MCMCirtHier1d {MCMCpack} | 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
Normal-Inverse Gamma hierarchical prior on subject ideal points (abilities). The user
supplies item-response 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.
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
.
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=.01, c0=.001, d0=.001, ab0=0, AB0=.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 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 |
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
eigenvalue-eigenvector decomposition of the agreement score matrix
formed from the |
a.start |
The starting values for the a 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 a. 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. |
b.start |
The starting values for the b 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 b. 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
regression coefficients that predict the means of ideal points
theta. This can either
be a scalar or a column vector with length equal to the
number of covariates. 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
linear regression of the covariates on (either provided or
generated) |
b0 |
The prior mean of beta. Can be either a scalar or a vector of length equal to the number of subject covariates. If a scalar all means with be set to the passed value. |
B0 |
The prior precision of beta. This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of beta. A default proper but diffuse value of .01 ensures finite marginal likelihood for model comparison. A value of 0 is equivalent to an improper uniform prior for beta. |
c0 |
c0/2 is the shape parameter for the inverse Gamma prior on sigma^2 (the variance of theta). The amount of information in the inverse Gamma prior is something like that from c0 pseudo-observations. |
d0 |
d0/2 is the scale parameter for the inverse Gamma prior on sigma^2 (the variance of theta). In constructing the inverse Gamma prior, d0 acts like the sum of squared errors from the c0 pseudo-observations. |
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
second-level 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 alpha for the residual variance in the latent data (Liu and Wu 1999). Default = TRUE |
px_a0 |
Prior shape parameter for the inverse-gamma distribution on alpha, the residual variance of the latent data. Default=10. |
px_b0 |
Prior scale parameter for the inverse-gamma distribution on alpha, the residual variance of the latent data. Default = 10 |
... |
further arguments to be passed |
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
(second-level) subject
ability predictors, an Inverse-Gamma draw for the (second-level)
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 * J). We assume that the choice is dictated by an unobserved utility:
z_ij = -a_i + b_i*theta_j + epsilon_ij
Where the errors are assumed to be
distributed standard Normal. This constitutes the measurement or level-1
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
second-level coefficients.
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.
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, malecki@wustl.edu, http://malecki.wustl.edu.
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): 1-21. http://www.jstatsoft.org/v42/i09/.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.wustl.edu.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.
Douglas Rivers. 2004. “Identification of Multidimensional Item-Response 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 ## 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)