ordIRT {emIRT}

Two-parameter Ordinal IRT estimation via EM

Description

ordIRT estimates an ordinal IRT model with three ordered response categories. Estimation is conducted using the EM algorithm described in the reference paper below. The algorithm will produce point estimates that are comparable to those of MCMCordfactanal, but will do so much more rapidly and also scale better with larger data sets.

Usage

  ordIRT(.rc, .starts = NULL, .priors = NULL, .D = 1L, .control = NULL) 

Arguments

.rc

matrix of numeric values containing the data to be scaled. Respondents are assumed to be on rows, and items assumed to be on columns, so the matrix is assumed to be of dimension (N x J). For each item, only 3 ordered category responses are accepted, and the only allowable responses are ‘1’, ‘2’, and ‘3’, with ‘0’ as a missing data record. If data of more than 3 categories are to be rescaled, they should be collapsed into 3 categories and recoded accordingly before proceeding.

.starts

a list containing several matrices of starting values for the parameters. Note that the parameters here correspond to the re-parameterized version of the model (i.e. alpha is alpha^*, not the original alpha_{1j} The list should contain the following matrices:

beta

A (J x 1) matrix of starting values for the reparameterized item discrimination parameter \beta^*.

x

An (N x 1) matrix of starting values for the respondent ideal points x_t.

tau

A (J x 1) matrix of starting values for the bill cutpoint \tau_j.

DD

A (J x 1) matrix of starting values for the squared bill cutpoint difference \tau_j^2.

.priors

list, containing several matrices of starting values for the parameters. Note that the parameters here correspond to the re-parameterized version of the model (i.e. alpha is \alpha^*, not the original \alpha_{1j} The list should contain the following matrices:

x$mu

A (1 x 1) prior means matrix for respondent ideal points x_t.

x$sigma

A (1 x 1) prior covariance matrix for respondent ideal points x_t.

beta$mu

A (2 x 1) prior means matrix for \tau_j and \beta^*.

beta$sigma

A (2 x 2) prior covariance matrix for \tau_j and \beta^*.

.D

integer, indicates number of dimensions. Only one dimension is implemented and this argument is ignored.

.control

list, specifying some control functions for estimation. Options include the following:

threads

integer, indicating number of cores to use. Default is to use a single core, but more can be supported if more speed is desired.

verbose

boolean, indicating whether output during estimation should be verbose or not. Set FALSE by default.

thresh

numeric. Algorithm will run until all parameters correlate at 1 - thresh across consecutive iterations. Set at 1e-6 by default.

maxit

integer. Sets the maximum number of iterations the algorithm can run. Set at 500 by default.

checkfreq

integer. Sets frequency of verbose output by number of iterations. Set at 50 by default.

Value

An object of class ordIRT.

means

list, containing several matrices of point estimates for the parameters corresponding to the inputs for the priors. The list should contain the following matrices.

x

A (N x 1) matrix of point estimates for the respondent ideal points x_t.

beta

A (J x 1) matrix of point estimates for the reparameterized item discrimination parameter \beta^*.

tau

A (J x 1) matrix of point estimates for the bill cutpoint \alpha^*.

Delta_sq

A (J x 1) matrix of point estimates for the squared bill cutpoint difference \tau_j^2.

Delta

A (J x 1) matrix of point estimates for the bill cutpoint difference \tau_j.

vars

list, containing several matrices of variance estimates for parameters corresponding to the inputs for the priors. Note that these variances are those recovered via variational approximation, and in most cases they are known to be far too small and generally unusable. Better estimates of variances can be obtained manually via the parametric bootstrap. The list should contain the following matrices:

x

A (N x 1) matrix of variances for the respondent ideal points x_t.

beta

A (J x 1) matrix of variances for the reparameterized item discrimination parameter \beta^*.

tau

A (J x 1) matrix of variances for the bill cutpoint \tau_j.

runtime

A list of fit results, with elements listed as follows:

iters

integer, number of iterations run.

conv

integer, convergence flag. Will return 1 if threshold reached, and 0 if maximum number of iterations reached.

threads

integer, number of threads used to estimated model.

tolerance

numeric, tolerance threshold for convergence. Identical to thresh argument in input to .control list.

n

Number of respondents in estimation, should correspond to number of rows in roll call matrix.

j

Number of items in estimation, should correspond to number of columns in roll call matrix.

call

Function call used to generate output.

Author(s)

Kosuke Imai [e-mail]

James Lo [e-mail]

Jonathan Olmsted [e-mail]

References

Kosuke Imai, James Lo, and Jonathan Olmsted (2016). “Fast Estimation of Ideal Points with Massive Data.” American Political Science Review, Vol. 110, No. 4 (December), pp. 631-656.

See Also

'AsahiTodai'.

Examples

## Not run: 
### Real data example: Asahi-Todai survey (not run)
## Collapses 5-category ordinal survey items into 3 categories for estimation
data(AsahiTodai)
out.varinf <- ordIRT(.rc = AsahiTodai$dat.all, .starts = AsahiTodai$start.values,
					.priors = AsahiTodai$priors, .D = 1,
					.control = {list(verbose = TRUE,
                     thresh = 1e-6, maxit = 500)})

## Compare against MCMC estimates using 3 and 5 categories
cor(ideal3, out.varinf$means$x)
cor(ideal5, out.varinf$means$x)  

## End(Not run)


### Monte Carlo simulation of ordIRT() model vs. known parameters
## Set number of legislators and items
set.seed(2)
NN <- 500
JJ <- 100

## Simulate true parameters from original model
x.true <- runif(NN, -2, 2)
beta.true <- runif(JJ, -1, 1)
tau1 <- runif(JJ, -1.5, -0.5)
tau2 <- runif(JJ, 0.5, 1.5)
ystar <- x.true %o% beta.true + rnorm(NN *JJ)

## These parameters are not needed, but correspond to reparameterized model
#d.true <- tau2 - tau1
#dd.true <- d.true^2
#tau_star <- -tau1/d.true
#beta_star <- beta.true/d.true

## Generate roll call matrix using simulated parameters
newrc <- matrix(0, NN, JJ)
for(j in 1:JJ) newrc[,j] <- cut(ystar[,j], c(-100, tau1[j], tau2[j],100), labels=FALSE)

## Generate starts and priors
cur <- vector(mode = "list")
cur$DD <- matrix(rep(0.5,JJ), ncol=1)
cur$tau <- matrix(rep(-0.5,JJ), ncol=1)
cur$beta <- matrix(runif(JJ,-1,1), ncol=1) 
cur$x <- matrix(runif(NN,-1,1), ncol=1) 
priors <- vector(mode = "list")
priors$x <- list(mu = matrix(0,1,1), sigma = matrix(1,1,1) )
priors$beta <- list(mu = matrix(0,2,1), sigma = matrix(diag(25,2),2,2))

## Call ordIRT() with inputs
time <- system.time({
    lout <- ordIRT(.rc = newrc,
                    .starts = cur,
                    .priors = priors,
                    .control = {list(
                        threads = 1,
                        verbose = TRUE,
                        thresh = 1e-6,
			maxit=300,
			checkfreq=50
                        )})
})

## Examine runtime and correlation of recovered ideal points vs. truth
time
cor(x.true,lout$means$x)