Analysis of rollcall data via the spatial voting model;
equivalent to a 2 parameter item-response model to educational testing data. Model fitting via Markov chain Monte Carlo (MCMC).
ideal(object, codes = object$codes,
dropList = list(codes = "notInLegis", lop = 0),
d = 1, maxiter = 10000, thin = 100, burnin = 5000,
impute = FALSE,
normalize = FALSE,
meanzero = normalize,
priors = NULL, startvals = "eigen",
store.item = FALSE, file = NULL,
verbose=FALSE, use.voter=NULL)a list of class ideal with named components
numeric, integer, number of legislators in the
analysis, after any subsetting via processing the dropList.
numeric, integer, number of rollcalls in roll
call matrix, after any subsetting via processing the dropList.
numeric, integer, number of dimensions
fitted.
a three-dimensional array containing the MCMC
output with respect to the
the ideal point of each legislator in each dimension.
The three-dimensional array is in iteration-legislator-dimension
order. The iterations run from burnin to maxiter, at
an interval of thin.
a three-dimensional array containing the
MCMC output for the item parameters. The three-dimensional array
is in iteration-rollcall-parameter order. The iterations run from
burnin to maxiter, at an interval of thin.
Each rollcall has d+1 parameters, with the
item-discrimination parameters stored first, in the first d
components of the 3rd dimension of the beta array; the
item-difficulty parameter follows in the final d+1
component of the 3rd dimension of the beta array.
a n by d matrix containing the means of the
MCMC samples for the ideal point of each legislator in each dimension,
using iterations burnin to maxiter, at an interval of
thin.
a m by d+1 matrix containing the means of
the MCMC samples for the item-specific parameters, using iterations
burnin to maxiter, at an interval of thin.
calling arguments, evaluated in the frame calling ideal.
an object of class call, containing
the arguments passed to ideal as unevaluated expressions or values (for functions arguments that evaluate to scalar integer or logical such as maxiter, burnin, etc).
an object of class rollcall
a list describing the types of voting
decisions in the roll call matrix (the votes component of the
rollcall object); defaults to
object$codes, the codes in the rollcall object.
a list (or alist)
listing voting decisions, legislators and/or votes to be dropped
from the analysis; see dropRollCall for details.
numeric, (small) positive integer (default = 1), dimensionality of the ability space (or "policy space" in the rollcall context).
numeric, positive integer, multiple of thin, number of MCMC iterations
numeric, positive integer, thinning interval used for recording MCMC iterations.
number of MCMC iterations to run before recording. The
iteration numbered burnin will be recorded. Must be a
multiple of thin.
logical, whether to treat missing entries
of the rollcall matrix as missing at random, sampling from the
predictive density of the missing entries at each MCMC iteration.
logical, impose identification with
the constraint that the ideal points have mean zero and
standard deviation one, in each dimension. For one dimensional models this option is sufficient to
locally identify the model parameters.
See Details.
to be deprecated/ignored; use normalize instead.
a list of parameters (means and variances)
specifying normal priors for the legislators' ideal points. The
default is NULL, in which case the normal priors used have mean zero and
precision 1 for the ideal points (ability parameters) and mean zero and
precision .04 (variance 25) for the bill parameters (item discrimination and difficulty parameters). If not NULL, priors must be a
list with as many as four named components xp, xpv, bp,
bpv:
xpa n by d matrix
of prior means for the legislators' ideal points;
or alternatively, a scalar, which will be replicated to fill a n by d matrix.
xpva n by d matrix of prior
precisions (inverse variances);
or alternatively, a scalar, which will be replicated to fill a n
by d matrix.
bpa m by d+1 matrix of prior means for the
item parameters (with the item difficulty parameter coming last);
or alternatively, a scalar, which will be replicated to fill a m
by d+1 matrix.
bpva m by d+1 matrix of prior
precisions for the item parameters;
or alternatively, a scalar, which will be replicated to fill a m
by d+1 matrix.
None of the components should contain NA. If any
of the four possible components are not provided, then the
corresponding component of priors is assigned using the default
values described above.
either a string naming a method for generating start
values, valid options are "eigen" (the default),
"random" or a list containing start values for
legislators' ideal points and item parameters. See Details.
logical, whether item discrimination
parameters should be stored. Storing item discrimination parameters
can consume a large amount of memory. These need to be stored for
prediction; see predict.ideal.
string, file to write MCMC output. Default is
NULL, in which case MCMC output is stored in memory. Note
that post-estimation commands like plot will not work unless
MCMC output is stored in memory.
logical, default
is FALSE, which generates relatively little output to the R
console during execution.
A vector of logicals of length n controlling
which legislators' vote data informs item parameter
estimates. Legislators corresponding to FALSE entries will
not have their voting data included in updates of the item
parameters. The default value of NULL will run the standard
ideal-point model, which uses all legislators in updating item
parameters. See Jessee (2016).
Simon Jackman simon.jackman@sydney.edu.au, with help from Christina Maimone and Alex Tahk.
The function fits a d+1 parameter item-response model to
the roll call data object, so in one dimension the model reduces
to the two-parameter item-response model popular in educational testing.
See References.
Identification: The model parameters are not identified without the user supplying some restrictions on the model parameters; i.e., translations, rotations and re-scalings of the ideal points are observationally equivalent, via offsetting transformations of the item parameters. It is the user's responsibility to impose these identifying restrictions if desired. The following brief discussion provides some guidance.
For one-dimensional models (i.e., d=1), a simple route to
identification is the normalize option, by imposing the restriction that the means of the posterior densities of the ideal points (ability parameters) have mean zero and standard deviation one, across legislators (test-takers). This normalization supplies
local identification (that is, identification up to a 180 degree rotation of
the recovered dimension).
Near-degenerate “spike” priors
(priors with arbitrarily large precisions) or the
constrain.legis option on any two legislators' ideal points
ensures global identification in one dimension.
Identification in higher dimensions can be obtained by supplying
fixed values for d+1 legislators' ideal points, provided the
supplied fixed points span a d-dimensional space (e.g., three
supplied ideal points form a triangle in d=2 dimensions), via
the constrain.legis option. In this case the function
defaults to vague normal priors on the unconstrained ideal points, but at each iteration the sampled
ideal points are transformed back into the space of identified
parameters, applying the linear transformation that maps the
d+1 fixed ideal points from their sampled values to their
fixed values. Alternatively, one can impose
restrictions on the item parameters via
constrain.items. See the examples in the documentation
for the constrain.legis and
constrain.items.
Another route to identification is via post-processing. That
is, the user can run ideal without any identification
constraints. This does not pose any formal/technical problem in a
Bayesian analysis. The fact that the posterior density may have
multiple modes doesn't imply that the posterior is improper or that
it can't be explored via MCMC methods. -- but then use the function
postProcess to map the MCMC output from the space of
unidentified parameters into the subspace of identified parameters.
See the example in the documentation for the
postProcess function.
When the normalize option is set to TRUE, an
unidentified model is run, and the ideal object is
post-processed with the normalize option, and then returned
to the user (but again, note that the normalize option is
only implemented for unidimensional models).
Start values. Start values can be supplied by the user, or generated by the function itself.
The default method, corresponding to startvals="eigen", first
forms a n-by-n correlation matrix from the
double-centered roll call matrix (subtracting row means, and column
means, adding in the grand mean), and then extracts the first
d principal components (eigenvectors), scaling the
eigenvectors by the square root of their corresponding eigenvector.
If the user is imposing constraints on ideal points (via
constrain.legis), these constraints are applied to the
corresponding elements of the start values generated from the
eigen-decomposition. Then, to generate start
values for the rollcall/item parameters, a series of
binomial glms are
estimated (with a probit link), one for
each rollcall/item, \(j = 1, \ldots, m\). The votes on the
\(j\)-th rollcall/item are binary responses (presumed to be
conditionally independent given each legislator's latent
preference), and the (constrained or unconstrained) start values for
legislators are used as predictors. The estimated coefficients from
these probit models are used as start values for the item
discrimination and difficulty parameters (with the intercepts from
the probit GLMs multiplied by -1 so as to make those coefficients
difficulty parameters).
The default eigen method generates extremely good start
values for low-dimensional models fit to recent U.S. congresses,
where high rates of party line voting result in excellent fits from
low dimensional models. The eigen method may be
computationally expensive or lead to memory errors for
rollcall objects with large numbers of legislators.
The random method generates start values via iid sampling
from a N(0,1) density, via rnorm, imposing any
constraints that may have been supplied via
constrain.legis, and then uses the probit method
described above to get start values for the rollcall/item
parameters.
If startvals is a list, it must contain the named
components x and/or b, or named components that
(uniquely) begin with the letters x and/or b. The
component x must be a vector or a matrix of dimensions equal to
the number of individuals (legislators) by d. If supplied,
startvals$b must be a matrix with dimension number of items
(votes) by d+1. The x and b components cannot
contain NA. If x is not supplied when startvals
is a list, then start values are generated using the default
eiegn method described above, and start values for the
rollcall/item parameters are regenerated using the probit method,
ignoring any user-supplied values in startvals$b. That is,
user-supplied values in startvals$b are only used when
accompanied by a valid set of start values for the ideal points in
startvals$x.
Implementation via Data Augmentation. The MCMC algorithm for this problem consists of a Gibbs sampler for the ideal points (latent traits) and item parameters, conditional on latent data \(y^*\), generated via a data augmentation (DA) step. That is, following Albert (1992) and Albert and Chib (1993), if \(y_{ij} = 1\) we sample from the truncated normal density $$y_{ij}^* \sim N(x_i' \beta_j - \alpha_j, 1)\mathcal{I}(y_{ij}^* \geq 0)$$ and for \(y_{ij}=0\) we sample $$y_{ij}^* \sim N(x_i' \beta_j - \alpha_j, 1)\mathcal{I}(y_{ij}^* < 0)$$ where \(\mathcal{I}\) is an indicator function evaluating to one if its argument is true and zero otherwise. Given the latent \(y^*\), the conditional distributions for \(x\) and \((\beta,\alpha)\) are extremely simple to sample from; see the references for details.
This data-augmented Gibbs sampling strategy is easily implemented, but can sometimes require many thousands of samples in order to generate tolerable explorations of the posterior densities of the latent traits, particularly for legislators with short and/or extreme voting histories (the equivalent in the educational testing setting is a test-taker who gets almost every item right or wrong).
Albert, James. 1992. Bayesian Estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational Statistics. 17:251-269.
Albert, James H. and Siddhartha Chib. 1993. Bayesian Analysis of Binary and Polychotomous Response Data. Journal of the American Statistical Association. 88:669-679.
Clinton, Joshua, Simon Jackman and Douglas Rivers. 2004. The Statistical Analysis of Roll Call Data. American Political Science Review. 98:335-370.
Jackman, Simon. 2009. Bayesian Analysis for the Social Sciences. Wiley: Hoboken, New Jersey.
Jessee, Stephen. 2016. (How) Can We Estimate the Ideology of Citizens and Political Elites on the Same Scale? American Journal of Political Science.
Patz, Richard J. and Brian W. Junker. 1999. A Straightforward Approach to Markov Chain Monte Carlo Methods for Item Response Models. Journal of Education and Behavioral Statistics. 24:146-178.
Rivers, Douglas. 2003. “Identification of Multidimensional Item-Response Models.” Typescript. Department of Political Science, Stanford University.
van Dyk, David A and Xiao-Li Meng. 2001. The art of data augmentation (with discussion). Journal of Computational and Graphical Statistics. 10(1):1-111.
rollcall, summary.ideal,
plot.ideal, predict.ideal.
tracex for graphical display of MCMC iterative
history.
idealToMCMC converts the MCMC iterates in an
ideal object to a form that can be used by the coda library.
constrain.items and
constrain.legis for implementing identifying
restrictions.
postProcess for imposing identifying restrictions
ex post.
MCMCirt1d and
MCMCirtKd in the MCMCpack
package provide similar functionality to ideal.
if (FALSE) {
## long run, many iterations
data(s109)
n <- dim(s109$legis.data)[1]
x0 <- rep(0,n)
x0[s109$legis.data$party=="D"] <- -1
x0[s109$legis.data$party=="R"] <- 1
id1 <- ideal(s109,
d=1,
startvals=list(x=x0),
normalize=TRUE,
store.item=TRUE,
maxiter=260E3,
burnin=10E3,
thin=100)
}
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