Flag redundant or artifactual factors (Forbes 2023 extension)

prune() never removes anything from an ackwards object. It only annotates factors with pruned and prune_reason flags in x$prune$nodes. A factor is a summary variable standing in for a group of items that move together. A redundant factor is one that persists across levels without changing. Pruning is flag-only and never removes, so 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

The function 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", which runs no auto rule. Combine it 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 factor-score correlations |r| >= redundancy_r (and optionally phi > redundancy_phi), using redundancy_criterion (default "direct", which follows Forbes's reference code). A factor score is each person's estimated standing on a factor. Applies Forbes's (2023) retention rule: keep the bottom node when the chain reaches level k_max (most specific), and 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 it in x$prune$phi. Congruence is a 0 to 1 index of how similar two loading patterns are. A loading is the correlation between an item and a factor under varimax, and the pattern coefficient under an oblique rotation. This mode also stores structural signals (few_items, orphan, and 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, because 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, and 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$engine == "pca": no phi filter. PCA is principal component analysis, and a component is a weighted sum of the items. Component scores are determinate, that is, exact linear functions of the data, with no factor-score indeterminacy. So the score correlation |r| is the correlation between the components themselves under the correlation matrix the fit uses, and phi adds nothing that |r| does not already capture.
x$engine is "efa" or "esem": automatically set to 0.95 (Lorenzo-Seva & ten Berge, 2006). EFA is exploratory factor analysis and ESEM is exploratory structural equation modeling. Factor-score indeterminacy off-PCA means |r| alone is liberal, so 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. At each level it takes the factor with the largest direct |r|. It continues while |r| >= redundancy_r, and while phi is above redundancy_phi when redundancy_phi is set (by you or automatically). The chase stops at the first level that fails. A level further up that passes again does not restart it. This contiguous stop follows Forbes's (2023) reference code, ChaseCorrPaths, and the chase reproduces her AMH applied example exactly. The direct correlation is the honest way to operationalize "the same construct", because two factor scores share >= redundancy_r^2 variance directly. To bridge such a gap, judge the case yourself, including which factor of the bridged chain to drop, and pass that factor to manual. A factor that no rule flagged then gets prune_reason = "manual", and a factor already flagged as redundant keeps that reason. manual adds no row to x$prune$chains.
"adjacent": trace adjacent primary-parent links only, so each consecutive level must meet |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 or 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, because 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. Such a factor 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, because 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). That is, it sits 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

Oblique objects. When ackwards() was fit with an oblique rotation, each edge r is a total correlation, which includes overlap through correlated factors at the same level. Every rule still runs unchanged: the edge criteria on r, and the congruence (phi) checks and item-assignment signals on the loadings, which are then pattern coefficients. A warning names the rules that run and says that they assume orthogonal levels. Their default thresholds, such as redundancy_r = 0.9, were calibrated on varimax edges.

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). It also does not screen every ancestor pair against every other, so there is no all-pairs test. An ancestor can join on a strong direct link to the leaf even where its adjacent hop to the next chain member is weak. That 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, because 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, meaning pairs at or above the thresholds. Forbes (2023) uses the artifact flags mainly for the messier band just below them. Say a pair correlates at |r| = 0.89 and shares a loading pattern at phi = 0.93. It is not quite redundant, but it may be the same factor re-rotated (re-oriented without changing fit), so it is worth a second look. Artifact mode surfaces this as x$prune$near_redundant. That data frame holds every cross-level pair that is not itself fully redundant. Such a pair has its direct (skip-level) |r| or its Tucker phi within near_margin below the corresponding threshold (redundancy_r or redundancy_phi). Its columns are from, to, level_from, level_to, r (direct score correlation), phi (loading congruence), and the logical band flags near_r and near_phi. Under the PCA engine redundancy_phi is NULL (component scores are determinate), so only the |r| band applies. Under EFA and 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.