VSS(x, n = 8, rotate = "varimax", diagonal = FALSE, pc = "pa", n.obs=NULL,plot=TRUE,title="Very Simple Structure",...)Techniques most commonly used include
1) Extracting factors until the chi square of the residual matrix is not significant.
2) Extracting factors until the change in chi square from factor n to factor n+1 is not significant.
3) Extracting factors until the eigen values of the real data are less than the corresponding eigen values of a random data set of the same size (parallel analysis) fa.parallel.
4) Plotting the magnitude of the successive eigen values and applying the scree test (a sudden drop in eigen values analogous to the change in slope seen when scrambling up the talus slope of a mountain and approaching the rock face.
5) Extracting principal components until the eigen value <1. <="" p="">
6) Extracting factors as long as they are interpetable. 7) Using the Very Structure Criterion (VSS).
8) Using Wayne Velicer's Minimum Average Partial (MAP) criterion.
Each of the procedures has its advantages and disadvantages. Using either the chi square test or the change in square test is, of course, sensitive to the number of subjects and leads to the nonsensical condition that if one wants to find many factors, one simlpy runs more subjects. Parallel analysis is partially sensitive to sample size in that for large samples the eigen values of random factors will be very small. The scree test is quite appealling but can lead to differences of interpretation as to when the scree "breaks". The eigen value of 1 rule, although the default for many programs, seems to be a rough way of dividing the number of variables by 3. Extracting interpretable factors means that the number of factors reflects the investigators creativity more than the data. VSS, while very simple to understand, will not work very well if the data are very factorially complex. (Simulations suggests it will work fine if the complexities of some of the items are no more than 2).
Most users of factor analysis tend to interpret factor output by focusing their attention on the largest loadings for every variable and ignoring the smaller ones. Very Simple Structure operationalizes this tendency by comparing the original correlation matrix to that reproduced by a simplified version (S) of the original factor matrix (F). R = SS' + U2. S is composed of just the c greatest (in absolute value) loadings for each variable. C (or complexity) is a parameter of the model and may vary from 1 to the number of factors.
The VSS criterion compares the fit of the simplified model to the original correlations: VSS = 1 -sumsquares(r*)/sumsquares(r) where R* is the residual matrix R* = R - SS' and r* and r are the elements of R* and R respectively.
VSS for a given complexity will tend to peak at the optimal (most interpretable) number of factors (Revelle and Rocklin, 1979).
Although originally written in Fortran for main frame computers, VSS has been adapted to micro computers (e.g., Macintosh OS 6-9) using Pascal. We now release R code for calculating VSS.
Note that if using a correlation matrix (e.g., my.matrix) and doing a factor analysis, the parameters n.obs should be specified for the factor analysis: e.g., the call is VSS(my.matrix,n.obs=500). Otherwise it defaults to 1000.
Wayne Velicer's MAP criterion has been added as an additional test for the optimal number of components to extract. Note that VSS and MAP will not always agree as to the optimal number.
see also Revelle, W. and Rocklin, T. 1979, Very Simple Structure: an Alternative Procedure for Estimating the Optimal Number of Interpretable Factors, Multivariate Behavioral Research, 14, 403-414.
Velicer, W. (1976) Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321-327.
VSS.plot, ICLUST, omega, fa.paralleltest.data <- Harman74.cor$cov
my.vss <- VSS(test.data,title="VSS of 24 mental tests")
#print(my.vss[,1:12],digits =2)
#VSS.plot(my.vss, title="VSS of 24 mental tests")
#now, some simulated data with two factors
VSS(circ.sim(nvar=24),pc="mle" ,title="VSS of 24 circumplex variables")
VSS(item.sim(nvar=24),pc="mle" ,title="VSS of 24 simple structure variables")Run the code above in your browser using DataLab