sfa_parallel: Embedding-Adapted Parallel Analysis

In semanticfa: Semantic Factor Analysis of Language Model Embeddings

Embedding-Adapted Parallel Analysis

Description

Determines the number of factors to retain from an embedding similarity matrix using random unit vectors as the null distribution, avoiding the need for a participant-level sample size.

Usage

sfa_parallel(
  sim_matrix,
  embeddings,
  n_iter = 100L,
  percentile = 95,
  seed = 42L
)

Arguments

sim_matrix	Numeric similarity matrix (n_items x n_items).
embeddings	Numeric embedding matrix (n_items x embedding_dim).
n_iter	Number of random iterations (default 100).
percentile	Percentile of null eigenvalues to use as threshold (default 95).
seed	Random seed, used via withr::with_seed() without touching the global RNG state.

Details

The adaptation keeps Horn's (1965) logic — retain leading eigenvalues that exceed those of structureless data of the same size — but replaces the respondent-level null with similarity matrices of random Gaussian unit vectors in the item count and embedding dimension of the data. Retention follows Horn's sequential rule: leading eigenvalues are counted until the first falls at or below its null percentile (against the 95th null percentile by default, a common modern choice; Horn compared against the null mean). Two caveats follow from the null. First, random unit vectors in a high-dimensional space are nearly orthogonal, so the null similarity matrix is near-identity and the eigenvalue thresholds concentrate just above one. Second, the null carries none of the general positive similarity component that real item embeddings share, so it is a structureless baseline, not a matched one. Benchmarking on embedding similarity matrices, Garrido et al. (2025) found conventional parallel analysis systematically overextracted; corroborate retention with the other criteria in sfa_nfactors().

Value

A list with components:

n_factors

Integer: suggested number of factors.

observed

Numeric vector: observed eigenvalues (descending).

percentiles

Numeric vector: threshold eigenvalues from the null.

References

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

Garrido, L. E., Russell-Lasalandra, L. L., & Golino, H. (2025). Estimating dimensional structure in generative psychometrics: Comparing PCA and network methods using large language model item embeddings. PsyArXiv preprint. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.31234/osf.io/2s7pw_v1")}

semanticfa documentation built on Sept. 2, 2026, 1:07 a.m.