BARTLETT: 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

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")
)

Value

A list containing

chisq: The chi square statistic, or NA 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.

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. Passed to stats::cor() if raw data is given as input. 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 with no empty-cell continuity correction). 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 tests requires multivariate normality. If this condition is not met, the Kaiser-Meyer-Olkin criterion (KMO()) can still be used.

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

The BARTLETT function can also be called together with the (KMO()) function and with factor retention criteria in the N_FACTORS() function.

Examples

Run this code

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