MCMCfactanal            package:MCMCpack            R Documentation

_M_a_r_k_o_v _C_h_a_i_n _M_o_n_t_e _C_a_r_l_o _f_o_r _N_o_r_m_a_l _T_h_e_o_r_y _F_a_c_t_o_r _A_n_a_l_y_s_i_s _M_o_d_e_l

_D_e_s_c_r_i_p_t_i_o_n:

     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.

_U_s_a_g_e:

     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, ... )
      

_A_r_g_u_m_e_n_t_s:

       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

_D_e_t_a_i_l_s:

     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.

_V_a_l_u_e:

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

_R_e_f_e_r_e_n_c_e_s:

     Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein.  2004.  
     _Scythe Statistical Library 1.0._ <URL: http://scythe.wustl.edu>.

     Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002.
     _Output Analysis and Diagnostics for MCMC (CODA)_. <URL:
     http://www-fis.iarc.fr/coda/>.

_S_e_e _A_l_s_o:

     'plot.mcmc','summary.mcmc','factanal'

_E_x_a_m_p_l_e_s:

        ## 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)

