MCMCirtKdRob | 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,
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 = 0.1,
k1 = 0.1,
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.mp |
The parameter governing the maximum possible width of the
slice interval for each ability parameter (the elements of
If |
alphabeta.w |
The initial width of the slice sampling interval for each
item parameter (the elements of |
alphabeta.mp |
The parameter governing the maximum possible width of
the slice interval for each item parameters (the elements of
If |
delta0.w |
The initial width of the slice sampling interval for
|
delta0.mp |
The parameter governing the maximum possible width of the
slice interval for |
delta1.w |
The initial width of the slice sampling interval for
|
delta1.mp |
The parameter governing the maximum possible width of the
slice interval for |
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
|
alphabeta.start |
The starting values for the |
delta0.start |
The starting value for the |
delta1.start |
The starting value for the |
b0 |
The prior means of the |
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 |
k0 |
|
k1 |
|
c0 |
Parameter governing the prior for |
d0 |
Parameter governing the prior for |
c1 |
Parameter governing the prior for |
d1 |
Parameter governing the prior for |
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 \times
1)
, and that each item has a scalar difficulty parameter
\alpha_i
and discrimination parameter \beta_i
(K \times 1)
. The observed choice by subject j
on
item i
is the observed data matrix which is (I \times J)
.
The probability that subject j
answers item i
correctly is
assumed to be:
\pi_{ij} = \delta_0 + (1 - \delta_0 - \delta_1) / (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} \sim \mathcal{N}(0,1)
These cannot be changed by the user. For each item parameter, we assume independent Normal priors:
\left[\alpha_i, \beta_i \right]' \sim \mathcal{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 \delta_0 / k_0 \sim
\mathcal{B}eta(c_0, d_0)
and
\delta_1 / k_1 \sim
\mathcal{B}eta(c_1, d_1)
.
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. \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.wustl.edu.s3-website-us-east-1.amazonaws.com/.
Radford Neal. 2003. “Slice Sampling” (with discussion). Annals of Statistics, 31: 705-767.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. “Output Analysis and Diagnostics for MCMC (CODA)”, R News. 6(1): 7-11. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf.
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)
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