Multivariate, Factor & Latent-Variable Analysis
Covers exploratory and confirmatory dimensionality reduction, clustering, reliability, latent structure discovery, and item response modeling — from classical PCA/factor analysis through modern IRT, SEM, and categorical methods.
35 procedures · 2 with documented limitations. Every result is computed by the open Python engine and is exportable to APA tables and reports.
| Procedure | What it does | Status | Key reference |
|---|---|---|---|
| Principal Component Analysis | Eigenvalue-based dimensionality reduction via correlation-matrix SVD with Kaiser criterion for retention. | ✅ Full | Hotelling (1933) |
| Exploratory Factor Analysis | ML factor extraction with Varimax/Promax rotation, KMO and Bartlett's sphericity test. | ✅ Full | Spearman (1904); Thurstone (1947) |
| K-Means Clustering | Standardized iterative partition with silhouette score; sample-based for n > 10k. | ✅ Full | MacQueen (1967) |
| Hierarchical Agglomerative Clustering | Distance-based clustering with Ward/complete/average linkage; limited to n ≤ 20k (O(n²) memory). | ⚠️ Limited | Ward (1963); Everitt et al. (2011) |
| Cronbach's Alpha | Internal consistency reliability via item correlations with item-total and alpha-if-deleted. | ✅ Full | Cronbach (1951) |
| McDonald's Omega (Total) | ML single-factor CFA reliability coefficient without tau-equivalence assumption; requires k ≥ 3 items. | ✅ Full | McDonald (1999); Hayes & Coutts (2020) |
| Split-Half Reliability | Odd-even split with Spearman-Brown and Guttman corrections. | ✅ Full | Spearman (1910); Brown (1910) |
| Parallel Analysis | Horn's method: compares observed eigenvalues against 95th percentile of random-data eigenvalues. | ✅ Full | Horn (1965) |
| Multivariate Analysis of Variance | One-way MANOVA with Wilks' lambda, Pillai trace, Hotelling-Lawley, Roy's root and F-approximations. | ✅ Full | Wilks (1932); Pillai (1955) |
| Linear Discriminant Analysis | Fisher discriminant functions and linear classification functions with resubstitution accuracy. | ✅ Full | Fisher (1936) |
| Canonical Correlation Analysis | Maximum correlation between two variable sets via CCA with Rao's F-approximation for Wilks' lambda. | ✅ Full | Hotelling (1936) |
| Confirmatory Factor Analysis | Measurement model via semopy with factor loadings, residuals, and model fit (CFI, TLI, RMSEA, SRMR). | ✅ Full | Jöreskog (1969) |
| Structural Equation Modeling | Full latent-variable SEM (measurement + structural paths) via semopy with standardized estimates. | ✅ Full | Jöreskog (1970); Jöreskog (1969) |
| Path Analysis | Single-level directional relationships with path coefficients, R² per endogenous, and overall fit. | ✅ Full | Wright (1934); Jöreskog (1970) |
| Rasch Model (1PL IRT) | Joint MLE for dichotomous items; person abilities, item difficulties, infit/outfit, reliability. | ✅ Full | Rasch (1960) |
| 2PL IRT Model | Two-parameter logistic with discrimination + difficulty; uses girth if available, else manual L-BFGS. | ✅ Full | Pedregosa et al. (2011) |
| 3PL IRT Model | Three-parameter logistic (discrimination, difficulty, guessing) via girth or manual EM. | ✅ Full | Pedregosa et al. (2011) |
| Graded Response Model (Samejima) | Ordinal polytomous items with cumulative thresholds per item; category response curves and item info. | ✅ Full | Samejima (1969) |
| Partial Credit Model (Masters) | Polytomous step parameters without item discrimination; flexible by-item category counts. | ✅ Full | Masters (1982) |
| Rating Scale Model (Andrich) | Polytomous with common thresholds across items and item-specific location parameters. | ✅ Full | Andrich (1978) |
| Nominal Response Model (Bock) | Unordered categorical responses with per-category slopes and intercepts per item. | ✅ Full | Bock (1972) |
| Differential Item Functioning Analysis | Mantel-Haenszel, logistic, or Lord's chi-squared DIF with ETS delta classification. | ✅ Full | Holland & Thayer (1988); Swaminathan & Rogers (1990) |
| Classical Test Theory Analysis | Item difficulty, corrected item-total correlation, and alpha-if-deleted per item. | ✅ Full | Cronbach (1951); Lord & Novick (1968) |
| IRT Fit Indices (Approximate) | Limited-information chi-square, approximate RMSEA/CFI, Q3 local independence (NOT M2-based). | ⚠️ Limited | Maydeu-Olivares & Joe (2005); Yen (1984) |
| Multidimensional Scaling | Metric or non-metric embedding via Euclidean distances; stress and Shepard diagram. | ✅ Full | Kruskal (1964); Torgerson (1952) |
| Simple Correspondence Analysis | SVD on contingency-table chi-square distances for two categorical variables. | ✅ Full | Greenacre (1984) |
| Multiple Correspondence Analysis | Indicator-matrix CA for k ≥ 2 categorical variables; uses prince or manual SVD. | ✅ Full | Greenacre (1984) |
| Gaussian Finite Mixture Model | k-component Gaussian mixture with BIC/AIC selection, posteriors, and component weights. | ✅ Full | McLachlan & Peel (2000); Dempster et al. (1977) |
| Latent Class Analysis | EM-estimated discrete latent classes for binary/categorical indicators; entropy and model fit. | ✅ Full | Goodman (1974); Lazarsfeld & Henry (1968) |
| Latent Profile Analysis | Diagonal-covariance GMM for continuous indicators with entropy, classification table. | ✅ Full | Vermunt & Magidson (2002); McLachlan & Peel (2000) |
| Categorical Regression (CATREG) | ALS optimal scaling for nominal/ordinal predictors with regression fit on quantified variables. | ✅ Full | Young et al. (1976); Gifi (1990) |
| Categorical PCA (CATPCA) | Optimal scaling PCA with ALS alternation; preserves ordinal/nominal structure. | ✅ Full | Gifi (1990) |
| OVERALS (Non-linear CCA) | Multi-set non-linear canonical correlation via ALS with set-wise quantifications. | ✅ Full | Van der Burg et al. (1988); Gifi (1990) |
| Optimal Scaling Transform | Applies nominal/ordinal/spline quantifications to variables without modeling. | ✅ Full | Virtanen et al. (2020) |
| Categorical Correlation | Pearson correlations on optimally-scaled categorical variables. | ✅ Full | Virtanen et al. (2020) |
Notes & limitations
- Hierarchical Agglomerative Clustering — O(n²) memory constraint; refuses n > 20,000 rows.
- IRT Fit Indices (Approximate) — Approximate method: fixed-1PL reference and diagonal weight matrix, not full Maydeu-Olivares M2.