Create a population or sample correlation matrix, perhaps with hierarchical structure.

Description

Create a population orthogonal or hierarchical correlation matrix from a set of factor loadings and factor intercorrelations. Samples of size n may be then be drawn from this population. Return either the sample data, sample correlations, or population correlations. This is used to create sample data sets for instruction and demonstration.

Usage

sim.hierarchical(gload=NULL, fload=NULL, n = 0, raw = FALSE,mu = NULL)
make.hierarchical(gload=NULL, fload=NULL, n = 0, raw = FALSE)  #deprecated

Arguments

Details

Many personality and cognitive tests have a hierarchical factor structure. For demonstration purposes, it is useful to be able to create such matrices, either with population values, or sample values.

Given a matrix of item factor loadings (fload) and of loadings of these factors on a general factor (gload), we create a population correlation matrix by using the general factor law (R = F' theta F where theta = g'g).

To create sample values, we use code adapted from the mvrnorm function in MASS.

The default is to return population correlation matrices. Sample correlation matrices are generated if n >0. Raw data are returned if raw = TRUE.

The default values for gload and fload create a data matrix discussed by Jensen and Weng, 1994.

Although written to create hierarchical structures, if the gload matrix is all 0, then a non-hierarchical structure will be generated.

Value

a matrix of correlations or a data matrix

Author(s)

William Revelle

References

https://personality-project.org/r/r.omega.html
Jensen, A.R., Weng, L.J. (1994) What is a Good g? Intelligence, 18, 231-258.

See Also

omega, schmid, ICLUST, VSS for ways of analyzing these data. Also see sim.structure to simulate a variety of structural models (e.g., multiple correlated factor models). The simulation uses the mvrnorm function from the MASS package.

Examples

gload <-  gload<-matrix(c(.9,.8,.7),nrow=3)    # a higher order factor matrix
fload <-matrix(c(                    #a lower order (oblique) factor matrix
           .8,0,0,
           .7,0,.0,
           .6,0,.0,
            0,.7,.0,
            0,.6,.0,
            0,.5,0,
            0,0,.6,
            0,0,.5,
            0,0,.4),   ncol=3,byrow=TRUE)
            
jensen <- sim.hierarchical(gload,fload)    #the test set used by omega
round(jensen,2)     

#simulate a non-hierarchical structure
fload <- matrix(c(c(c(.9,.8,.7,.6),rep(0,20)),c(c(.9,.8,.7,.6),rep(0,20)),
    c(c(.9,.8,.7,.6),rep(0,20)),c(c(c(.9,.8,.7,.6),rep(0,20)),c(.9,.8,.7,.6))),ncol=5)
gload <- matrix(rep(0,5))
five.factor <- sim.hierarchical(gload,fload,500,TRUE) #create sample data set
#do it again with a hierachical structure
gload <- matrix(rep(.7,5)  )
five.factor.g <- sim.hierarchical(gload,fload,500,TRUE) #create sample data set
#compare these two with omega
#not run
#om.5 <- omega(five.factor$observed,5)
#om.5g <- omega(five.factor.g$observed,5)