efa_bartlett: Bartlett's test of sphericity

In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Bartlett's test of sphericity

Description

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett's test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

Usage

efa_bartlett(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

x	data.frame or matrix. Dataframe or matrix of raw data or matrix with correlations.
N	numeric. The number of observations. Needs only be specified if a correlation matrix is used.
use	character. The missing-data policy for raw data. Passed to stats::cor() for "pearson", "spearman", and "kendall"; for "poly" / "tetra" the same policies are applied to the raw data before the polychoric estimation, where "all.obs" and "everything" abort on a missing value instead of returning NA correlations. Default is "pairwise.complete.obs".
cor_method	character. Correlation computed from raw data: "pearson", "spearman", or "kendall" (passed to stats::cor()), or "poly" / "tetra" for polychoric / tetrachoric correlations of ordinal / binary data (a two-step estimator). Default is "pearson".

Details

Bartlett (1951) proposed this statistic to determine a correlation matrix' suitability for factor analysis. The statistic is approximately chi square distributed with df = \frac{p(p - 1)}{2} and is given by

chi^2 = -log(det(R)) (N - 1 - (2 * p + 5)/6)

where det(R) is the determinant of the correlation matrix, N is the sample size, and p is the number of variables.

This test requires multivariate normality. If this condition is not met, the Kaiser-Meyer-Olkin criterion (efa_kmo()) can still be used.

This function was heavily influenced by the psych::cortest.bartlett() function from the psych package.

The efa_bartlett function can also be called together with the (efa_kmo()) function and with factor retention criteria in the efa_retain() function.

Value

A list containing

chisq	The chi square statistic, or NA, with a warning, if N is too small for the Bartlett correction (i.e. N - 1 - (2p + 5)/6 \le 0).
p_value	The p value of the chi square statistic, or NA when chisq is NA.
df	The degrees of freedom for the chi square statistic.
settings	A list of the settings used.

Source

Bartlett, M. S. (1951). The effect of standardization on a Chi-square approximation in factor analysis. Biometrika, 38, 337-344.

See Also

efa_kmo() for another measure to determine suitability for factor analysis.

efa_retain() as a wrapper function for this function, efa_kmo() and several factor retention criteria.

Other factor analysis suitability: efa_kmo(), efa_screen(), print.efa_screen()

Examples

efa_bartlett(test_models$baseline$cormat, N = 500)

EFAtools documentation built on Aug. 21, 2026, 5:16 p.m.