Exploring the Sensitivity of Horn's Parallel Analysis to the Distributional Form of Random Data

Abstract

Horn's parallel analysis (PA) is the method of consensus in the literature on empirical methods for deciding how many components/factors to retain. Different authors have proposed various implementations of PA. Horn's seminal 1965 article, a 1996 article by Thompson and Daniel, and a 2004 article by Hayton, Allen, and Scarpello all make assertions about the requisite distributional forms of the random data generated for use in PA. Readily available software is used to test whether the results of PA are sensitive to several distributional prescriptions in the literature regarding the rank, normality, mean, variance, and range of simulated data on a portion of the National Comorbidity Survey Replication (Pennell et al., 2004) by varying the distributions in each PA. The results of PA were found not to vary by distributional assumption. The conclusion is that PA may be reliably performed with the computationally simplest distributional assumptions about the simulated data.

Figure 1

Graphical illustration of parallel analysis on a simulated data set of 50 observations, across 20 variables, with two uncorrelated factors, and %50 total variance. The dashed line connects unadjusted eigenvalues of the observed data, the dotted line connects mean eigenvalues of 600 random 50*20 data sets, and the solid line connects adjusted eigenvalues (i.e. subtracting the mean eigenvalues minus one from the observed eigenvalues). The retention criterion is the point at which the adjusted eigenvalues cross the horizontal line at y=1, which is the same point at which the unadjusted eigenvalues cross the line of mean eigenvalues of the random data sets. The solid adjusted eigenvalue markers are those components (or factors, if using factor analysis) that are retained.

Figure 2

Histograms showing the distributions of the first variable from three of nine simulated data sets. All variables have five values (the integers from 1 to 5), and variable distributions based on different parameterizations of the Beta distribution plus an amount of uniform noise.

Figure 3

Figures 3a and 3b. Plot connecting the means (black) and 95% quantiles (grey) of 5000 random eigenvalues for simulated data sets with 75 observations and 50 variables for parallel analyses conducted with ten different random data distributions for principal components analysis (3a) and factor analysis (3b). The near perfect overlap of the means and quantiles across the entire range of factors with such a small sample size illustrates the absolute or virtual insensitivity of parallel analysis to the distributional form of simulated data.