MCMCirtKdRob {MCMCpack} | R Documentation |
This function generates a posterior sample from a Robust K-dimensional item response theory (IRT) model with logistic link, independent standard normal priors on the subject abilities (ideal points), and independent 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.
MCMCirtKdRob(datamatrix, dimensions, item.constraints=list(), ability.constraints=list(), burnin = 500, mcmc = 5000, thin=1, interval.method="step", theta.w=0.5, theta.mp=4, alphabeta.w=1.0, alphabeta.mp=4, delta0.w=NA, delta0.mp=3, delta1.w=NA, delta1.mp=3, verbose = FALSE, seed = NA, theta.start = NA, alphabeta.start = NA, delta0.start = NA, delta1.start = NA, b0 = 0, B0=0, k0=.1, k1=.1, c0=1, d0=1, c1=1, d1=1, store.item=TRUE, store.ability=FALSE, drop.constant.items=TRUE, ... )
datamatrix |
The matrix of data. Must be 0, 1, or missing values. It is of dimensionality subjects by items. |
dimensions |
The number of dimensions in the latent space. |
item.constraints |
List of lists specifying possible equality
or simple inequality constraints on the item parameters. A typical
entry in the list has one of three forms: |
ability.constraints |
List of lists specifying possible equality
or simple inequality constraints on the ability parameters. A typical
entry in the list has one of three forms: |
burnin |
The number of burn-in iterations for the sampler. |
mcmc |
The number of iterations for the sampler after burn-in. |
thin |
The thinning interval used in the simulation. The number of iterations must be divisible by this value. |
interval.method |
Method for finding the slicing interval. Can
be equal to either |
theta.w |
The initial width of the slice sampling interval for each ability parameter (the elements of theta) |
theta.mp |
The parameter governing the maximum possible width of
the slice interval for each ability parameter (the elements of
theta). If If |
alphabeta.w |
The initial width of the slice sampling interval for each item parameter (the elements of alpha and beta) |
alphabeta.mp |
The parameter governing the maximum possible width of
the slice interval for each item parameters (the elements of
alpha and beta). If
If |
delta0.w |
The initial width of the slice sampling interval for delta0 |
delta0.mp |
The parameter governing the maximum possible width of
the slice interval for delta0. If
|
delta1.w |
The initial width of the slice sampling interval for delta1 |
delta1.mp |
The parameter governing the maximum possible width of
the slice interval for delta1. If
|
verbose |
A switch which determines whether or not the progress of the sampler is printed to the screen. If verbose > 0, the iteration number with be printed to the screen every verbose'th iteration. |
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 ability parameters
theta. Can be either a scalar or a matrix with
number of rows equal to the number of subjects and number of
columns equal to the dimension K of the latent space. If
|
alphabeta.start |
The starting values for the
alpha and beta difficulty and
discrimination parameters. If |
delta0.start |
The starting value for the delta0 parameter. |
delta1.start |
The starting value for the delta1 parameter. |
b0 |
The prior means of the alpha and beta difficulty and discrimination parameters, stacked for all items. If a scalar is passed, it is used as the prior mean for all items. |
B0 |
The prior precisions (inverse variances) of the independent Normal prior on the item parameters. Can be either a scalar or a matrix of dimension (K+1) x items. |
k0 |
delta0 is constrained to lie in the interval
between 0 and |
k1 |
delta1 is constrained to lie in the interval
between 0 and |
c0 |
Parameter governing the prior for
delta0. delta0 divided by |
d0 |
Parameter governing the prior for
delta0. delta0 divided by |
c1 |
Parameter governing the prior for
delta1. delta1 divided by |
d1 |
Parameter governing the prior for
delta1. delta1 divided by |
store.item |
A switch that determines whether or not to store the item parameters for posterior analysis. 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. |
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. |
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 |
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 theta_j (K x 1), and that each item has a scalar difficulty parameter alpha_i and discrimination parameter beta_i (K x 1). The observed choice by subject j on item i is the observed data matrix which is (I * J).
The probability that subject j answers item i correctly is assumed to be:
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:
theta_j,k ~ N(0, 1)
These cannot be changed by the user. For each item parameter, we assume independent Normal priors:
[alpha_i beta_i]' ~ N_(K+1) (b_0,i, B_0,i)
Where B_0,i is a diagonal matrix. One can specify a separate prior mean and precision for each item parameter. We also assume delta0/k0 ~ Beta(c0, d0) and 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.
An mcmc object that contains the posterior sample. 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: 251-269.
Joseph Bafumi, Andrew Gelman, David K. Park, and Noah Kaplan. 2005. “Practical Issues in Implementing and Understanding Bayesian Ideal Point Estimation.” Political Analysis.
Joshua Clinton, Simon Jackman, and Douglas Rivers. 2004. “The Statistical Analysis of Roll Call Data." American Political Science Review. 98: 355-370.
Simon Jackman. 2001. “Multidimensional Analysis of Roll Call Data via Bayesian Simulation.” Political Analysis. 9: 227-241.
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): 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.
Radford Neal. 2003. “Slice Sampling” (with discussion). Annals of Statistics, 31: 705-767.
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
,
MCMCirt1d
, MCMCirtKd
## Not run: ## 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) ## End(Not run)