ordrating {Ratings}R Documentation

MCMC for Unidimensional Ordinal IRT Model

Description

This function generates a sample from the posterior distribution of a unidimensional ordinal item response theory (IRT) model, with Gaussian priors on the ability and 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

ordrating(Y, beta.constraint = NULL, theta.neg.index = NULL, 
             theta.pos.index = NULL, vinva = 0.2, vinvb = 0.2, ma = 0, 
             mb = 1, theta.start = NULL, gamma.start = NULL, 
             burnin = 1000, mcmc = 10000, thin = 1, tune = 1, 
             verbose = 0, seed = NA)

Arguments

Y Matrix of data to be analyzed. Entries must be integers from 1, ..., C or NA where C is the number of ordinal categories. Items are on the rows and subjects are on the columns.
beta.constraint Possible inequality constraint on all beta parameters. beta.constraint = NULL (default) implies no constraint, beta.constraint > 0 implies all beta values are constrained to be positive, and beta.constraint < 0 implies all beta values are constrained to be negative.
theta.neg.index An index of theta that is constrained to be negative. To constrain the theta for the subject whose data is in the jth column of Y to be negative one would set theta.neg.index = j.
theta.pos.index An index of theta that is constrained to be positive. To constrain the theta for the subject whose data is in the jth column of Y to be positive one would set theta.pos.index = j.
vinva The prior inverse variance for each alpha_r.
vinvb The prior inverse variance for each beta_r.
ma The prior mean for each alpha_r.
mb The prior mean for each beta_r.
theta.start Starting values for theta. If NULL starting values are calculated based on the sample means of Y.
gamma.start Starting values for gamma. If NULL starting values are chosen to be equally spaced between 1 and the number of ordinal categories C.
burnin The number of burn-in iterations for the sampler.
mcmc The number of MCMC iterations for the sampler.
thin The thinning interval used in the simulation. The number of Gibbs iterations must be divisible by this value.
tune The scalar tuning parameter for the Metropolis-Hastings sampling. Must be strictly positive.
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.

Details

Let r = 1, ..., R index items, p = 1, ..., P index subjects, and c = 1, ..., C index ordinal rating categories.

Y is an R x P matrix with elements in {1, ..., C, NA}. NA denotes missing data that are assumed to be missing at random (MAR).

The distribution of Y is governed by a R x P matrix of latent variables Y* and a series of cutpoints gamma.

More specifically,

y*_{rp} = alpha_r + beta_r theta_p + epsilon_{rp}

where epsilon_{rp} ~iid N(0, 1). It is assumed that y_{rp = c} if and only if

y*_{rp} in (gamma_{c-1}, gamma_c]

.

The priors used for this model are that each alpha_r is iid Gaussian, each beta_r is iid Gaussian, each theta_p is standard normal, and the elements of gamma are improper uniform with all parameters assumed to be a priori independent.

ordrating simulates from the posterior distribution using a Metropolis-Hastings within Gibbs sampling algorithm. The algorithm employed is based on work by Cowles (1996). Note that the first element gamma_1 is normalized to zero, and thus not returned in the mcmc object.

ordrating fits a model that is a special case of the model fit by MCMCordfactanal in the MCMCpack package. The primary differences are the types of identifying constraints employed, the dimensionality of the theta and beta parameters, and the computational speed. Because ordrating fits a narrower class of models it can be optimized for speed much more effectively.

Value

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

References

Ho, Daniel E. and Kevin M. Quinn. forthcoming. "Improving the Presentation and Interpretation of Online Ratings Data with Model-based Figures." The American Statistician.

M. K. Cowles. 1996. ``Accelerating Monte Carlo Markov Chain Convergence for Cumulative-link Generalized Linear Models.'' Statistics and Computing. 6: 101-110.

Valen E. Johnson and James H. Albert. 1999. ``Ordinal Data Modeling.'' Springer: New York.

Kevin M. Quinn. 2004. ``Bayesian Factor Analysis for Mixed Ordinal and Continuous Responses.'' Political Analysis. 12: 338-353.

Shawn Treier and Simon Jackman. 2003. ``Democracy as a Latent Variable.'' Paper presented at the Midwest Political Science Association Annual Meeting.

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

See Also

plot.mcmc,summary.mcmc, MCMCordfactanal

Examples

## Not run: 
## Mondo Times example from Ho & Quinn (nd).
## may have to increase stack limit to run this example on some machines

data(Mondo)

ord.out <- ordrating(Mondo, beta.constraint=1, tune=.035, 
                     ma=1, mb=-5, vinva=1, vinvb=0.05,
                     gamma.start=c(-300, 0, 1.5, 3.0, 4.5, 300),
                     thin=20, burnin=20000, mcmc=100000, verbose=1000)

plot(ord.out)
summary(ord.out) 



## subsetting the Mondo data to include only raters who rated 5 or more 
## outlets (should avoid any stacksize problems)

Mondo.sub <- Mondo[apply(!is.na(Mondo), 1, sum) >= 5, ]
## also getting rid of outlets that are not rated now
Mondo.sub <- Mondo.sub[,apply(is.na(Mondo.sub), 2, mean) != 1] 

ord.out <- ordrating(Mondo.sub, beta.constraint=1, tune=.035, 
                     ma=1, mb=-5, vinva=1, vinvb=0.05,
                     gamma.start=c(-300, 0, 1.5, 3.0, 4.5, 300),
                     thin=20, burnin=20000, mcmc=100000, verbose=1000)

plot(ord.out)
summary(ord.out) 

## End(Not run)

[Package Ratings version 0.1-1 Index]