MCMCfactanal {MCMCpack} | R Documentation |
This function generates a sample from the posterior distribution of a normal theory factor analysis model. Normal priors are assumed on the factor loadings and factor scores while inverse Gamma priors are assumed for the uniquenesses. The user supplies data and parameters for the prior distributions, and a sample from the posterior distribution is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.
MCMCfactanal(x, factors, lambda.constraints=list(), data=parent.frame(), burnin = 1000, mcmc = 20000, thin=1, verbose = 0, seed = NA, lambda.start = NA, psi.start = NA, l0=0, L0=0, a0=0.001, b0=0.001, store.scores = FALSE, std.var=TRUE, ... )
x |
Either a formula or a numeric matrix containing the manifest variables. |
factors |
The number of factors to be fitted. |
lambda.constraints |
List of lists specifying possible simple equality
or inequality constraints on the factor loadings. A typical
entry in the list has one of three forms: varname=list(d,c) which
will constrain the dth loading for the variable named varname to
be equal to c, varname=list(d,"+") which will constrain the dth
loading for the variable named varname to be positive, and
varname=list(d, "-") which will constrain the dth loading for the
variable named varname to be negative. If x is a matrix without
column names defaults names of ``V1",``V2", ... , etc will be
used. |
data |
A data frame. |
burnin |
The number of burn-in iterations for the sampler. |
mcmc |
The number of iterations for the sampler. |
thin |
The thinning interval used in the simulation. The number of iterations must be divisible by this value. |
verbose |
A switch which determines whether or not the progress of
the sampler is printed to the screen. If verbose is greater
than 0 the iteration number and
the factor loadings and uniquenesses are printed to the screen every
verbose th iteration. |
seed |
The seed for the random number generator. If NA, the Mersenne
Twister generator is used with default seed 12345; if an integer is
passed it is used to seed the Mersenne twister. The user can also
pass a list of length two to use the L'Ecuyer random number generator,
which is suitable for parallel computation. The first element of the
list is the L'Ecuyer seed, which is a vector of length six or NA (if NA
a default seed of rep(12345,6) is used). The second element of
list is a positive substream number. See the MCMCpack
specification for more details. |
lambda.start |
Starting values for the factor loading matrix
Lambda. If lambda.start is set to a scalar the starting value for
all unconstrained loadings will be set to that scalar. If
lambda.start is a matrix of the same dimensions as Lambda then the
lambda.start matrix is used as the starting values (except
for equality-constrained elements). If lambda.start is set to
NA (the default) then starting values for unconstrained
elements are set to 0, and starting values for inequality
constrained elements are set to either 0.5 or -0.5 depending on the
nature of the constraints. |
psi.start |
Starting values for the uniquenesses. If
psi.start is set to a scalar then the starting value for all
diagonal elements of Psi are set to this value. If
psi.start is a k-vector (where k is the
number of manifest variables) then the staring value of Psi
has psi.start on the main diagonal. If psi.start is
set to NA (the default) the starting values of all the
uniquenesses are set to 0.5. |
l0 |
The means of the independent Normal prior on the factor
loadings. Can be either a scalar or a matrix with the same
dimensions as Lambda . |
L0 |
The precisions (inverse variances) of the independent Normal
prior on the factor loadings. Can be either a scalar or a matrix with
the same dimensions as Lambda . |
a0 |
Controls the shape of the inverse Gamma prior on the
uniqueness. The actual shape parameter is set to a0/2 . Can be
either a scalar or a k-vector. |
b0 |
Controls the scale of the inverse Gamma prior on the
uniquenesses. The actual scale parameter is set to b0/2 . Can
be either a scalar or a k-vector. |
store.scores |
A switch that determines whether or not to store the factor scores for posterior analysis. NOTE: This takes an enormous amount of memory, so should only be used if the chain is thinned heavily, or for applications with a small number of observations. By default, the factor scores are not stored. |
std.var |
If TRUE (the default) the manifest variables are
rescaled to have zero mean and unit variance. Otherwise, the manifest
variables are rescaled to have zero mean but retain their observed
variances. |
... |
further arguments to be passed |
The model takes the following form:
x_i = Lambda phi_i + epsilon_i
epsilon_i ~ N(0, Psi)
where x_i is the k-vector of observed variables specific to observation i, Lambda is the k by d matrix of factor loadings, phi_i is the d-vector of latent factor scores, and Psi is a diagonal, positive definite matrix. Traditional factor analysis texts refer to the diagonal elements of Psi as uniquenesses.
The implementation used here assumes independent conjugate priors for each element of Lambda, each phi_i, and each diagonal element of Psi. More specifically we assume:
Lambda_ij ~ N(l0_ij, L0_ij^-1), i=1,...,k, j=1,...,d
phi_i ~ N(0, I), i=1,...,n
Psi_ii ~ IG(a0_i/2, b0_i/2), i=1,...,k
MCMCfactanal
simulates from the posterior distribution using
standard Gibbs sampling. The simulation proper is done in
compiled C++ code to maximize efficiency. Please consult the
coda documentation for a comprehensive list of functions that
can be used to analyze the posterior sample.
As is the case with all measurement models, make sure that you have plenty of free memory, especially when storing the scores.
An mcmc object that contains the sample from the posterior distribution. This object can be summarized by functions provided by the coda package.
Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2004. Scythe Statistical Library 1.0. http://scythe.wustl.edu.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.
plot.mcmc
,summary.mcmc
,factanal
## Not run: ### An example using the formula interface data(swiss) posterior <- MCMCfactanal(~Agriculture+Examination+Education+Catholic +Infant.Mortality, factors=2, lambda.constraints=list(Examination=list(1,"+"), Examination=list(2,"-"), Education=c(2,0), Infant.Mortality=c(1,0)), verbose=0, store.scores=FALSE, a0=1, b0=0.15, data=swiss, burnin=5000, mcmc=50000, thin=20) plot(posterior) summary(posterior) ### An example using the matrix interface Y <- cbind(swiss$Agriculture, swiss$Examination, swiss$Education, swiss$Catholic, swiss$Infant.Mortality) colnames(Y) <- c("Agriculture", "Examination", "Education", "Catholic", "Infant.Mortality") post <- MCMCfactanal(Y, factors=2, lambda.constraints=list(Examination=list(1,"+"), Examination=list(2,"-"), Education=c(2,0), Infant.Mortality=c(1,0)), verbose=0, store.scores=FALSE, a0=1, b0=0.15, burnin=5000, mcmc=50000, thin=20) ## End(Not run)