ordrating              package:Ratings              R Documentation

_M_C_M_C _f_o_r _U_n_i_d_i_m_e_n_s_i_o_n_a_l _O_r_d_i_n_a_l _I_R_T _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 unidimensional ordinal item response theory (IRT) model, with
     Gaussian priors on the ability and 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:

     ordrating(Y, beta.constraint = NULL, theta.neg.index = NULL, 
                  theta.pos.index = NULL, vinva = 0.2, vinvb = 0.2, ma = 0, 
                  mb = 1, theta.start = NULL, gamma.start = NULL, 
                  burnin = 1000, mcmc = 10000, thin = 1, tune = 1, 
                  verbose = 0, seed = NA)

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

       Y: Matrix of data to be analyzed. Entries must be integers from 
          1, ..., C or 'NA' where C  is the number of ordinal
          categories. Items are on the rows and  subjects are on the
          columns. 

beta.constraint: Possible inequality constraint on all  beta
          parameters. 'beta.constraint = NULL' (default)  implies no
          constraint, 'beta.constraint > 0' implies all  beta values
          are constrained to be positive, and  'beta.constraint < 0'
          implies all beta values are  constrained to be negative.  

theta.neg.index: An index of theta that is constrained  to be negative.
          To constrain the theta for the subject whose  data is in the
          jth column of 'Y' to be negative one would set 
          'theta.neg.index = j'. 

theta.pos.index: An index of theta that is constrained  to be positive.
          To constrain the theta for the subject whose  data is in the
          jth column of 'Y' to be positive one would set 
          'theta.pos.index = j'. 

   vinva: The prior inverse variance for each alpha_r. 

   vinvb: The prior inverse variance for each beta_r. 

      ma: The prior mean for each alpha_r. 

      mb: The prior mean for each beta_r. 

theta.start: Starting values for theta. If 'NULL'  starting values are
          calculated based on the sample means of 'Y'. 

gamma.start: Starting values for gamma. If 'NULL' starting values are
          chosen to be equally spaced between 1 and the number of
          ordinal categories C.  

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

    mcmc: The number of MCMC iterations for the sampler. 

    thin: The thinning interval used in the simulation.  The number of
          Gibbs iterations must be divisible by this value. 

    tune: The scalar tuning parameter for the Metropolis-Hastings
          sampling. 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 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. 

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

     Let r = 1, ..., R index items,  p = 1, ..., P index subjects, and 
     c = 1, ..., C index ordinal rating categories. 

     Y is an R x P matrix with elements in  {1, ..., C, NA}. 'NA'
     denotes missing data  that are assumed to be missing at random
     (MAR). 

     The distribution of Y is governed by a R x P matrix  of latent
     variables Y* and a series of cutpoints  gamma. 

     More specifically, 


          y*_{rp} = alpha_r + beta_r theta_p + epsilon_{rp}


     where  epsilon_{rp} ~iid N(0, 1). It is assumed that y_{rp = c} if
     and only if 


                  y*_{rp} in (gamma_{c-1}, gamma_c]

     . 

     The priors used for this model are that each alpha_r is iid
     Gaussian, each beta_r is iid Gaussian, each theta_p is standard
     normal, and the elements of gamma are improper uniform with all
     parameters assumed to be a priori independent.

     'ordrating' 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  gamma_1 is normalized to zero, and thus not  returned in
     the mcmc object.

     'ordrating' fits a model that is a special case of the model fit
     by 'MCMCordfactanal' in the 'MCMCpack' package. The primary
     differences are the types of identifying constraints employed, the
     dimensionality of the theta and beta parameters, and the
     computational speed. Because 'ordrating' fits a narrower class of
     models it can be optimized for speed much more effectively.

_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:

     Ho, Daniel E. and Kevin M. Quinn. forthcoming. "Improving the
     Presentation  and Interpretation of Online Ratings Data with
     Model-based Figures."  The American Statistician.   

     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.

     Kevin M. Quinn. 2004. ``Bayesian Factor Analysis for Mixed Ordinal
     and Continuous Responses.'' _Political Analysis_. 12: 338-353.

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

     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', 'MCMCordfactanal'

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

     ## Not run: 
     ## Mondo Times example from Ho & Quinn (nd).
     ## may have to increase stack limit to run this example on some machines

     data(Mondo)

     ord.out <- ordrating(Mondo, beta.constraint=1, tune=.035, 
                          ma=1, mb=-5, vinva=1, vinvb=0.05,
                          gamma.start=c(-300, 0, 1.5, 3.0, 4.5, 300),
                          thin=20, burnin=20000, mcmc=100000, verbose=1000)

     plot(ord.out)
     summary(ord.out) 



     ## subsetting the Mondo data to include only raters who rated 5 or more 
     ## outlets (should avoid any stacksize problems)

     Mondo.sub <- Mondo[apply(!is.na(Mondo), 1, sum) >= 5, ]
     ## also getting rid of outlets that are not rated now
     Mondo.sub <- Mondo.sub[,apply(is.na(Mondo.sub), 2, mean) != 1] 

     ord.out <- ordrating(Mondo.sub, beta.constraint=1, tune=.035, 
                          ma=1, mb=-5, vinva=1, vinvb=0.05,
                          gamma.start=c(-300, 0, 1.5, 3.0, 4.5, 300),
                          thin=20, burnin=20000, mcmc=100000, verbose=1000)

     plot(ord.out)
     summary(ord.out) 

     ## End(Not run)

