MCMCirt1d  R Documentation 
This function generates a sample from the posterior distribution of a one dimensional 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.
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 = 0.25,
store.item = FALSE,
store.ability = TRUE,
drop.constant.items = TRUE,
...
)
datamatrix 
The matrix of data. Must be 0, 1, or missing values. The
rows of 
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: 
burnin 
The number of burnin 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 
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

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 eigenvalueeigenvector
decomposition of the aggreement score matrix formed from the

alpha.start 
The starting values for the 
beta.start 
The starting values for the 
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 
AB0 
The prior precision of 
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.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 
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 
If you are interested in fitting Kdimensional item response theory models,
or would rather identify the model by placing constraints on the item
parameters, please see MCMCirtKd
.
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 \times
J)
. We assume that the choice is dictated by an unobserved
utility:
z_{i,j} = \alpha_i + \beta_i \theta_j + \varepsilon_{i,j}
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 \sim \mathcal{N}(t_{0},T_{0}^{1})
For the item parameters, the prior is:
\left[\alpha_i, \beta_i \right]' \sim \mathcal{N}_2 (ab_{0},AB_{0}^{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.
An mcmc object that contains the sample from the posterior distribution. This object can be summarized by functions provided by the coda package.
James H. Albert. 1992. “Bayesian Estimation of Normal Ogive Item Response Curves Using Gibbs Sampling." Journal of Educational Statistics. 17: 251269.
Joshua Clinton, Simon Jackman, and Douglas Rivers. 2004. “The Statistical Analysis of Roll Call Data." American Political Science Review. 98: 355370.
Valen E. Johnson and James H. Albert. 1999. “Ordinal Data Modeling." Springer: New York.
Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. “MCMCpack: Markov Chain Monte Carlo in R.”, Journal of Statistical Software. 42(9): 121. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v042.i09")}.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.lsa.umich.edu.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. “Output Analysis and Diagnostics for MCMC (CODA)”, R News. 6(1): 711. https://CRAN.Rproject.org/doc/Rnews/Rnews_20061.pdf.
Douglas Rivers. 2004. “Identification of Multidimensional ItemResponse Models." Stanford University, typescript.
plot.mcmc
,summary.mcmc
,
MCMCirtKd
## 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)
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