MCMCirt1d              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 _O_n_e _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 one dimentional item response theory (IRT) model, with Normal
     priors on the subject abilities (ideal points), and multivariate
     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.

     If you are interested in fitting  K-dimensional item response
     theory models, or would rather identify the model by placing
     constraints on the item parameters, please see 'MCMCirtKd'.

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

     MCMCirt1d(datamatrix, theta.constraints=list(), burnin = 1000,
        mcmc = 20000, thin=1, verbose = 0, seed = NA, theta.start = NA, 
        alpha.start = NA, beta.start = NA, t0 = 0, T0 = 1, ab0=0, AB0=.25,
        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.   The
          rows of 'datamatrix' correspond to subjects and the columns
          correspond to items.

theta.constraints: A list specifying possible simple equality or
          inequality constraints on the ability parameters. A typical
          entry in the list has one of three forms: 'varname=c' which
          will constrain the ability parameter for the subject named
          'varname' to be equal to c, 'varname="+"' which will
          constrain the ability parameter for the subject named
          'varname' to be positive, and  'varname="-"' which will
          constrain the ability parameter for the subject named
          'varname' to be negative. If x is a matrix without row names
          defaults names of ``V1",``V2", ... , etc will be used. See
          Rivers (2003) for a thorough discussion of identification of
          IRT models.

  burnin: The number of burn-in iterations for the sampler.

    mcmc: The number of Gibbs iterations for the sampler.

    thin: The thinning interval used in the simulation.  The number of
          Gibbs 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.

theta.start: The starting values for the subject abilities (ideal
          points). This can either be a scalar or a column vector with
          dimension equal to the number of voters.   If this takes a
          scalar value, then that value will serve as the starting
          value for all of the thetas.  The default value of NA will
          choose the starting values based on an eigenvalue-eigenvector
          decomposition of the aggreement score matrix formed from the
          'datamatrix'.

alpha.start: The starting values for the alpha difficulty parameters.
          This can either be a scalar or a column vector with dimension
          equal to the number of items.   If this takes a scalar value,
          then that value will serve as the starting value for all of
          the alphas. The default value of NA will set the starting
          values based on a series of probit regressions that condition
          on the starting values of theta.

beta.start: The starting values for the beta discrimination parameters.
          This can either be a scalar or a column vector with dimension
          equal to the number of items.   If this takes a scalar value,
          then that value will serve as the starting value for all of
          the betas.  The default value of NA will set the starting
          values based on a series of probit regressions that condition
          on the starting values of theta.

      t0: A scalar parameter giving the prior mean of the subject
          abilities (ideal points).

      T0: A scalar parameter giving the prior precision (inverse
          variance) of the subject abilities (ideal points).

     ab0: The prior mean of '(alpha, beta)'. Can be either a scalar or
          a 2-vector. If a scalar both means will be set to the passed
          value. The prior mean is assumed to be the same across all
          items.

     AB0: The prior precision of '(alpha, beta)'.This can either be
          ascalar or a 2 by 2 matrix. If this takes a scalar value,
          then that value times an identity matrix serves as the prior
          precision. The prior precision is assumed to be the same
          across all items.

store.item: A switch that determines whether or not to store the item
          parameters for posterior analysis.  _NOTE: In situations with
          many items storing the item parameters takes an enormous
          amount of memory, so 'store.item' should only be 'FALSE' if
          the chain is thinned heavily, or for applications with a
          small number of items_. By default, the item parameters are
          not stored.

store.ability: A switch that determines whether or not to store the
          ability parameters for posterior analysis.  _NOTE: In
          situations with many individuals storing the ability
          parameters takes an enormous amount of memory, so
          'store.ability' should only be 'TRUE' if the chain is thinned
          heavily, or for applications with a small number of
          individuals_. By default, the item parameters are 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:

     'MCMCirt1d' 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 model takes the following form.  We assume that each subject
     has an subject ability (ideal point) denoted theta_j and that each
     item has a difficulty parameter alpha_i and discrimination
     parameter beta_i.  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 errors 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):

                       theta_j ~ N(t0, T0^{-1})

     For the item parameters, the prior is:

               [alpha_i beta_i]' ~ N_2 (ab0, AB0^{-1})


     The model is identified by the proper priors on the item
     parameters and constraints placed on the ability parameters.

     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 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:

     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. 2004. ``The
     Statistical  Analysis of Roll Call Data."  _American Political
     Science Review_. 98: 355-370.

     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', 'MCMCirtKd'

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

        ## Not run: 
        ## US Supreme Court Example with inequality constraints
        data(SupremeCourt)
        posterior1 <- MCMCirt1d(t(SupremeCourt),
                        theta.constraints=list(Scalia="+", Ginsburg="-"),
                        B0.alpha=.2, B0.beta=.2,
                        burnin=500, mcmc=100000, thin=20, verbose=500,
                        store.item=TRUE)
        geweke.diag(posterior1)
        plot(posterior1)
        summary(posterior1)

        ## US Senate Example with equality constraints
        data(Senate)
        Sen.rollcalls <- Senate[,6:677]
        posterior2 <- MCMCirt1d(Sen.rollcalls,
                         theta.constraints=list(KENNEDY=-2, HELMS=2),
                         burnin=2000, mcmc=100000, thin=20, verbose=500)
        geweke.diag(posterior2)
        plot(posterior2)
        summary(posterior2)
        ## End(Not run)

