cfa {lazy.fa}

R Documentation

Simple Iterative Solution of Confirmatory Factor Analysis
Sigma = Lambda %*% Phi %*% t(Lambda) + Psi
with Prescribed Zeros in Lambda.

Description

Simple Iterative Solution of Confirmatory Factor Analysis
Sigma = Lambda %*% Phi %*% t(Lambda) + Psi
with Prescribed Zeros in Lambda.

Usage

cfa(
  S,
  ndim = 3,
  n = 1,
  Lambda0 = NULL,
  psi0 = NULL,
  Phi0 = NULL,
  Lambda = NULL,
  psi = NULL,
  Phi = NULL,
  estLambda = 1,
  estpsi = 1,
  estPhi = 0,
  nodiag = 1,
  maxiter = 1000,
  epsd = 1e-07,
  minpsi = 0.01,
  maxPhi = 0.9,
  derivN = 1,
  SQUAREM = 3,
  nSQUAREM = 1,
  minalpha = -99,
  maxalpha = 1,
  always = 1,
  reset1 = 1,
  reset2 = 2,
  print = 1
)

Arguments

S	Input dispersion matrix
ndim	# of dimensions
n	# of observations
Lambda0	Lambda matrix to specify the fixed elements. The elements of Lambda corresponding to the NA elements of this matrix will be estimated.
psi0	psi vector to specify the fixed elements. The elements of psi corresponding to the NA elements of this vector will be estimated.
Phi0	Phi matrix to specify the fixed elements. The elements of Phi corresponding to the NA elements of this matrix will be estimated.
Lambda	Initial value of Lambda. (cannot have NAs.)
psi	Initial value of c(Psi). (cannot have NAs.)
Phi	Initial value of Phi. (cannot have NAs.)
estLambda	= 0 to skip the estimation of Lambda.
estpsi	= 0 to skip the estimatio of Psi.
estPhi	= 1 to estimate the off diagonal elements of Phi.
nodiag	= 0 to estimate the diagonal elements of Phi. (Always use nodiag=1.)
maxiter	Maximum # of iterations
epsd	The convergence criterion for the difference of param values.
minpsi	The minimum value of psi.
maxPhi	The maximum value of off diag elements of Phi.
derivN	= 0 to skip the calculation of the final derivative numerically.
SQUAREM	= 3 : See the help of iSQUAREM in lazy.accel package.
nSQUAREM	>= 1 # of iterations a which SQUAREM begins.
minalpha	= -999 : See the help of iSQUAREM in lazy.accel package.
maxalpha	= -1 : See the help of iSQUAREM in lazy.accel package.
always	= 1 : See the help of iSQUAREM in lazy.accel package.
reset1	= 0 : See the help of iSQUAREM in lazy.accel package.
reset2	= 1 : See the help of iSQUAREM in lazy.accel package.
print	= 1 to print the result.

Details

This function minimizes -2/n times the log likelihood ratio criterion:
crit = tr(S%*%invSigma)+log(det(Sigma)) - log(det(S)) - nvar
where S is the observed dispersion matrix, Sigma is the model dispersion matrix, invSigma is the inverse of Sigma, nvar is the # of variables, and n is the # of observations.
Note that n is not used during the course of estimation.
The EM algorithm is used to minimize crit with some restrictions on the elements of the paramters.
When estLambda=1, either Lambda or Lambda0 must contain a valid initial. The same goes with psi and Phi.
In this version, Phi is scaled so that diag(Phi)=I.

Value

A list of
Lambda Factor Loadings
Psi Scalar error variance
Phi Factor correlation matrix
crit The value of criterion function.
nparam, aic # of parameters and aic
dcritN and dcritA The derivatives of the criterion.
dcritA and dcritN analutic and numeric derivative of the criterion function.

References

Rubin, D.B., Thayer, T.T.(1982) EM algorithms for ML factor analysis. Psychometrika 47, 69-76.

Examples

# Orthogonal Independent Cluster
seed=1701
set.seed(seed)

nvar=20
ndim0=2
ps=0.2
df=500
phi=0

A=gendatafa_A( nvar, ndim0, large=0.8,small=0.01, pc=0, sd=0.01 )$loadings

Phi=(1-phi)*diag(ndim0)+phi*matrix(1,ndim0,ndim0)
Sigma=A%*%Phi%*%t(A)+ps*diag(nvar)
dS=sqrt(diag(Sigma))
Sigma=diag(1/dS)%*%Sigma%*%diag(1/dS)
# Print(Sigma)
S=rWishart( 1, df, Sigma )
S=S[,,1]/df
dS=sqrt(diag(S))
S=diag(1/dS)%*%S%*%diag(1/dS)


ndim=2
Lambda0=gendatafa_A( nvar, ndim, large=1,small=0, pc=0, sd=0 )$loadings
Lambda0[Lambda0==1]=NA

res2=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1 )
res2=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1, estPhi=1 )



# Oblique Independent Cluster
seed=1701
set.seed(seed)

nvar=20
ndim0=4
ps=0.2
df=500
phi=0.3

A=gendatafa_A( nvar, ndim0, large=0.8,small=0.01, pc=0, sd=0.01 )$loadings

Phi=(1-phi)*diag(ndim0)+phi*matrix(1,ndim0,ndim0)
Sigma=A%*%Phi%*%t(A)+ps*diag(nvar)
dS=sqrt(diag(Sigma))
Sigma=diag(1/dS)%*%Sigma%*%diag(1/dS)
# Print(Sigma)
S=rWishart( 1, df, Sigma )
S=S[,,1]/df
dS=sqrt(diag(S))
S=diag(1/dS)%*%S%*%diag(1/dS)


ndim=3
Lambda0=gendatafa_A( nvar, ndim, large=1,small=0, pc=0, sd=0 )$loadings
Lambda0[Lambda0==1]=NA

res2=cfa( S, ndim=ndim, n=df, Lambda0=Lambda0, print=1 )
res2=cfa( S, ndim=ndim, n=df, Lambda0=Lambda0, print=1, estPhi=1 )






# Bifactor Model
seed=1701
set.seed(seed)

nvar=20
ndim0=3
ps=0.2
df=500
phi=0

A=gendatafa_A( nvar, ndim0-1, large=0.8,small=0.01, pc=0, sd=0.01 )$loadings
A=cbind(0.9,A)

Phi=(1-phi)*diag(ndim0)+phi*matrix(1,ndim0,ndim0)
Sigma=A%*%Phi%*%t(A)+ps*diag(nvar)
dS=sqrt(diag(Sigma))
Sigma=diag(1/dS)%*%Sigma%*%diag(1/dS)
# Print(Sigma)
S=rWishart( 1, df, Sigma )
S=S[,,1]/df
dS=sqrt(diag(S))
S=diag(1/dS)%*%S%*%diag(1/dS)


ndim=3
Lambda0=gendatafa_A( nvar, ndim-1, large=1,small=0, pc=0, sd=0 )$loadings
Lambda0=cbind(1,Lambda0)
Lambda0[Lambda0==1]=NA

Phi0=diag(ndim)
Phi0[3,2]=NA; Phi0[2,3]=NA
Phi1=diag(ndim)
Phi1[2:ndim,1]=NA; Phi1[1,2:ndim]=NA
Phi2=diag(ndim)
Phi2[3,2]=0.4; Phi2[2,3]=0.4


# crit  =   0.2926539
res1=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1, estPhi=0 )
# crit  =   0.288043
res11=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1, estPhi=0, Phi0=Phi2 )
# crit  =   0.2891818
res2=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1, estPhi=1 )
# crit  =   0.3323131
res3=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1, estPhi=1, Phi0=Phi0 )
# crit  =    0.2873195
res4=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1, estPhi=1
, Phi0=Phi0, Phi=res11$Phi, Lambda=res11$Lambda, psi=res11$psi )
# crit  =   0.28719
res5=cfa( S, ndim=ndim, Lambda0=Lambda0, print=1, estPhi=1, Phi0=NULL
    , Phi=res11$Phi, Lambda=res11$Lambda, psi=res11$psi, maxiter=4000 )

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