R: Markov Chain Monte Carlo for Dynamic One Dimensional Item...

MCMCdynamicIRT1d_b {MCMCpack} | R Documentation

## Markov Chain Monte Carlo for Dynamic One Dimensional Item Response Theory Model

### Description

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.

### Usage

    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,
      ...
    )

### 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.
`item.time.map` | 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.
`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 `verbose`th 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 α 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 β 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.
`tau2.start` | 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`.
`a0` | A vector containing the prior mean of each of the difficulty parameters α. 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.
`A0` | A vector containing the prior precision (inverse variance) of each of the difficulty parameters α. 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.
`b0` | A vector containing the prior mean of each of the discrimination parameters β. 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.
`B0` | A vector containing the prior precision (inverse variance) of each of the discrimination parameters β. 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.
`c0` | c_{0/2} is the shape parameter for the inverse Gamma prior on τ^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 τ^2 is not estimated– the values in `tau2.start` are used throughout the sampling.
`d0` | d_{0/2} is the scale parameter for the inverse Gamma prior on τ^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 τ^2 is not estimated– the values in `tau2.start` are used throughout the sampling.
`e0` | 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.
`E0` | 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.
`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.
`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.
`...` | further arguments to be passed

### Details

`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 θ_{j,t} (where j indexes subjects and t indexes time periods) and that each item has a difficulty parameter α_i and discrimination parameter β_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} = -α_i + β_i θ_{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):

θ_{j,t} \sim \mathcal{N}(θ_{j,t-1}, τ^2_j)

with

θ_{j,0} \sim \mathcal{N}(e0, E0)

.

The evolution variance has the following prior:

τ^2_j \sim \mathcal{IG}(c0/2, d0/2)

.

For the item parameters in the standard model, the prior is:

α_i \sim \mathcal{N}(a0, A0^{-1})

and

β_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.

### Value

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

### Author(s)

Kevin M. Quinn

### References

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. doi: 10.18637/jss.v042.i09.

### See Also

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

### Examples

      ## Not run:
    	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)

    ## End(Not run)

[Package MCMCpack version 1.6-3 Index]