Bartlett's test of sphericity

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")
)

Source

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

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".

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.

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.

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)
#> 
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#> 
#> χ²(153) = 2173.28, p < .001