MCMCordfactanal           package:MCMCpack           R Documentation

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

     This function generates a sample from the posterior distribution
     of an ordinal data factor analysis model. Normal priors are
     assumed on the factor loadings and factor scores while improper
     uniform priors are assumed on the cutpoints. 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:

     MCMCordfactanal(x, factors, lambda.constraints=list(),
                     data=parent.frame(), burnin = 1000, mcmc = 20000,
                     thin=1, tune=NA, verbose = 0, seed = NA,
                     lambda.start = NA, l0=0, L0=0,
                     store.lambda=TRUE, store.scores=FALSE,
                     drop.constantvars=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 equality or
          simple 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. Note that, unlike 'MCMCfactanal', the
          Lambda matrix used here has 'factors'+1 columns. The first
          column of Lambda corresponds to negative item difficulty
          parameters and should generally not be constrained.  

    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.

    tune: The tuning parameter for the Metropolis-Hastings sampling.
          Can be either a scalar or a k-vector. Must be strictly
          positive.

 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 Metropolis-Hastings
          acceptance rate 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 in the first column of Lambda are
          based on the observed response pattern, the remaining
          unconstrained elements of Lambda are set to , and starting
          values for inequality constrained elements are set to either
          1.0 or -1.0 depending on the nature of the constraints.

      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'.

store.lambda: A switch that determines whether or not to store the
          factor loadings for posterior analysis. By default, the
          factor loadings are all stored.

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.

drop.constantvars: A switch that determines whether or not manifest
          variables that have no variation should be deleted before
          fitting the model. Default = TRUE.

     ...: further arguments to be passed

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

     The model takes the following form:

     Let 1=1,...,n index observations and j=1,...,K index response
     variables within an observation. The typical observed variable
     x_ij is ordinal with a total of C_j   categories. The distribution
     of X is governed by a N by K matrix of latent variables Xstar and
     a series of cutpoints gamma. Xstar is assumed to be generated
     according to:


                  xstar_i = Lambda phi_i + epsilon_i


                         epsilon_i ~ N(0, I)


     where xstar_i is the k-vector of latent variables specific to
     observation i, Lambda is the k by d matrix of factor loadings, and
     phi_i is the d-vector of latent factor scores. It is assumed that
     the first element of phi_i is equal to 1 for all i. 

     The probability that the jth variable in observation i takes the
     value c is:


 pi_ijc = pnorm(gamma_jc - Lambda'_j phi_i) - pnorm(gamma_j(c-1) - Lambda'_j phi_i)


     The implementation used here assumes independent conjugate priors
     for each element of Lambda and each phi_i. More specifically we
     assume:


        Lambda_ij ~ N(l0_ij,  L0_ij^-1), i=1,...,k, j=1,...,d



                   phi_i(2:d) ~ N(0, I), i=1,...,n


     The standard two-parameter item response theory model with probit
     link is a special case of the model sketched above. 

     'MCMCordfactanal' simulates from the posterior distribution using
     a Metropolis-Hastings within Gibbs sampling algorithm. The
     algorithm employed is based on work by Cowles (1996).  Note that
     the first element of phi_i is a 1. As a result, the first column
     of Lambda can be interpretated as item difficulty parameters. 
     Further, the first element  gamma_1 is normalized to zero, and
     thus not  returned in the mcmc object. 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 posterior sample.  This  object
     can be summarized by functions provided by the coda package.

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

     Shawn Treier and Simon Jackman. 2003. ``Democracy as a Latent
     Variable."  Paper presented at the Midwest Political Science
     Association Annual Meeting.

     M. K. Cowles. 1996. ``Accelerating Monte Carlo Markov Chain
     Convergence for Cumulative-link Generalized Linear Models."
     _Statistics and Computing._ 6: 101-110.

     Valen E. Johnson and James H. Albert. 1999. ``Ordinal Data
     Modeling."  Springer: New York.

     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', 'MCMCfactanal',
     'MCMCirt1d', 'MCMCirtKd'

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

        ## Not run: 
        data(painters)
        new.painters <- painters[,1:4]
        cuts <- apply(new.painters, 2, quantile, c(.25, .50, .75))
        for (i in 1:4){
           new.painters[new.painters[,i]<cuts[1,i],i] <- 100
          new.painters[new.painters[,i]<cuts[2,i],i] <- 200
          new.painters[new.painters[,i]<cuts[3,i],i] <- 300
          new.painters[new.painters[,i]<100,i] <- 400
        }

        posterior <- MCMCordfactanal(~Composition+Drawing+Colour+Expression,
                             data=new.painters, factors=1,
                             lambda.constraints=list(Drawing=list(2,"+")),
                             burnin=5000, mcmc=500000, thin=200, verbose=500,
                             L0=0.5, store.lambda=TRUE,
                             store.scores=TRUE, tune=1.2)
        plot(posterior)
        summary(posterior)
        ## End(Not run)

