MCMCirtKd              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 _K-_D_i_m_e_n_s_i_o_n_a_l _I_t_e_m _R_e_s_p_o_n_s_e _T_h_e_o_r_y
_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 K-dimensional item response theory (IRT) model, with standard
     normal priors on the subject abilities (ideal points), and normal
     priors on the item parameters.  The user supplies data and priors,
     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:

     MCMCirtKd(datamatrix, dimensions, item.constraints=list(),
        burnin = 1000, mcmc = 10000, thin=1, verbose = 0, seed = NA,
        alphabeta.start = NA, b0 = 0, B0=0, store.item = FALSE,
        store.ability=TRUE, drop.constant.items=TRUE, ... )  

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

datamatrix: The matrix of data.  Must be 0, 1, or missing values.   It
          is of dimensionality subjects by items.

dimensions: The number of dimensions in the latent space.

item.constraints: List of lists specifying possible equality or simple
          inequality constraints on the item parameters. A typical
          entry in the list has one of three forms: 'rowname=list(d,c)'
          which will constrain the dth item parameter for the item
          named rowname to be equal to c, 'rowname=list(d,"+")' which
          will constrain the dth item parameter for the item named
          rowname to be positive, and'rowname=list(d, "-")' which will
          constrain the dth item parameter for the item named rowname
          to be negative. If x is a matrix without row names defaults
          names of ``V1", ``V2", ... , etc will be used. In a K
          dimensional model, the first item parameter for item i is the
          difficulty parameter (alpha_i), the second item parameter is
          the discrimation parameter on dimension 1 (beta_{i,1}), the
          third item parameter is the discrimation parameter on
          dimension 2 (beta_{i,2}), ...,  and the (K+1)th item
          parameter is the discrimation parameter on dimension K
          (beta_{i,1}).  The item difficulty parameters (alpha) should
          generally not be constrained.  

  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 then every 'verbose'th iteration will be 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.

alphabeta.start: The starting values for the alpha and beta difficulty
          and discrimination parameters. If 'alphabeta.start' is set to
          a scalar the starting value for all unconstrained item
          parameters will be set to that scalar. If 'alphabeta.start'
          is a matrix of dimension (K+1) x items then the
          'alphabeta.start' matrix is used as the starting values
          (except for equality-constrained elements). If
          'alphabeta.start' is set to 'NA' (the default) then starting
          values for unconstrained elements are set to values generated
          from a series of proportional odds logistic regression fits,
          and starting values for inequality constrained elements are
          set to either 1.0 or -1.0 depending on the nature of the
          constraints. 

      b0: The prior means of the alpha and beta difficulty and
          discrimination parameters, stacked for all items. If a scalar
          is passed, it is used as the prior mean for all items.

      B0: The prior precisions (inverse variances) of the independent
          normal prior on the item parameters. Can be either a scalar
          or a matrix of dimension (K+1) x items.

store.item: A switch that determines whether or not to store the item
          parameters for posterior analysis.  _NOTE: In applications
          with many items this takes an enormous amount of memory. If
          you have many items and want to want to store the item
          parameters you may want to thin the chain heavily_.  By
          default, the item parameters are not stored.

store.ability: A switch that determines whether or not to store the
          subject abilities for posterior analysis. _NOTE: In
          applications with many subjects this takes an enormous amount
          of memory. If you have many subjects and want to want to
          store the ability parameters you may want to thin the chain
          heavily_. By default, the ability parameters are all stored.

drop.constant.items: A switch that determines whether or not items 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:

     'MCMCirtKd' simulates from the posterior distribution using
     standard Gibbs sampling using data augmentation (a normal draw for
     the subject abilities, a multivariate normal draw for the item
     parameters, and a truncated normal draw for the latent utilities).
     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.

     The default number of burnin and mcmc iterations is much smaller
     than the typical default values in MCMCpack.  This is because
     fitting this model is extremely computationally expensive.  It
     does not mean that this small of a number of scans will yield good
     estimates. The priors of this model need to be proper for
     identification purposes.  The user is asked to provide prior means
     and precisions _(not variances)_ for the item parameters and the
     subject parameters.

     The model takes the following form.  We assume that each subject
     has an ability (ideal point) denoted theta_j (K x 1), and that
     each item has a difficulty parameter alpha_i and discrimination
     parameter beta_i (K x 1). The observed choice by subject j on item
     i is the observed data matrix which is (I * J).  We assume that
     the choice is dictated by an unobserved utility: 

            z_ij = alpha_i + beta_i'*theta_j + epsilon_ij

     Where the epsilon_ijs are assumed to be distributed standard
     normal.  The parameters of interest are the subject abilities
     (ideal points) and the item parameters.

     We assume the following priors.  For the subject abilities (ideal
     points) we assume independent standard normal priors:

                         theta_j,k ~ N(0, 1)

     These cannot be changed by the user. For each item parameter, we
     assume independent normal priors:

              [alpha_i beta_i]' ~ N_(K+1) (b_0,i, B_0,i)

     Where B_0,i is a diagonal matrix. One can specify a separate prior
     mean and precision for each item parameter.

     The model is identified by the constraints on the item parameters
     (see Jackman 2001).  The user cannot place constraints on the
     subect abilities.  This identification scheme differs from that in
     'MCMCirt1d', which uses constraints on the subject abilities to
     identify the model. In our experience, using subject  ability
     constraints for models in greater than one dimension does not work
      particularly well.

     As is the case with all measurement models, make sure that you
     have plenty of free memory, especially when storing the item
     parameters.

_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:

     James H. Albert. 1992. ``Bayesian Estimation of Normal Ogive Item
     Response  Curves Using Gibbs Sampling." _Journal of Educational
     Statistics_.   17: 251-269.

     Joshua Clinton, Simon Jackman, and Douglas Rivers. 2000. ``The
     Statistical  Analysis of Legislative Behavior: A Unified
     Approach." Paper presented at  the Annual Meeting of the Political
     Methodology Society.

     Simon Jackman. 2001. ``Multidimensional Analysis of Roll Call Data
     via Bayesian Simulation.'' _Political Analysis._ 9: 227-241.

     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/>.

     Douglas Rivers.  2004.  ``Identification of Multidimensional
     Item-Response Models."  Stanford University, typescript.

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

     'plot.mcmc','summary.mcmc', 'MCMCirt1d', 'MCMCordfactanal'

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

        ## Not run: 
        data(SupremeCourt)
        # note that the rownames (the item names) are "1", "2", etc
        posterior1 <- MCMCirtKd(t(SupremeCourt), dimensions=1,
                        burnin=5000, mcmc=50000, thin=10,
                        B0=.25, store.item=TRUE,
                        item.constraints=list("1"=list(2,"-")))
        plot(posterior1)
        summary(posterior1)

        data(Senate)
        Sen.rollcalls <- Senate[,6:677]
        posterior2 <- MCMCirtKd(Sen.rollcalls, dimensions=2,
                        burnin=5000, mcmc=50000, thin=10,
                        item.constraints=list(rc2=list(2,"-"), rc2=c(3,0),
                                              rc3=list(3,"-")),
                        B0=.25)
        plot(posterior2)
        summary(posterior2)
        ## End(Not run)

