It compares an observed correlation matrix to the identity matrix in order to check if there is a certain redundancy between the variables that we can summarize with a few number of factors. A statistically significant test shows that the variables (columns) in x are correlated, thus it makes sense to perform some dimensionality reduction of the data in x.

ndim = barttest (x, alpha) returns the number of dimensions necessary to explain the nonrandom variation in the data matrix x at the alpha significance level. alpha is an optional input argument and, when not provided, it is 0.05 by default.

[ndim, pval, chisq] = barttest (…) also returns the significance values pval for the hypothesis test for each dimension as well as the associated chi^2 values in chisq

Source Code: barttest