mcmcirtkdrob.rd

使用R语言的马尔科夫链蒙特卡洛模拟（MCMC）源代码程序。

    \eqn{(K+1) \times items}{(K+1) x items}.}
  
  \item{k0}{\eqn{\delta_0}{delta0} is constrained to lie in the interval
    between 0 and \code{k0}.}

  \item{k1}{\eqn{\delta_1}{delta1} is constrained to lie in the interval
    between 0 and \code{k1}.}

  \item{c0}{Parameter governing the prior for
    \eqn{\delta_0}{delta0}. \eqn{\delta_0}{delta0} divided by \code{k0} is
    assumed to be follow a beta distribution with first parameter
    \code{c0}.}

  \item{d0}{Parameter governing the prior for
    \eqn{\delta_0}{delta0}. \eqn{\delta_0}{delta0} divided by \code{k0} is
    assumed to be follow a beta distribution with second parameter
    \code{d0}.}

  \item{c1}{Parameter governing the prior for
    \eqn{\delta_1}{delta1}. \eqn{\delta_1}{delta1} divided by \code{k1} is
    assumed to be follow a beta distribution with first parameter
    \code{c1}.}

  \item{d1}{Parameter governing the prior for
    \eqn{\delta_1}{delta1}. \eqn{\delta_1}{delta1} divided by \code{k1} is
    assumed to be follow a beta distribution with second parameter
    \code{d1}.}
  
  \item{store.item}{A switch that determines whether or not to
    store the item parameters for posterior analysis. 
    \emph{NOTE: This typically 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 items}.  By default, the
    item parameters are not stored.}
  
   \item{store.ability}{A switch that determines whether or not to store
    the subject abilities for posterior analysis.  By default, the
    item parameters are all stored.}

  \item{drop.constant.items}{A switch that determines whether or not
        items that have no variation
	should be deleted before fitting the model. Default = TRUE.}
  
    \item{...}{further arguments to be passed}       
}

\value{
   An mcmc object that contains the posterior sample.  This 
   object can be summarized by functions provided by the coda package.
}

\details{
  \code{MCMCirtKdRob} simulates from the posterior using
  the slice sampling algorithm of Neal (2003).
  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
  \eqn{\theta_j}{theta_j} \eqn{(K \times 1)}{(K x 1)},
  and that each item has a scalar difficulty
  parameter \eqn{\alpha_i}{alpha_i} and discrimination parameter
  \eqn{\beta_i}{beta_i} \eqn{(K \times 1)}{(K x 1)}.
  The observed choice by subject
  \eqn{j}{j} on item \eqn{i}{i} is the observed data matrix which
  is \eqn{(I \times J)}{(I * J)}.

  The probability that subject \eqn{j}{j} answers item \eqn{i}{i}
  correctly is assumed to be:

  \deqn{\pi_{ij} = \delta_0 + (1 - \delta_0 - \delta_1)
    /(1+\exp(\alpha_i - \beta_i \theta_j))}{pi_{ij} =
    delta0 + (1 - delta0 - delta1)  / (1 + exp(alpha_i - beta_i * theta_j))}

  This model was discussed in Bafumi et al. (2005). 
  
  We assume the following priors.  For the subject abilities (ideal points)
  we assume independent standard Normal priors:
  \deqn{\theta_{j,k} \sim \mathcal{N}(0,1)}{theta_j,k ~ N(0, 1)}
  These cannot be changed by the user.
  For each item parameter, we assume independent Normal priors:
  \deqn{\left[\alpha_i, \beta_i \right]' \sim \mathcal{N}_{(K+1)} 
  (b_{0,i},B_{0,i})}{[alpha_i beta_i]' ~ N_(K+1) (b_0,i, B_0,i)}
  Where \eqn{B_{0,i}}{B_0,i} is a diagonal matrix.
  One can specify a separate prior mean and precision
  for each item parameter. We also assume \eqn{\delta_0 / k_0 \sim
    \mathcal{B}eta(c_0, d_0)}{delta0/k0 ~ Beta(c0, d0)} and
  \eqn{\delta_1 / k_1 \sim
    \mathcal{B}eta(c_1, d_1)}{delta1/k1 ~ Beta(c1, d1)}. 
  
  The model is identified by constraints on the item parameters and / or
  ability parameters. See Rivers (2004) for a discussion of
  identification of IRT models.
    
  As is the case with all measurement models, make sure that you have plenty
  of free memory, especially when storing the item parameters.
}

\references{
   James H. Albert. 1992. ``Bayesian Estimation of Normal Ogive Item Response 
   Curves Using Gibbs Sampling." \emph{Journal of Educational Statistics}.  
   17: 251-269.

   Joseph Bafumi, Andrew Gelman, David K. Park, and Noah
   Kaplan. 2005. ``Practical Issues in Implementing and Understanding
   Bayesian Ideal Point Estimation.'' \emph{Political Analysis}. 
   
   Joshua Clinton, Simon Jackman, and Douglas Rivers. 2004. ``The Statistical 
   Analysis of Roll Call Data."  \emph{American Political Science Review}.
   98: 355-370.
   
   Simon Jackman. 2001. ``Multidimensional Analysis of Roll Call Data
   via Bayesian Simulation.'' \emph{Political Analysis.} 9: 227-241.
   
   Valen E. Johnson and James H. Albert. 1999. \emph{Ordinal Data Modeling}. 
   Springer: New York.

   Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin.  2007.  
   \emph{Scythe Statistical Library 1.0.} \url{http://scythe.wustl.edu}.

   Radford Neal. 2003. ``Slice Sampling'' (with discussion). \emph{Annals of
   Statistics}, 31: 705-767. 
  
   Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002.
   \emph{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.
}


\examples{
   \dontrun{
   ## Court example with ability (ideal point) and
   ##  item (case) constraints
   data(SupremeCourt)
   post1 <- MCMCirtKdRob(t(SupremeCourt), dimensions=1,
                   burnin=500, mcmc=5000, thin=1,
                   B0=.25, store.item=TRUE, store.ability=TRUE,
                   ability.constraints=list("Thomas"=list(1,"+"),
                   "Stevens"=list(1,-4)),
                   item.constraints=list("1"=list(2,"-")),
                   verbose=50)
   plot(post1)
   summary(post1)

   ## Senate example with ability (ideal point) constraints
   data(Senate)
   namestring <- as.character(Senate$member)
   namestring[78] <- "CHAFEE1"
   namestring[79] <- "CHAFEE2"
   namestring[59] <- "SMITH.NH"
   namestring[74] <- "SMITH.OR"
   rownames(Senate) <- namestring
   post2 <- MCMCirtKdRob(Senate[,6:677], dimensions=1,
                         burnin=1000, mcmc=5000, thin=1,
                         ability.constraints=list("KENNEDY"=list(1,-4),
                                  "HELMS"=list(1, 4), "ASHCROFT"=list(1,"+"),
                                  "BOXER"=list(1,"-"), "KERRY"=list(1,"-"),
                                  "HATCH"=list(1,"+")),
                         B0=0.1, store.ability=TRUE, store.item=FALSE,
                         verbose=5, k0=0.15, k1=0.15,
                         delta0.start=0.13, delta1.start=0.13)

   plot(post2)
   summary(post2)
   }
}

\keyword{models}

\seealso{\code{\link[coda]{plot.mcmc}},\code{\link[coda]{summary.mcmc}},
\code{\link[MCMCpack]{MCMCirt1d}}, \code{\link[MCMCpack]{MCMCirtKd}}
}