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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

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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