MCMCfactanal {MCMCpack} | R Documentation

## 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

`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

### Details

The model takes the following form:

x_i = Λ φ_i + ε_i

ε_i \sim \mathcal{N}(0,Ψ)

where x_i is the k-vector of observed variables specific to observation i, Λ is the k \times d matrix of factor loadings, φ_i is the d-vector of latent factor scores, and Ψ is a diagonal, positive definite matrix. Traditional factor analysis texts refer to the diagonal elements of Ψ as uniquenesses.

The implementation used here assumes independent conjugate priors for each element of Λ each φ_i, and each diagonal element of Ψ. More specifically we assume:

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

φ_i \sim \mathcal{N}(0, I), i=1,…,n

Ψ_{ii} \sim \mathcal{IG}(a_{0_i}/2, b_{0_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.

### Value

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

### 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. https://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.

### See Also

`plot.mcmc`,`summary.mcmc`,`factanal`

### Examples

       ## 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)

[Package MCMCpack version 1.5-0 Index]