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.
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 = 0.01,
c0 = 0.001,
d0 = 0.001,
ab0 = 0,
AB0 = 0.25,
store.item = FALSE,
store.ability = TRUE,
drop.constant.items = TRUE,
marginal.likelihood = c("none", "Chib95"),
px = TRUE,
px_a0 = 10,
px_b0 = 10,
...
)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.
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 data.frame containing second-level predictor
covariates for ideal points \(\theta\). Predictors are modeled as a
linear regression on the mean vector of \(\theta\); the posterior
sample contains regression coefficients \(\beta\) and common
variance \(\sigma^2\). 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 agreement score matrix formed from the
datamatrix.
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.
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.
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) theta.start.
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.
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.
\(c_0/2\) 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 \(c_0\) pseudo-observations.
\(d_0/2\) is the scale parameter for the inverse Gamma prior on \(\sigma^2\) (the variance of \(\theta\)). In constructing the inverse Gamma prior, \(d_0\) acts like the sum of squared errors from the \(c_0\) pseudo-observations.
The prior mean of (a, b). Can be either a scalar or a
2-vector. If a scalar both means will be set to the passed value. The prior
mean is assumed to be the same across all items.
The prior precision of (a, b).This can either be ascalar
or a 2 by 2 matrix. If this takes a scalar value, then that value times an
identity matrix serves as the prior precision. The prior precision is
assumed to be the same across all items.
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 TRUE if the chain is thinned
heavily, or for applications with a small number of items. By default, the
item parameters are not stored.
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, ability parameters are stored.
A switch that determines whether or not items that have no variation should be deleted before fitting the model. Default = TRUE.
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 mcmc object and
suitable for comparison using BayesFactor.
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
Prior shape parameter for the inverse-gamma distribution on \(alpha\), the residual variance of the latent data. Default=10.
Prior scale parameter for the inverse-gamma distribution on \(alpha\), the residual variance of the latent data. Default = 10
further arguments to be passed
Michael Malecki, mike@crunch.io, https://github.com/malecki.
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 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 \((I \times J)\). We assume that the choice is dictated by an unobserved utility:
$$z_{i,j} = -\alpha_i + \beta_i \theta_j + \varepsilon_{i,j}$$
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):
$$\theta_j \sim \mathcal{N}(\mu_{\theta} ,T_{0}^{-1})$$
For the item parameters, the prior is:
$$\left[a_i, b_i \right]' \sim \mathcal{N}_2 (ab_{0},AB_{0}^{-1})$$
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.
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. 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/.
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,
MCMCirtKd
if (FALSE) {
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% Missing -- this affects the expansion parameter, increasing
## 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)
}
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