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: rowname=list(d,c)
which will constrain the dth item parameter for the item named
rowname to be equal to c, rowname=list(d,"+") which will
constrain the dth item parameter for the item named rowname to be
positive, and rowname=list(d, "-") which will constrain the dth
item parameter for the item named rowname to be negative. If
datamatrix is a
matrix without row names defaults names of ``V1", ``V2", ... , etc
will be used. In a K-dimensional model,
the first item parameter for
item i is the difficulty parameter (alpha_i),
the second item parameter is the discrimation parameter on dimension
1 (beta_{i,1}), the third item parameter is the
discrimation parameter on dimension 2
(beta_{i,2}), ..., and the (K+1)th
item parameter
is the discrimation parameter on dimension K
(beta_{i,K}).
The item difficulty parameters (alpha) should
generally not be constrained.
|
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: colname=list(d,c)
which will constrain the dth ability parameter for the subject named
colname to be equal to c, colname=list(d,"+") which will
constrain the dth ability parameter for the subject named colname to be
positive, and colname=list(d, "-") which will constrain the dth
ability parameter for the subject named colname to be negative. If
datamatrix is a
matrix without column names defaults names of ``V1", ``V2", ... , etc
will be used. |
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 step in which case the stepping out
algorithm of Neal (2003) is used or doubling in which case the
doubling procedure of Neal (2003) is used. The stepping out method
tends to be faster on a per-iteration basis as it typically requires few
function calls. The doubling method expands the initial interval
more quickly which makes the Markov chain mix somewhat more
quickly– at least in some situations. |
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 interval.method="step" then the maximum
width is theta.w * theta.mp .
If interval.method="doubling"
then the maximum width is theta.w * 2^theta.mp . |
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
interval.method="step" then the maximum width is
alphabeta.w * alphabeta.mp .
If interval.method="doubling"
then the maximum width is alphabeta.w * 2^alphabeta.mp . |
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
interval.method="step" then the maximum width is
delta0.w * delta0.mp . If interval.method="doubling"
then the maximum width is delta0.w * 2^delta0.mp . |
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
interval.method="step" then the maximum width is
delta1.w * delta1.mp . If interval.method="doubling"
then the maximum width is delta1.w * 2^delta1.mp . |
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 rep(12345,6) is used). The second element of
list is a positive substream number. See the MCMCpack
specification for more details. |
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
theta.start=NA then starting values will be chosen that are
0 for unconstrained subjects, -0.5 for subjects with negative
inequality constraints and 0.5 for subjects with positive inequality
constraints. |
alphabeta.start |
The starting values for the
alpha and beta difficulty and
discrimination parameters. If alphabeta.start is set to a
scalar the starting value for all unconstrained item parameters will
be set to that scalar. If alphabeta.start is a matrix of
dimension (K+1) x items then the
alphabeta.start matrix is used as the starting values (except
for equality-constrained elements). If alphabeta.start is set
to NA (the default) then starting values for unconstrained
elements are set to values generated from a series of proportional
odds logistic regression fits, and starting values for inequality
constrained elements are set to either 1.0 or -1.0 depending on the
nature of the constraints. |
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 k0 . |
k1 |
delta1 is constrained to lie in the interval
between 0 and k1 . |
c0 |
Parameter governing the prior for
delta0. delta0 divided by k0 is
assumed to be follow a beta distribution with first parameter
c0 . |
d0 |
Parameter governing the prior for
delta0. delta0 divided by k0 is
assumed to be follow a beta distribution with second parameter
d0 . |
c1 |
Parameter governing the prior for
delta1. delta1 divided by k1 is
assumed to be follow a beta distribution with first parameter
c1 . |
d1 |
Parameter governing the prior for
delta1. delta1 divided by k1 is
assumed to be follow a beta distribution with second parameter
d1 . |
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 |
MCMCirtKd
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. 2000. ``The Statistical Analysis of Legislative Behavior: A Unified Approach." Paper presented at the Annual Meeting of the Political Methodology Society.
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 Daniel Pemstein. 2004. 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)