# sim.congeneric: Simulate a congeneric data set In psych: Procedures for Psychological, Psychometric, and Personality Research

## Description

Classical Test Theory (CTT) considers four or more tests to be congenerically equivalent if all tests may be expressed in terms of one factor and a residual error. Parallel tests are the special case where (usually two) tests have equal factor loadings. Tau equivalent tests have equal factor loadings but may have unequal errors. Congeneric tests may differ in both factor loading and error variances.

## Usage

 ```1 2``` ```sim.congeneric(loads = c(0.8, 0.7, 0.6, 0.5),N = NULL, err=NULL, short = TRUE, categorical=FALSE, low=-3,high=3,cuts=NULL) ```

## Arguments

 `N` How many subjects to simulate. If NULL, return the population model `loads` A vector of factor loadings for the tests `err` A vector of error variances – if NULL then error = 1 - loading 2 `short` short=TRUE: Just give the test correlations, short=FALSE, report observed test scores as well as the implied pattern matrix `categorical` continuous or categorical (discrete) variables. `low` values less than low are forced to low `high` values greater than high are forced to high `cuts` If specified, and categorical = TRUE, will cut the resulting continuous output at the value of cuts

## Details

When constructing examples for reliability analysis, it is convenient to simulate congeneric data structures. These are the most simple of item structures, having just one factor. Mainly used for a discussion of reliability theory as well as factor score estimates.

The implied covariance matrix is just pattern %*% t(pattern).

## Value

 `model` The implied population correlation matrix if N=NULL or short=FALSE, otherwise the sample correlation matrix `pattern ` The pattern matrix implied by the loadings and error variances `r` The sample correlation matrix for long output `observed` a matrix of test scores for n tests `latent` The latent trait and error scores

William Revelle

## References

Revelle, W. (in prep) An introduction to psychometric theory with applications in R. To be published by Springer. (working draft available at https://personality-project.org/r/book/

`item.sim` for other simulations, `fa` for an example of factor scores, `irt.fa` and `polychoric` for the treatment of item data with discrete values.
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```test <- sim.congeneric(c(.9,.8,.7,.6)) #just the population matrix test <- sim.congeneric(c(.9,.8,.7,.6),N=100) # a sample correlation matrix test <- sim.congeneric(short=FALSE, N=100) round(cor(test\$observed),2) # show a congeneric correlation matrix f1=fa(test\$observed,scores=TRUE) round(cor(f1\$scores,test\$latent),2) #factor score estimates are correlated with but not equal to the factor scores set.seed(42) #500 responses to 4 discrete items items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) d4 <- irt.fa(items\$observed) #item response analysis of congeneric measures ```