tradition. The Big Five are common factors: the claim is that adjective ratings covary because a small number of latent dispositions drive them, not that five linear combinations reconstruct the ratings. PCA on the same data will give a similar-looking answer with inflated loadings.

What this project uses. qwen35/analyse_fa_qwen35.py implements PAF with SMC starts, iterated to convergence. Both are reported where they differ: paper_notes section 3.8's comparison table lists the project's method as "PCA and factor analysis on deltas" against Persona Cartography's "PCA on flattened delta-W". Persona Cartography's own unsupervised pipeline also uses PAF -- k = 4, oblimin rotation -- but on questionnaire responses.

Parallel analysis (Horn, 1965)

Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179-185. [verified]

How many factors to keep. Horn's answer: compare each observed eigenvalue against the eigenvalues you would get from random data of the same shape -- same number of variables, same number of observations -- and retain factors whose eigenvalue exceeds the random benchmark, conventionally at the 95th percentile of the permutation or simulation null. It is the correction to Kaiser's eigenvalue-greater-than-one rule, which systematically over-extracts because even random data produces eigenvalues above one.

Why it matters here more than usual. A set of near-orthogonal points has a near-flat spectrum, and a near-flat spectrum can be mistaken for structure. Persona Cartography's eleven-point PCA over flattened OCEAN deltas runs 15.26%, 13.38%, 12.21%, 12.17%, 11.60%, ..., 1.70% -- which paper_notes section 3.8 reads as "what ~11 near-orthogonal points look like" and says should be treated as the null, not as structure. A permutation null is exactly the instrument that turns that intuition into a test, which is why the project runs one.

Persona Cartography's own use of it is also instructive about its limits: in their questionnaire factor analysis "real eigenvalues remain above the 95th-percentile permutation null out to k = 11, while a clean scree elbow sits at k = 4", and they take 4 on three convergent grounds -- parallel analysis, the scree elbow, Cronbach's alpha, and cross-model congruence -- rather than letting parallel analysis decide alone. Parallel analysis says how many factors are above noise, not how many are interpretable.

The scree elbow

Cattell's scree test: plot the eigenvalues in descending order and keep the factors before the plot flattens into scree. Subjective, old, and still the second opinion everyone consults, because parallel analysis over-retains on large n and the elbow catches where the substantively large factors stop. This project reports the scree alongside the permutation null for the same reason Persona Cartography does; see Weight-space geometry of the 134-adapter zoo for the zoo's spectrum and Goldberg (2006) on reading a whole hierarchy instead of picking one k.

Rotation: varimax and direct oblimin

A factor solution is only determined up to rotation, so the solution reported is a choice of rotation, and the choice is between orthogonal and oblique.

Varimax (Kaiser, 1958) is the standard orthogonal criterion: rotate to maximise the variance of squared loadings within factors, driving each variable toward loading strongly on one factor and near zero on the rest -- simple structure, with the factors held mutually uncorrelated.

Direct oblimin (Jennrich & Bentler, 2002, as implemented here) is the oblique counterpart: it allows the factors to correlate, which is more honest whenever the constructs are believed to be correlated -- as the Big Five, and personality factors generally, are.

What this project does. qwen35/analyse_fa_qwen35.py runs two algorithmically independent varimax implementations and cross-checks them against each other: Kaiser's (1958) cyclic pairwise rotation, which has a closed form per plane, and Jennrich's (2001) gradient-projection varimax. Agreement between two independent implementations on both a known simple structure and random loadings is the self-test. Direct oblimin is then run as the oblique solution. Persona Cartography uses oblimin for its questionnaire factors on the explicit grounds that correlated factors should be allowed.

Tucker's congruence coefficient

Tucker, L. R. (1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Washington, DC: Department of the Army. [not verified]

Standard modern treatment: Lorenzo-Seva, U., & ten Berge, J. M. F. (2006). Tucker's congruence coefficient as a meaningful index of factor similarity. Methodology, 2(2), 57-64. [not verified]

The measure of whether two factor solutions found in different samples, by different methods, or on different variables are the same factor. It is the cosine between two loading vectors: phi = / sqrt(), which is exactly how qwen35/analyse_fa_qwen35.py defines it. It is scale-invariant and insensitive to a proportional rescaling of loadings, which is what makes it the right instrument here and a plain correlation the wrong one.

The conventional reading, from Lorenzo-Seva and ten Berge, is that |phi| in 0.85-0.94 indicates fair similarity and |phi| above 0.95 indicates the two factors can be considered equal. Those thresholds are the ones the project's factor-recovery claims are judged against.

What this project does with it. Two ways. Congruence of the recovered factors against the Goldberg marker targets -- a target vector per factor with +1 on each positively keyed marker of that factor, -1 on each negatively keyed marker, 0 elsewhere -- which is the direct test of whether the weight-space factors are the Big Five. And congruence across replications. qwen35/build_findings_page.py reports that the zoo does "not recover Goldberg's structure: Tucker congruence peaks at 0.68 and nothing clears the" thresholds, against a stated 0.447 construction baseline for a perfectly clean Goldberg solution. The number and its interpretation belong to Weight-space geometry of the 134-adapter zoo; what belongs here is that 0.68 is below the 0.85 fair-similarity line, so the instrument is being read the way its own literature says to read it.

Persona Cartography uses the same coefficient for split-half stability (median |phi| above 0.97 over 100 random half-splits) and cross-model agreement (Hungarian-matched per-pair |phi| 0.54-0.80, mean 0.66).

Procrustes rotation

Schonemann, P. H. (1966). A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31(1), 1-10. [not verified]

Given two loading matrices, find the orthogonal rotation of one that brings it as close as possible to the other in least squares -- solved in closed form by the SVD of the cross-product. In factor analysis it is the standard way to make two solutions comparable before measuring congruence, since rotational indeterminacy would otherwise make any two solutions look different for no substantive reason.

The caution that matters when it is used to test whether a solution replicates: Procrustes rotation toward a target will always improve the fit, including for random data, so a congruence measured after Procrustes rotation needs its own null -- rotate random loadings toward the same target and see what congruence that buys. The zoo's factor pages should state which of their congruences are post-Procrustes.

Verification note

Horn (1965) was confirmed by web search on 2026-09-07 (Psychometrika 30(2), 179-185) and is marked [verified]. Tucker (1951), Lorenzo-Seva & ten Berge (2006) and Schonemann (1966) are marked [not verified]: they are standard citations written out from general reference and were not confirmed against a publisher record in this pass, and none of the three is named in the project's own code -- qwen35/analyse_fa_qwen35.py implements the congruence coefficient without citing Tucker's report. Kaiser (1958), Jennrich (2001) and Jennrich & Bentler (2002) are named in that file's docstrings and are quoted from it here; their full citations were not looked up. No work on this page was read in full. The Lorenzo-Seva and ten Berge thresholds (0.85 fair, 0.95 equal) are stated from general reference and should be checked before the blog post quotes them as a bar.