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

A 1 000-row, fully continuous dataset simulated from a population model with a known 1 -> 2 -> 4 bass-ackwards hierarchy. A factor is a summary variable that stands in for a group of items that move together. The dataset showcases the default cor = "pearson" extraction path without the ordinal-detection warning that bfi25's Likert items trigger. Ordinal items have a few ordered categories, such as a 1 to 5 rating.

Usage

sim16

Format

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

Source

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

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, with no MASS dependency):

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

All items load 0.75 on their true factor, with no cross-loadings. A loading is the correlation between an item and a factor. Factor correlations are 0.45 within a metatrait (f1-f2, f3-f4) and 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 reach a consensus of k = 4, from five criteria, because VSS reports at complexities 1 and 2.

Idealized by design. The planted signal is strong and clean, so all six suggest_k() recommendations converge on k = 4. That is 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 to 6. The two datasets are complementary teaching foils. Use sim16 for "watch the method recover a structure we planted," and 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 (exploratory factor analysis) 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. The true factors, the ones that do not split, persist almost unchanged from k=3 onward. So 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). Two factors are redundant when one adds nothing over the other. This is a textbook overextraction artifact, included so the Forbes and 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