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.
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,
...
)An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
The matrix of data. Must be 0, 1, or missing values. It is of dimensionality subjects by items.
The number of dimensions in the latent space.
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 (\(\alpha_i\)), the second item parameter
is the discrimation parameter on dimension 1 (\(\beta_{i,1}\)),
the third item parameter is the discrimation parameter on
dimension 2 (\(\beta_{i,2}\)), ..., and the \((K+1)\)th item
parameter is the discrimation parameter on dimension \(K\)
(\(\beta_{i,K}\)). The item difficulty parameters
(\(\alpha\)) should generally not be constrained.
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.
The number of burn-in iterations for the sampler.
The number of iterations for the sampler after burn-in.
The thinning interval used in the simulation. The number of iterations must be divisible by this value.
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.
The initial width of the slice sampling interval for each ability parameter (the elements of \(\theta\))
The parameter governing the maximum possible width of the
slice interval for each ability parameter (the elements of
\(\theta\)). 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.
The initial width of the slice sampling interval for each item parameter (the elements of \(\alpha\) and \(\beta\))
The parameter governing the maximum possible width of
the slice interval for each item parameters (the elements of
\(\alpha\) and \(\beta\)). 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.
The initial width of the slice sampling interval for \(\delta_0\)
The parameter governing the maximum possible width of the
slice interval for \(\delta_0\). 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.
The initial width of the slice sampling interval for \(\delta_1\)
The parameter governing the maximum possible width of the
slice interval for \(\delta_1\). 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.
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.
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 ability parameters
\(\theta\). 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.
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 \((K+1) \times
items\) 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.
The starting value for the \(\delta_0\) parameter.
The starting value for the \(\delta_1\) parameter.
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.
The prior precisions (inverse variances) of the independent Normal prior on the item parameters. Can be either a scalar or a matrix of dimension \((K+1) \times items\).
\(\delta_0\) is constrained to lie in the interval
between 0 and k0.
\(\delta_1\) is constrained to lie in the interval
between 0 and k1.
Parameter governing the prior for \(\delta_0\).
\(\delta_0\) divided by k0 is assumed to be follow a beta
distribution with first parameter c0.
Parameter governing the prior for \(\delta_0\).
\(\delta_0\) divided by k0 is assumed to be follow a beta
distribution with second parameter d0.
Parameter governing the prior for \(\delta_1\).
\(\delta_1\) divided by k1 is assumed to be follow a beta
distribution with first parameter c1.
Parameter governing the prior for \(\delta_1\).
\(\delta_1\) divided by k1 is assumed to be follow a beta
distribution with second parameter d1.
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.
A switch that determines whether or not to store the subject abilities for posterior analysis. By default, the item parameters are all stored.
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
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 \(\theta_j\) \((K \times 1)\), and that each item has a scalar difficulty parameter \(\alpha_i\) and discrimination parameter \(\beta_i\) \((K \times 1)\). 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:
$$\pi_{ij} = \delta_0 + (1 - \delta_0 - \delta_1) / (1+\exp(\alpha_i - \beta_i \theta_j))$$
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:
$$\theta_{j,k} \sim \mathcal{N}(0,1)$$
These cannot be changed by the user. For each item parameter, we assume independent Normal priors:
$$\left[\alpha_i, \beta_i \right]' \sim \mathcal{N}_{(K+1)} (b_{0,i},B_{0,i})$$
Where \(B_{0,i}\) is a diagonal matrix. One can specify a separate prior mean and precision for each item parameter. We also assume \(\delta_0 / k_0 \sim \)\( \mathcal{B}eta(c_0, d_0)\) and \(\delta_1 / k_1 \sim \)\( \mathcal{B}eta(c_1, d_1)\).
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.
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. 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/.
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.
plot.mcmc,summary.mcmc,
MCMCirt1d, MCMCirtKd
if (FALSE) {
## 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)
}
Run the code above in your browser using DataLab