MCMCirtHier1d  R Documentation 
This function generates a sample from the posterior distribution of a one
dimensional item response theory (IRT) model, with multivariate Normal
priors on the item parameters, and a NormalInverse Gamma hierarchical prior
on subject ideal points (abilities). The user supplies itemresponse data,
subject covariates, and priors. Note that this identification strategy
obviates the constraints used on theta in MCMCirt1d
.
A sample from the posterior distribution is returned as an mcmc object,
which can be subsequently analyzed with functions provided in the coda
package.
MCMCirtHier1d(
datamatrix,
Xjdata,
burnin = 1000,
mcmc = 20000,
thin = 1,
verbose = 0,
seed = NA,
theta.start = NA,
a.start = NA,
b.start = NA,
beta.start = NA,
b0 = 0,
B0 = 0.01,
c0 = 0.001,
d0 = 0.001,
ab0 = 0,
AB0 = 0.25,
store.item = FALSE,
store.ability = TRUE,
drop.constant.items = TRUE,
marginal.likelihood = c("none", "Chib95"),
px = TRUE,
px_a0 = 10,
px_b0 = 10,
...
)
datamatrix 
The matrix of data. Must be 0, 1, or missing values. The
rows of 
Xjdata 
A 
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 agreement score matrix formed from the

a.start 
The starting values for the 
b.start 
The starting values for the 
beta.start 
The starting values for the 
b0 
The prior mean of 
B0 
The prior precision of 
c0 

d0 

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. 
marginal.likelihood 
Should the marginal likelihood of the
secondlevel model on ideal points be calculated using the method of Chib
(1995)? It is stored as an attribute of the posterior 
px 
Use Parameter Expansion to reduce autocorrelation in the chain?
PX introduces an unidentified parameter 
px_a0 
Prior shape parameter for the inversegamma distribution on

px_b0 
Prior scale parameter for the inversegamma distribution on

... 
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
.
MCMCirtHier1d
simulates from the posterior distribution using
standard Gibbs sampling using data augmentation (a Normal draw for the
subject abilities, a multivariate Normal draw for (secondlevel) subject
ability predictors, an InverseGamma draw for the (secondlevel) variance of
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 a_i
and discrimination parameter
b_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.
This constitutes the measurement or level1 model. The subject abilities
(ideal points) are modeled by a second level Normal linear predictor for
subject covariates Xjdata
, with common variance
\sigma^2
. The parameters of interest are the subject
abilities (ideal points), item parameters, and secondlevel coefficients.
We assume the following priors. For the subject abilities (ideal points):
\theta_j \sim \mathcal{N}(\mu_{\theta} ,T_{0}^{1})
For the item parameters, the prior is:
\left[a_i, b_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. If marginal.likelihood = "Chib95"
the object will have
attribute logmarglike
.
Michael Malecki, mike@crunch.io, https://github.com/malecki.
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.
Liu, Jun S. and Ying Nian Wu. 1999. “Parameter Expansion for Data Augmentation.” Journal of the American Statistical Association 94: 1264–1274.
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:
data(SupremeCourt)
Xjdata < data.frame(presparty= c(1,1,0,1,1,1,1,0,0),
sex= c(0,0,1,0,0,0,0,1,0))
## Parameter Expansion reduces autocorrelation.
posterior1 < MCMCirtHier1d(t(SupremeCourt),
burnin=50000, mcmc=10000, thin=20,
verbose=10000,
Xjdata=Xjdata,
marginal.likelihood="Chib95",
px=TRUE)
## But, you can always turn it off.
posterior2 < MCMCirtHier1d(t(SupremeCourt),
burnin=50000, mcmc=10000, thin=20,
verbose=10000,
Xjdata=Xjdata,
#marginal.likelihood="Chib95",
px=FALSE)
## Note that the hierarchical model has greater autocorrelation than
## the naive IRT model.
posterior0 < MCMCirt1d(t(SupremeCourt),
theta.constraints=list(Scalia="+", Ginsburg=""),
B0.alpha=.2, B0.beta=.2,
burnin=50000, mcmc=100000, thin=100, verbose=10000,
store.item=FALSE)
## Randomly 10% Missing  this affects the expansion parameter, increasing
## the variance of the (unidentified) latent parameter alpha.
scMiss < SupremeCourt
scMiss[matrix(as.logical(rbinom(nrow(SupremeCourt)*ncol(SupremeCourt), 1, .1)),
dim(SupremeCourt))] < NA
posterior1.miss < MCMCirtHier1d(t(scMiss),
burnin=80000, mcmc=10000, thin=20,
verbose=10000,
Xjdata=Xjdata,
marginal.likelihood="Chib95",
px=TRUE)
## End(Not run)
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