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Political Analysis, 9:3
Multidimensional Analysis of Roll
Call Data via Bayesian Simulation:
Identification, Estimation, Inference,
and Model Checking
Simon Jackman
Department of Political Science,
Stanford University, Stanford, California 94305-6044
e-mail: jackman@stanford.edu
Vote-specific parameters are often by-products of roll call analysis, the primary goal be-
ing the measurement of legislators’ ideal points. But these vote-specific parameters are
more important in higher-dimensional settings: prior restrictions on vote parameters help
identify the model, and researchers often have prior beliefs about the nature of the dimen-
sions underlying the proposal space. Bayesian methods provide a straightforward and
rigorous way for incorporating these prior beliefs into roll call analysis. I demonstrate this
by exploiting the close connections among roll call analysis, item–response models, and
“full-information” factor analysis. Vote-specific discrimination parameters are equivalent
to factor loadings, and as in factor analysis, they (1) enable researchers to discern the
substantive content of the recovered dimensions, (2) can be used for assessing dimen-
sionality and model checking, and (3) are an obvious vehicle for introducing and testing
researchers’ prior beliefs about the dimensions. Bayesian simulation facilitates these uses
of discrimination parameters, by simplifying estimation and inference for the massive num-
ber of parameters generated by roll call analysis.
1 Introduction
IT IS WELL KNOWN that the analysis of roll call data generally results in statistical models
with many parameters. Operationalizing the D-dimensional Euclidean spatial voting model
(Enelow and Hinich 1984) with roll call data from n legislators over m roll calls generates
a statistical model with nD +m(D + 1) parameters. For instance, fitting a unidimensional
model to data from a recent U.S. Senate (n = 100, m ≈ 500) creates a 1100-parameter
problem (100 ideal points and 500×2 proposal parameters), while a two-dimensional model
has 1700 parameters. Likelihood-based estimation and inference with this many parameters
remain formidable even with the computing power now available to social scientists.1
Author’s note: I thank John Londregan, Adam Mierowitz, Keith Poole, and, especially, my collaborators on this
project, Joshua Clinton and Doug Rivers, for helpful discussion and comments. Errors and omissions remain my
own responsibility.
1Direct MLE may be feasible with data sets that are small relative to the data sets generated by the contemporary
U.S. Congress. For instance, see Londregan’s (2000b) analyses of committees in the Chilean legislature, where
Copyright 2001 by the Society for Political Methodology
227

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228 Simon Jackman
In a recent article in Political Analysis (Jackman 2000) I reported on recent work with
Joshua Clinton and Doug Rivers, where we use Bayesian simulation (Markov chain Monte
Carlo methods) to simplify estimation and inference for the large number of parameters
arising in roll call analysis. In my earlier article I focused on the issue of inferences for
legislators’ ideal points in a unidimensional setting. Here I show how the Bayesian approach
helps us deal with the added complexities of moving to higher-dimensional contexts. First,
identification, estimation, and inference for ideal points become more complicated in the
higher-dimensional setting, and I show how each task is accomplished in a Bayesian setting.
Second, I show how the proposal parameters assume more importance when we shift
to higher dimensional settings. In most roll call analyses, proposal parameters are often
considered nuisances, since the usual goal is measuring legislator’s ideal points, so much
so that the statistical analysis of roll call data is often referred to as “legislative scaling.”
In the Bayesian approach there is no real distinction between either type of parameter
(legislators’ ideal points or proposal-specific parameters), and Bayesian simulation methods
easily provide estimates and inferences for both sets of parameters; contrast likelihood-based
approaches that marginalize with respect to one set of parameters so as to obtain estimates
and inference for the others (e.g., Bock and Aitken 1981). I show below that proposal
parameters are analogous to factor loadings and can be put to the same uses as factor loadings.
These include determining the qualitative nature of recovered dimensions (as in exploratory
factor analyses) or a means for researchers either to impose or to test conjectures about the
nature of the underlying dimensions (as in confirmatory factor analyses). I develop some
diagnostics for assessing dimensionality based on the proposal parameters. In short, my
goal here is to use Bayesian simulation to make the analysis of roll call data less a technical
“scaling” exercise and more genuinely data analytic, in which researchers’ conjectures or
substantive expertise can alternately be tested, or used to guide the data analysis.
2 Operationalizing the Euclidean Spatial Voting Model
Assume a D-dimensional Euclidean proposal space. Each bill j = 1, . . . , m presents each
legislator i = 1, . . . , n with a choice between a Yea position, ζ j , and a Nay position ψ j .
The recorded votes (roll calls) are binary indicators: yi j = 1 if legislator i votes Aye on the
j th vote and yi j = 0 if legislator i votes Nay. The Euclidean spatial voting model drives
the development of a statistical model for these data: legislators receive utilities from ζ j
and ψ j declining in the squared distance of the these points from each ideal point xi . It is
well known that the statistical model implied by the Euclidean spatial voting model is the
following two-parameter item-response model, used extensively in the educational testing
literature:2
y∗i j = Ui (ζ j ) − Ui (ψ j ) = β′j xi − α j + εi j , (1)
with the censoring rule yi j = 1 ⇐⇒ y∗i j > 0, otherwise yi j = 0. With the further assump-
tion εi j ∼ N (0, 1) we have a hierarchical probit model with the complication that the ideal
points xi appear as unobserved predictors in Eq. (1), to be estimated along with the proposal
parameters β j and α j .
n < 10 and (by assumption) the m(D + 1) proposal parameters are reduced to a dramatically smaller set of
proposer parameters.
2See Jackman (2000, pp. 317–323) and Londregan (2000a).