Choosing an Engine: PCA, EFA, and ESEM

ackwards supports three factor extraction engines. A factor is an unobserved dimension that explains why a set of items correlate. The engines share the same downstream machinery: the same rotation, the same tenBerge scoring weights, and the same between-level correlation algebra. A rotation re-orients the factors within a level without changing how well they fit. The engines differ in their statistical model and in what they report. This vignette explains when each one is appropriate and what the differences look like in practice.

The three engines at a glance

A few terms recur in the table below. PCA (principal component analysis) summarizes the items with weighted sums called components. EFA (exploratory factor analysis) models the items as a small number of shared factors plus item-specific noise. ESEM (exploratory structural equation modeling) fits that same factor model in lavaan and adds standard errors. A polychoric correlation estimates the correlation between two ordinal items, items recorded on an ordered scale with few categories, as if each were a continuous variable cut into categories. FIML (full-information maximum likelihood) uses every partially observed row of the data instead of dropping it. A loading is the correlation between an item and a factor.

	`"pca"`	`"efa"`	`"esem"`
What it models	Total item variance	Common (latent) variance	Common (latent) variance
Engine substrate	`psych::principal()`	`psych::fa()`	`lavaan`
Communalities	All 1.0 (by definition)	Estimated from data	Estimated from data
Correlations	Pearson or polychoric	Pearson or polychoric	Pearson or polychoric
Estimators	Eigen-decomposition	`minres` (OLS), `ml`, or `pa`, via `psych`’s `fm=`	ML or MLR for continuous data (plus FIML for missing data). WLSMV or ULSMV for ordinal data
Fit indices	Eigenvalues only	χ², RMSEA, TLI, BIC	CFI, TLI, RMSEA, SRMR, χ²
Loading SEs	No	No	Yes
Speed	Fast	Moderate	Slowest
Best for	Exploration, large k	Latent-factor inference	Model evaluation, loading SEs, ordinal or continuous

All three produce the same labels (m{k}f{j}), the same tidy() / glance() / augment() interface, and comparable between-level edges for well-structured data. The hierarchy they reveal is usually the same. The statistical guarantees differ.

Setup

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

We use the BFI-25 with polychoric correlations throughout, so that differences in output reflect the engine and not the correlation basis.

PCA: components from total variance

PCA extracts principal components, linear combinations of the observed variables that capture maximum variance, including measurement error. Every item is modeled with communality 1.0: the components account for 100% of each item’s variance. This is not a true latent variable model. It is a data reduction method.

In the bass-ackwards context, PCA is the natural default. It is fast, always converges, and produces eigenvalues that can guide the choice of k. Waller (2007) showed that the between-level algebra (W'RW) holds exactly for components. The edges are therefore algebraically exact rather than approximated from materialized scores.

x_pca <- ackwards(bfi, k_max = 3, cor = "polychoric")
x_pca
#> 
#> ── Bass-Ackwards Analysis (ackwards) ───────────────────────────────────────────
#> Engine: pca
#> Rotation: varimax
#> Basis: polychoric
#> n: 875
#> k (max): 3
#> 
#> ── Levels ──
#> 
#> ✔ k = 1: 1 factor, 23.2% variance
#> ✔ k = 2: 2 factors, 35.5% variance
#> ✔ k = 3: 3 factors, 44.6% variance
#> 
#> ── Edges ──
#> 
#> 5 of 8 edges have |r| ≥ 0.3
#> ────────────────────────────────────────────────────────────────────────────────
#> 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 “fit” for PCA is just the eigenvalue of each component, the amount of variance it captures. There are no chi-square tests, no RMSEA, no model rejection.

tidy(x_pca, what = "fit")
#>   level       statistic    value
#> 1     1 eigenvalue.m1f1 5.802803
#> 2     2 eigenvalue.m2f1 5.802803
#> 3     2 eigenvalue.m2f2 3.067627
#> 4     3 eigenvalue.m3f1 5.802803
#> 5     3 eigenvalue.m3f2 3.067627
#> 6     3 eigenvalue.m3f3 2.275419

EFA: factors from common variance

EFA extracts latent factors that model only the variance shared among items. Each item retains a unique variance (communality < 1.0) that the factors do not explain. This is the classical common-factor model. It is more appropriate than PCA when you believe the items are fallible indicators of latent constructs rather than the constructs themselves.

x_efa <- ackwards(bfi, k_max = 3, engine = "efa", cor = "polychoric")
x_efa
#> 
#> ── Bass-Ackwards Analysis (ackwards) ───────────────────────────────────────────
#> Engine: efa
#> Rotation: varimax
#> Basis: polychoric
#> n: 875
#> k (max): 3
#> 
#> ── Levels ──
#> 
#> ✔ k = 1: 1 factor, 20.3% variance
#> ✔ k = 2: 2 factors, 30.8% variance
#> ✔ k = 3: 3 factors, 37.7% variance
#> 
#> ── Edges ──
#> 
#> 5 of 8 edges have |r| ≥ 0.3
#> ────────────────────────────────────────────────────────────────────────────────
#> 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.

EFA produces genuine goodness-of-fit indices. These tell you whether the k factors are sufficient to reproduce the observed correlation matrix within sampling error.

tidy(x_efa, what = "fit")
#>    level statistic        value
#> 1      1       chi 5327.2295200
#> 2      1       dof  275.0000000
#> 3      1   p_value    0.0000000
#> 4      1     RMSEA    0.1448964
#> 5      1       TLI    0.3429279
#> 6      1       BIC 3464.3179512
#> 7      2       chi 3520.3788570
#> 8      2       dof  251.0000000
#> 9      2   p_value    0.0000000
#> 10     2     RMSEA    0.1220036
#> 11     2       TLI    0.5337688
#> 12     2       BIC 1820.0486615
#> 13     3       chi 2545.6948352
#> 14     3       dof  228.0000000
#> 15     3   p_value    0.0000000
#> 16     3     RMSEA    0.1077785
#> 17     3       TLI    0.6358514
#> 18     3       BIC 1001.1717891

The RMSEA values here are large (> 0.10). They indicate that 1 to 3 factors do not fully account for the BFI item correlations, which is no surprise, because the true structure is 5 factors. Fit improves steadily from k = 1 to k = 3. That is exactly the kind of evidence bass-ackwards analysis is designed to make visible.

How close are EFA and PCA loadings?

For clean, continuous data with moderate-to-strong factor structure, EFA and PCA loadings are highly correlated but not identical. EFA loadings are systematically somewhat smaller because they model only the common variance. PCA inflates loadings by fitting noise alongside signal.

The table below compares primary loadings, the loading of each item on its dominant factor, for six representative items at k = 3. The items are two each from the Neuroticism, Extraversion, and Conscientiousness families. The Δ column is the teaching point: how much smaller EFA loadings are in absolute value once measurement error is partitioned into uniqueness. Using |EFA| − |PCA| keeps the attenuation consistently negative regardless of loading sign.

Item	Factor	Loading
PCA vs EFA: primary loadings for anchor items (k = 3)
Item	Factor	PCA	EFA	Δ (\|EFA\| − \|PCA\|)¹
E1	m3f1	−0.61	−0.55	−0.06
E2	m3f1	−0.70	−0.67	−0.03
N1	m3f2	−0.78	−0.75	−0.03
N2	m3f2	−0.79	−0.76	−0.03
C1	m3f3	0.66	0.62	−0.04
C2	m3f3	0.64	0.61	−0.03
¹ Factor assignment and sign verified to match between engines. Delta uses \|EFA\| - \|PCA\| so attenuation is always negative regardless of loading sign.

EFA loadings for the same items are consistently a few points lower. The PCA loadings include some noise variance that EFA partitions into uniqueness. The factor structure (which items define which factor) is unchanged.

ESEM: EFA with full model diagnostics

ESEM (exploratory structural equation modeling, Asparouhov & Muthén, 2009) fits the same common-factor model as EFA but uses lavaan as the engine. ESEM is not an ordinal-only tool. It handles continuous items with maximum-likelihood estimators (ML or MLR, the default for continuous data) and ordinal items with WLSMV. It unlocks three capabilities that EFA cannot provide:

Standard errors for every loading, which enable confidence intervals and significance tests, for continuous and ordinal data.
Full maximum-likelihood estimation for continuous data (ML/MLR), including FIML for missing data (missing = "fiml"), which uses all partially observed rows rather than deleting them.
The WLSMV estimator for ordinal data, the appropriate maximum-likelihood-adjacent estimator for categorical indicators. When cor = "polychoric" is set with engine = "esem", WLSMV is used automatically.

Those three are the only reasons to pay ESEM’s cost (a lavaan fit per level, occasional convergence trouble). EFA is otherwise a first-class reporting engine. It returns the same loadings, variance, and between-level edges plus per-level RMSEA and TLI, which is enough to report a hierarchy. Reach for ESEM when you need loading standard errors (especially at smaller n), the field-standard WLSMV estimator for ordinal indicators, or true FIML for missing data. Otherwise EFA is the simpler, faster choice.

x_esem <- ackwards(bfi, k_max = 3, engine = "esem", cor = "polychoric")
x_esem
#> 
#> ── Bass-Ackwards Analysis (ackwards) ───────────────────────────────────────────
#> Engine: esem
#> Rotation: varimax
#> Basis: polychoric
#> n: 875
#> k (max): 3
#> 
#> ── Levels ──
#> 
#> ✔ k = 1: 1 factor, 23.5% variance
#> ✔ k = 2: 2 factors, 32.9% variance
#> ✔ k = 3: 3 factors, 39.5% variance
#> 
#> ── Edges ──
#> 
#> 5 of 8 edges have |r| ≥ 0.3
#> ────────────────────────────────────────────────────────────────────────────────
#> 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.

ESEM fit indices include CFI and SRMR in addition to RMSEA and TLI, which gives a richer picture of model adequacy. See the “Per-level fit” section below for how to report and interpret these indices.

Loading standard errors and confidence intervals

The unique output from ESEM is the rotation-aware standard error of every loading. These SEs are returned as part of tidy(what = "loadings"), alongside ci_lower and ci_upper columns:

ld <- tidy(x_esem, what = "loadings")
head(ld)
#>   level factor item    loading         se   ci_lower   ci_upper
#> 1     1   m1f1   A1 -0.3155807 0.02831633 -0.3710797 -0.2600817
#> 2     1   m1f1   A2  0.5584169 0.02316777  0.5130089  0.6038249
#> 3     1   m1f1   A3  0.6424786 0.01971862  0.6038308  0.6811264
#> 4     1   m1f1   A4  0.4261675 0.02777029  0.3717387  0.4805962
#> 5     1   m1f1   A5  0.6588505 0.01870826  0.6221830  0.6955180
#> 6     1   m1f1   C1  0.4138252 0.02754885  0.3598304  0.4678200

The intervals are computed as loading ± z × SE (default 95%). Set conf_level = 0.99 for wider intervals. With the 875 complete cases used here the SEs are fairly small. With smaller samples they become important for judging which loadings are meaningfully non-zero. For PCA and EFA objects the se, ci_lower, and ci_upper columns are present but NA, because those engines carry no loading SEs.

Per-level fit: what it tells you (and what it does not)

The key distinction

Bass-ackwards produces a series of independent factor solutions, not a fitted hierarchical model. The between-level edges are descriptive correlations between factor scores, each person’s estimated position on a factor. They have no sampling distribution of their own. Per-level fit indices therefore describe something narrower: does a k-factor model adequately reproduce the items at this level?

That is a real, bounded question. A level that fits terribly is one you must not over-interpret. Its k factors are not cleanly separating the items. A level that fits well tells you the factor structure at that depth is stable. But good fit at k = 3 does not validate the edges connecting k = 3 to k = 2. It only says the k = 3 solution itself is trustworthy. Keep that boundary in mind whenever you report or interpret fit.

The converse question comes up often: if the k = 3 solution fits badly, should I distrust the edges connecting k = 2 to k = 3? The honest answer has two parts. The edge correlation is still a faithful description of the relationship between the k = 2 and k = 3 factor scores as extracted. The arithmetic is not wrong. What poor fit undermines is the interpretation of the k = 3 factors themselves. If three factors do not cleanly reproduce the items, then “factor m3f2” is a shakier construct, and any edge incident to it inherits that shakiness. So a badly-fitting level does weaken the edges touching it. The correlation is not miscomputed. Rather, one of the things it connects is poorly defined. The edges between two well-fitting levels are on firmer ground than edges touching a poorly-fitting one.

Reporting fit with `tidy()` and `autoplot()`

tidy(what = "fit") returns the raw long table. For reporting, format = "wide" gives one row per level:

tidy(x_esem, what = "fit", format = "wide")
#>   level      chi dof p_value       CFI       TLI     RMSEA       SRMR BIC
#> 1     2 3616.834 251       0 0.7117569 0.6554864 0.1238665 0.09547997  NA
#> 2     3 2448.356 228       0 0.8098533 0.7498069 0.1055573 0.07172502  NA

tidy() does not flag rows against a threshold. The Hu & Bentler (1999) conventional cutoffs (CFI/TLI ≥ .95, RMSEA ≤ .06, SRMR ≤ .08) are conventional and contested, so a pass/fail column would overstate their authority. Instead they appear only as visual or inline reference points in autoplot() and summary():

Thresholds are conventional and contested. They were derived from specific simulation conditions (continuous, well-distributed items and balanced designs). WLSMV fit for ordinal data tends to produce lower CFI and higher RMSEA than ML on the same underlying structure. Do not interpret WLSMV cutoffs as strictly as ML-based rules. Use them as a rough orientation, not a gatekeeping criterion.

autoplot(x, what = "fit") visualises the trajectory across levels, with cutoff reference lines:

autoplot(x_esem, what = "fit")

plot of chunk fit-plot

The shape of the trajectory matters as much as the absolute values. A sharp improvement from k = 2 to k = 3 suggests the third factor is capturing genuine signal. Flat or worsening indices suggest adding another level is splitting noise.

The examples in this vignette stop at k_max = 3 on purpose, so the ESEM chunks build quickly and the tables stay legible. The BFI hierarchy does not end there. For these data suggest_k() points to roughly k = 5 (see vignette("ackwards-suggest-k")), so a real analysis would extend the trajectory further and read the fit curve across all five levels. The truncated plot here shows the mechanics of reading a fit trajectory, not the recommended depth for the BFI.

glance() also carries the deepest-level fit for quick inspection:

glance(x_esem)
#>   engine rotation        cor k_max n_obs deepest_converged n_edges       CFI
#> 1   esem  varimax polychoric     3   875                 3       8 0.8098533
#>         TLI     RMSEA       SRMR BIC
#> 1 0.7498069 0.1055573 0.07172502  NA

Should you care about fit in a bass-ackwards workflow?

It depends on your goal:

Exploratory (finding the hierarchical structure): fit is a secondary check. Start with PCA for speed, confirm with EFA or ESEM. If a level’s fit is poor, consider whether you have too many factors at that depth, or whether the items at that level are genuinely multidimensional.
Confirmatory / publication: fit is table-stakes for ESEM or EFA. Report per-level CFI, TLI, RMSEA (and SRMR for ESEM) alongside the hierarchy. The wide table and autoplot(what = "fit") are designed for this.
Ordinal data: WLSMV (ESEM) gives fit indices appropriate for categorical items. EFA’s RMSEA/TLI under Pearson correlation is a rougher diagnostic.

The bottom line: per-level fit qualifies each level of the hierarchy. It does not bless the hierarchy as a whole. Use it to decide how deep the structure is credibly resolved, not to claim the overall model is “good”.

How much do the edges differ?

The primary output of bass-ackwards analysis is the between-level edges. For well-structured, continuous data, all three engines should agree closely on the hierarchy.

The table below compares the primary-parent edge strength for every adjacent level transition. The Δ column is the shift in connection strength (|EFA| − |PCA|). It is a direct, sign-safe measure of how much the latent-variable model changes your inference about the hierarchy.

From	To	Edge strength (r)
Primary-parent edges: PCA vs EFA
From	To	PCA	EFA	Δ (EFA − PCA)¹
m1f1	m2f1	0.89	0.91	0.02
m1f1	m2f2	0.46	0.42	−0.04
m2f1	m3f1	0.87	0.89	0.02
m2f2	m3f2	0.99	0.98	−0.01
m2f1	m3f3	0.48	0.44	−0.04
¹ NA in either column means the engines disagree on the primary parent for that factor.

The r values are very close between engines. The hierarchy that PCA reveals is in effect the same hierarchy that EFA reveals. This convergence across methods is reassuring. It suggests the structure is real and not an artifact of the extraction method.

How clean this convergence looks depends on how strong the underlying structure is. The simulated sim16 dataset (?sim16) is an idealized case. Its planted 1 → 2 → 4 hierarchy is strong enough that engines and suggest_k() criteria agree almost perfectly. Real data is messier. For bfi25 the suggest_k() criteria span k = 4 to 6 even though the engines agree on the edges. Treat clean cross-method consensus as the best case, not the norm. The vignette("ackwards-suggest-k") develops this idealized-versus-realistic contrast in full.

When the engines disagree on edges, that is itself informative. It usually indicates factors whose definition depends on whether you account for measurement error (EFA/ESEM) or not (PCA).

Choosing an engine

Situation	Recommendation
Exploratory, large k, unknown structure	Start with `"pca"`
Report a hierarchy with per-level fit (RMSEA/TLI)	`"efa"`, a complete reporting engine
You specifically need loading SEs / CIs (especially smaller n)	`"esem"`
Ordinal items and you want the field-standard WLSMV estimator	`"esem"` with `cor = "polychoric"`
Missing data you want handled by true FIML	`"esem"` (ML/MLR) or PCA/EFA with `missing = "fiml"`
Replicating Goldberg (2006) or `psych::bassAckward()`	`"pca"` (the default engine. `fm` applies only to `"efa"`)

A practical workflow: start with PCA to get a feel for the hierarchy and choose k. EFA is enough to confirm and report. It returns loadings, variance, edges, and per-level RMSEA/TLI. Reach for ESEM only when you need one of the three things it adds (loading SEs, WLSMV, or FIML). It is not a required “publication” upgrade. If PCA and EFA/ESEM edges agree, you have strong evidence for the hierarchy. If they disagree, investigate why.

Missing data

ackwards() accepts a missing argument that controls how incomplete rows are handled before the correlation matrix, engine fit, and edges are computed. The full per-engine semantics, including the minor ESEM ML/MLR pairwise fit-versus-edges inconsistency and the $meta fields that record it, are documented in ?ackwards. In brief:

"pairwise" (default): all available observations, pairwise. MCAR-valid, full N. Warns when NAs are present.
"listwise": complete cases only, applied before all steps, so the correlation matrix, fit, and edges are fully consistent. Valid for all engines.
"fiml": full-information ML, on two routes. For engine = "esem" (with estimator = "ML" or "MLR"), lavaan estimates under FIML and the edges derive from its FIML saturated model. For engine = "pca" or "efa" on the Pearson basis, the correlation matrix is estimated by psych::corFiml() (full-information ML under multivariate normality, MAR-valid) and fed to the usual between-level algebra. The route announces itself with a message. missing = "fiml" errors for WLSMV/ULSMV (limited-information estimators have no FIML extension) and for a non-Pearson PCA/EFA basis (corFiml() estimates a multivariate-normal matrix). FIML improves estimation but does not impute items, so keep_scores = TRUE still yields NA for incomplete rows.

FIML for continuous PCA/EFA

For continuous data with MAR missingness, pass missing = "fiml" directly. It is more principled than pairwise deletion, which is only MCAR-valid:

# sim16 is continuous, so FIML's normality assumption is appropriate here.
set.seed(1)
sim_na <- sim16
for (j in seq_len(ncol(sim_na))) sim_na[sample(nrow(sim_na), 60L), j] <- NA

x_fiml <- ackwards(sim_na, k_max = 4, engine = "efa", missing = "fiml")
#> ℹ `missing = "fiml"`: correlation matrix estimated via `psych::corFiml()`
#>   (full-information ML).
#> ℹ Fit indices use N = 1000 (`n_obs = "total"`); point estimates (loadings,
#>   edges) are unaffected by this choice.
#> ! Fit indices are approximate: a FIML correlation matrix is fed into a
#>   normal-theory EFA (a two-step procedure). See `?ackwards` (`n_obs`).
x_fiml
#> 
#> ── Bass-Ackwards Analysis (ackwards) ───────────────────────────────────────────
#> Engine: efa
#> Rotation: varimax
#> Basis: pearson
#> n: 1,000
#> k (max): 4
#> 
#> ── Levels ──
#> 
#> ✔ k = 1: 1 factor, 23.3% variance
#> ✔ k = 2: 2 factors, 38.7% variance
#> ✔ k = 3: 3 factors, 48.2% variance
#> ✔ k = 4: 4 factors, 56.8% variance
#> 
#> ── Edges ──
#> 
#> 9 of 20 edges have |r| ≥ 0.3
#> ────────────────────────────────────────────────────────────────────────────────
#> 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.

Under the hood this estimates the correlation matrix with psych::corFiml() and runs the normal W'RW algebra on it. So the loadings and edges are exactly what the manual ackwards(psych::corFiml(sim_na), …) correlation-matrix call would give. That seam remains available for non-standard cases, e.g. a FIML matrix you have already computed elsewhere.

Two caveats matter. First, the fit-index N is your call. FIML draws information from every partially observed row, so there is no single “correct” N for the EFA fit indices, and the n_obs argument selects it on this route. "total" is the default: every row contributing to the FIML likelihood, the convention a FIML analysis reports. It is mildly anti-conservative, because χ² and RMSEA then treat partial rows as if complete. "complete" (the complete-case count) is conservative. The loading and edge point estimates are unaffected by this choice. Only the fit indices depend on it, and those are approximate under this two-step route (FIML matrix into normal-theory EFA) regardless of N. Second, the route assumes multivariate normality, so it is for continuous data only. For ordinal items use engine = "esem" with cor = "polychoric" instead.

Which option to use?

Situation	Recommendation
Continuous data, little missingness	`"pairwise"` (default)
Ordinal data + WLSMV, any missingness	`"pairwise"` (uses `available.cases`, MCAR-valid, full N)
Want consistent fit statistics and edges (continuous ML/MLR)	`"listwise"`
ESEM ML/MLR, meaningful missingness, want all rows used in estimation	`"fiml"`
Continuous PCA/EFA, MAR missingness	`"fiml"` (via `psych::corFiml()`. See above)
MAR-valid with ordinal (not yet built-in)	MI via `lavaan.mi` or `mirt`

Correlation-matrix input

You may have a pre-computed correlation matrix: a published table, a polychoric matrix computed externally, a FIML estimate for missing data, or a subset you want to analyse without refitting. You can pass it directly to ackwards() or suggest_k(). The matrix is auto-detected from its shape (square, symmetric, unit diagonal).

R <- cor(bfi25, use = "pairwise.complete.obs")

# PCA from a correlation matrix (n_obs optional for PCA; required for EFA)
x_R <- ackwards(R, k_max = 5)
# EFA requires n_obs for fit indices:
x_efa_R <- ackwards(R, k_max = 5, engine = "efa", n_obs = 875L)

# Edges are identical to the raw-data run (same W'RW algebra):
x_d <- ackwards(bfi25, k_max = 5)
all.equal(tidy(x_R)$r, tidy(x_d)$r) # TRUE within floating-point tolerance

Constraints

Constraint	Detail
Engine	`"pca"` and `"efa"` only. `"esem"` errors (lavaan needs raw data)
`n_obs`	Required for `"efa"`. Optional for `"pca"` (stored as `NA`)
`cor` argument	Ignored (basis is fixed). Warns if set explicitly
`missing` argument	Ignored. Warns if set explicitly
Factor scores	`keep_scores = TRUE`, `augment()`, `tidy(what = "scores")` all error
`$cor` field	Stored as `NA`. Shown as `"(user-supplied matrix)"` in print

suggest_k() with a correlation matrix

sk_R <- suggest_k(R, n_obs = 875L)
# CD is skipped (resampling requires raw item distributions)
# PA, MAP, and VSS run normally

Performance with many items (ESEM)

Bass-ackwards analyses often involve large item pools. ESEM is the most expensive engine because it fits a separate lavaan model at every level. Two automatic optimisations keep this manageable:

Sample statistics are computed once. For ordinal data (cor = "polychoric", WLSMV), lavaan’s thresholds, polychoric matrix, and asymptotic weight matrix depend only on the data, not on the number of factors. So ackwards() computes them at the first level and reuses them for every deeper level. This is the single biggest saving at large item counts.
Levels can be fit in parallel. The per-level fits are independent. Install future.apply and set a future plan before your call. The default plan is sequential (no change in behaviour).

library(future)
plan(multisession, workers = 4) # parallel across background R sessions

x <- ackwards(items, k_max = 8, engine = "esem", cor = "polychoric", seed = 1)

plan(sequential) # restore when done

plan() comes from future (it is not re-exported by future.apply), so load future directly rather than reaching for future.apply::plan(). Parallelism pays off only when the per-level fits are genuinely heavy. For small problems the worker startup cost can outweigh the gain. Results are reproducible across plans when you pass seed. PCA and EFA compute their correlation matrix once and do not need this.

You may only need the hierarchy (loadings and edges) and not ESEM’s rotation-aware standard errors and per-level fit indices. In that case engine = "efa" with cor = "polychoric" computes the polychoric matrix once and runs psych::fa() at each level. That is substantially cheaper than k WLSMV fits, and it recovers the same structure.

References

Asparouhov, T., & Muthén, B. (2009). Exploratory structural equation modeling. Structural Equation Modeling, 16(3), 397–438.

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.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55.

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