This function generates a sample from the posterior distribution of a dynamic one dimensional item response theory (IRT) model, with Normal random walk priors on the subject abilities (ideal points), and multivariate 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.
MCMCdynamicIRT1d_b(
datamatrix,
item.time.map,
theta.constraints = list(),
burnin = 1000,
mcmc = 20000,
thin = 1,
verbose = 0,
seed = NA,
theta.start = NA,
alpha.start = NA,
beta.start = NA,
tau2.start = 1,
a0 = 0,
A0 = 0.1,
b0 = 0,
B0 = 0.1,
c0 = -1,
d0 = -1,
e0 = 0,
E0 = 1,
store.ability = TRUE,
store.item = TRUE,
...
)MCMCdynamicIRT1d(
datamatrix,
item.time.map,
theta.constraints = list(),
burnin = 1000,
mcmc = 20000,
thin = 1,
verbose = 0,
seed = NA,
theta.start = NA,
alpha.start = NA,
beta.start = NA,
tau2.start = 1,
a0 = 0,
A0 = 0.1,
b0 = 0,
B0 = 0.1,
c0 = -1,
d0 = -1,
e0 = 0,
E0 = 1,
store.ability = TRUE,
store.item = TRUE,
...
)
An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
The matrix of data. Must be 0, 1, or missing
values. The rows of datamatrix correspond to subjects and
the columns correspond to items.
A vector that relates each item to a time
period. Each element of item.time.map gives the time
period of the corresponding column of datamatrix. It is
assumed that the minimum value of item.time.map is 1.
A list specifying possible simple equality
or inequality constraints on the ability parameters. A typical
entry in the list has one of three forms: varname=c which
will constrain the ability parameter for the subject named
varname to be equal to c, varname="+" which will
constrain the ability parameter for the subject named
varname to be positive, and varname="-" which will
constrain the ability parameter for the subject named
varname to be negative. If x is a matrix without row names
defaults names of ``V1",``V2", ... , etc will be used. See Rivers
(2003) for a thorough discussion of identification of IRT models.
The number of burn-in iterations for the sampler.
The number of Gibbs iterations for the sampler.
The thinning interval used in the simulation. The number of Gibbs iterations must be divisible by this value.
A switch which determines whether or not the
progress of the sampler is printed to the screen. If
verbose is greater than 0 then every verboseth
iteration will be printed to the screen.
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.
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 aggreement score matrix formed from the datamatrix.
The starting values for the \(\alpha\) 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 of the alphas. 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.
The starting values for the \(\beta\) 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 of the betas. 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.
The starting values for the evolution variances
(the variance of the random walk increments for the ability
parameters / ideal points. Order corresponds to the rows of
datamatrix.
A vector containing the prior mean of each of the
difficulty parameters \(\alpha\). Should have as many
elements as items / roll calls. Order corresponds to the columns
of datamatrix. If a scalar is passed it is assumed that
all elements of a0 are equal to the scalar.
A vector containing the prior precision (inverse
variance) of each of the difficulty parameters \(\alpha\).
Should have as many elements as items / roll calls. Order
corresponds to the columns of datamatrix. If a scalar is
passed it is assumed that all elements of A0 are equal to
the scalar.
A vector containing the prior mean of each of the
discrimination parameters \(\beta\). Should have as many
elements as items / roll calls. Order corresponds to the columns
of datamatrix. If a scalar is passed it is assumed that
all elements of b0 are equal to the scalar.
A vector containing the prior precision (inverse
variance) of each of the discrimination parameters
\(\beta\). Should have as many elements as items / roll
calls. Order corresponds to the columns of datamatrix. If
a scalar is passed it is assumed that all elements of B0
are equal to the scalar.
\(c_{0/2}\) is the shape parameter for the inverse
Gamma prior on \(\tau^2\) (the variance of the random walk
increments). The amount of information in the inverse Gamma
prior is something like that from \(c_0\)
pseudo-observations. c0 can be either a vector with an
element for each subject or a scalar. If c0 is negative
then \(\tau^2\) is not estimated-- the values in
tau2.start are used throughout the sampling.
\(d_{0/2}\) is the scale parameter for the inverse
Gamma prior on \(\tau^2\) (the variance of the random walk
increments). In constructing the inverse Gamma prior, \(d_0\)
acts like the sum of squared errors from the \(c_0\)
pseudo-observations. d0 can be either a vector with an
element for each subject or a scalar. If d0 is negative
then \(\tau^2\) is not estimated-- the values in
tau2.start are used throughout the sampling.
A vector containing the prior mean of the initial ability
parameter / ideal point for each subject. Should have as many
elements as subjects. Order corresponds to the rows of
datamatrix. If a scalar is passed it is assumed that all
elements of e0 are equal to the scalar.
A vector containing the prior variance of the initial
ability parameter / ideal point for each subject. Should have as
many elements as subjects. Order corresponds to the rows of
datamatrix. If a scalar is passed it is assumed that all
elements of E0 are equal to the scalar.
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
store.ability should only be TRUE if the chain is
thinned heavily, or for applications with a small number of
individuals. By default, the item parameters are stored.
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.item should only be
FALSE if the chain is thinned heavily, or for applications
with a small number of items. By default, the item parameters
are not stored.
further arguments to be passed
Kevin M. Quinn
MCMCdynamicIRT1d simulates from the posterior distribution
using the algorithm of Martin and Quinn (2002). 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 \(\theta_{j,t}\) (where \(j\) indexes subjects and \(t\) indexes time periods) and that each item has a difficulty parameter \(\alpha_i\) and discrimination parameter \(\beta_i\). 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:
$$z_{i,j,t} = -\alpha_i + \beta_i \theta_{j,t} + \varepsilon_{i,j,t}$$
Where the disturbances are assumed to be distributed standard Normal. The parameters of interest are the subject abilities (ideal points) and the item parameters.
We assume the following priors. For the subject abilities (ideal points):
$$\theta_{j,t} \sim \mathcal{N}(\theta_{j,t-1}, \tau^2_j)$$
with
$$\theta_{j,0} \sim \mathcal{N}(e0, E0)$$.
The evolution variance has the following prior:
$$\tau^2_j \sim \mathcal{IG}(c0/2, d0/2)$$.
For the item parameters in the standard model, the prior is:
$$\alpha_i \sim \mathcal{N}(a0, A0^{-1})$$
and
$$\beta_i \sim \mathcal{N}(b0, B0^{-1})$$.
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.
Andrew D. Martin and Kevin M. Quinn. 2002. "Dynamic Ideal Point Estimation via Markov Chain Monte Carlo for the U.S. Supreme Court, 1953-1999." Political Analysis. 10: 134-153. <doi:10.1093/pan/10.2.134>
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. tools:::Rd_expr_doi("10.18637/jss.v042.i09").
plot.mcmc,summary.mcmc,
MCMCirt1d
if (FALSE) {
data(Rehnquist)
## assign starting values
theta.start <- rep(0, 9)
theta.start[2] <- -3 ## Stevens
theta.start[7] <- 2 ## Thomas
out <- MCMCdynamicIRT1d(t(Rehnquist[,1:9]),
item.time.map=Rehnquist$time,
theta.start=theta.start,
mcmc=50000, burnin=20000, thin=5,
verbose=500, tau2.start=rep(0.1, 9),
e0=0, E0=1,
a0=0, A0=1,
b0=0, B0=1, c0=-1, d0=-1,
store.item=FALSE,
theta.constraints=list(Stevens="-", Thomas="+"))
summary(out)
}
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