suggest_k: Suggest a maximum number of factors for bass-ackwards...

In ackwards: Bass-Ackwards Hierarchical Structural Analysis

Suggest a maximum number of factors for bass-ackwards analysis

Description

Runs complementary factor-retention criteria and reports their recommendations. No single criterion is definitive; the goal is a consensus range to inform your choice of k in ackwards().

Usage

suggest_k(
  data,
  k_max = NULL,
  criteria = c("pa_pc", "pa_fa", "map", "vss", "cd"),
  cor = "pearson",
  n_obs = NULL,
  n_iter = 20L,
  seed = NULL,
  ...
)

Arguments

data	A data frame or numeric matrix (items in columns, observations in rows). Alternatively, a pre-computed correlation matrix may be supplied (a square, symmetric, numeric matrix with unit diagonal). When a correlation matrix is supplied, n_obs is required (PA and VSS need N), the cor argument is ignored, and the Comparison Data (CD) criterion is skipped (CD requires raw item distributions for resampling).
k_max	Maximum number of factors/components to evaluate when recommending a depth – not a depth itself. Defaults to min(ncol(data) - 1, 8). Increase if you expect a deeper hierarchy. (Note: this is a distinct meaning from ackwards()'s k_max, which is the extraction depth; the two share a name because they're the same dial in the suggest_k() -> ackwards() workflow, just applied at different stages.)
criteria	Character vector of criteria to compute. Any subset of c("pa_pc", "pa_fa", "map", "vss", "cd"); default is all five. "pa_pc" and "pa_fa" share one parallel-analysis call (both or neither are fast); "map" and "vss" share one VSS call. Criteria not requested are skipped entirely (no computation) and their ⁠k_*⁠ fields in the result are NA. "vss" selects both VSS-1 and VSS-2 as a unit.
cor	Correlation basis: "pearson" (default) or "spearman". Should match the cor argument you plan to use in ackwards(). Ignored when data is a correlation matrix (the basis is already fixed).
n_obs	Number of observations. Required when data is a pre-computed correlation matrix (PA and VSS need N). Ignored when raw data are supplied (N is determined from nrow(data)).
n_iter	Number of Monte Carlo iterations for parallel analysis. Default 20. Reduce to 5 for fast/exploratory runs; increase to ⁠100+⁠ for publication.
seed	Integer seed passed to set.seed() before the Comparison Data (CD) step. NULL (default) uses the current RNG state. Note: the parallel-analysis step uses psych::fa.parallel(), which does not respond reliably to set.seed() – PA simulation results will vary across calls regardless of seed.
...	Reserved for future arguments.

Details

Criteria available (controlled by the criteria argument):

• PA-PC ("pa_pc") – Horn (1965) parallel analysis on PC eigenvalues. Compares observed eigenvalues to those from random correlation matrices; suggests retaining components whose eigenvalues exceed the 95th percentile of chance. Tends to overextract; treat as an upper bound.

• PA-FA ("pa_fa") – Horn (1965) parallel analysis using common-factor eigenvalues. More conservative than PA-PC and the better match for the EFA and ESEM engines in ackwards(). Shares one psych::fa.parallel() call with PA-PC.

• MAP ("map") – Velicer (1976) Minimum Average Partial criterion. Finds the k that minimises the average squared partial correlation remaining after extracting k components. Usually conservative. Shares one psych::vss() call with VSS.

• VSS ("vss") – Revelle & Rocklin (1979) Very Simple Structure fit at complexities 1 and 2 (VSS-1 and VSS-2). Finds the k maximising the fit of a very simple loading structure. Shares one psych::vss() call with MAP.

• CD ("cd") – Ruscio & Roche (2012) Comparison Data. Resamples from the observed item distributions to generate comparison eigenvalue profiles; retains factors until adding one no longer improves RMSE beyond chance. Requires the EFAtools package (install separately). Skipped with an informational message when EFAtools is absent or when a correlation matrix is supplied.

PA (both bases), MAP, and VSS share the same correlation matrix as ackwards(). CD operates on the raw data matrix directly (required for resampling).

Value

An object of class "suggest_k". Print it for a formatted summary; call autoplot() on it for a diagnostic scree/criteria plot. The list contains:

k_parallel_pc	Recommended k from PC-based parallel analysis (NA_integer_ when "pa_pc" not in criteria).
k_parallel_fa	Recommended k from FA-based parallel analysis (NA_integer_ if no FA factor exceeded the random threshold, or when "pa_fa" not in criteria).
k_map	Recommended k from MAP (NA_integer_ when "map" not in criteria).
k_vss1	Recommended k from VSS complexity-1 (NA_integer_ when "vss" not in criteria).
k_vss2	Recommended k from VSS complexity-2 (NA_integer_ when "vss" not in criteria).
k_cd	Recommended k from Comparison Data (NA_integer_ when "cd" not in criteria, EFAtools is not installed, or CD fails).
cd_available	Logical; TRUE when "cd" was requested, EFAtools was found, and CD ran successfully.
criteria	Data frame with one row per k: k, ev_obs (observed PC eigenvalue), ev_obs_fa (observed FA eigenvalue), pa_pc_quant / pa_fa_quant (95th-pct simulated eigenvalue for each basis), pa_pc_suggested / pa_fa_suggested (logical retention), map, vss1, vss2. Columns for non-requested criteria are NA.
criteria_requested	Character vector of the criteria that were requested (and therefore computed).
k_max, n_obs, n_vars, cor	Metadata.

Interpreting the output

k in ackwards() is a maximum depth, not a claim that exactly k factors exist. Users commonly set k one or two levels above the consensus to watch higher-level factors fragment – this is a feature of the method, not overextraction.

Treating retention estimates as a range rather than a verdict has direct support in the parallel-analysis literature: Lim and Jahng (2019) recommend interpreting the PA estimate as a range of roughly plus or minus one factor, resolved by interpretability, and Achim (2021) argues that even that overstates PA's precision – the disagreement itself is why suggest_k() reports several criteria and a consensus range, never a single number.

Ordinal (Likert) data

suggest_k() screens on the Pearson or Spearman basis by design and never computes polychoric correlations itself, so when the raw data look ordinal (at most 7 distinct integer values per column) it emits a one-per-session warning – the same detect_ordinal() signal ackwards() and comparability() use (Invariant 6: announce consequential defaults loudly). Crucially, the advice points at the final ackwards() fit (cor = "polychoric"), not at suggest_k() itself: the screening basis is intentional. The plausible-k range is robust to the Pearson-vs-polychoric choice, so screening on Pearson and fitting the final model polychoric is the recommended workflow – computing polychoric correlations at the screening stage would only add cost and NPD risk without changing the range. The warning is skipped for correlation-matrix input (there are no items to inspect).

A note on overextraction

PA-PC in particular tends to recommend more factors than replicate across independent samples, especially with correlated items (Forbes, 2023). This is a long-standing observation in practice: Saucier (1997, footnote 14) reported parallel analysis suggesting as many as 30 factors in wide lexical item sets. PA-FA and CD are more conservative. Treat the full set of criteria as a range: the true k is likely somewhere in the middle.

References

Achim, A. (2021). Determining the number of factors using parallel analysis and its recent variants: Comment on Lim and Jahng (2019). Psychological Methods, 26(1), 69–73. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/met0000269")}

Forbes, M. K. (2023). Improving hierarchical models of individual differences: An extension of Goldberg's bass-ackward method. Psychological Methods. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/met0000546")}

Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179–185.

Lim, S., & Jahng, S. (2019). Determining the number of factors using parallel analysis and its recent variants. Psychological Methods, 24(4), 452–467. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/met0000230")}

Revelle, W., & Rocklin, T. (1979). Very simple structure: An alternative procedure for estimating the optimal number of interpretable factors. Multivariate Behavioral Research, 14(4), 403–414.

Ruscio, J., & Roche, B. (2012). Determining the number of factors to retain in an exploratory factor analysis using comparison data of a known factorial structure. Psychological Assessment, 24(2), 282–292.

Saucier, G. (1997). Effects of variable selection on the factor structure of person descriptors. Journal of Personality and Social Psychology, 73(6), 1296–1312. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/0022-3514.73.6.1296")}

Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321–327.

See Also

ackwards(); factorability(), check_items(), and comparability(), the other pre-analysis diagnostics.

Examples

sk <- suggest_k(sim16)
sk
autoplot(sk)

# Run only MAP (fast; skips parallel analysis and CD)
suggest_k(sim16, criteria = "map")

# Run only the parallel-analysis criteria
suggest_k(sim16, criteria = c("pa_pc", "pa_fa"), n_iter = 5)

# Faster exploratory run
suggest_k(sim16, k_max = 6, n_iter = 5)

# Correlation-matrix input (CD is skipped; n_obs required)
R <- cor(sim16)
suggest_k(R, n_obs = nrow(sim16))

ackwards documentation built on July 25, 2026, 1:08 a.m.