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.