MCMCfactanal {MCMCpack} | R Documentation |

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.

MCMCfactanal(x, factors, lambda.constraints=list(), data=parent.environment(), burnin = 1000, mcmc = 20000, thin=1, verbose = FALSE, 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 TRUE, the iteration number and the factor loadings and uniquenesses are printed to the screen. |

`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 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.

An mcmc object that contains the posterior density sample. 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=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 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=FALSE, store.scores=FALSE, a0=1, b0=0.15, burnin=5000, mcmc=50000, thin=20) ## End(Not run)

[Package *MCMCpack* version 0.5-2 Index]