# sim.hierarchical: Create a population or sample correlation matrix, perhaps...

Description Usage Arguments Details Value Author(s) References See Also Examples

### 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

 ```1 2``` ```sim.hierarchical(gload=NULL, fload=NULL, n = 0, raw = FALSE,mu = NULL) make.hierarchical(gload=NULL, fload=NULL, n = 0, raw = FALSE) #deprecated ```

### Arguments

 `gload` Loadings of group factors on a general factor `fload` Loadings of items on the group factors `n` Number of subjects to generate: N=0 => population values `raw` raw=TRUE, report the raw data, raw=FALSE, report the sample correlation matrix. `mu` means for the individual variables

### 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

William Revelle

### References

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

`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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27``` ```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) ```

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