sim16: Simulated continuous bass-ackwards teaching example (16...

In ackwards: Bass-Ackwards Hierarchical Structural Analysis

Simulated continuous bass-ackwards teaching example (16 items, known hierarchy)

Description

A 1 000-row, fully continuous dataset simulated from a population model with a known 1 -> 2 -> 4 bass-ackwards hierarchy, for showcasing the default cor = "pearson" extraction path without the ordinal-detection warning that bfi25's Likert items trigger.

Usage

sim16

Format

A data frame with 1 000 rows and 16 numeric columns (i1–i16, continuous, no missing values).

Details

Population model. Sigma = Lambda %*% Phi %*% t(Lambda) + Psi, an oblique common-factor model with 4 true group factors, sampled via a Cholesky factorization (base R; no MASS dependency):

Factor	Items	Metatrait
f1	i1--i4	1 (with f2)
f2	i5--i8	1 (with f1)
f3	i9--i12	2 (with f4)
f4	i13--i16	2 (with f3)

All items load 0.75 on their true factor (no cross-loadings). Factor correlations: 0.45 within a metatrait (f1-f2, f3-f4), 0.15 between metatraits. Uniquenesses are 1 - communality (uniform 0.4375 by the symmetry of the design above).

Ground-truth hierarchy (verified against ackwards(engine = "efa")): k=1 recovers a single general factor across all 16 items; k=2 splits along the metatrait line (i1-i8 vs. i9-i16); k=4 recovers the 4 true group factors exactly; all six suggest_k() recommendations (five criteria – VSS reports at complexities 1 and 2) reach a consensus of k = 4.

Idealized by design. The planted signal is strong and clean, so all six suggest_k() recommendations converge on k = 4 – deliberately the easy case, for building intuition about what recovering a known hierarchy looks like. Real data rarely agree this cleanly: on bfi25 the same criteria span k = 4–6. The two datasets are complementary teaching foils – sim16 for "watch the method recover a structure we planted," bfi25 for "reason about a hierarchy when the criteria disagree." Present sim16's consensus as the ideal, not the norm.

Deliberate overextraction artifact at k=5. The population has exactly 4 factors, so requesting a 5th finds no real dimension: EFA produces an orphan factor with zero primary-loading items. With prune(x, "artifact") (default min_items = 3, orphan_r = 0.5), that factor is flagged both few_items and orphan. Because the true (non- splitting) factors persist essentially unchanged from k=3 onward, their parent-child score correlations approach 1 and are flagged by prune(x, "redundant") (⁠|r| >= .9⁠ and, under the EFA auto-default, Tucker's phi ⁠> .95⁠). This is a textbook overextraction artifact, included so the Forbes/redundancy examples have a guaranteed finding to teach against (unlike bfi25, which does not reliably trigger one).

To regenerate this dataset, run source("data-raw/sim16.R") from the package root (set.seed(42)).

Source

Simulated; see data-raw/sim16.R for the full generative model.

Examples

dim(sim16)
head(sim16)

ackwards documentation built on July 25, 2026, 1:08 a.m.