Interpreting and Labeling Factors

Fitting an ackwards model gives you a hierarchy of factors with stable but opaque IDs — m1f1, m2f1, m2f2, and so on. Turning those IDs into something you can reason about is interpretive work: you read each factor’s loadings, decide what construct it represents, and give it a name. This is the part of the analysis that requires judgment, and it is harder in a bass-ackwards hierarchy than in a single flat factor solution, because the same construct can appear, split, and merge across levels.

Reading a factor with `top_items()`

A factor is defined by the items that load strongly on it. top_items() lists, for each factor, the items whose absolute loading meets a threshold, sorted from strongest to weakest. This is far easier to read than a full item-by-factor matrix, especially at deeper levels.

top_items(x, level = 5, cut = 0.5)
#> 
#> ── Salient items by factor (ackwards) ──────────────────────────────────────────
#> Engine: pca
#> Cut: |loading| >= 0.5
#> Top-n: all
#> 
#> ── Level 5 (5 factors) ──
#> 
#> m5f1
#> E2: Find it difficult to approach others [-0.752]
#> E4: Make friends easily [0.747]
#> E1: Don't talk a lot [-0.701]
#> E3: Know how to captivate people [0.677]
#> E5: Take charge [0.597]
#> 
#> m5f2
#> N3: Have frequent mood swings [-0.825]
#> N1: Get angry easily [-0.810]
#> N2: Get irritated easily [-0.805]
#> N5: Panic easily [-0.688]
#> N4: Often feel blue [-0.646]
#> 
#> m5f3
#> C2: Continue until everything is perfect [0.735]
#> C4: Do things in a half-way manner [-0.716]
#> C1: Am exacting in my work [0.690]
#> C3: Do things according to a plan [0.679]
#> C5: Waste my time [-0.652]
#> 
#> m5f4
#> A1: Am indifferent to the feelings of others [-0.704]
#> A3: Know how to comfort others [0.703]
#> A2: Inquire about others' well-being [0.692]
#> A5: Make people feel at ease [0.580]
#> A4: Love children [0.522]
#> 
#> m5f5
#> O5: Will not probe deeply into a subject [-0.705]
#> O3: Carry the conversation to a higher level [0.655]
#> O1: Am full of ideas [0.604]
#> O2: Avoid difficult reading material [-0.595]
#> O4: Spend time reflecting on things [0.551]
#> ────────────────────────────────────────────────────────────────────────────────
#> Loadings reflect primary-parent sign alignment. Use tidy(x, what = "loadings")
#> for the full matrix.

The cut threshold controls how inclusive the listing is. Here we raise it to 0.5 to keep each factor to its defining items; the default is 0.3. Raise it further to isolate only the strongest markers, or lower it to surface weaker cross-loadings (see below). When a factor has many salient items, n caps the list at the strongest few, and sort = FALSE keeps items in their original order instead of sorting by |loading| — useful when your items follow a meaningful sequence.

top_items(x, level = 5, cut = 0.3, n = 4)
#> 
#> ── Salient items by factor (ackwards) ──────────────────────────────────────────
#> Engine: pca
#> Cut: |loading| >= 0.3
#> Top-n: 4
#> 
#> ── Level 5 (5 factors) ──
#> 
#> m5f1
#> E2: Find it difficult to approach others [-0.752]
#> E4: Make friends easily [0.747]
#> E1: Don't talk a lot [-0.701]
#> E3: Know how to captivate people [0.677]
#> 
#> m5f2
#> N3: Have frequent mood swings [-0.825]
#> N1: Get angry easily [-0.810]
#> N2: Get irritated easily [-0.805]
#> N5: Panic easily [-0.688]
#> 
#> m5f3
#> C2: Continue until everything is perfect [0.735]
#> C4: Do things in a half-way manner [-0.716]
#> C1: Am exacting in my work [0.690]
#> C3: Do things according to a plan [0.679]
#> 
#> m5f4
#> A1: Am indifferent to the feelings of others [-0.704]
#> A3: Know how to comfort others [0.703]
#> A2: Inquire about others' well-being [0.692]
#> A5: Make people feel at ease [0.580]
#> 
#> m5f5
#> O5: Will not probe deeply into a subject [-0.705]
#> O3: Carry the conversation to a higher level [0.655]
#> O1: Am full of ideas [0.604]
#> O2: Avoid difficult reading material [-0.595]
#> ────────────────────────────────────────────────────────────────────────────────
#> Loadings reflect primary-parent sign alignment. Use tidy(x, what = "loadings")
#> for the full matrix.

Cross-loadings are signal

Items that load on more than one factor are not noise to be suppressed — they often tell you how two factors relate. The by = "item" mode inverts the grouping: instead of listing the items under each factor, it lists, for each item, the factors it loads on. Lowering the cut then makes cross-loadings easy to read item by item:

top_items(x, level = 3, cut = 0.25, by = "item")
#> 
#> ── Salient factors by item (ackwards) ──────────────────────────────────────────
#> Engine: pca
#> Cut: |loading| >= 0.25
#> Top-n: all
#> 
#> ── Level 3 (3 factors) ──
#> 
#> A1: Am indifferent to the feelings of others
#> m3f1 [-0.354]
#> 
#> A2: Inquire about others' well-being
#> m3f1 [0.650]
#> 
#> A3: Know how to comfort others
#> m3f1 [0.750]
#> 
#> A4: Love children
#> m3f1 [0.490]
#> 
#> A5: Make people feel at ease
#> m3f1 [0.735]
#> 
#> C1: Am exacting in my work
#> m3f3 [0.658]
#> 
#> C2: Continue until everything is perfect
#> m3f3 [0.642]
#> 
#> C3: Do things according to a plan
#> m3f3 [0.476]
#> 
#> C4: Do things in a half-way manner
#> m3f3 [-0.609]
#> m3f2 [-0.348]
#> 
#> C5: Waste my time
#> m3f3 [-0.453]
#> m3f2 [-0.428]
#> m3f1 [-0.252]
#> 
#> E1: Don't talk a lot
#> m3f1 [-0.611]
#> 
#> E2: Find it difficult to approach others
#> m3f1 [-0.698]
#> 
#> E3: Know how to captivate people
#> m3f1 [0.664]
#> 
#> E4: Make friends easily
#> m3f1 [0.784]
#> 
#> E5: Take charge
#> m3f1 [0.507]
#> m3f3 [0.392]
#> 
#> N1: Get angry easily
#> m3f2 [-0.777]
#> 
#> N2: Get irritated easily
#> m3f2 [-0.791]
#> 
#> N3: Have frequent mood swings
#> m3f2 [-0.812]
#> 
#> N4: Often feel blue
#> m3f2 [-0.680]
#> 
#> N5: Panic easily
#> m3f2 [-0.650]
#> 
#> O1: Am full of ideas
#> m3f3 [0.493]
#> m3f1 [0.253]
#> 
#> O2: Avoid difficult reading material
#> m3f3 [-0.520]
#> 
#> O3: Carry the conversation to a higher level
#> m3f3 [0.504]
#> m3f1 [0.355]
#> 
#> O4: Spend time reflecting on things
#> m3f2 [-0.414]
#> m3f3 [0.319]
#> 
#> O5: Will not probe deeply into a subject
#> m3f3 [-0.553]
#> ────────────────────────────────────────────────────────────────────────────────
#> Loadings reflect primary-parent sign alignment. Use tidy(x, what = "loadings")
#> for the full matrix.

An item that appears under two factors at the same level marks a point where the constructs overlap. Whether that overlap is substantively meaningful or a sign of overextraction is a judgment call — see vignette("ackwards-suggest-k") for the overextraction discussion.

Showing item wording instead of codes

Item codes like A1 or N3 are compact but opaque. The wording you have been seeing above — E4: Make friends easily — comes for free: bfi25 ships each item’s IPIP stem as a variable label (a "label" column attribute), ackwards() captures those labels at fit time, and top_items() prints them as code: label.

Your own data often carries the same attribute (packages like labelled and haven set it, and survey exports frequently include it). If it does not, attach the wording yourself — with labelled::var_label() or base attr() — on the data you actually fit (base row-subsetting such as na.omit() drops plain attributes, which is why we fit the raw bfi25 above):

# tidyverse idiom:
labelled::var_label(my_data$item1) <- "Full wording of item 1"

# or base R, no extra package:
attr(my_data$item1, "label") <- "Full wording of item 1"

Labelled items show their wording; unlabelled ones fall back to the bare code, so a partially labelled data set still prints cleanly. Pass show_labels = FALSE to force the bare codes even when labels are present.

These variable labels (item wording) are distinct from the factor labels (the names you give m1f1, m2f1, … below): one describes your measured items, the other names the latent factors the hierarchy discovers. Keep the two ideas — and the word “label” — separate.

The sign convention: negative does not mean “low”

Before naming anything, understand how ackwards orients factors. Loadings are sign-aligned to the primary parent (see ?ackwards): the level-1 factor is anchored so its loadings sum positive, and every deeper factor is flipped so its correlation with its primary parent is positive. This makes the diagram readable, but it has a consequence for interpretation.

A factor’s sign is arbitrary in the sense that flipping every loading and the factor’s orientation describes the same dimension. So a column of negative loadings does not mean “low” on that construct — it means the construct’s positive pole was oriented the other way by the alignment step. Read the pattern of items, not the bare sign:

top_items(x, level = 2)
#> 
#> ── Salient items by factor (ackwards) ──────────────────────────────────────────
#> Engine: pca
#> Cut: |loading| >= 0.3
#> Top-n: all
#> 
#> ── Level 2 (2 factors) ──
#> 
#> m2f1
#> A3: Know how to comfort others [0.730]
#> A5: Make people feel at ease [0.694]
#> E3: Know how to captivate people [0.684]
#> A2: Inquire about others' well-being [0.661]
#> E4: Make friends easily [0.638]
#> E5: Take charge [0.630]
#> E2: Find it difficult to approach others [-0.614]
#> O3: Carry the conversation to a higher level [0.565]
#> E1: Don't talk a lot [-0.525]
#> A4: Love children [0.471]
#> C2: Continue until everything is perfect [0.468]
#> O1: Am full of ideas [0.467]
#> C1: Am exacting in my work [0.421]
#> C5: Waste my time [-0.411]
#> C4: Do things in a half-way manner [-0.389]
#> C3: Do things according to a plan [0.365]
#> A1: Am indifferent to the feelings of others [-0.335]
#> 
#> m2f2
#> N3: Have frequent mood swings [-0.806]
#> N1: Get angry easily [-0.790]
#> N2: Get irritated easily [-0.789]
#> N4: Often feel blue [-0.677]
#> N5: Panic easily [-0.659]
#> C5: Waste my time [-0.491]
#> C4: Do things in a half-way manner [-0.438]
#> O4: Spend time reflecting on things [-0.358]
#> ────────────────────────────────────────────────────────────────────────────────
#> Loadings reflect primary-parent sign alignment. Use tidy(x, what = "loadings")
#> for the full matrix.

If a Neuroticism factor shows up with negative loadings on the anxiety items, that is an orientation artifact of the alignment, not a “low anxiety” factor. Name it for the construct (Neuroticism), and if the sign matters for your downstream use, flip the scores yourself. The values shown by top_items() and tidy(what = "loadings") are the aligned values, so they are consistent with the diagram and the edge table.

Naming in a hierarchy

In a flat factor solution you name each factor once. In a bass-ackwards hierarchy you are naming factors at every level, and the levels are related — so the names should be related too. The lineage tells you how.

summary(x)
#> 
#> ── Summary: Bass-Ackwards Analysis (ackwards) ──────────────────────────────────
#> Engine: pca
#> Rotation: varimax
#> Basis: polychoric
#> n: 875
#> k (max): 5
#> 
#> ── Levels ──
#> 
#> k = 1: 1 factor (23.2% cumulative variance)
#> m1f1 23.2% eigenvalue 5.80
#> 
#> k = 2: 2 factors (35.5% cumulative variance)
#> m2f1 20.9% eigenvalue 5.80
#> m2f2 14.5% eigenvalue 3.07
#> 
#> k = 3: 3 factors (44.6% cumulative variance)
#> m3f1 18.0% eigenvalue 5.80
#> m3f2 13.9% eigenvalue 3.07
#> m3f3 12.7% eigenvalue 2.28
#> 
#> k = 4: 4 factors (52.2% cumulative variance)
#> m4f1 17.5% eigenvalue 5.80
#> m4f2 13.6% eigenvalue 3.07
#> m4f3 11.8% eigenvalue 2.28
#> m4f4 9.2% eigenvalue 1.90
#> 
#> k = 5: 5 factors (58.4% cumulative variance)
#> m5f1 13.8% eigenvalue 5.80
#> m5f2 13.6% eigenvalue 3.07
#> m5f3 11.9% eigenvalue 2.28
#> m5f4 10.1% eigenvalue 1.90
#> m5f5 9.1% eigenvalue 1.56
#> 
#> ── Lineage (primary parents) ──
#> 
#> m1f1 → m2f1, m2f2
#> m2f1 → m3f1, m3f3
#> m2f2 → m3f2
#> m3f1 → m4f1
#> m3f2 → m4f2
#> m3f3 → m4f3, m4f4
#> m4f1 → m5f1, m5f4
#> m4f2 → m5f2
#> m4f3 → m5f3
#> m4f4 → m5f5
#> ────────────────────────────────────────────────────────────────────────────────
#> 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.

The lineage list (m1f1 → m2f1, m2f2 → ...) shows each factor’s primary children. Use it to name top-down:

Upper-level factors are broader. A level-2 factor that is the primary parent of two level-3 factors is the construct they share. Name it for the blend, not for one child. In the bfi25 data above, the level-1 factor is a single broad dimension; at level 2 it separates into a Neuroticism factor (m2f2, defined by the N items) and a broad factor blending the remaining four trait families (m2f1). The Big Five themselves do not appear cleanly until level 5 — so a substantive name for m2f1 (“broad well-adjustment”, say) is necessarily coarser than the names you give its descendants.
A split is a refinement, not a contradiction. When a parent factor splits into two children, the children carve up the parent’s content. Their names should read as specializations of the parent’s name, so that reading down a branch tells a coherent story.
Watch for factors that reorganize. A child whose strongest parent is at the other side of the level above (a crossing edge), or a factor whose primary edge is weak (|r| well below the near-1.0 values of stable dimensions), is a place where the structure is genuinely rearranging. The edge table makes these visible:

# Primary-parent edges, weakest last: the bottom rows are where structure shifts
tidy(x, what = "edges", primary_only = TRUE, sort = "strength") |> tail()
#>    from   to level_from level_to         r is_primary above_cut
#> 9  m4f1 m5f1          4        5 0.8377659       TRUE      TRUE
#> 10 m3f3 m4f3          3        4 0.7316162       TRUE      TRUE
#> 11 m3f3 m4f4          3        4 0.6802343       TRUE      TRUE
#> 12 m4f1 m5f4          4        5 0.5458269       TRUE      TRUE
#> 13 m2f1 m3f3          2        3 0.4814452       TRUE      TRUE
#> 14 m1f1 m2f2          1        2 0.4558587       TRUE      TRUE

Edges with |r| near 1.0 are factors that pass through nearly unchanged — name the child the same as the parent. The smaller |r| values at the bottom flag where a new, distinct construct is emerging and deserves its own name.

Borrowing names for the upper levels

The hardest factors to name are usually the broad ones near the top, precisely because they blend several familiar constructs. Rather than inventing an ad-hoc label, it often helps to borrow a name from a theory that has already charted the level above your usual constructs. Two are especially handy for personality and psychopathology data:

Big Five metatraits. Above the five factors sit two higher-order dimensions — Stability (the shared variance of Agreeableness, Conscientiousness, and low Neuroticism) and Plasticity (Extraversion and Openness) — from DeYoung’s work on the metatraits. In a bfi25 hierarchy these are frequently what a two- or three-factor level is, so “Stability” / “Plasticity” are ready-made names for factors that would otherwise be an awkward “broad well-adjustment”.
HiTOP spectra. In clinical data, the Hierarchical Taxonomy of Psychopathology names the broad bands directly — internalizing, externalizing (disinhibited and antagonistic), thought disorder, detachment, and somatoform — with a general p-factor at the apex. A level whose items span several disorders is usually one of these spectra, and naming it as such keeps your hierarchy legible to readers who already think in HiTOP terms.

The point is not to force your data onto either scheme, but to recognise that an upper-level factor is often a known superordinate construct — and that reusing its established name communicates far more than a bespoke one.

Applying your names

Once you have decided on names, attach them to the diagram. autoplot() takes a node_labels argument: a named character vector mapping factor IDs to display strings.

Typing that vector out by hand is tedious and error-prone, so label_template() generates it for you, in the same order the diagram uses, and prints an editable c(...) literal you can paste straight into your script:

label_template(x)
#> `label_template()` scaffold (id style):
#> c(
#>   "m1f1" = "m1f1",
#>   "m2f1" = "m2f1",
#>   "m2f2" = "m2f2",
#>   "m3f1" = "m3f1",
#>   "m3f2" = "m3f2",
#>   "m3f3" = "m3f3",
#>   "m4f1" = "m4f1",
#>   "m4f2" = "m4f2",
#>   "m4f3" = "m4f3",
#>   "m4f4" = "m4f4",
#>   "m5f1" = "m5f1",
#>   "m5f2" = "m5f2",
#>   "m5f3" = "m5f3",
#>   "m5f4" = "m5f4",
#>   "m5f5" = "m5f5"
#> )

Copy that literal, fill in your names, and pass it to autoplot(). Unspecified IDs keep their default m{k}f{j} label, so you can label just the level you care about:

autoplot(x, node_labels = c(
  m5f1 = "Neuroticism",
  m5f2 = "Extraversion",
  m5f3 = "Conscientiousness",
  m5f4 = "Agreeableness",
  m5f5 = "Openness"
))

Making the names stick with `set_factor_labels()`

node_labels styles a single plot. When you want the same names to follow the object everywhere — in print(), summary(), tidy(), and top_items(), not just one diagram — attach them once with set_factor_labels(). It takes the same named vector label_template() scaffolds, and returns the object so it pipes:

x <- set_factor_labels(x, c(
  m5f1 = "Neuroticism",
  m5f2 = "Extraversion",
  m5f3 = "Conscientiousness",
  m5f4 = "Agreeableness",
  m5f5 = "Openness"
))

Now the labels appear wherever factors are listed. summary() shows them as label (id), keeping the stable ID visible so you can still cross-reference the edge and loading tables:

summary(x)
#> 
#> ── Summary: Bass-Ackwards Analysis (ackwards) ──────────────────────────────────
#> Engine: pca
#> Rotation: varimax
#> Basis: polychoric
#> n: 875
#> k (max): 5
#> 
#> ── Levels ──
#> 
#> k = 1: 1 factor (23.2% cumulative variance)
#> m1f1 23.2% eigenvalue 5.80
#> 
#> k = 2: 2 factors (35.5% cumulative variance)
#> m2f1 20.9% eigenvalue 5.80
#> m2f2 14.5% eigenvalue 3.07
#> 
#> k = 3: 3 factors (44.6% cumulative variance)
#> m3f1 18.0% eigenvalue 5.80
#> m3f2 13.9% eigenvalue 3.07
#> m3f3 12.7% eigenvalue 2.28
#> 
#> k = 4: 4 factors (52.2% cumulative variance)
#> m4f1 17.5% eigenvalue 5.80
#> m4f2 13.6% eigenvalue 3.07
#> m4f3 11.8% eigenvalue 2.28
#> m4f4 9.2% eigenvalue 1.90
#> 
#> k = 5: 5 factors (58.4% cumulative variance)
#> Neuroticism (m5f1) 13.8% eigenvalue 5.80
#> Extraversion (m5f2) 13.6% eigenvalue 3.07
#> Conscientiousness (m5f3) 11.9% eigenvalue 2.28
#> Agreeableness (m5f4) 10.1% eigenvalue 1.90
#> Openness (m5f5) 9.1% eigenvalue 1.56
#> 
#> ── Lineage (primary parents) ──
#> 
#> m1f1 → m2f1, m2f2
#> m2f1 → m3f1, m3f3
#> m2f2 → m3f2
#> m3f1 → m4f1
#> m3f2 → m4f2
#> m3f3 → m4f3, m4f4
#> m4f1 → Neuroticism (m5f1), Agreeableness (m5f4)
#> m4f2 → Extraversion (m5f2)
#> m4f3 → Conscientiousness (m5f3)
#> m4f4 → Openness (m5f5)
#> ────────────────────────────────────────────────────────────────────────────────
#> 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.

top_items(by = "factor") uses them on its group headers, and tidy() adds a factor_label column (and from_label/to_label for edges) — but only when labels are set, so unlabelled objects keep their exact previous output:

head(tidy(x, what = "loadings"))
#>   level factor item    loading se ci_lower ci_upper factor_label
#> 1     1   m1f1   A1 -0.3440908 NA       NA       NA         <NA>
#> 2     1   m1f1   A2  0.5977210 NA       NA       NA         <NA>
#> 3     1   m1f1   A3  0.6511387 NA       NA       NA         <NA>
#> 4     1   m1f1   A4  0.4837850 NA       NA       NA         <NA>
#> 5     1   m1f1   A5  0.6848262 NA       NA       NA         <NA>
#> 6     1   m1f1   C1  0.4502778 NA       NA       NA         <NA>

autoplot() uses stored labels as the node text automatically, so you no longer need to pass node_labels for a labelled object; a call-time node_labels entry still overrides a stored label for that one node. Labels are display only — factor IDs never change — and they ride along through prune(), boot_edges(), augment(), and predict(). Read them back with factor_labels(x); clear one by setting it to NA, or all of them by passing NULL.

The Forbes letter convention

Forbes (2023) labels nodes by level-letter and within-level index — A1 for the single level-1 factor, B1/B2 at level 2, and so on. label_template() produces this convention directly with style = "forbes":

autoplot(x, node_labels = label_template(x, style = "forbes"))

This is useful when you want to refer to nodes by position rather than by substantive name — for example, in a methods section that walks through the hierarchy before interpreting it.

Starting from a blank slate

If you would rather supply every label yourself with no defaults showing through, style = "blank" gives you an all-empty scaffold to fill in:

labs <- label_template(x, style = "blank")
labs["m5f1"] <- "N"
labs["m5f2"] <- "E"
labs["m5f3"] <- "C"
labs["m5f4"] <- "A"
labs["m5f5"] <- "O"
autoplot(x, node_labels = labs)

This article is about what to put on the diagram.

Note: For how the diagram looks — colours, edge thresholds, monochrome and publication styling, level labels, and the Forbes pruned-diagram mode — see vignette("ackwards-visualization").