name: multivariate-outputs
description: Long-form output and parameter detail for the multivariate operators — how rotated PCA and factor loadings, the maximum-likelihood fit indices, centroid distances, row scores and the vector tests lay out their figures. Topical design; the atomic op-mat-*, op-attr-* and op-test-* skills keep the short form.
type: guide
kind: design
applies_to: process, compose, predict
covers: [PCA rotation, factor fit indices, RMSEA, centroid distance, factor score, Hotelling, MANOVA, vector test details]
requires: [capability:matrices]

Multivariate outputs in detail

multivariate-design picks the operator; the atomic skill lists its params and a one-line output. This file holds the longer reading of the figures. Everything here is descriptive layout, not new params.

PCA

primary holds the loadings (eigenvector · √λ), rectangular, columns PC1…PCk, always the full p rows. auxiliary.eigenvectors has the same shape.

With rotation: varimax (stats::varimax, Kaiser-normalized, k ≥ 2) primary holds the rotated loadings: columns ordered by descending variance accounted, each summing positive. auxiliary.unrotated keeps the originals and vectors.rotated_variance gives each rotated column’s Σ loading². Eigenvalues, shares, communalities and eigenvectors stay unrotated, so explained_variance is still per eigenvalue: read rotated_variance for the rotated split, and expect PC1 to be the column with the most variance, not the first eigenvector.

Factor analysis

primary holds the loadings (F1…Fk, rotated when set); auxiliary.unrotated appears on rotated runs and factor_correlation (k × k) on promax / oblimin. kappa is the promax power (default 4), gamma the oblimin γ (default 0); any other rotation refuses them. extraction follows psych fm: minres, principal_axis (pa) or ml (factanal, ψ held in [0.005, 1]).

vectors carry communalities, uniquenesses, variance_accounted; pairwise runs add auxiliary.n. scalars always carry rmsr, empirical_chisq, df, empirical_p, kmo, bartlett_*, iterations, converged (1/0) and factors_retained. empirical_chisq measures the residual size; it is not a likelihood test.

ml adds the likelihood fit: likelihood_chisq (Bartlett-corrected, on df), likelihood_p, tli, rmsea with rmsea_lower / rmsea_upper (psych RMSEA lower / upper; conf_level sets the coverage, default 0.9) and bic, null when df ≤ 0. These need N* > 1 + (2p + 5)/6 + 2k/3 (PULSE_MATRIX_INSUFFICIENT_N) and a positive-definite R (PULSE_MATRIX_SINGULAR). Identification: (p − k)² ≥ p + k, so 1 factor needs 3 members and 2 need 5 (PULSE_MATRIX_NOT_IDENTIFIED). A uniqueness held at 0.005 is a Heywood case, kept with PULSE_MATRIX_HEYWOOD.

Centroid distances

between takes a grouper spec as in groups ({type, field | vector | fields, params, include, interval}): any built-in, extension groupers too; no own weight, and tz only on a date-family grouper (else the request time_zone; predict echoes it at matrices[i].params.between). primary is k × k, square_symmetric, keyed by bucket label in the grouper’s order (a clustering grouper labels buckets cluster_1… by size), zero diagonal. auxiliary.centroids is k × p; centroids_standardized appears with standardize, pooled_covariance with mahalanobis (computed over the pooled within-bucket covariance, on raw centroids). vectors.sizes is Σw per bucket and counts the rows. metric names come from the shared registry (manifest matrix.metrics, consumer MAT_DISTANCE). cosine is 1 − cos; the rest follow stats::dist, an undefined coordinate dropped and the sums rescaled by p/p_observed.

Components add metric, standardize, buckets and n_null_key (rows in no bucket, excluded). A massless bucket gives a null centroid and PULSE_MATRIX_INSUFFICIENT_N scope buckets; mahalanobis with N* ≤ k uses scope pooled.

Factor scores

One f64 per record: zᵀw on members standardized by the fit. Regression weights are w = R⁻¹LΦ, Bartlett weights w = U⁻²L(LᵀU⁻²L)⁻¹ (psych factor.scores). Default label ATTR_FACTOR_SCORE_<vector or first field>_F<factor>. The score has mean 0 over the fit rows; a null member gives a null score. The fit takes the factor knobs (factors, extraction, rotation, kappa, gamma; no repair).

All scores null with a warning: a singular fit (PULSE_MATRIX_SINGULAR), N* ≤ p (PULSE_MATRIX_INSUFFICIENT_N), k not identified or "kaiser" keeping fewer than factor (PULSE_MATRIX_NOT_IDENTIFIED), Bartlett over a uniqueness ≤ 0 (PULSE_MATRIX_HEYWOOD). Scored with a warning: a Heywood case under regression, or an iteration cap (PULSE_MATRIX_NOT_CONVERGED, kept beside a refusal too).

Hotelling’s T²

Statistic = T², DF = p. Details: f, df1, df2, n, n_dropped, members; one-sample adds mean, mu; paired members2, mean_diff, mu; two-sample groups (sorted), means, mean_diff. Variant: one_sample, paired, two_sample_pooled, two_sample_unequal. variance: unequal is Krishnamoorthy–Yu (fractional df2), Welch’s t at p = 1, and refused on one-sample / paired. Needs n > p (one-sample, paired), n₁+n₂ > p+1 (pooled), n > p per level (unequal). unequal at p = 1 is Welch’s t. Weighted: moments on w*, N* = Σw or n_eff, and Details add sum_weights (+ n_eff) shaped like n.

MANOVA

Statistic = Pillai’s V, DF its df1, PValue its p. Details: f, df1, df2, members, groups (sorted), n, n_dropped, means, df_hypothesis (k − 1), df_residual (N* − k), eigenvalues (of E⁻¹H) and pillai / wilks / hotelling_lawley / roy, each {statistic, f, df1, df2, p_value}. roy.upper_bound is true: Roy’s p is a lower bound. A null p_value means R would print NA. Needs n ≥ p + k complete rows. Weighted: N*_g = Σw or n_eff, and Details add per-level sum_weights (+ n_eff).

See

• multivariate-design — which operator answers which question.
• matrix-results — the matrix slot, encodings and refusals.