# The Forbes Extension: Skip-Level Connections and Pruning

The classic Goldberg (2006) method computes between-level factor-score correlations only for adjacent levels: 1↔︎2, 2↔︎3, 3↔︎4, and so on. Forbes (2023) extended this in two ways: computing correlations across all level pairs (not just adjacent), and using those extra connections to identify and flag redundant or artifactual factors in the hierarchy.

This vignette covers both extensions: `pairs = "all"` and `prune`. It teaches the concepts on a small didactic dataset; for a full reproduction of the paper’s 155-variable applied example on the bundled `forbes2023` dataset, see `vignette("ackwards-forbes2023")`.

## The limitation of adjacent-only edges

An adjacent-only hierarchy shows you the immediate parent–child relationships. What it cannot tell you is whether a factor at level k is essentially the same construct as a factor several levels up — a sign that the intermediate levels are adding noise rather than resolution.

Consider a factor that appears at k = 2, k = 3, and k = 4 and correlates > 0.97 with its counterpart at every adjacent level. The adjacent-only diagram shows three consecutive arrows, each nearly perfect. But nothing in the diagram directly flags that the k = 3 factor is redundant: you could skip straight from k = 2 to k = 4 without losing any information.

Skip-level correlations make this visible by computing the correlation between every pair of levels, not just neighbors.

## Setup

    library(ackwards)
    bfi <- na.omit(bfi25)

## Computing every between-level correlation with `pairs = "all"`

Adding `pairs = "all"` extends the edge table from adjacent pairs only to every combination of levels.

    # Classic adjacent-only
    x_adj <- ackwards(bfi, k_max = 5, cor = "polychoric")

    # All pairs
    x_all <- ackwards(bfi, k_max = 5, cor = "polychoric", pairs = "all")

    # How many edges?
    nrow(tidy(x_adj, what = "edges")) # adjacent only
    #> [1] 40
    nrow(tidy(x_all, what = "edges")) # all pairs
    #> [1] 85

With k = 5, the adjacent-only model has 40 edges (1×2 + 2×3 + 3×4 + 4×5). The all-pairs model adds every non-adjacent pair — 1↔︎3, 1↔︎4, 1↔︎5, 2↔︎4, 2↔︎5, 3↔︎5 — for 85 edges total.

### 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")`:

Strongest skip-level edges (|r| ≥ 0.5, non-adjacent levels)
Sorted by |r|; shows at most 12 rows
From | To | Level (from) | Level (to) | r
--- | --- | --- | --- | ---
m3f2 | m5f2 | 3 | 5 | 0.98
m2f2 | m4f2 | 2 | 4 | 0.97
m2f2 | m5f2 | 2 | 5 | 0.97
m2f1 | m4f1 | 2 | 4 | 0.85
m3f1 | m5f1 | 3 | 5 | 0.82
m1f1 | m3f1 | 1 | 3 | 0.77
m1f1 | m4f1 | 1 | 4 | 0.75
m3f3 | m5f3 | 3 | 5 | 0.73
m2f1 | m5f1 | 2 | 5 | 0.69
m3f3 | m5f5 | 3 | 5 | 0.68
m1f1 | m5f1 | 1 | 5 | 0.61
m3f1 | m5f4 | 3 | 5 | 0.56

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.

## Pruning: identifying redundant factors

The `prune()` verb uses the skip-level correlations to automatically flag factors that may not be adding genuine information to the hierarchy.

### 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"`) the criterion is a star anchored on the chain’s deepest factor, not a walk. Every candidate ancestor is judged by its own score correlation directly with that deepest leaf: the chain forms while each shallower factor correlates `|r| ≥ 0.9` with the leaf (sharing ≥ 81% of its variance directly) — the rule Forbes’s own code uses, and the honest reading of “the same construct”. Two things it is not: it does not require each consecutive pair to correlate (that adjacent-hop rule is `redundancy_criterion = "adjacent"`), and it does not screen every ancestor pair against every other. Because correlation is non-transitive, the star and the adjacent-hop walk can disagree in deep hierarchies — an ancestor can join on a strong direct link to the leaf even where its hop to the next chain member is weak; 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`.

Node-level pruning annotation
6 of 15 factors flagged as redundant
Factor | Level | Flagged? | Reason
--- | --- | --- | ---
m1f1 | 1 | FALSE | —
m2f1 | 2 | FALSE | —
m2f2 | 2 | TRUE | redundant
m3f1 | 3 | FALSE | —
m3f2 | 3 | TRUE | redundant
m3f3 | 3 | FALSE | —
m4f1 | 4 | TRUE | redundant
m4f2 | 4 | TRUE | redundant
m4f3 | 4 | TRUE | redundant
m4f4 | 4 | TRUE | redundant
m5f1 | 5 | FALSE | —
m5f2 | 5 | FALSE | —
m5f3 | 5 | FALSE | —
m5f4 | 5 | FALSE | —
m5f5 | 5 | FALSE | —

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 drops anything automatically. The reason is not that flagging would add investigator degrees of freedom — a standardized flag that is always computed the same way removes researcher discretion rather than adding it. It is that the decision to drop a factor as an artifact is substantive: Forbes (2023) is explicit that identifying an artifact requires researcher judgment. 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` — and leaves the drop decision to you.

    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 Near-redundant band: 5 cross-level pairs just below the redundancy thresholds
    #>   (|r| in [0.8, 0.9)).
    #> ℹ Inspect `x$prune$phi`, `x$prune$structural`, and `x$prune$near_redundant`;
    #>   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:

Strongest non-adjacent loading congruences
Top 8 pairs by |φ|, from x$prune$phi — evidence, not flags
From | To | Level (from) | Level (to) | φ
--- | --- | --- | --- | ---
m3f2 | m5f2 | 3 | 5 | 0.99
m2f2 | m4f2 | 2 | 4 | 0.98
m2f2 | m5f2 | 2 | 5 | 0.98
m2f1 | m4f1 | 2 | 4 | 0.93
m3f1 | m5f1 | 3 | 5 | 0.92
m1f1 | m3f1 | 1 | 3 | 0.88
m1f1 | m4f1 | 1 | 4 | 0.86
m2f1 | m5f1 | 2 | 5 | 0.85

For these data the strongest non-adjacent congruence is |φ| = 0.99. Values near 1 mean the deeper factor recycles an earlier loading pattern — but a congruence this high travels with a near-perfect score correlation too, so these top pairs are already the province of `prune(x, "redundant")`: the factor they name is flagged (and dropped from the publication diagram) by the redundancy rule above. The distinctive contribution of artifact mode is not re-flagging what redundancy already caught; it is surfacing the messier band just below the redundancy thresholds.

### The near-redundant band

Full redundancy — `|r| ≥ redundancy_r` — is dropped in the redundancy stage. Forbes (2023) uses the artifact flags mainly for the pairs that sit just under that line: a pair that correlates, say, `|r| = 0.89` and shares a loading pattern `φ = 0.94` is not quite redundant, but it is a candidate re-rotation worth a second look rather than a clean new level of resolution. `prune(x, "artifact")` collects exactly these in `x$prune$near_redundant`: every cross-level pair that is not itself fully redundant yet has its direct score correlation `|r|` (or, off PCA, its Tucker `φ`) within `near_margin` (default `0.1`) below the corresponding threshold.

    #> 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`.

Near-redundant band
Cross-level pairs just below the redundancy thresholds — evidence, not flags
From | To | Level (from) | Level (to) | r | φ
--- | --- | --- | --- | --- | ---
m1f1 | m2f1 | 1 | 2 | 0.89 | 0.94
m2f1 | m3f1 | 2 | 3 | 0.87 | 0.94
m2f1 | m4f1 | 2 | 4 | 0.85 | 0.93
m4f1 | m5f1 | 4 | 5 | 0.84 | 0.93
m3f1 | m5f1 | 3 | 5 | 0.82 | 0.92

The strongest near-redundant pair here is m1f1 ↔︎ m2f1 (r = 0.89, φ = 0.94) — and, unlike the top-φ pairs above, neither factor is flagged by prune(x, “redundant”): this pair is a genuine near-miss, not something redundancy already drops. The examples here use the default PCA engine, so `redundancy_phi` is `NULL` and only the `|r|` band is active; on an EFA or ESEM fit `redundancy_phi` auto-resolves to `0.95` and the `φ` band switches on too (`prune()` announces this). Widen or narrow the band with `near_margin`. Like the φ table, the band is report-only — no factor is dropped on its basis.

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.

Structural artifact signals
2 of 15 factors raise a structural signal
Factor | Level | Few items | Orphan | Split/merge
--- | --- | --- | --- | ---
m1f1 | 1 | FALSE | FALSE | FALSE
m2f1 | 2 | FALSE | FALSE | FALSE
m2f2 | 2 | FALSE | FALSE | FALSE
m3f1 | 3 | FALSE | FALSE | FALSE
m3f2 | 3 | FALSE | FALSE | FALSE
m3f3 | 3 | FALSE | FALSE | TRUE
m4f1 | 4 | FALSE | FALSE | FALSE
m4f2 | 4 | FALSE | FALSE | FALSE
m4f3 | 4 | FALSE | FALSE | FALSE
m4f4 | 4 | FALSE | FALSE | TRUE
m5f1 | 5 | FALSE | FALSE | FALSE
m5f2 | 5 | FALSE | FALSE | FALSE
m5f3 | 5 | FALSE | FALSE | FALSE
m5f4 | 5 | FALSE | FALSE | FALSE
m5f5 | 5 | FALSE | FALSE | FALSE

Like Tucker’s φ, these signals are flag-and-report only — `prune()` never removes a factor on their basis. As before, a standardized signal removes researcher discretion rather than adding it; what requires judgment is the substantive decision to treat a factor as an artifact, which the signals simply point you toward. The two thresholds, `min_items` and `orphan_r`, are arguments to `prune()`.

## Tuning the thresholds

The redundancy criterion has an adjustable `redundancy_r` threshold (default `0.90`) matching Forbes (2023). The artifact criterion has no auto-flag threshold — `prune(x, "artifact")` computes Tucker’s φ for researcher inspection; no factors are auto-flagged.

The `redundancy_phi` companion criterion. Redundancy can optionally require that linked factors also share a loading pattern (Tucker’s φ above a threshold), not just a high score correlation. The `redundancy_phi` argument controls this, and its default (`NULL`) auto-resolves based on the engine:

- PCA — no φ filter. Component scores are determinate: unlike factor scores they are exact linear functions of the observed data, so the score correlation `|r|` is the correlation between the components themselves and suffices as the redundancy signal.
- EFA / ESEM — φ is required to exceed `0.95` (Lorenzo-Seva & ten Berge, 2006). Factor scores are indeterminate — any factor admits infinitely many score series consistent with the model — which makes an `|r|`-only rule too liberal; the loading-congruence guard makes the criterion conservative. `prune()` announces this auto-resolution in the console.

The examples here use the default PCA engine, so no φ filter is applied. To disable the φ guard on an EFA/ESEM run, pass `redundancy_phi = NA`; to set your own threshold, pass a number in `(0, 1]`.

For the BFI, the result is the same across a wide range of thresholds because the redundant chains all have correlations > 0.97 — the flagging is unambiguous. With your own data you may find borderline cases where the threshold matters:

Because `prune()` is a cheap, standalone step, checking a few `redundancy_r` thresholds does not require refitting `ackwards()` each time — the already-fit `x_all` object is re-pruned directly:

    thresholds <- c(0.80, 0.85, 0.90, 0.95)
    counts <- sapply(thresholds, function(thr) {
      x <- prune(x_all, "redundant", redundancy_r = thr)
      sum(tidy(x, what = "nodes")$pruned)
    })
    #> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.8) flagged 9 nodes.
    #> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and
    #>   `x$prune$chains`.
    #> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.85) flagged 8 nodes.
    #> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and
    #>   `x$prune$chains`.
    #> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.9) flagged 6 nodes.
    #> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and
    #>   `x$prune$chains`.
    #> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.95) flagged 6 nodes.
    #> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and
    #>   `x$prune$chains`.
    thr_df <- data.frame(redundancy_r = thresholds, n_flagged = counts)

Factors flagged redundant at each redundancy_r threshold
redundancy_r | Factors flagged
--- | ---
0.80 | 9
0.85 | 8
0.90 | 6
0.95 | 6

For the BFI all thresholds agree: the flagged factors are robustly redundant, not borderline cases. In noisier datasets or smaller samples you will typically see the count increase as you lower the threshold.

## Practical interpretation

The Forbes extension does not change the core bass-ackwards analysis. It enriches it with two questions:

1. Do any factors persist unchanged across multiple levels? (`pairs = "all"`) Skip-level correlations near 1.0 indicate stable dimensions that survive changes in k — exactly the kind of robust construct you want to report.

2. Are there levels where the factor structure is just reorganizing rather than genuinely differentiating? (`prune(x, "redundant")`) Flagged levels can often be removed from the k range without losing interpretive content.

A common workflow: fit with `pairs = "all"` first to examine the full picture, then pipe the result through `prune(x, "redundant")` to identify which levels add the most new information, and use that to guide your focus in reporting. For where redundancy pruning sits in the full recommended workflow — alongside a split-half replicability gate on hierarchy depth (`comparability()`) — see `vignette("ackwards-girard")`.

## References

Forbes, M. K. (2023). Improving hierarchical models of individual differences: An extension of Goldberg’s bass-ackward method. Psychological Methods. https://doi.org/10.1037/met0000546

Goldberg, L. R. (2006). Doing it all Bass-Ackwards: The development of hierarchical factor structures from the top down. Journal of Research in Personality, 40(4), 347–358.

Lorenzo-Seva, U., & ten Berge, J. M. F. (2006). Tucker’s congruence coefficient as a meaningful index of factor similarity. Methodology, 2(2), 57–64. https://doi.org/10.1027/1614-2241.2.2.57

Tucker, L. R. (1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Department of the Army.