R: Markov Chain Monte Carlo for Robust K-Dimensional Item...

MCMCirtKdRob {MCMCpack} | R Documentation

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

### Description

This function generates a posterior sample from a Robust K-dimensional item response theory (IRT) model with logistic link, independent standard normal priors on the subject abilities (ideal points), and independent 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

    MCMCirtKdRob(
      datamatrix,
      dimensions,
      item.constraints = list(),
      ability.constraints = list(),
      burnin = 500,
      mcmc = 5000,
      thin = 1,
      interval.method = "step",
      theta.w = 0.5,
      theta.mp = 4,
      alphabeta.w = 1,
      alphabeta.mp = 4,
      delta0.w = NA,
      delta0.mp = 3,
      delta1.w = NA,
      delta1.mp = 3,
      verbose = FALSE,
      seed = NA,
      theta.start = NA,
      alphabeta.start = NA,
      delta0.start = NA,
      delta1.start = NA,
      b0 = 0,
      B0 = 0,
      k0 = 0.1,
      k1 = 0.1,
      c0 = 1,
      d0 = 1,
      c1 = 1,
      d1 = 1,
      store.item = TRUE,
      store.ability = FALSE,
      drop.constant.items = TRUE,
      ...
    )

### Arguments

`datamatrix` | The matrix of data. Must be 0, 1, or missing values. It is of dimensionality subjects by items.
`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 rowname to be negative. If datamatrix is a matrix without row names defaults names of “V1", “V2", ... , etc will be used. In a K-dimensional model, the first item parameter for item i is the difficulty parameter (α_i), the second item parameter is the discrimation parameter on dimension 1 (β_{i,1}), the third item parameter is the discrimation parameter on dimension 2 (β_{i,2}), ..., and the (K+1)th item parameter is the discrimation parameter on dimension K (β_{i,K}). The item difficulty parameters (α) should generally not be constrained.
`ability.constraints` | List of lists specifying possible equality or simple inequality constraints on the ability parameters. A typical entry in the list has one of three forms: `colname=list(d,c)` which will constrain the dth ability parameter for the subject named colname to be equal to c, `colname=list(d,"+")` which will constrain the dth ability parameter for the subject named colname to be positive, and `colname=list(d, "-")` which will constrain the dth ability parameter for the subject named colname to be negative. If datamatrix is a matrix without column names defaults names of “V1", “V2", ... , etc will be used.
`burnin` | The number of burn-in iterations for the sampler.
`mcmc` | The number of iterations for the sampler after burn-in.
`thin` | The thinning interval used in the simulation. The number of iterations must be divisible by this value.
`interval.method` | Method for finding the slicing interval. Can be equal to either `step` in which case the stepping out algorithm of Neal (2003) is used or `doubling` in which case the doubling procedure of Neal (2003) is used. The stepping out method tends to be faster on a per-iteration basis as it typically requires few function calls. The doubling method expands the initial interval more quickly which makes the Markov chain mix somewhat more quickly– at least in some situations.
`theta.w` | The initial width of the slice sampling interval for each ability parameter (the elements of θ)
`theta.mp` | The parameter governing the maximum possible width of the slice interval for each ability parameter (the elements of θ). If `interval.method="step"` then the maximum width is `theta.w * theta.mp`. If `interval.method="doubling"` then the maximum width is `theta.w * 2^theta.mp`.
`alphabeta.w` | The initial width of the slice sampling interval for each item parameter (the elements of α and β)
`alphabeta.mp` | The parameter governing the maximum possible width of the slice interval for each item parameters (the elements of α and β). If `interval.method="step"` then the maximum width is `alphabeta.w * alphabeta.mp`. If `interval.method="doubling"` then the maximum width is `alphabeta.w * 2^alphabeta.mp`.
`delta0.w` | The initial width of the slice sampling interval for δ_0
`delta0.mp` | The parameter governing the maximum possible width of the slice interval for δ_0. If `interval.method="step"` then the maximum width is `delta0.w * delta0.mp`. If `interval.method="doubling"` then the maximum width is `delta0.w * 2^delta0.mp`.
`delta1.w` | The initial width of the slice sampling interval for δ_1
`delta1.mp` | The parameter governing the maximum possible width of the slice interval for δ_1. If `interval.method="step"` then the maximum width is `delta1.w * delta1.mp`. If `interval.method="doubling"` then the maximum width is `delta1.w * 2^delta1.mp`.
`verbose` | A switch which determines whether or not the progress of the sampler is printed to the screen. If verbose > 0, the iteration number with be printed to the screen every verbose'th iteration.
`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 ability parameters θ. Can be either a scalar or a matrix with number of rows equal to the number of subjects and number of columns equal to the dimension K of the latent space. If `theta.start=NA` then starting values will be chosen that are 0 for unconstrained subjects, -0.5 for subjects with negative inequality constraints and 0.5 for subjects with positive inequality constraints.
`alphabeta.start` | The starting values for the α and β 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) \times 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.
`delta0.start` | The starting value for the δ_0 parameter.
`delta1.start` | The starting value for the δ_1 parameter.
`b0` | The prior means of the α and β 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) \times items.
`k0` | δ_0 is constrained to lie in the interval between 0 and `k0`.
`k1` | δ_1 is constrained to lie in the interval between 0 and `k1`.
`c0` | Parameter governing the prior for δ_0. δ_0 divided by `k0` is assumed to be follow a beta distribution with first parameter `c0`.
`d0` | Parameter governing the prior for δ_0. δ_0 divided by `k0` is assumed to be follow a beta distribution with second parameter `d0`.
`c1` | Parameter governing the prior for δ_1. δ_1 divided by `k1` is assumed to be follow a beta distribution with first parameter `c1`.
`d1` | Parameter governing the prior for δ_1. δ_1 divided by `k1` is assumed to be follow a beta distribution with second parameter `d1`.
`store.item` | A switch that determines whether or not to store the item parameters for posterior analysis. NOTE: This typically 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.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

`MCMCirtKdRob` simulates from the posterior using the slice sampling algorithm of Neal (2003). 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 (K \times 1), and that each item has a scalar difficulty parameter α_i and discrimination parameter β_i (K \times 1). The observed choice by subject j on item i is the observed data matrix which is (I \times J).

The probability that subject j answers item i correctly is assumed to be:

π_{ij} = δ_0 + (1 - δ_0 - δ_1) / (1+\exp(α_i - β_i θ_j))

This model was discussed in Bafumi et al. (2005).

We assume the following priors. For the subject abilities (ideal points) we assume independent standard Normal priors:

θ_{j,k} \sim \mathcal{N}(0,1)

These cannot be changed by the user. For each item parameter, we assume independent Normal priors:

≤ft[α_i, β_i \right]' \sim \mathcal{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. We also assume δ_0 / k_0 \sim \mathcal{B}eta(c_0, d_0) and δ_1 / k_1 \sim \mathcal{B}eta(c_1, d_1).

The model is identified by constraints on the item parameters and / or ability parameters. See Rivers (2004) for a discussion of identification of IRT models.

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.

### References

James H. Albert. 1992. “Bayesian Estimation of Normal Ogive Item Response Curves Using Gibbs Sampling." Journal of Educational Statistics. 17: 251-269.

Joseph Bafumi, Andrew Gelman, David K. Park, and Noah Kaplan. 2005. “Practical Issues in Implementing and Understanding Bayesian Ideal Point Estimation.” Political Analysis.

Joshua Clinton, Simon Jackman, and Douglas Rivers. 2004. “The Statistical Analysis of Roll Call Data." American Political Science Review. 98: 355-370.

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 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.

Radford Neal. 2003. “Slice Sampling” (with discussion). Annals of Statistics, 31: 705-767.

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`, `MCMCirt1d`, `MCMCirtKd`

### Examples

       ## Not run:
       ## Court example with ability (ideal point) and
       ##  item (case) constraints
       data(SupremeCourt)
       post1 <- MCMCirtKdRob(t(SupremeCourt), dimensions=1,
                       burnin=500, mcmc=5000, thin=1,
                       B0=.25, store.item=TRUE, store.ability=TRUE,
                       ability.constraints=list("Thomas"=list(1,"+"),
                       "Stevens"=list(1,-4)),
                       item.constraints=list("1"=list(2,"-")),
                       verbose=50)
       plot(post1)
       summary(post1)

       ## Senate example with ability (ideal point) constraints
       data(Senate)
       namestring <- as.character(Senate$member)
       namestring[78] <- "CHAFEE1"
       namestring[79] <- "CHAFEE2"
       namestring[59] <- "SMITH.NH"
       namestring[74] <- "SMITH.OR"
       rownames(Senate) <- namestring
       post2 <- MCMCirtKdRob(Senate[,6:677], dimensions=1,
                             burnin=1000, mcmc=5000, thin=1,
                             ability.constraints=list("KENNEDY"=list(1,-4),
                                      "HELMS"=list(1, 4), "ASHCROFT"=list(1,"+"),
                                      "BOXER"=list(1,"-"), "KERRY"=list(1,"-"),
                                      "HATCH"=list(1,"+")),
                             B0=0.1, store.ability=TRUE, store.item=FALSE,
                             verbose=5, k0=0.15, k1=0.15,
                             delta0.start=0.13, delta1.start=0.13)

       plot(post2)
       summary(post2)

    ## End(Not run)

[Package MCMCpack version 1.6-3 Index]