FAPA
Factor Analytic Profile Analysis of Ipsatized Data
Implements Factor Analytic Profile Analysis of Ipsatized Data
('FAPA'), a metric inferential framework for pattern detection and
person-level reconstruction in multivariate profile data. After
row-centering (ipsatization) to remove profile elevation, 'FAPA' applies
singular value decomposition ('SVD') to recover shared core profiles and
individual pattern weights. Dimensionality is determined by a
variance-matched Horn's parallel analysis. A three-stage bootstrap
verification framework assesses (1) dimensionality via parallel analysis,
(2) subspace stability via Procrustes principal angles, and (3) profile
replicability via Tucker's congruence coefficients. BCa bootstrap
confidence intervals for core-profile coordinates are computed via the
canonical 'boot' package implementation of Davison and Hinkley (1997)
FAPA: Factor Analytic Profile Analysis of Ipsatized Data
FAPA implements a metric inferential framework for pattern detection and person-level reconstruction in multivariate profile data. After removing overall profile elevation via row-centering (ipsatization), FAPA applies singular value decomposition (SVD) to recover shared core profiles and individual pattern weights, and provides a unified three-stage bootstrap verification framework.
Motivation
Many existing approaches to profile analysis — including PCA applied to
ipsatized data — lack an integrated, workflow-level treatment of
dimensionality selection, subspace stability, and profile replicability.
FAPA addresses this by combining well-validated tools (Horn's parallel
analysis, Procrustes principal angles, Tucker's congruence coefficients, BCa
bootstrap CIs via the boot package) into a coherent, reproducible pipeline
designed specifically around the ipsatized structure of the data.
Three-stage verification framework
| Stage | Method | Criterion |
|---|---|---|
| 1 | Horn's parallel analysis (variance-matched) | Retain components where σ²_k exceeds the (1−α) quantile of a variance-matched null |
| 2 | Procrustes / principal angles | All K principal angles < 30° across bootstrap resamples |
| 3 | Tucker's congruence coefficients | Mean CC ≥ 0.85 (fair) or ≥ 0.95 (high) per Lorenzo-Seva & ten Berge (2006) |
Installation
# Install from GitHub (development version):
# install.packages("remotes")
remotes::install_github("sekangakim/FAPA")
Quick start
library(FAPA)
# 1. Load and ipsatize
dat <- load_and_ipsatize("Calibration.csv", col_labels = paste0("V", 1:22))
Xtilde <- dat$ipsatized
# 2. Stage 1: parallel analysis
pa <- fapa_pa(Xtilde, B = 2000, seed = 1)
print_pa(pa)
K <- pa$n_retain
# 3. Core solution
fit <- fapa_core(Xtilde, K = K)
# 4. Stages 2 & 3: stability and replicability
pr <- fapa_procrustes(Xtilde, K = K + 3, B = 2000, seed = 1)
tc <- fapa_tucker(Xtilde, K = K + 3, B = 2000, seed = 1)
print_procrustes(pr, K_pa = K)
print_tucker(tc, cc_thresh = 0.85, K_pa = K)
# 5. BCa confidence intervals
bca <- fapa_bca(Xtilde, K = K, B = 2000, seed = 1)
plot_fapa_core(bca, i = 1, split_at = 11)
# 6. Person-level reconstruction
prs <- fapa_person(Xtilde, fit, participants = 1:5, seed = 1)
cat(sprintf("Mean R² = %.4f\n", prs$R2_mean))
See vignette("fapa_example", package = "FAPA") for a fully documented
walkthrough using EDI-2 eating-disorder symptom data.
Key functions
| Function | Description |
|---|---|
load_and_ipsatize() |
Load CSV and row-centre |
fapa_core() |
SVD-based core profile estimation |
fapa_pa() |
Stage 1: variance-matched parallel analysis |
fapa_procrustes() |
Stage 2: subspace stability via principal angles |
fapa_tucker() |
Stage 3: Tucker's congruence coefficients |
fapa_bca() |
BCa bootstrap CIs for core profiles |
fapa_person() |
Person-level reconstruction and weights |
print_pa/procrustes/tucker() |
Formatted console summaries |
plot_fapa_core() |
Core profile with BCa CI bands |
plot_pa_scree() |
PA scree plot |
plot_principal_angles() |
Stage 2 angle distributions |
plot_tucker_cc() |
Stage 3 CC distributions |
write_fapa_results() |
Write core-profile CSVs |
write_verification() |
Write three-stage verification CSVs |
Citation
If you use FAPA in your research, please cite the accompanying manuscript:
Kim, S.-K. (in preparation). Factor Analytic Profile Analysis of Ipsatized Data (FAPA): An inferential framework for pattern detection and person reconstruction.
and the software:
Kim, S.-K. (2026). FAPA: Factor Analytic Profile Analysis of Ipsatized Data. R package version 0.1.0. https://github.com/sekangakim/FAPA
References
- Davison, A. C., & Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge University Press.
- Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179–185.
- Kim, S.-K. (2023). Factorization of person response profiles to identify summative profiles carrying central response patterns. Psychological Methods. https://doi.org/10.1037/met0000568
- 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.
- Ten Berge, J. M. F. (1999). A legitimate case of component analysis of ipsative measures. Multivariate Behavioral Research, 34(1), 89–102.
License
MIT © Se-Kang Kim