MCMCirt1d {MCMCpack}

Markov Chain Monte Carlo for One Dimensional Item Response Theory Model

Description

This function generates a sample from the posterior distribution of a one dimensional item response theory (IRT) model, with Normal 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.

Usage

MCMCirt1d(
  datamatrix,
  theta.constraints = list(),
  burnin = 1000,
  mcmc = 20000,
  thin = 1,
  verbose = 0,
  seed = NA,
  theta.start = NA,
  alpha.start = NA,
  beta.start = NA,
  t0 = 0,
  T0 = 1,
  ab0 = 0,
  AB0 = 0.25,
  store.item = FALSE,
  store.ability = TRUE,
  drop.constant.items = TRUE,
  ...
)

Arguments

datamatrix

The matrix of data. Must be 0, 1, or missing values. The rows of datamatrix correspond to subjects and the columns correspond to items.

theta.constraints

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.

burnin

The number of burn-in iterations for the sampler.

mcmc

The number of Gibbs iterations for the sampler.

thin

The thinning interval used in the simulation. The number of Gibbs iterations must be divisible by this value.

verbose

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.

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 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.

alpha.start

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.

beta.start

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.

t0

A scalar parameter giving the prior mean of the subject abilities (ideal points).

T0

A scalar parameter giving the prior precision (inverse variance) of the subject abilities (ideal points).

ab0

The prior mean of (alpha, beta). 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.

AB0

The prior precision of (alpha, beta).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.

store.item

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.

store.ability

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.

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

Details

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.

MCMCirt1d simulates from the posterior distribution using standard Gibbs sampling using data augmentation (a Normal draw for the 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 \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} = -\alpha_i + \beta_i \theta_j + \varepsilon_{i,j}

Where the errors 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 \sim \mathcal{N}(t_{0},T_{0}^{-1})

For the item parameters, the prior is:

\left[\alpha_i, \beta_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.

Value

An mcmc object that contains the sample from the posterior distribution. This object can be summarized by functions provided by the coda package.

References

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.

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. doi:10.18637/jss.v042.i09.

Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.lsa.umich.edu.

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.

See Also

plot.mcmc,summary.mcmc, MCMCirtKd

Examples

   ## Not run: 
   ## US Supreme Court Example with inequality constraints
   data(SupremeCourt)
   posterior1 <- MCMCirt1d(t(SupremeCourt),
                   theta.constraints=list(Scalia="+", Ginsburg="-"),
                   B0.alpha=.2, B0.beta=.2,
                   burnin=500, mcmc=100000, thin=20, verbose=500,
                   store.item=TRUE)
   geweke.diag(posterior1)
   plot(posterior1)
   summary(posterior1)

   ## US Senate Example with equality constraints
   data(Senate)
   Sen.rollcalls <- Senate[,6:677]
   posterior2 <- MCMCirt1d(Sen.rollcalls,
                    theta.constraints=list(KENNEDY=-2, HELMS=2),
                    burnin=2000, mcmc=100000, thin=20, verbose=500)
   geweke.diag(posterior2)
   plot(posterior2)
   summary(posterior2)
   
## End(Not run)