Screen a dataset for factorability and sampling adequacy

Before extracting a hierarchy, it is worth asking two prior questions: is this correlation matrix even worth factoring, and is the sample large enough to trust the answer. factorability() reports the standard diagnostics for both, one object you can print or pull values from:

KMO – the Kaiser-Meyer-Olkin measure of sampling adequacy, overall and per variable (via psych::KMO()). It contrasts correlations with partial correlations: a low value means the variables share little common variance once other variables are partialled out, so a factor model has little to recover.
Bartlett's test of sphericity – tests whether the correlation matrix differs from the identity (via psych::cortest.bartlett()). Needs N; a non-significant result says the correlations are too weak to factor.
Sample size – N, the number of variables p, and the N:p ratio.
Ledermann bound – the largest number of common factors identifiable from p variables (a hard limit for EFA/ESEM, though not for PCA).

Read these as conventions, not verdicts. Every cutoff here – Kaiser's KMO bands, the 5:1 / 10:1 N:p rules of thumb, Bartlett significance at .05 – is a widely repeated rule of thumb, not a settled threshold, and each is contested in the methodological literature (the required N in particular depends on communalities and factor overdetermination far more than on any fixed ratio; MacCallum et al. 1999). factorability() deliberately reports the numbers and their conventional bands rather than returning a pass/fail flag. ackwards() runs the same screen internally and warns only at the genuinely consequential extreme (KMO < 0.5, N:p < 5).

Usage

factorability(data, cor = c("pearson", "spearman", "polychoric"), n_obs = NULL)

Arguments

data: A data frame or numeric matrix of item responses, or a correlation matrix (detected automatically). KMO and Bartlett are computed on the correlation matrix in the basis you name via cor.
cor: The correlation basis to screen on: "pearson" (default), "spearman", or "polychoric". Ignored when data is already a correlation matrix. Use the same basis you plan to pass to ackwards().
n_obs: Number of observations. Taken from nrow(data) for raw data (a supplied value is ignored with a warning). Required for the N-based diagnostics (Bartlett, N:p) when data is a correlation matrix; those are reported as NA when it is absent.

Value

An object of class factorability (a named list): cor, n_obs, n_vars, np_ratio, kmo_overall, kmo_items (a data frame of per-item MSA), bartlett (a list of chisq/df/p_value, or NULL when N is unknown), and ledermann (the bound). Print it for a banded summary; index the list for the raw values.

References

Kaiser, H. F. (1974). An index of factorial simplicity. Psychometrika, 39(1), 31–36.

MacCallum, R. C., Widaman, K. F., Zhang, S., & Hong, S. (1999). Sample size in factor analysis. Psychological Methods, 4(1), 84–99.

Examples

# Continuous data, Pearson basis
factorability(sim16)
#> 
#> ── Factorability screen (ackwards) ─────────────────────────────────────────────
#> Basis: pearson
#> Observations: 1,000
#> Variables: 16
#> N:p ratio: 62.5:1
#> 
#> ── Sampling adequacy ──
#> 
#> Overall KMO: 0.86 (meritorious)
#> 
#> Bartlett's test of sphericity: chi-square(120) = 6476.3, p < .001
#> 
#> ── Identifiability ──
#> 
#> Ledermann bound: at most 10 common factors are identifiable from 16 variables
#> (EFA/ESEM; PCA is unbounded).
#> ────────────────────────────────────────────────────────────────────────────────
#> KMO bands (Kaiser 1974), the N:p >= 5/10 rules, and Bartlett at .05 are widely
#> used *rules of thumb*, not settled thresholds -- required N depends on
#> communalities and factor overdetermination (MacCallum et al. 1999). Read the
#> numbers, not a pass/fail.

# From a correlation matrix: supply n_obs for the N-based diagnostics
R <- cor(sim16)
factorability(R, n_obs = nrow(sim16))
#> 
#> ── Factorability screen (ackwards) ─────────────────────────────────────────────
#> Basis: pearson
#> Observations: 1,000
#> Variables: 16
#> N:p ratio: 62.5:1
#> 
#> ── Sampling adequacy ──
#> 
#> Overall KMO: 0.86 (meritorious)
#> 
#> Bartlett's test of sphericity: chi-square(120) = 6476.3, p < .001
#> 
#> ── Identifiability ──
#> 
#> Ledermann bound: at most 10 common factors are identifiable from 16 variables
#> (EFA/ESEM; PCA is unbounded).
#> ────────────────────────────────────────────────────────────────────────────────
#> KMO bands (Kaiser 1974), the N:p >= 5/10 rules, and Bartlett at .05 are widely
#> used *rules of thumb*, not settled thresholds -- required N depends on
#> communalities and factor overdetermination (MacCallum et al. 1999). Read the
#> numbers, not a pass/fail.