Parallel Analysis (PA) to for determining the number of components to retain from PCA.

% Parallel Analysis (PA) to for determining the number of components to retain from PCA. component is retained if the associated eigenvalue is bigger than the 95th of the distribution of eigenvalues derived from the random data.
% Syntax:
% ======
% pa_test(x, nShuffle, alpha, princomp_parameters[ ])
% x - the data matrix (nXp where n is the number of observation and p is dimension of each observation)
% nShuffle - number of shuffles. optional, default = 100
% alpha - significance level. optional, default 0.05
% princomp_parameters - parameters to pass to the princomp function (see help princomp). optional, default ={true,'Centered',false}

% Background:
% ==========
% From Wikipedia: http://en.wikipedia.org/wiki/Factor_analysis
% Horn's Parallel Analysis (PA):
% A Monte-Carlo based simulation method that compares the observed eigenvalues with those obtained from uncorrelated normal variables.
% A factor or component is retained if the associated eigenvalue is bigger than the 95th of the distribution of eigenvalues derived from the random data.
% PA is one of the most recommendable rules for determining the number of components to retain, but only few programs include this option.

% References:
% * Ledesma, R.D.; Valero-Mora, P. (2007). "Determining the Number of Factors to Retain in EFA: An easy-to-use computer program for carrying out Parallel Analysis". Practical Assessment Research & Evaluation 12 (2): 1–11.