MCMCpack (version 0.4-5)

MCMCfactanal: Markov chain Monte Carlo for Normal Theory Factor Analysis Model

Description

This function generates a posterior density sample from 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 density is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.

Usage

MCMCfactanal(x, factors, lambda.constraints=list(),
             data=list(), burnin = 1000, mcmc = 10000,
             thin=5, verbose = FALSE, seed = 0,
             lambda.start = NA, psi.start = NA,
             l0=0, L0=0, a0=0.001, b0=0.001,
             store.scores = FALSE, std.var=TRUE, ... )

Arguments

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

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 TRUE, the iteration number and the factor loadings and uniquenesses are printed to the screen.

seed

The seed for the random number generator. The code uses the Mersenne Twister, which requires an integer as an input. If nothing is provided, the Scythe default seed is used.

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

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 num

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. Can be either a scalar or a $k$-vector.

b0

Controls the scale of the inverse Gamma prior on the uniquenesses. 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 num

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

Value

• An mcmc object that contains the posterior density sample. This object can be summarized by functions provided by the coda package.

Details

The model takes the following form:

$$x_i = \Lambda \phi_i + \epsilon_i$$ $$\epsilon_i \sim \mathcal{N}(0,\Psi)$$

where $x_i$ is the $k$-vector of observed variables specific to observation $i$, $\Lambda$ is the $k \times d$ 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:

$$\Lambda_{ij} \sim \mathcal{N}(l_{0_{ij}}, L_{0_{ij}}^{-1}), i=1,\ldots,k, j=1,\ldots,d$$

$$\phi_i \sim \mathcal{N}(0, I), i=1,\dots,n$$

$$\Psi_{ii} \sim \mathcal{IG}(a_{0_i}/2, b_{0_i}/2), i=1,\ldots,k$$ MCMCfactanal simulates from the posterior density 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 density sample.

References

Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2003. Scythe Statistical Library 0.4. 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/.

See Also

plot.mcmc,summary.mcmc,factanal

Examples

### 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=FALSE, 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
   Lambda <- matrix(runif(45,-.5,.5), 15, 3)
   Psi <- diag(1 - apply(Lambda ^2, 1, sum))
   Sigma <- Lambda %*% t(Lambda) + Psi 
   Y <- t(t(chol(Sigma)) %*% matrix(rnorm(500*15), 15, 500))

   posterior <- MCMCfactanal(Y, factors=3,
                    lambda.constraints=list(V1=c(2,0),
                       V1=c(3,0), V2=c(3,0), V3=list(1,"+"),
                       V3=list(2,"+"), V3=list(3,"+")),
                    verbose=FALSE)
   plot(posterior)
   summary(posterior)