sim for a factor simplex, and sim.simplex for a data simplex, sim.circ for a circumplex structure, sim.congeneric for a one factor factor congeneric model, sim.dichot to simulate dichotomous items, sim.hierarchical to create a hierarchical factor model, sim.item a more general item simulation,
sim.minor to simulate major and minor factors,
sim.omega to test various examples of omega,
sim.parallel to compare the efficiency of various ways of deterimining the number of factors,
sim.rasch to create simulated rasch data,
sim.irt to create general 1 to 4 parameter IRT data by calling
sim.npl 1 to 4 parameter logistic IRT or
sim.npn 1 to 4 paramater normal IRT,
sim.structural a general simulation of structural models, and sim.anova for ANOVA and lm simulations, and sim.VSS. Some of these functions are separately documented and are listed here for ease of the help function. See each function for more detailed help.sim(fx=NULL,Phi=NULL,fy=NULL,n=0,mu=NULL,raw=TRUE)
sim.simplex(nvar =12, r=.8,mu=NULL, n=0)
sim.minor(nvar=12,nfact=3,n=0,fbig=NULL,fsmall = c(-.2,.2),bipolar=TRUE)
sim.omega(nvar=12,nfact=3,n=0,fbig=NULL,fsmall = c(-.2,.2),bipolar=TRUE,om.fact=3,flip=TRUE,option="equal",ntrials=10)
sim.parallel(ntrials=10,nvar = c(12,24,36,48),nfact = c(1,2,3,4,6),
n = c(200,400))
sim.rasch(nvar = 5,n = 500, low=-3,high=3,d=NULL, a=1,mu=0,sd=1)
sim.irt(nvar = 5, n = 500, low=-3, high=3,a=NULL,c=0,z=1,d=NULL,mu=0,sd=1,mod="logistic")
sim.npl(nvar = 5, n = 500, low=-3,high=3,a=NULL,c=0,z=1,d=NULL,mu=0,sd=1)
sim.npn(nvar = 5, n = 500, low=-3,high=3,a=NULL,c=0,z=1,d=NULL,mu=0,sd=1)The default values for sim.structure is to generate a 4 factor, 12 variable data set with a simplex structure between the factors.
Two data structures that are particular challenges to exploratory factor analysis are the simplex structure and the presence of minor factors. Simplex structures sim.simplex will typically occur in developmental or learning contexts and have a correlation structure of r between adjacent variables and r^n for variables n apart. Although just one latent variable (r) needs to be estimated, the structure will have nvar-1 factors.
Many simulations of factor structures assume that except for the major factors, all residuals are normally distributed around 0. An alternative, and perhaps more realistic situation, is that the there are a few major (big) factors and many minor (small) factors. The challenge is thus to identify the major factors. sim.minor generates such structures. The structures generated can be thought of as havinga a major factor structure with some small correlated residuals.
Although coefficient $$\omega$$ is a very useful indicator of the general factor saturation of a unifactorial test (one with perhaps several sub factors), it has problems with the case of multiple, independent factors. In this situation, one of the factors is labelled as ``general'' and the omega estimate is too large. This situation may be explored using the sim.omega function.
The four irt simulations, sim.rasch, sim.irt, sim.npl and sim.npn, simulate dichotomous items following the Item Response model. sim.irt just calls either sim.npl (for logistic models) or sim.npn (for normal models) depending upon the specification of the model.
The logistic model is $$P(i,j) = \gamma + \frac{\zeta-\gamma}{1+ e^{\alpha(\delta-\theta)}}$$ where $\gamma$ is the lower asymptote or guesssing parameter, $\zeta$ is the upper asymptote (normally 1), $\alpha$ is item discrimination and $\delta$ is item difficulty. For the 1 Paramater Logistic (Rasch) model, gamma=0, zeta=1, alpha=1 and item difficulty is the only free parameter to specify.
For the 2PL and 2PN models, a = $\alpha$ and d = $\delta$ are specified.
For the 3PL or 3PN models, items also differ in their guessing parameter c =$\gamma$.
For the 4PL and 4PN models, the upper asymptote, z= $\zeta$ is also specified.
(Graphics of these may be seen in the demonstrations for the logistic function.)
The normal model (irt.npn calculates the probability using pnorm instead of the logistic function used in irt.npl, but the meaning of the parameters are otherwise the same. With the a = $\alpha$ parameter = 1.702 in the logitistic model the two models are practically identical.
Other simulation functions in psych are:
sim.structure A function to combine a measurement and structural model into one data matrix. Useful for understanding structural equation models. Combined with structure.diagram to see the proposed structure.
sim.congeneric A function to create congeneric items/tests for demonstrating classical test theory. This is just a special case of sim.structure.
sim.hierarchical A function to create data with a hierarchical (bifactor) structure.
sim.item A function to create items that either have a simple structure or a circumplex structure.
sim.circ Create data with a circumplex structure.
sim.dichot Create dichotomous item data with a simple or circumplex structure.
sim.minor Create a factor structure for nvar variables defined by nfact major factors and nvar/2 ``minor" factors for n observations.
Although the standard factor model assumes that K major factors (K << nvar) will account for the correlations among the variables
$$R = FF' + U^2$$ where R is of rank P and F is a P x K matrix of factor coefficients and U is a diagonal matrix of uniquenesses. However, in many cases, particularly when working with items, there are many small factors (sometimes referred to as correlated residuals) that need to be considered as well. This leads to a data structure such that $$R = FF' + MM' + U^2$$ where R is a P x P matrix of correlations, F is a P x K factor loading matrix, M is a P x P/2 matrix of minor factor loadings, and U is a diagonal matrix (P x P) of uniquenesses.
Such a correlation matrix will have a poor $\chi^2$ value in terms of goodness of fit if just the K factors are extracted, even though for all intents and purposes, it is well fit.
sim.minor will generate such data sets with big factors with loadings of .6 to .8 and small factors with loadings of -.2 to .2. These may both be adjusted.
sim.parallel Create a number of simulated data sets using sim.minor to show how parallel analysis works. The general observation is that with the presence of minor factors, parallel analysis is probably best done with component eigen values rather than factor eigen values, even when using the factor model.
sim.anova Simulate a 3 way balanced ANOVA or linear model, with or without repeated measures. Useful for teaching research methods and generating teaching examples.
simplex <- sim.simplex() #create the default structure
round(simplex,2) #the correlation matrix
congeneric <- sim.congeneric()
round(congeneric,2)
R <- sim.hierarchical()
R
fx <- matrix(c(.9,.8,.7,rep(0,6),c(.8,.7,.6)),ncol=2)
fy <- c(.6,.5,.4)
Phi <- matrix(c(1,0,.5,0,1,.4,0,0,0),ncol=3)
R <- sim.structure(fx,Phi,fy)
cor.plot(R$model) #show it graphically
simp <- sim.simplex()
#show the simplex structure using cor.plot
cor.plot(simp,colors=TRUE)Run the code above in your browser using DataLab