Flag redundant or artifactual factors (Forbes 2023 extension)

prune() never removes anything from an ackwards object – it only annotates factors with pruned/prune_reason flags in x$prune$nodes (flag-only, never removes; the object keeps every level). Because it is a separate, cheap step from extraction, you can re-prune with new thresholds without re-running ackwards():


  x <- ackwards(bfi25, k_max = 6, engine = "esem")  # expensive
  x |> prune("redundant")                            # cheap, repeatable
  x |> prune("redundant", redundancy_r = 0.95)        # no re-extraction

prune() is an S3 generic (rather than a plain function) so it coexists with the prune generics already defined by recursive-partitioning packages (e.g. rpart::prune) regardless of package load order.

Usage

prune(x, ...)

# S3 method for class 'ackwards'
prune(
  x,
  rules = "none",
  manual = NULL,
  redundancy_r = 0.9,
  redundancy_phi = NULL,
  redundancy_criterion = c("direct", "adjacent"),
  min_items = 3L,
  orphan_r = 0.5,
  near_margin = 0.1,
  ...
)

Arguments

x

An ackwards object.

...

Reserved for future methods/arguments.

rules

Character vector controlling which auto-rules run. Default "none" (no auto rule; combine with manual for pure manual pruning, or call prune(x) with no arguments to clear any existing pruning). Options:

"redundant" – identify chains of factors connected by score correlations |r| >= redundancy_r (and optionally phi > redundancy_phi), using redundancy_criterion (default "direct", faithful to Forbes). Applies Forbes's (2023) retention rule: keep the bottom node when the chain reaches level k_max (most specific); keep the top node otherwise. Flagged nodes get pruned = TRUE and prune_reason = "redundant" in x$prune$nodes.
"artifact" (or the alias "artefact", normalized to "artifact") – compute Tucker's congruence coefficient (phi) for all cross-level factor pairs and store in x$prune$phi, structural signals (few_items/orphan/split_merge) in x$prune$structural, and the near-redundant band (near_margin, see Details) in x$prune$near_redundant. No factors are auto-flagged; artifact identification requires judgment (Forbes, 2023; Wicherts et al., 2016).

manual

Character vector of factor labels (e.g. c("m4f3", "m4f4")) to flag directly, in addition to or instead of an auto rule. Standalone manual pruning is supported: prune(x, manual = c("m4f3")). Unknown labels error. A node already flagged by an auto rule keeps that prune_reason; only otherwise-unflagged manual nodes get prune_reason = "manual".

redundancy_r

Scalar in (0, 1]. Score-correlation |r| threshold for redundancy chains. Default 0.9 (Forbes, 2023).

redundancy_phi

Scalar in (0, 1], NULL (default, auto), or NA (explicit opt-out). When NULL:

x$engine == "pca" – no phi filter. Component scores are determinate (exact linear functions of the data, with no factor-score indeterminacy), so the score correlation |r| is the true correlation between the components themselves; phi adds nothing that |r| does not already capture.
x$engine is "efa" or "esem" – automatically set to 0.95 (Lorenzo-Seva & ten Berge, 2006). Factor-score indeterminacy off-PCA means |r|-alone is liberal; the conjunctive phi criterion is the conservative default. A cli message announces the resolved value. Pass NA to disable phi filtering regardless of engine. Pass a numeric value to override on any engine.

redundancy_criterion

How redundancy chains are traced. One of:

"direct" (default) – chase upward via the direct (skip-level) correlation between a factor and each ancestor level, continuing while |r| >= redundancy_r contiguously. This is Forbes's (2023) published ChaseCorrPaths rule and the honest operationalization of "the same construct" (two factor scores share >= redundancy_r^2 variance directly). It reproduces her AMH applied example exactly.
"adjacent" – trace adjacent primary-parent links only (each consecutive level |r| >= redundancy_r). This was the pre-M53 default; because correlation is non-transitive it can both over- and under-flag versus "direct" in deep (many-level) hierarchies, so it is retained only as an opt-in. On shallow/transitive hierarchies the two agree.

min_items

Minimum number of items for which a factor must be the primary loader (highest |loading|). Factors with fewer than min_items primary items are flagged few_items = TRUE in x$prune$structural. Only used when rules includes "artifact". Default 3L – a factor defined by one or two items is under-identified and frequently an extraction artifact rather than a replicable construct (the classic "three-indicator rule"; Forbes, 2023, Fig. 2).

orphan_r

Threshold for the orphan structural signal. A factor whose maximum adjacent-level |r| (to the immediately shallower and deeper levels) falls below orphan_r is flagged orphan = TRUE in x$prune$structural – it does not connect to the neighbouring solutions and so does not replicate across the hierarchy. Only used when rules includes "artifact". Default 0.5 – a moderate correlation; a factor that shares less than a quarter of its variance with every neighbour is a structural outlier worth inspecting.

near_margin

Scalar in (0, 1]. Width of the near-redundant band reported by "artifact" mode (see Details). A cross-level pair is flagged near-redundant when its direct |r| sits in [redundancy_r - near_margin, redundancy_r) or its Tucker phi sits in [redundancy_phi - near_margin, redundancy_phi) – i.e. within near_margin below a redundancy threshold – while the pair is not itself fully redundant. Only used when rules includes "artifact". Default 0.1.

Value

x, with $prune populated (replacing any prior pruning).

Details

The "direct" criterion is a star anchored on the leaf, not a walk. Take a three-level chain candidate with deepest leaf m3f1 and shallower factors m2f1 (level 2) and m1f1 (level 1). Under the default "direct" criterion the chain m1f1 -> m2f1 -> m3f1 forms when both direct-to-leaf correlations |r(m1f1, m3f1)| and |r(m2f1, m3f1)| meet redundancy_r – every member is judged by its own direct correlation to the same deepest factor (a star centred on the leaf). It does not require the adjacent hop |r(m1f1, m2f1)| to meet the threshold (that is the "adjacent" criterion), and it does not screen every ancestor pair against every other (there is no all-pairs test). So an ancestor can join on a strong direct link to the leaf even where its adjacent hop to the next chain member is weak – which is exactly why r_to_prev (below) can sit under redundancy_r.

Reading x$prune$chains under redundancy_criterion = "direct". The r_to_prev and phi_to_prev columns report the adjacent-level correlation and congruence between consecutive chain members (for continuity of display), but chain membership is decided by the direct (skip-level) correlation to the chain's deepest factor. A direct chain can therefore legitimately contain a link whose r_to_prev is below redundancy_r – the stronger direct link is what justified it. The endpoint_r column gives the direct root-to-leaf correlation as an at-a-glance cross-check. Under redundancy_criterion = "adjacent", r_to_prev is the criterion and always meets redundancy_r.

The near-redundant band ("artifact" mode). prune("redundant") drops full redundancy – pairs at or above the thresholds. Forbes (2023) uses the artifact flags mainly for the messier band just below them: a pair that correlates, say, |r| = 0.89 and shares a loading pattern phi = 0.93 is not quite redundant but is a candidate re-rotation worth a second look. Artifact mode surfaces this as x$prune$near_redundant – a data frame of every cross-level pair that is not itself fully redundant yet has its direct (skip-level) |r| or its Tucker phi within near_margin below the corresponding threshold (redundancy_r / redundancy_phi). Columns: from, to, level_from, level_to, r (direct score correlation), phi (loading congruence), and the logical band flags near_r / near_phi. Under the PCA engine redundancy_phi is NULL (component scores are determinate), so only the |r| band applies; under EFA/ESEM redundancy_phi auto-resolves to 0.95 (announced via cli) and the phi band is active too. Like phi and the structural signals, the band is report-only – it never drops or mutates the kept node set.

References

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

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. doi:10.1027/1614-2241.2.2.57

Examples

# sim16 has a planted redundant chain + overextraction artifact at k = 5,
# so the prune rules always have a finding to show (and no ordinal warning).
x <- ackwards(sim16, k_max = 5)

xp <- prune(x, "redundant")
#> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.9) flagged 7 nodes.
#> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and
#>   `x$prune$chains`.
xp$prune$nodes
#>      id level pruned prune_reason
#> 1  m1f1     1  FALSE         <NA>
#> 2  m2f1     2  FALSE         <NA>
#> 3  m2f2     2  FALSE         <NA>
#> 4  m3f1     3   TRUE    redundant
#> 5  m3f2     3   TRUE    redundant
#> 6  m3f3     3   TRUE    redundant
#> 7  m4f1     4   TRUE    redundant
#> 8  m4f2     4   TRUE    redundant
#> 9  m4f3     4   TRUE    redundant
#> 10 m4f4     4   TRUE    redundant
#> 11 m5f1     5  FALSE         <NA>
#> 12 m5f2     5  FALSE         <NA>
#> 13 m5f3     5  FALSE         <NA>
#> 14 m5f4     5  FALSE         <NA>
#> 15 m5f5     5  FALSE         <NA>

# Re-prune with a new threshold -- no re-extraction needed
prune(x, "redundant", redundancy_r = 0.95)
#> 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`.
#> 
#> ── Bass-Ackwards Analysis (ackwards) ───────────────────────────────────────────
#> Engine: pca
#> Rotation: varimax
#> Basis: pearson
#> n: 1,000
#> k (max): 5
#> 
#> ── Levels ──
#> 
#> ✔ k = 1: 1 factor, 28.2% variance
#> ✔ k = 2: 2 factors, 46.5% variance
#> ✔ k = 3: 3 factors, 57.5% variance
#> ✔ k = 4: 4 factors, 67.7% variance
#> ✔ k = 5: 5 factors, 70.8% variance
#> 
#> ── Edges ──
#> 
#> 13 of 40 edges have |r| ≥ 0.3
#> 
#> ── Pruning ──
#> 
#> Redundancy (direct, |r| ≥ 0.95): 6 nodes flagged
#> ────────────────────────────────────────────────────────────────────────────────
#> Note: Pruning is interpretive relabeling, not re-estimation. Flagged nodes
#> remain in the object; all edges are preserved. Inspect with `x$prune$nodes` and
#> `tidy(x, what = "nodes")`.
#> Note: This is a series of linked solutions, not a fitted hierarchical model.
#> Cross-level edges are descriptive score correlations. Per-level fit indices
#> (EFA/ESEM) describe how well a k-factor model fits the items at that level --
#> they do not validate the edges or the hierarchy itself.

# Manual pruning: standalone, or mixed with an auto rule
prune(x, manual = "m4f2")
#> 
#> ── Bass-Ackwards Analysis (ackwards) ───────────────────────────────────────────
#> Engine: pca
#> Rotation: varimax
#> Basis: pearson
#> n: 1,000
#> k (max): 5
#> 
#> ── Levels ──
#> 
#> ✔ k = 1: 1 factor, 28.2% variance
#> ✔ k = 2: 2 factors, 46.5% variance
#> ✔ k = 3: 3 factors, 57.5% variance
#> ✔ k = 4: 4 factors, 67.7% variance
#> ✔ k = 5: 5 factors, 70.8% variance
#> 
#> ── Edges ──
#> 
#> 13 of 40 edges have |r| ≥ 0.3
#> 
#> ── Pruning ──
#> 
#> Manual: 1 node explicitly flagged (m4f2)
#> ────────────────────────────────────────────────────────────────────────────────
#> Note: Pruning is interpretive relabeling, not re-estimation. Flagged nodes
#> remain in the object; all edges are preserved. Inspect with `x$prune$nodes` and
#> `tidy(x, what = "nodes")`.
#> Note: This is a series of linked solutions, not a fitted hierarchical model.
#> Cross-level edges are descriptive score correlations. Per-level fit indices
#> (EFA/ESEM) describe how well a k-factor model fits the items at that level --
#> they do not validate the edges or the hierarchy itself.
prune(x, "redundant", manual = "m4f2")
#> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.9) flagged 7 nodes.
#> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and
#>   `x$prune$chains`.
#> 
#> ── Bass-Ackwards Analysis (ackwards) ───────────────────────────────────────────
#> Engine: pca
#> Rotation: varimax
#> Basis: pearson
#> n: 1,000
#> k (max): 5
#> 
#> ── Levels ──
#> 
#> ✔ k = 1: 1 factor, 28.2% variance
#> ✔ k = 2: 2 factors, 46.5% variance
#> ✔ k = 3: 3 factors, 57.5% variance
#> ✔ k = 4: 4 factors, 67.7% variance
#> ✔ k = 5: 5 factors, 70.8% variance
#> 
#> ── Edges ──
#> 
#> 13 of 40 edges have |r| ≥ 0.3
#> 
#> ── Pruning ──
#> 
#> Redundancy (direct, |r| ≥ 0.9): 7 nodes flagged
#> Manual: 1 node explicitly flagged (m4f2)
#> ────────────────────────────────────────────────────────────────────────────────
#> Note: Pruning is interpretive relabeling, not re-estimation. Flagged nodes
#> remain in the object; all edges are preserved. Inspect with `x$prune$nodes` and
#> `tidy(x, what = "nodes")`.
#> Note: This is a series of linked solutions, not a fitted hierarchical model.
#> Cross-level edges are descriptive score correlations. Per-level fit indices
#> (EFA/ESEM) describe how well a k-factor model fits the items at that level --
#> they do not validate the edges or the hierarchy itself.