Create correlation matrices or data matrices with a particular measurement and structural model

Description

Structural Equation Models decompose correlation or correlation matrices into a measurement (factor) model and a structural (regression) model. sim.structural creates data sets with known measurement and structural properties. Population or sample correlation matrices with known properties are generated. Optionally raw data are produced.

It is also possible to specify a measurement model for a set of x variables separately from a set of y variables. They are then combined into one model with the correlation structure between the two sets.

Finally, the general case is given a population correlation matrix, generate data that will reproduce (with sampling variability) that correlation matrix. sim.correlation.

Usage

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sim.structure(fx=NULL,Phi=NULL, fy=NULL, f=NULL, n=0, uniq=NULL, raw=TRUE, 
  items = FALSE, low=-2,high=2,d=NULL,cat=5, mu=0)
sim.structural(fx=NULL, Phi=NULL, fy=NULL, f=NULL, n=0, uniq=NULL, raw=TRUE,
      items = FALSE, low=-2,high=2,d=NULL,cat=5, mu=0)  #deprecated
sim.correlation(R,n=1000,data=FALSE)

Arguments

fx

The measurement model for x

Phi

The structure matrix of the latent variables

fy

The measurement model for y

f

The measurement model

n

Number of cases to simulate. If n=0, the population matrix is returned.

uniq

The uniquenesses if creating a covariance matrix

raw

if raw=TRUE, raw data are returned as well for n > 0.

items

TRUE if simulating items, FALSE if simulating scales

low

Restrict the item difficulties to range from low to high

high

Restrict the item difficulties to range from low to high

d

A vector of item difficulties, if NULL will range uniformly from low to high

cat

Number of categories when creating binary (2) or polytomous items

mu

A vector of means, defaults to 0

R

The correlation matrix to reproduce

data

if TRUE, return the raw data, otherwise return the sample correlation matrix.

Details

Given the measurement model, fx and the structure model Phi, the model is f %*% Phi %*% t(f). Reliability is f %*% t(f). f φ f' and the reliability for each test is the items communality or just the diag of the model.

If creating a correlation matrix, (uniq=NULL) then the diagonal is set to 1, otherwise the diagonal is diag(model) + uniq and the resulting structure is a covariance matrix.

Given the model, raw data are generated using the mvnorm function.

A special case of a structural model are one factor models such as parallel tests, tau equivalent tests, and congeneric tests. These may be created by letting the structure matrix = 1 and then defining a vector of factor loadings. Alternatively, make.congeneric will do the same.

sim.correlation will create data sampled from a specified correlation matrix for a particular sample size. If desired, it will just return the sample correlation matrix. With data=TRUE, it will return the sample data as well.

Value

model

The implied population correlation or covariance matrix

reliability

The population reliability values

r

The sample correlation or covariance matrix

observed

If raw=TRUE, a sample data matrix

Author(s)

William Revelle

References

Revelle, W. (in preparation) An Introduction to Psychometric Theory with applications in R. Springer. at http://personality-project.org/r/book/

See Also

make.hierarchical for another structural model and make.congeneric for the one factor case. structure.list and structure.list for making symbolic structures.

Examples

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fx <-matrix(c( .9,.8,.6,rep(0,4),.6,.8,-.7),ncol=2)  
fy <- matrix(c(.6,.5,.4),ncol=1)
rownames(fx) <- c("V","Q","A","nach","Anx")
rownames(fy)<- c("gpa","Pre","MA")
Phi <-matrix( c(1,0,.7,.0,1,.7,.7,.7,1),ncol=3)
gre.gpa <- sim.structural(fx,Phi,fy)
print(gre.gpa,2)  
#correct for attenuation to see structure

round(correct.cor(gre.gpa$model,gre.gpa$reliability),2)  
congeneric <- sim.structure(f=c(.9,.8,.7,.6)) # a congeneric model 
congeneric 

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