Simulated continuous bass-ackwards teaching example (16 items, known hierarchy)
Source:R/data.R
sim16.RdA 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.
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)).
Examples
dim(sim16)
#> [1] 1000 16
head(sim16)
#> i1 i2 i3 i4 i5 i6
#> 1 1.3709584 2.6935162 1.65649737 1.0043474 0.4985536 0.62179404
#> 2 -0.5646982 0.1157001 -0.37601948 -0.8080455 -0.9279296 0.23675384
#> 3 0.3631284 1.0068595 -0.83720071 -0.1617170 -0.2746798 0.12758795
#> 4 0.6328626 0.6676658 -1.07970658 -0.7959827 0.3055024 0.01789392
#> 5 0.4042683 -0.5960341 -1.06548809 0.4971529 -1.9685936 -1.31694755
#> 6 -0.1061245 -0.5536923 0.04465981 -0.8170172 -1.0854028 -0.72248740
#> i7 i8 i9 i10 i11 i12 i13
#> 1 0.7158967 1.6662461 0.1762268 -0.5591902 0.8260793 -0.05320752 0.2350657
#> 2 -1.2592888 -0.1193249 -0.3315514 -0.6307922 -0.5144422 -0.06145734 0.5210552
#> 3 -0.7035972 0.5190214 -1.5114876 -3.4055614 -1.9790895 -2.95739348 -0.6546594
#> 4 0.5725013 -0.2194903 0.4623181 -0.3135998 -0.2535567 0.70600027 -0.3650701
#> 5 -1.2225681 -0.4665789 -1.3092214 -0.5390024 -0.1509965 -0.08517385 -0.3197751
#> 6 -0.8549671 -0.7564596 0.3796133 0.1051229 -1.1088126 -0.14489947 -0.5835958
#> i14 i15 i16
#> 1 -1.25462666 -2.0439388 -1.4869628
#> 2 0.08671852 0.1475108 -0.4135835
#> 3 0.72176685 -1.5012947 -1.4849784
#> 4 0.67296942 1.4489548 -0.2946622
#> 5 -0.30006027 -0.8133962 0.8467458
#> 6 1.48728557 0.1890597 2.4643392