The Forbes Extension: Skip-Level Connections and Pruning

Reading the skip-level edge table

The table below keeps the non-adjacent edges (levels more than one apart) with |r| >= 0.5, strongest first — drawn from tidy(x_all, what = "edges"):

Several factors connect across two or more levels with correlations above 0.90. m3f2 (level 3, factor 2) correlates 0.98 with m5f2 (level 5, factor 2), jumping two levels. This tells you that m3f2 and m5f2 are essentially the same construct — the intermediate levels are just refinements within a stable dimension.

Reading the strongest edge with care

Reading the strongest edge off a table of many correlations is itself a form of selection. With k = 5 the all-pairs table holds 85 edges, and the maximum of that many correlations is biased upward even when every individual estimate is honest. Treat a “strongest link” claim as descriptive rather than inferential: if it is load-bearing, pre-specify which pair of factors you care about rather than reporting whichever correlation came out largest.

Chains of near-identical factors with prune(x, "redundant")

A redundant chain is a sequence of factors connected by near-perfect correlations (|r| ≥ 0.9 by default) across levels. If m2f2 → m3f2 → m4f2 → m5f1 all share r > 0.97, the chain reaches the deepest level, so its bottom node m5f1 — the most specific, best-defined manifestation — is retained and the others (m2f2, m3f2, m4f2) are flagged as redundant: they repeat rather than refine the same dimension. A chain that stops short of the deepest level instead keeps its top node, the broadest manifestation (Forbes, 2023).

By default (redundancy_criterion = "direct") a factor is joined to an ancestor when their score correlation is high directly — the rule Forbes’s own code uses, and the honest reading of “the same construct” (the two factors share ≥ 81% of their variance directly). Because correlation is non-transitive, this can differ from following one high-correlation hop at a time in deep hierarchies; redundancy_criterion = "adjacent" selects that older, adjacent-hop behavior. On a shallow hierarchy like this one the two agree.

x_prune <- ackwards(bfi, k_max = 5, cor = "polychoric", pairs = "all") |>
  prune("redundant")
#> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.9) flagged 6 nodes.
#> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and
#>   `x$prune$chains`.

6 factors are flagged as redundant (m2f2, m3f2, m4f1, m4f2, m4f3, m4f4): 1 factor at k = 2, 1 factor at k = 3, the entire k = 4 level. This is a striking finding: for this dataset and this k, the four-factor level adds little beyond what you already know from k = 3 and k = 5.

The flagged factors are not removed from the object — prune() is purely a diagnostic annotation, not a deletion. You can still inspect their loadings, use their scores, and include them in the diagram. Pruning flags guide interpretation; they do not alter the model.

The pruned-factor diagram

For presentations and publications it is cleaner to omit the flagged factors entirely and draw direct connections from each retained factor to its single strongest kept ancestor — even when that ancestor is several levels away. This is the Forbes (2023) pruned-factor diagram, activated by drop_pruned = TRUE.

Forbes (2023) presents two variants: one with correlation labels on each arrow and one without. The first is useful when the strength of each spanning connection matters to the interpretation; the second is cleaner for presentations. Both are reproduced below using the same publication style — black lines, uniform width, plain line ends, and no legend.

With correlation labels (show_r = TRUE):

autoplot(x_prune,
  drop_pruned = TRUE, show_r = TRUE,
  color_pos = "black", color_neg = "black",
  edge_linewidth = 0.6, show_arrows = FALSE, legend = FALSE
)

plot of chunk drop-pruned

Without labels (cleaner for slides or when the exact values are not the focus):

autoplot(x_prune,
  drop_pruned = TRUE,
  color_pos = "black", color_neg = "black",
  edge_linewidth = 0.6, show_arrows = FALSE, legend = FALSE
)

plot of chunk drop-pruned-unlabeled

Level 4 is entirely pruned, leaving a visible gap in the y-axis. The gap is intentional: it shows which level was removed. Spanning arrows bridge directly from level 3 factors to level 5 factors (and from level 2 to level 5 where intermediate levels were flagged). In this publication style every drawn line has the same uniform weight (edge_linewidth = 0.6); only connections with |r| at or above the display threshold (cut_show, default 0.3) are drawn at all.

To close the gaps and compact the layout while retaining the original level numbers on the axis:

autoplot(x_prune,
  drop_pruned = TRUE, compress_levels = TRUE,
  color_pos = "black", color_neg = "black",
  edge_linewidth = 0.6, show_arrows = FALSE, legend = FALSE
)

plot of chunk drop-pruned-compressed

The level labels still read “1 factor”, “2 factors”, “3 factors”, “5 factors” so readers know which levels were retained; the uniform vertical spacing makes the diagram easier to read in constrained page layouts.

For further cosmetic customization — colours, node labels, arrowheads, and more — see vignette("ackwards-visualization").

Inspecting structural similarity with prune(x, "artifact")

An artifact factor is one that looks like a copy of a factor from another level rather than a genuine refinement — its loading pattern closely resembles a factor elsewhere in the hierarchy. Similarity is measured by Tucker’s congruence coefficient (φ):

\[\phi(F_a, F_b) = \frac{\sum_i \lambda_{ia}\lambda_{ib}} {\sqrt{\sum_i \lambda_{ia}^2 \cdot \sum_i \lambda_{ib}^2}}\]

φ ranges from −1 to +1, with values above 0.95 conventionally read as near-identical loading patterns regardless of sign (Lorenzo-Seva & ten Berge, 2006).

Unlike "redundant", the artifact mode never flags anything automatically. Forbes (2023) is explicit that identifying an artifact requires researcher judgment — automating it would manufacture investigator degrees of freedom — so prune(x, "artifact") computes and stores the evidence for you to weigh: Tucker’s φ for every cross-level factor pair in x$prune$phi, plus the structural signals of the next section in x$prune$structural.

x_art <- ackwards(bfi, k_max = 5, cor = "polychoric", pairs = "all") |>
  prune("artifact")
#> ℹ Artifact mode: Tucker's φ computed for all cross-level factor pairs.
#> ℹ Structural signals computed: 2 factors flagged (few_items / orphan /
#>   split_merge).
#> ℹ Inspect `x$prune$phi` and `x$prune$structural`; removal is a researcher
#>   judgment (Forbes, 2023).

The table to read is x$prune$phi. The natural first cut is the non-adjacent pairs with the highest |φ| — a deep factor whose loading pattern nearly duplicates a factor two or more levels up is the classic candidate:

For these data the strongest non-adjacent congruence is |φ| = 0.99. Values near 1 mean the deeper factor recycles an earlier loading pattern; whether that makes it a rotation artifact — or a faithfully persisting construct, which is the redundancy view of the same fact — is a judgment made with the substantive content of the items in view, not a threshold the package applies for you.

The two modes surface different fingerprints of the same underlying question:

• "redundant": this factor appears at multiple levels with near-identical score correlations — it persists unchanged as k increases. Auto-flagged (with Forbes’s retention rule), because the score-correlation chain is a sharp, replicable criterion.
• "artifact": this factor’s loading pattern closely resembles a factor elsewhere in the hierarchy, or its structure looks under-identified (next section). Reported only — the call is yours.

Structural artifact signals

Congruence (φ) is not the only fingerprint of an artifactual factor. Forbes (2023, Fig. 2) describes several structural signatures, and prune(x, "artifact") reports three of them per factor in x$prune$structural:

• few_items — the factor is the primary (highest-loading) home for fewer than min_items items (default 3). A factor anchored by one or two items is under-identified and often an extraction artifact rather than a replicable construct.
• orphan — the factor’s strongest correlation to the immediately neighbouring levels is below orphan_r (default 0.5). It does not connect to the solutions on either side, so it does not replicate across the hierarchy.
• split_merge — the factor’s primary items were spread across two or more different parent factors at the level above. Items that were separated at the coarser solution have been merged under one factor at the finer solution — the split-then-merge anomaly of Forbes Fig. 2.

Like Tucker’s φ, these signals are flag-and-report only — prune() never removes a factor on their basis. Identifying an artifact requires researcher judgment (Forbes is explicit that this step introduces investigator degrees of freedom); the signals simply point you to the factors worth a closer look. The two thresholds, min_items and orphan_r, are arguments to prune().