MCMCirtKd {MCMCpack} R Documentation

## Markov chain Monte Carlo for K-Dimensional Item Response Theory Model

### Description

This function generates a posterior density sample from a K-dimensional item response theory (IRT) model, with standard Normal priors on the subject abilities (ideal points), and Normal priors on the item parameters. The user supplies data and priors, and a sample from the posterior density is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.

### Usage

```MCMCirtKd(datamatrix, dimensions, item.constraints=list(),
burnin = 1000, mcmc = 10000, thin=1, verbose = FALSE, seed = NA,
alphabeta.start = NA, b0 = 0, B0=0, store.item = FALSE,
store.ability=TRUE, drop.constantvars=TRUE, ... )  ```

### Arguments

 `datamatrix` The matrix of data. Must be 0, 1, or missing values. It is of dimensionality items by subjects. `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 varname to be negative. If x is a matrix without row names defaults names of ``V1", ``V2", ... , etc will be used. In a d 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 (β_{i,1}{beta_{i,1}}), the third item parameter is the discrimation parameter on dimension 2 (β_{i,2}{beta_{i,2}}), ..., and the (d+1)th item parameter is the discrimation parameter on dimension d (β_{i,1}{beta_{i,1}}). The item difficulty parameters (alpha) should generally not be constrained. `burnin` The number of burn-in iterations for the sampler. `mcmc` The number of iterations for the sampler. `thin` The thinning interval used in the simulation. The number of 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 TRUE, the iteration number and the subject abilities (ideal points) are 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. `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. `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. `store.item` A switch that determines whether or not to store the item parameters for posterior analysis. NOTE: This 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.constantvars` A switch that determines whether or not items and subjects that have no variation should be deleted before fitting the model. Default = TRUE. `...` further arguments to be passed

### Details

`MCMCirtKd` simulates from the posterior density 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 density sample.

The default number of burnin and mcmc iterations is much smaller than the typical default values in MCMCpack. This is because fitting this model is extremely computationally expensive. It does not mean that this small of a number of scans will yield good estimates. If the verbose option is chosen, output will be printed to the screen every fifty iterations. The priors of this model need to be proper for identification purposes. The user is asked to provide prior means and precisions (not variances) for the item parameters and the subject parameters.

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 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). We assume that the choice is dictated by an unobserved utility:

z_ij = alpha_i + beta_i'*theta_j + epsilon_ij

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) 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.

The model is identified by the constraints on the item parameters (see Jackman 2001). The user cannot place constraints on the subect abilities. This identification scheme differs from that in `MCMCirt1d`, which uses a single directional constraint on one subject ability. However, in our experience, using subject ability constraints for models in greater than one dimension does not work particularly well.

### Value

An mcmc object that contains the posterior density sample. 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. 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.

Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.

`plot.mcmc`,`summary.mcmc`, `MCMCirt1d`, `MCMCordfactanal`

### Examples

```   ## Not run:
data(SupremeCourt)
# note that the rownames (the item names) are "1", "2", etc
posterior1 <- MCMCirtKd(SupremeCourt, dimensions=1,
burnin=5000, mcmc=50000, thin=10,
B0=.25, store.item=TRUE,
item.constraints=list("1"=list(2,"-")))
plot(posterior1)
summary(posterior1)

data(Senate)
rownames(Senate) <- Senate\$member
# note that we need to transpose the data to get
# the bills on the rows
posterior2 <- MCMCirtKd(t(Senate[,6:677]), dimensions=2,
burnin=5000, mcmc=50000, thin=10,
item.constraints=list(rc2=list(2,"-"), rc2=c(3,0),
rc3=list(3,"-")),
B0=.25)
plot(posterior2)
summary(posterior2)
## End(Not run)
```

[Package MCMCpack version 0.5-2 Index]