Two questions come before extracting a hierarchy. Is this correlation matrix even worth factoring, and is the sample large enough to trust the answer? A factor is a summary variable standing in for a group of items that move together. The function reports the standard diagnostics for both, in 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, which tests whether the correlation matrix differs from the identity (via
psych::cortest.bartlett()). It needsN. A non-significant result says the correlations are too weak to factor.Sample size:
N, the number of variablesp, and theN:pratio.Ledermann bound, the largest number of common factors identifiable from
pvariables. It is a hard limit for EFA (exploratory factor analysis) and ESEM (exploratory structural equation modeling), though not for PCA (principal component analysis).
Read these as conventions, not verdicts. Three cutoffs appear here:
Kaiser's KMO bands, the 5:1 and 10:1 N:p rules of thumb, and Bartlett
significance at .05. Each 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 far more on two other things than on
any fixed ratio (MacCallum et al. 1999). One is the communalities (the share
of each item's variance the factors explain). The other is factor
overdetermination (how many items load strongly on each factor). This
function deliberately reports the numbers and their conventional bands rather than
returning a pass/fail flag. ackwards() runs the same screen internally.
It 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", which estimates the correlation between the continuous traits assumed to underlie ordered responses. Ignored whendatais already a correlation matrix. Use the same basis you plan to pass toackwards().- 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) whendatais a correlation matrix. Those are reported asNAwhen it is absent.
Value
An object of class factorability, a named list. Its entries are
cor, n_obs,
n_vars, np_ratio, kmo_overall, kmo_items (a data frame of per-item
MSA), bartlett (a list of chisq, df, and 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.
See also
check_items() (per-item screening), suggest_k() (how many
factors), and comparability(), which checks split-half replicability by
splitting the sample into two random halves. Those are the other
pre-analysis diagnostics. See also ackwards(), which runs this screen
internally.
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.