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

Description

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.

Usage

MCMCfactanal(
  x,
  factors,
  lambda.constraints = list(),
  data = NULL,
  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,
  ...
)

Arguments

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 verboseth 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.

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.

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.

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.

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

Value

An mcmc object that contains the sample from the posterior distribution. 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 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.

References

Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. ``MCMCpack: Markov Chain Monte Carlo in R.'', Journal of Statistical Software. 42(9): 1-21. http://www.jstatsoft.org/v42/i09/.

Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.lsa.umich.edu.

Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. ``Output Analysis and Diagnostics for MCMC (CODA)'', R News. 6(1): 7-11. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf.

Examples

Run this code

# NOT RUN {
   
# }
# 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)
   
# }
# NOT RUN {
# }

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