# Explaining Differences in Voting Patterns Across Voting Domains Using
Hierarchical Bayesian Models

Erin Lipman  University of Washington  Scott Moser  University of Nottingham,
Associate professor in the School of Politics and International Relations
Abel Rodriguez  University of Washington, Professor of Statistics

###### Abstract

Spatial voting models of legislators’ preferences are used in political
science to test theories about their voting behavior. These models posit that
legislators’ ideologies as well as the ideologies reflected in votes for and
against a bill or measure exist as points in some low dimensional space, and
that legislators vote for positions that are close to their own ideologies.
Bayesian spatial voting models have been developed to test sharp hypotheses
about whether a legislator’s revealed ideal point differs for two distinct
sets of bills. This project extends such a model to identify covariates that
explain whether legislators exhibit such differences in ideal points. We use
our method to examine voting behavior on procedural versus final passage votes
in the U.S. house of representatives for the 93rd through 113th congresses.
The analysis provides evidence that legislators in the minority party as well
as legislators with a moderate constituency are more likely to have different
ideal points for procedural versus final passage votes.

##  1 Introduction

Estimating members’ preferences from roll-called votes in parliaments and
legislatures has been crucial for theory-testing (Krehbiel (1990), Theriault
(2006)). Significant progress has been made in the methodology of ideal point
estimation since the seminal work of Poole & Rosenthal (1985) and Poole &
Rosenthal (2000), with early contributions from Aldrich & McKelvey (1977) and
others. However, methodological challenges remain when comparing ideal points
over time, across domains, or between chambers. Many of these challenges have
been addressed in the literature (e.g., DW-NOMINATE, bridging, Jessee (2016)).
Such studies typically either treat all votes the same (Poole & Rosenthal
(2000)) or hand-pick particular domains in which to use roll-call votes (e.g.
Jeong (2009); Murillo & Pinto (2022)). There are two components relevant to
this enterprise: _scaling_ ideal points to allow for comparisons and
_explaining_ variation in estimated ideal points. The focus of this paper is
on devising methodology that enables both components to be carried out
simultaneously.

We are interested in settings where legislators vote on motions in various
domains. These domains could be, for example issue areas (Jones & Baumgartner
(2005), Moser et al. (2021)); or dimensions, e.g. social and economic
policies, as in Poole & Rosenthal (2000), Poole & Rosenthal (2011); or type of
vote (e.g. procedural, amendment, final passage as in Crespin & Rohde (2007)
or Jessee & Theriault (2014)).111There are many other ways to categorise roll-
called votes, e.g., Rohde (2004), along with Clausen Categories (a 6 category
classification of votes), Peltzman Categories, and “Specific Issue Codes,”
which is a 109 category classification of votes all provided by
https://voteview.com/articles/issue_codes. Legislators may have different
revealed preferences across these domains. For instance, Jessee & Theriault
(2014) found that party polarization is greater in the procedural domain,
meaning that when using roll call data only from votes on procedural motions,
party polarization is greater than when legislators vote on amendments or
final passage motions. This pattern is similar to the findings of Kirkland &
Slapin (2017) when looking at defections in the minority party. To ensure
common scaling of ideal points across domains, previous literature assumed
that a small number of legislators (often referred to as “bridges”) had the
same revealed preferences across domains (Shor et al. (2010); Shor & McCarty
(2011); Treier (2011)). This assumption was relaxed by Lofland et al. (2017)
and Moser et al. (2021), who estimated the identity of the bridges. These
models produce a posterior distributions for the identity of each legislator
as a bridge (expected bridge probability) and, in the case more than two
voting domains, a clustering of legislators with similar patterns of changing
revealed preferences across domains.

While the previous models enable us to identify bridges, they does not
directly allow us to incorporate covariates that can potentially explain
changes in revealed preferences across domains. Understanding factors that
influence voting behavior in different domains and changes in voting patterns
across domains has the potential to contribute to studies on party influence
on legislative voting (Snyder Jr. & Groseclose, 2001), dynamics of
polarization (Jessee & Theriault, 2014; Sulkin & Schmitt, 2014), and party
defection (Finocchiaro & Heimann, 1999). Additionally, answering this question
sheds light on the broader literature on determinants of legislative voting.
Such studies typically either treat all votes the same or hand-pick particular
domains in which to use roll-call votes (e.g. Jeong, 2009; Murillo & Pinto,
2022). Our approach extends these approaches by allowing for domain-specific
voting patterns and simultaneously estimating factors that influence changes
in voting behavior across the two domains.

This paper develops a fully Bayesian hierarchical model that jointly addresses
the questions of how to properly scale ideal points across two voting domains
in order to estimate the identity of the bridge legislators and of how to
identify factors that might explain the fact that preferences vary across
domains. This allows for testing hypotheses such as “Does covariate $X$ matter
for explaining differences in voting patterns across domains?” For example,
our approach allows for testing claims such as “constituency characteristics
have no effect on the probability that a member will exhibit different
revealed preferences across domains” or “being in the minority party has no
effect on the probability of having the same revealed preference across all
domains.” The use of a joint model has a number of advantages over a two-step
procedure, in which a point estimate of the bridge identities is first
estimated from the voting data and then a regression model is fitted using
these point estimates as the response variables. Two-step procedures are
potentially sensitive to the way the point estimator of the bridge identities
is constructed (e.g., the threshold used for the posterior probabilities that
an individual legislator is a bridge). Furthermore, because the ideal points
of the legislators and the identity of the bridges are estimated from the
voting records, and therefore and subject to uncertainty, two-step procedures
can be expected to underestimate the uncertainty associated with the variable
selection step. Joint models avoid both of these issues.

We illustrate our joint model through a study of the factors that drive
legislators to vote differently on procedural and final passage votes in the
modern U.S. House of Representatives. The literature views procedural votes
separately from substantive ones (e.g., see Theriault, 2006, Jessee &
Theriault, 2014, Patty, 2010, Carson et al., 2014; Goodman & Nokken, 2004 and
Kirkland & Williams, 2014). Our empirical analysis builds on this literature
by simultaneously considering a panel of constituency-level, legislator-level,
and chamber-level factors over an extended period of 42 years ending in 2014.

##  2 The Model

###  2.1 Spatial voting in multiple domains

Let $y_{i,j}\in\\{0,1\\}$ correspond to the vote of legislator $i=1,\dots,I$
on measure $j=1,\dots,J$, with $y_{i,j}=0$ representing a negative (“nay”)
vote and $y_{i,j}=1$ representing a positive (“yay”) vote. In the spirit of
Jackman (2001) and Clinton et al. (2004), we assume that

$y_{i,j}\mid\mu_{j},\boldsymbol{\alpha}_{j},\boldsymbol{\beta}_{i,0},\boldsymbol{\beta}_{i,1},\gamma_{j}\sim\mathsf{Ber}\left(y_{i,j}\,\bigg{|}\,\frac{\exp\left\\{\mu_{j}+\boldsymbol{\alpha}_{j}^{T}\boldsymbol{\beta}_{i,\gamma_{j}}\right\\}}{1+\exp\left\\{\mu_{j}+\boldsymbol{\alpha}_{j}^{T}\boldsymbol{\beta}_{i,\gamma_{j}}\right\\}}\right),$
(1)

where $\mu_{j}\in\mathbb{R}$ and $\boldsymbol{\alpha}_{j}\in\mathbb{R}^{d}$
are unknown parameters that can be interpreted, respectively, as the baseline
probability of an affirmative vote and the discrimination associated with
measure $j$, $\gamma_{j}\in\\{0,1\\}$ is a known indicator variable of whether
the $j$-th vote corresponds to a procedural ($\gamma_{j}=0$) or final passage
($\gamma_{j}=1$) vote,
$\boldsymbol{\beta}_{i,0},\boldsymbol{\beta}_{i,1}\in\mathbb{R}^{d}$ are
unknown parameters representing the ideal point of legislator $i$ on
procedural and final passage votes, respectively, and $d$ is the (maximum)
dimension of the latent policy space, which is assumed to be known. This
statistical model can be derived from a spatial voting model (Enelow & Hinich,
1984; Davis et al., 1970) using the random utility framework of McFadden
(1973). It also mimics the structure of a logistic IRT model (Fox, 2010).
However, unlike traditional IRT model, it allows for each legislator to have
(in principle) different ideal points on each of the two voting domains.

In this paper we adopt a Bayesian approach to inference. Therefore, we need to
specify priors for the model parameters. Our approach to prior elicitation is
similar to that in Bafumi et al. (2005), who advocate the use of hierarchical
priors. In particular, for the intercepts $\mu_{1},\ldots,\mu_{J}$, we let

$\displaystyle\mu_{j}\mid\rho_{\mu},\kappa^{2}_{\mu}$
$\displaystyle{\,\,\overset{\text{iid}}{\sim}}\,\,\mathsf{N}(\mu_{j}\mid\rho_{\mu},\kappa^{2}_{\mu}),$
$\displaystyle j$ $\displaystyle=1,\ldots,J,$

where $\mathsf{N}(\cdot\mid a,b^{2})$ denotes the normal distribution with
mean $a$ and variance $b^{2}$. The hyperparameters $\rho_{\mu}$ and
$\kappa^{2}_{\mu}$ are then given normal and inverse gamma hyperpriors
respectively. On the other hand, for the discrimination parameters
$\boldsymbol{\alpha}_{1},\ldots,\boldsymbol{\alpha}_{J}$ we set

$\displaystyle\alpha_{j,k}\mid\omega_{\alpha},\kappa^{2}_{\alpha}$
$\displaystyle{\,\,\overset{\text{iid}}{\sim}}\,\,\omega_{\alpha,k}\delta_{0}(\alpha_{j,k})+(1-\omega_{\alpha,k})\mathsf{N}(\alpha_{j,k}\mid
0,\kappa^{2}_{\alpha}),$ $\displaystyle j$ $\displaystyle=1,\ldots,J,$
$\displaystyle k=1,\ldots,d,$

where $\delta_{0}$ is a point mass at $0$. This prior is completed by
assigning an inverse gamma hyperprior on $\kappa_{\alpha}^{2}$ and independent
beta hyperpriors on $\omega_{\alpha,1},\ldots,\omega_{\alpha,d}$. The use of
zero-inflated Gaussian distributions for the discrimination parameters allows
us to automatically handle unanimous votes without having to explicitly remove
them from the dataset.

Note that the likelihood defined in (1) is invariant to affine transformations
of the policy space. The literature on item response models discusses various
methods to deal with the ensuing lack of identifiability in the model
parameters. A popular alternative in the context of spatial voting models is
to “anchor” the position of $d+1$ legislators whose ideal points are well
separated in the policy space (Rivers, 2003). For example in the $d=1$ case,
we might fix the positions of the Democratic and Republican party leaders to
be $-1$ and $1$ respectively. In this manuscript, we adopt a version of this
approach to endow our scaling procedure with an absolute scale that is
interpretable. However, as we discuss below, this type of approach is not
appropriate to address all the identifiability issues that arise in our
setting.

###  2.2 Testing sharp hypothesis about differences in ideal points

To complete our model, we need to assign prior distributions to the ideal
points of each legislator and voting domain (procedural or final passage). The
simplest alternative, independent priors across both legislators and types,
has several shortcomings. Most importantly, such an approach is (roughly)
equivalent to fitting independent IRT models of the kind described in Jackman
(2001) for each group of votes. Under such an approach, the two latent spaces
are unlinked and are therefore incomparable. Indeed, while an absolute scale
can be created for each one of them independently by, for example, fixing the
position of a small number of legislators on each of the scales separately,
the only way to ensure comparability is to assume we know how the position of
these legislators changes (or not) when moving from voting on procedural to
final passage motions. While assumptions of this kind have been successfully
used in the past (for example, by Shor et al., 2010 to create a common scale
to map the relative ideological positions of various state legislatures), they
are very strong and often hard to justify. In particular, some of the natural
“bridge” legislators (such as the party leaders) are those whose behavior
might be most interesting to study.

Hence, in this paper we adopt an approach similar that of Lofland et al.
(2017) and elicit a joint prior on the pairs
$(\boldsymbol{\beta}_{i,0},\boldsymbol{\beta}_{i,1})$ that explicitly accounts
for the possibility that $\boldsymbol{\beta}_{i,0}=\boldsymbol{\beta}_{i,1}$.
More specifically, we introduce a set of binary indicators
$\zeta_{1},\ldots,\zeta_{I}$ such that $\zeta_{i}=1$ if and only if
$\boldsymbol{\beta}_{i,0}=\boldsymbol{\beta}_{i,1}$, i.e., legislator $i$
exhibits the same ideal point in both final passage and procedural votes, and
$\zeta_{i}=0$ otherwise. We call legislators for which $\zeta_{i}=1$ bridges.
Then, conditionally on $\zeta_{i}\in\\{0,1\\}$ we let

$\displaystyle(\boldsymbol{\beta}_{i,0},\boldsymbol{\beta}_{i,1})\mid\boldsymbol{\rho}_{\beta},\boldsymbol{\Sigma}_{\beta},\zeta_{i}$
$\displaystyle{\,\,\overset{\text{iid}}{\sim}}\,\,\begin{cases}\mathsf{N}\left(\boldsymbol{\beta}_{i,0}\mid\boldsymbol{\rho}_{\beta},\boldsymbol{\Sigma}_{\beta}\right)\,\delta_{\boldsymbol{\beta}_{i,0}}\left(\boldsymbol{\beta}_{i,1}\right)&\zeta_{i}=1,\\\
\mathsf{N}\left(\boldsymbol{\beta}_{i,0}\mid\boldsymbol{\rho}_{\beta},\boldsymbol{\Sigma}_{\beta}\right)\mathsf{N}\left(\boldsymbol{\beta}_{i,1}\mid\boldsymbol{\rho}_{\beta},\boldsymbol{\Sigma}_{\beta}\right)&\zeta_{i}=0.\\\
\end{cases}$ (2)

Note that, when $\zeta_{i}=0$ and therefore
$\boldsymbol{\beta}_{i,0}\neq\boldsymbol{\beta}_{i,1}$, the two ideal points
$\boldsymbol{\beta}_{i,0}$ and $\boldsymbol{\beta}_{i,1}$ are given
independent (albeit identical) priors. Using the same values of
$\boldsymbol{\rho}_{\beta}$ and $\boldsymbol{\Sigma}_{\beta}$ in every case is
particularly appropriate in this setting as it reflects the idea that all
these ideal points live in the same latent policy space. The hyperparameters
$\boldsymbol{\rho}_{\beta}$ and $\boldsymbol{\Sigma}_{\beta}$ are learned from
the data and given conditionally conjugate multivariate normal and inverse
Wishart hyperpriors.

When reliable prior information about the identity of the bridge legislators
is available, then the indicators $\zeta_{1},\ldots,\zeta_{I}$ can be treated
as known covariates. Furthermore, as long as at $d+1$ bridges are available,
the two latent scales (for procedural and for final passage votes) are
comparable, addressing the remaining identifiability issues that anchoring was
not able to address on its own. As we indicated before, this strategy is at
the core of Shor et al. (2010). Instead, in this paper we treat the indicators
$\zeta_{1},\ldots,\zeta_{I}$ as unknown and device a hierarchical prior that
allows us to learn the identity of the bridge legislators and, more
importantly for our goals, identify explanatory variables that are associated
with revealing identical preferences on both voting domains.

###  2.3 Explaining differences in ideal points

Previous approaches to identifying bridge legislators from data (e.g., Lofland
et al., 2017 and Moser et al., 2021) have used exchangeable priors on the
vector $\boldsymbol{\zeta}=(\zeta_{1},\ldots,\zeta_{I})$ to control for
multiplicities (Scott & Berger, 2006, 2010). Instead, in this paper we
consider a hierarchical prior that allows us to introduce explanatory variable
and formally test whether they are associated with the probability that a
particular legislator is a bridge. More specifically, let $\boldsymbol{X}$ be
an $I\times p$ design matrix of (centered) explanatory variables, with
$\boldsymbol{x}_{i}^{T}$ denoting the $i$-th row of $\boldsymbol{X}$, which in
turn corresponds to the vector of observed explanatory variables associated
with legislator $i=1,\ldots,I$. We model the individual $\zeta_{i}$s using a
logistic regression of the form

$\displaystyle\zeta_{i}\mid\boldsymbol{x}_{i},\eta_{0},\boldsymbol{\eta}\sim\mathsf{Ber}\left(\zeta_{i}\,\bigg{|}\,\frac{1}{1+\exp\left\\{-\left(\eta_{0}+\boldsymbol{x}_{i}^{T}\boldsymbol{\eta}\right)\right\\}}\right),$
(3)

where $\eta_{0}$ is an intercept and
$\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{p})^{T}$ is a $p$-dimensional vector
of unknown regression coefficients.

In order to identify explanatory variables that affect the probability that a
particular legislator is a bridge, we adopt a prior for $\boldsymbol{\eta}$
that allows for coefficients to be exactly zero. With this goal in mind, let
$\boldsymbol{\xi}=(\xi_{1},\ldots,\xi_{p})$ be a binary $p$-dimensional vector
such $\xi_{k}=1$ if variable $k$ is included in the model (i.e., if
$\eta_{k}\neq 0$) and $0$ otherwise (i.e., if $\eta_{k}=0$). Also, let
$\boldsymbol{X}_{\boldsymbol{\xi}}$ be the matrix created by removing the
columns of $\boldsymbol{X}$ for which the corresponding entries of
$\boldsymbol{\xi}$ are zero and, similarly, let
$\boldsymbol{\eta}_{\boldsymbol{\xi}}$ be the vector made of the entries of
$\boldsymbol{\eta}$ for which the corresponding entries of $\boldsymbol{\xi}$
are equal to 1. Then, conditional on $\boldsymbol{\xi}$, the prior for
$\boldsymbol{\eta}$ takes the form

$\displaystyle
f(\boldsymbol{\eta}\mid\boldsymbol{\xi},\boldsymbol{X})=f(\boldsymbol{\eta}_{\boldsymbol{\xi}}\mid{\boldsymbol{\xi}},\boldsymbol{X}_{\boldsymbol{\xi}})\prod_{\\{k:\xi_{k}=0\\}}\delta_{0}(\eta_{k})$

where $\delta_{0}(\cdot)$ again denotes a point mass at zero, and
$p(\boldsymbol{\eta}_{\boldsymbol{\xi}})$ corresponds to a g-prior of the form

$\displaystyle\eta_{\boldsymbol{\xi}}\mid
g,\boldsymbol{\xi},\boldsymbol{X}_{\boldsymbol{\xi}}\sim
N\left(\eta_{\boldsymbol{\xi}}\,\Big{|}\,0,4g\left(\boldsymbol{X}_{\boldsymbol{\xi}}^{T}\boldsymbol{X}_{\boldsymbol{\xi}}\right)^{-1}\right).$
(4)

In the sequel, we work with $g=I$, which means that (4) can be interpreted as
an (approximately) unit information prior under an imaginary training sample
(see Sabanés Bové & Held, 2011 for additional details).

Finally, we follow Scott & Berger (2010) and use a Beta-Binomial prior for
$\boldsymbol{\xi}$,

$\displaystyle
f(\boldsymbol{\xi})=\frac{\Gamma\left(p^{*}_{\boldsymbol{\xi}}+1\right)\Gamma\left(p-p^{*}_{\boldsymbol{\xi}}+1\right)}{\Gamma(p+2)}=\int_{0}^{1}\upsilon^{p^{*}_{\boldsymbol{\xi}}}(1-\upsilon)^{p-p^{*}_{\boldsymbol{\xi}}}d\upsilon,$
(5)

where $p^{*}_{\boldsymbol{\xi}}=\sum_{k=1}^{p}\xi_{k}$ is the size of the
model. This prior has several advantages. One of them is interpretability. As
the last equality in (5) shows, it can be interpreted as the result of
assuming a common (but unknown) prior probability of inclusion $\upsilon$ for
every variable in the model, and then assigning $\upsilon$ a uniform
distribution. Hence, it implies that the marginal probability of inclusion for
each variable is $1/2$. However, because the entries of $\boldsymbol{\xi}$ are
correlated, the prior induces multiplicity control by assigning a uniform
distribution on the size of the model $p^{*}_{\boldsymbol{\xi}}$ (see Scott &
Berger (2010) for additional details).

The model is completed by specifying the prior for the intercept $\eta_{0}$.
Because of the hierarchical nature of our model, we avoid the use of improper
flat priors and instead adopt a standard logistic prior with density

$f(\eta_{0})=\frac{\exp\left\\{\eta_{0}\right\\}}{\left(1+\exp\left\\{\eta_{0}\right\\}\right)^{2}}.$

Employing a prior for $\eta_{0}$ that is independent of the prior on
$\boldsymbol{\eta}_{\boldsymbol{\xi}}$ is well justified in this case because
we work with a centered design matrix $\boldsymbol{X}$. The choice of a
logistic prior implies that, under the null model where none of the variables
affect the bridging probabilities, the implied prior on
$1/(1+\exp\left\\{-(\eta_{0}+\boldsymbol{x}_{i}^{T}\boldsymbol{\eta})\right\\})$
reduces to a uniform distribution on the $[0,1]$ interval. Hence, under the
null model, this approach is equivalent to that of Lofland et al. (2017).
Another important feature of this prior is that it typically leads to stable
estimates in the presence of separation (e.g., see Boonstra et al., 2019).
This is important because separation is a potential concern in our setting
(particularly for years in which the the number of bridges is very high), and
because the latent nature of $\boldsymbol{\zeta}_{i}$ makes it impossible to
check for the presence of separation before fitting our model.222See Rainey
(2016) for a more detailed discussion of separation issues in logistic
regression.

##  3 Computation

Posterior inference is carried out using a Markov chain Monte Carlo (MCMC)
algorithm. Our approach relies heavily on the data augmentation approach
introduced in Polson et al. (2013). More specifically, we rely on the fact
that the Bernoulli likelihood can be written as a mixture of the form

$\displaystyle\frac{\exp\left\\{z_{i}\psi_{i}\right\\}}{1+\exp{\left\\{\psi_{i}\right\\}}}$
$\displaystyle\propto\exp\left\\{(z_{i}-1/2)\psi_{i}\right\\}\int_{0}^{\infty}\exp\left\\{-\frac{\omega_{i}}{2}\psi_{i}^{2}\right\\}f_{PG}(\omega_{i}\mid
1,0)d\omega_{i},$

where $f_{PG}(\omega\mid a,b)$ denotes the density of a Pòlya-Gamma random
variate with parameters $a$ and $b$. We use this data augmentation twice in
our algorithm: once for the likelihood for the observed data in (1), and then
again for the distribution of the bridge indicators in (3).

Introducing Pòlya-Gamma auxiliary variables dramatically simplifies the
formulation of our computational algorithm. As a consequence, most of the full
conditionals of interest can be sampled directly (i.e., most of the steps of
our MCMC algorithm reduce to Gibbs sampling steps). The only two exceptions
are the full conditional posterior distribution for $\eta_{0}$, which does not
belong to a known family, and the full conditional posterior distribution for
$(\boldsymbol{\xi},\boldsymbol{\eta})$, which is a mixture with an
unmanageable number of components. For $\eta_{0}$, we develop a Metropolis-
Hastings algorithm with heavy-tailed independent proposals (as opposed to a
random-walk Metropolis Hasting) that has a very high acceptance rate. For
$(\boldsymbol{\xi},\boldsymbol{\eta})$, we use a random walk Metropolis-
Hasting proposal for the marginal full conditional of $\boldsymbol{\xi}$
obtained after integrating out $\boldsymbol{\eta}$, and then sample
$\boldsymbol{\eta}\mid\boldsymbol{\xi}$ directly from its Gaussian full
conditional distribution. For additional details, please see the online
Supplementary Materials document and the implementation available at
https://github.com/e-lipman/IdealPointsCompare.

We use the empirical averages of the posterior samples to approximate
posterior summaries of interest. For example, we summarize the posterior
distribution on models through posterior inclusion probabilities (PIPs)
associated with each variable in the design matrix $\boldsymbol{X}$. The PIP
for variable $k$ is defined as

$\Pr(\xi_{k}=1\mid\mbox{data})=\mathbb{E}\left\\{\xi_{k}\mid\mbox{data}\right\\}\approx\frac{1}{S}\sum_{s=1}^{S}\xi_{k}^{(s)}$
(6)

where $S$ represents the total number of samples drawn from the posterior
distribution and $\xi_{k}^{(s)}$ denotes the $s$-th sampled value for the
parameter $\xi_{k}$. PIPs provide a measure of the uncertainty associated with
the influence of any given variable on the probability that a legislator is a
bridge. Point and interval estimates of other quantities of interest can be
obtained in a similar way.

###  3.1 Parameter identifiability

The $\mu$s, $\alpha_{j}$s, $\beta_{i,0}$s and $\beta_{i,1}$s are not
identifiable. Hence, if we are interested in performing inferences on them, we
need to be careful how the samples from the Markov chain Monte Carlo algorithm
are used construct the empirical averages that serve as posterior summaries.

In this manuscript, identifiability constraints are enforced after each
iteration of the Markov chain Monte Carlo algorithm rather than through
constrained priors. This approach is often referred to in the literature as
parameter expansion (Liu & Wu, 1999; Schliep & Hoeting, 2015). One such
constraint refers to the need to have enough bridges to make the scale
associated with the two voting domains comparable. Since in our application we
consider a one-dimensional latent space with $d=1$, this reduces to enforcing
that $\zeta_{\cdot}=\sum_{i=1}^{I}\zeta_{i}\geq d+1=2$. In practice, the
posterior distributions for $\zeta_{\cdot}$ puts all of its mass on much
larger values for all the datasets discussed in Section 4.2, which means that
the constraint is never binding and no adjustment to the posterior samples was
required in practice. A second identifiability constraint requires that both
scales should point in the “same direction” (e.g., Republican legislators
should tend to have positive values on both scales). This can be easily
accomplished by enforcing a common sign (but not necessarily a common value)
for the two ideal points (procedural and final passage) associated with one
carefully selected legislator (in our case, the Republican party leader). A
third constraint is associated with identifying the absolute scale, and is
enforced by translating and scaling the ideal points of the legislators so
that $d+1$ of them (the anchors) are fixed to specific values in one of the
voting domains. (in our case, procedural votes, but not final passage votes).
In our case, these correspond to the Democratic and Republican leaders. Then,
a corresponding transformation is applied to the $\mu_{j}$s and
$\boldsymbol{\alpha}_{j}$s so that the value of the linear predictor
$\mu_{j}+\boldsymbol{\alpha}^{T}_{j}\boldsymbol{\beta}_{i}$ remains the same.

##  4 Example Application: Voting in the U.S. House of Representatives,
1973-2014

In this section we apply the model we just described to voting in the U.S.
House of Representatives in two domains: procedural motions and final passage
motions. Procedural votes are broadly viewed separately from substantive votes
(Theriault, 2006). The determinants of legislative voting on procedural
motions in the House of Representatives include party affiliation and the
presence of other votes on the same day (Patty, 2010). Party matters for
procedural motions, but there are differences between the majority party and
the minority party (Carson et al., 2014). On the other hand, the determinants
of legislators’ voting on final passage motions include electoral concerns and
the possibility of receiving political appointments (Goodman & Nokken, 2007).
Legislative voting on final passage motions in the House of Representatives is
influenced by various factors such as the importance of electoral
considerations (Goodman & Nokken, 2004), change in parliamentary voting
behavior following electoral reforms, access to pork benefits (Shin, 2017),
and cross-pressured legislators who represent the interests of their
constituents against their party leaders (Kirkland & Williams, 2014).

Scholars have long been interested in the influences on legislators’ voting
both in the U.S. Congress, which is where our empirical example occurs, as
well in other parliaments and legislatures. We focus on three classes of
explanations. Broadly, scholars have studied three classes of covariates when
explaining legislative voting behavior: _constituency-level characteristics_
(how conservative/ liberal is a member’s district? Are there military bases
located in their district? How much agricultural activity takes place?);
_legislator-level characteristics_ (is the member male or female? Is the
member from a ‘safe district?’) and; what we call _chamber-level
characteristics_ (is the member in the majority party? Was electronic voting
used in the chamber or not? Under what rule was the roll call was
taken?).333This is by no means exhaustive. Other scholars have looked at the
effect of lame duck sessions on legislative voting, effective term limits,
etc. etc.

In general, legislators must balance the competing interests of their
constituents, their political parties, interest groups, and their own
ideological beliefs. Constituent interests play a significant role in
legislative voting (López & Jewell, 2007). Political parties also strongly
influence how legislators vote (Hix & Noury, 2016). Public interest lobbies,
ideology, and PAC contributions also predict legislative voting (Kau & Rubin,
2013). Constituency matters (Fleisher & Bond, 2004; Bond & Fleisher, 2002),
while gender does not seem to have an effect on legislative voting (Schwindt-
Bayer & Corbetta, 2004). We return to these general findings when discussing
our results (Section 4.2).

###  4.1 Data

This section describes the data that serves as the basis for our empirical
analysis and provides a brief descriptive analysis. Similar to Jessee &
Theriault (2014), we consider the roll-call voting records in the U.S. House
of Representatives starting with the 93rd Congress (in session between January
1973 and January 1975). This coincides with the introduction of electronic
voting in the House, which led to a marked increase in the number of roll-call
votes, as well as a change in their nature. Our analysis extends to the 113th
Congress, which is as far as our data sources on potential covariates extend
(please see below).

In our analysis, we consider 21 variables as potentially explaining the
tendency of individual legislators to reveal identical preferences across both
procedural and final passage votes.444See Appendix A for details. The
constituency-level covariates include measures of racial diversity and
economic inequality, as well as the political leaning of the constituancy.
Political leaning is measured by the percentage of the district’s vote that
went to the Republican candidate in the most recent presidential election. The
legislators-level covariates include age, gender, and race, as well as
measures of activity while in office, such as the number of sponsored bills.
The chamber-level covariates include whether the legislator’s party was in the
majority. Figure 1 presents the correlation matrices for the covariates
associated with four different Houses. Overall, correlations seem to be low.
In earlier Houses (e.g., the 93rd) there is a high correlation between some of
the constituency variables (such as the gini index, the socio-economic status
index, and the percentage of various minority groups in the district), but
these correlations tend to become lower in more recent Houses.

Figure 1: Correlation plots for the regression matrices associated with four
Houses under study: the 93rd, 98th, 103rd and 108th Houses.

Two variables that appears to be particularly important in explaining
differences in voting behavior between procedural and final passage votes are
(1) the percentage of the two-party presidential vote share received by the
Republican presidential candidate in the most recent presidential election,
and (2) the party of affiliation of the legislator and, in particular, whether
their party was in control of the House. We proceed with a brief descriptive
analysis of each of these variables. Figure 2 shows boxplots of the Republican
voteshare for each House, broken down by the party affiliation of the
legislator representing the districts. One interesting feature of these data
is that the range of districts held by Democrats tends to be much larger than
that of the districts held by Republicans. In particular, there are many
districts that lean heavily Democratic on presidential elections (with
Democratic vote share often above 90%), but there are few or no districts that
lean heavily Republican. Furthermore, Democrats seem able to consistently hold
districts that lean moderately Republican in presidential elections (with
Republican vote share above 60%), but Republicans seem to have a hard time
holding districts where the percentage of Democratic vote share in
presidential elections is above 55%. Because the interpretation of this
covariate appears to differ based on the party of the legislator representing
the district, we include separately the interaction of this covariate with the
Democratic party indicator and with the Republican party indicator.

Figure 2: Percentage of the two-party presidential vote share received by the
Republican presidential candidate in the legislator’s district in the most
recent presidential election. Districts are divided into Republican and
Democrat depending on the party of affiliation of the legislator representing
that district during the corresponding House. Figure 3: Percentage of
legislators in each party voting with the party majority. Shown separately for
procedural and final passage votes. Shaded regions represent sessions with a
Republican majority and unshaded regions represent sessions with a Democratic
majority.

In order to explore the role that party affiliation might play in voting
behavior on procedural and final passage votes, Figure 3 presents the median
(across legislators) of the proportion of times that each legislator voted
with their party, broken down by vote type and party. We can see that the
value has been historically high (generally over 80%), and there is also
evidence of an increasing trend over time. This increase is particularly
dramatic for the Republican party during the 104th House, when the percentage
jumps by about 10% for both types of votes. This observation is consistent
with previous literature pointing out that party influence on legislator’s
voting behaviors has been variable over time (e.g., see Aldrich et al., 1995
and Sinclair, 2006). We also observe that the percentage is often (but not
always) slightly higher for procedural than for final passage votes. Again,
this observation is consistent with previous literature suggesting that party
influence in the U.S. House of Representatives tends to be higher on
procedural votes (e.g., see Roberts & Smith, 2003, Roberts, 2007, Lee, 2009
and Gray & Jenkins, 2022)555Whether this is also true in the Senate is
debated, e.g., see Algara & Zamadics (2019) and Smith (2014)..

###  4.2 Results

In this section we report the results from applying the model described in
Section 2.We verify convergence of our algorithm by running multiple chains
(typically, four) started at over-dispersed initial values. For each chain, we
monitor the trace plot and the Gelman-Rubin statistic (Gelman & Rubin, 1992;
Vats & Knudson, 2021) of the (unnormalized) joint posterior distribution of
all parameters, the total number of bridge legislators,
$\zeta_{\cdot}=\sum_{i=1}^{I}\zeta_{i}$ (as well as the number of bridge
legislators broken down by party affiliation), the value of the linear
predictor $\eta_{0}+\boldsymbol{x}_{i}^{T}\boldsymbol{\eta}$ for each
legislator, and the number of predictors in the model, $p^{*}$. Most of the
results in Section 4.2 are based on 60,000 posterior samples for each House,
obtained by combining four independent runs of the Markov chain Monte Carlo
algorithm, each consisting of 15,000 samples obtained after a burn-in period
of 20,000 samples and thinning every 20 samples. The one exception is the
105th House. Unlike the other Houses under study, the posterior distribution
of $\boldsymbol{\xi}$ appears to be bimodal in this case. The dominant mode
favors relatively small models that include between 2 and 5 variables, while
the second mode favors large models that can have up to 15 explanatory
variables. These two modes also manifest themselves in the posterior
distribution of $\boldsymbol{\zeta}$, with the dominant mode supporting what
appears to be a superset of the bridges supported by the lower-probability
one. To ensure that both modes are explored thoroughly and that we can
accurately estimate their relative importance, we base our inference in this
House on eight chains, each of which consists of 20,000 iterations obtained
after thinning every 25 samples and a burn-in period of 20,000 iterations.

####  4.2.1 Aggregate analysis

We begin with a chamber-level analysis. Figure 4 shows the posterior mean and
95 percent credible intervals for the Bridging Frequency (BF) for each House.
The BF is defined as the proportion of bridges, i.e., legislators on each
House whose revealed preferences are the same across both procedural and final
passage votes. From equation (2) we have:

$BF=\frac{1}{I}\sum_{i=1}^{I}\zeta_{i}.$ (7)

Figure 4: Posterior mean and 95 percent credible intervals for the bridging
frequency in the U.S. House of Representatives from the 93rd through 113th
sessions of congress. Shaded regions represent sessions with a Republican
majority and unshaded regions represent sessions with a Democratic majority.

Figure 4 suggests a decline in bridging frequencies during the 70s and 80s,
followed by a steady increase starting perhaps with the 102nd House
(1991-1992). The decrease in the number of legislator that vote differently in
procedural and final passage votes has at least two potential explanations,
both of which are related to an increased tendency for legislators to vote
along party lines (recall Figure 3 and the associated discussion in Section
4.1). One explanation, which is consistent with the arguments in Jessee &
Theriault (2014), is that party influence on final passage votes has increased
since the early 1990s. Under this interpretation, our results provide support
for the “procedural cartel” theory of political parties (e.g., see Cox &
McCubbins, 2005 and Clark, 2012). An alternative explanation is provided by
constituency pressure. Increasing polarization within congressional districts
driven (among other factors) by the sorting of political identities along the
urban/rural divide and by aggressive gerrymandering, has resulted in fewer
competitive districts overall. There is evidence that fewer competitive
districts have meant fewer opportunities for moderate candidates (Barber &
McCarty, 2015; Thomsen, 2014, 2017). This has, in turn, led to primary
elections having an outsized impact on the final outcome of congressional
elections (Kaufmann et al., 2003; Abramowitz & Saunders, 2008; Bafumi &
Herron, 2010; Hall & Thompson, 2018). In this context, there are strong
incentives for legislators to vote along party lines on all votes in order to
avoid primary challenges, independently of any leadership pressure.

####  4.2.2 Legislator-level analysis: explaining the bridges

Figure 5: Number of covariates included in the posterior median model for each
House under consideration. Shading indicates congresses with a Republican
majority.

We turn our attention to the identification of variables that are correlated
with bridging behavior. Figure 5 shows the number of variables with a PIP
greater than 0.5 for each House (recall Equation (6)). In almost all cases,
this number is between 0 and 5 variables. The only outlier is the 109th House,
in which 17 variables appear to be significant. This result was surprising.
While the 109th House met for only 242 days (the fewest since World War II),
neither the absolute nor the relative number of procedural and final passage
bills appear to be particularly out of the ordinary. The one other notorious
fact about this House is the large number of political scandals that affected
its members, and particularly those in the Republican party. We speculate that
the Republican majority leader Tom DeLay’s campaign finance scandal and his
eventual resignation from Congress (along with the Bob Ney, Randy Cunningham
and Mark Foley scandals) might have led to a breakdown in Republican party
discipline in the House. This hypothesis seems consistent with the procedural
cartel theory we discussed in the previous section, and supports the idea that
party discipline might be the main driver behind the higher bridging
frequencies, and not the smaller number of competitive districts.

Figure 6: Heatmap showing posterior inclusion probabilities (PIP) for each
covariate in each House under study. Plus and minus signs indicate covariates
with PIPs with positive and negative point estimates, respectively. Horizontal
lines divide the covariates into groups: the three most commonly-selected
covariates are in the top group, followed by legislator characteristics
(middle), and the remaining constituancy characteristics (bottom).

Figure 6 shows the posterior inclusion probabilities (PIPs) for each covariate
in each House under study. The variable most commonly included in the model is
the indicator of whether a given legislator belongs to the party in control of
the House (its PIP is greater than 0.5 in 15 of the 21 Houses we studied, and
is greater than 0.9 in all of these). Except for the 95th House, the
coefficient associated with this variable is positive, with the model
indicating that belonging to the party in control of the House increases the
odds of being a bridge by a factor of anywhere between 3 and 435 (see Figure
7(a)). Another way to visualize this pattern is by reviewing raw bridging
frequencies (recall Equation (7)) computed separately for each party (see
Figure 8). Starting in 1985 (the 99th House), there is a clear pattern in
which the bridging frequency of the party in control of Congress is higher
(and, in some cases, much higher) than the bridging frequency for the minority
party. This result should not be surprising in light of our previous
discussions. For the majority party to have any hope of passing legislation,
it needs to achieve a certain level of party cohesion not only on procedural,
but also on final passage votes. Except perhaps in cases of very small
majorities, the level of pressure on legislators from the minority party to
toe the party line can be expected to be much lower (Pinnell, 2019; Ramey,
2015; Roberts, 2005).

(a) Influence of belonging to the majority party on odds of being a bridge
legislator

(b) Influence of proportion of constituency-level Republican vote in the most
recent presidential election on odds of being a bridge legislator, for
Democrats only

(c) Influence of proportion of constituency-level Republican vote in the most
recent presidential election on odds of being a bridge legislator, for
Republicans only.

Figure 7: Posterior mean and 95% percent credible intervals for the increase
in odds ratios of being a bridge legislator for the three most important
variables identified by our analysis. Figure 8: Posterior mean and 95 percent
credible intervals for the bridging frequency calculated separately for
legislators in each party. Shaded regions represent sessions with a Republican
majority and unshaded regions represent sessions with a Democratic majority.

The next two variables that are most commonly included in the model correspond
to interactions between the party of affiliation of a legislator and the
percentage of their district’s vote gathered by the Republican candidate in
the most recent presidential election (which we interpret as a proxy for the
ideological leanings of the constituency of that district). The interaction
associated with members of the Democratic party is included in the model in 9
of the 21 Houses under study and, in each of these cases, the associated
coefficient is negative (indicating that, as their district leans more
Republican, Democrats are less likely to be bridges). If we exclude the 109th
congress (which is an outlier) and the 111th congress (where this variable is
only marginally included), a 5% increase in the percentage of the Republican
vote in the most recent presidential election decrease the odds of a
Democratic legislator being a bridge by a factor of between 1.3 and 3.2 (see
Figure 7(b)). On the other hand, the coefficient for members of the Republican
party is included in the model for 8 Houses. In 7 of those cases, the
coefficient is now positive. The exception is the 109th House which, as we
mentioned above, seems to be an overall outlier. A 5% increase in the
percentage of the Republican vote in the most recent presidential election
increases the odds of a Republican legislator being a bridge by a factor of
between 1.5 and 3.3, roughly in line with what we observed for Democratic
legislators (see Figure 7(c)). Taken together, these two results suggests that
legislators representing swing districts are more likely to vote differently
in procedural and final passage votes. This observation is consistent with the
hypothesis that these legislators are subject to greater pressure from their
constituents than legislators in safe districts.

None of the other variables included in our study seem to consistently explain
bridging behavior by legislators. If we leave out the outlier 109th House, the
number of cosponsored bills appears to be significant during the 95th and the
108th House, but the signs of the coefficients are opposite. Similarly, the
percentage of the population in the district that is unemployed seems to be
marginally significant in the 94th and 107th Houses, but again with different
signs each time. Finally the number of sponsored bills, the indicator for
whether a legislator is Hispanic, the indicator for whether a legislator is
Black, the percent of foreign born residents, and the percent of recent
arrivals in the legislator’s district appear to be marginally significant in a
single House each (the 111th, 94nd, 108nd, 95nd, and 102nd, respectively).

##  5 Discussion and Conclusion

We have introduced a new class of hierarchical models that can be used to
identify covariates that might explain why legislators vote differently across
voting domains, with a focus on the differences in voting patterns between
procedural and final passage votes in the U.S. House of Representatives. The
paper makes both methodological and applied contributions. On the
methodological side, the hierarchical model is easy to interpreted and fully
accounts for all the uncertainty associated with learning legislator’s
revealed preferences across the multiple domains. The model is quite general
and can be applied to a number of other settings (e.g., committee vs. floor
voting, or voting on economic vs. social policy). On the application side, our
empirical results add to the literatures on competing principals in
legislative voting and on substantive vs. procedural voting. Like Carson et
al. (2014), we find that party matters. This is also consistent with Fleisher
& Bond (2004); Bond & Fleisher (2002). We also find evidence that the
constituency’s partisan leaning plays a role, which is consistent with Jessee
& Theriault (2014). In as much as this translates into electoral
considerations our results are also consistent with Goodman & Nokken (2004).
It is notable that we found that legislator-level characteristics such as
gender and ethnicity do not seem to explain bridging behavior. This is a novel
insight, which is nonetheless in line with the results of Schwindt-Bayer &
Corbetta (2004).

Our joint model seems to perform satisfactorily in most of the datasets we
considered in Section 4. The one possible exception is the 109th House, where
the model identifies a large number of variables as being potentially
influential on the bridging probability. In our discussion of the results, we
attributed this outlier to the unique nature of the political scandals that
arose during this period. Indeed, while scandals of some sort are (sadly) not
uncommon in the U.S. Congress, the nature of those ocurring during the 109th
House have a very particular significance in the context of our empirical
explanation, as they involved the highest echelons of the leadership of the
party in control of the House. Nonetheless, there are potential alternative
explanations for this outlier. One of them is model misspecification. Our
hierarchical model can be conceived as being made of two “modules” (roughly
speaking, one that learns the bridges, and one that determines which factors
explain those bridges). Over the last five years there has been growing
interest on the impact that the misspecification of one of the modules might
have on overall model performance (e.g., see Jacob et al., 2017 and Nott et
al., 2023). In our case, the most likely misspecified module is the one that
recovers the preferences of legislators from their voting records. Indeed,
recent evidence (e.g., see Yu & Rodriguez, 2021, Duck-Mayr & Montgomery, 2022
and Lei & Rodriguez, 2023) suggests that standard IRT models such as that in
Jackman (2001) might fail to accurately capture the preferences of some
legislators when legislators that are located at different ends of the
political spectrum tend to vote together against the rest. Solutions to this
issue include the use of more flexible models for capturing reveled
preferences, as well the use of so-called “cut” inference (Plummer, 2015;
Jacob et al., 2017). These approaches will be explored elsewhere.

The approach developed in this paper is appropriate for situations in which
voting happens across two domains. A future area of research is how to extend
the approach to situations in which there are more that two voting domains.
The most obvious approach (extending the latent logistic regression for binary
data to a multinomial regression) quickly becomes impractical, even for
relatively small numbers of voting domains. An alternative is to employ a
covariate-dependent prior on partitions (e.g., see Müller et al., 2011, Dahl
et al., 2017 or Page & Quintana, 2018) to implicitly define a prior
distribution on the probability that legislators reveal the same preferences
across any pair of voting domains. This kind of extension will be explored
elsewhere.

##### Acknowledgements

We would like to thank Stephen Jessee and Sean Theriault for providing access
to their data.

##### Funding Statement

This research was supported by grants from the National Science Foundation,
2114727 and 2023495.

##### Competing Interests

None.

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##  Appendix A Explanatory variables considered

Demographic and socioeconomic data for the legislators and constituencies was
obtained from Foster-Molina (2017). Data on the results of the presidential
elections between 1970 and 2008 was generously provided by Stephen Jessee and
Sean Theriault (personal communication), while data for the 2012 and 2016
elections was obtained from Daily Kos Staff (2022).

  1. 1.

Covariates related to party affiliation

belto.partyControl:

Indicator for whether the legislator is a member of the party that has the
majority in the current House.

p_R_R and p_R_D:

These two covariates are interactions between the two party membership
indicators and a measure of partisan political ideology for the legislator’s
district. The Republican share of the two-party presidential vote (p_R) is the
percentage of the two-party vote won by the Republican candidate in the most
recent presidential election (centered so that $0$ indicates that the two
parties received an equal percentage of the vote).
$p\\_R\\_R=I(Republican)\times p\\_R$ is equal to the Republican voteshare
$p\\_R$ for legislators belonging to the Republican party and $0$ for
legislators belonging to the Democratic party. The other interaction
$p\\_R\\_D=I(Democrat)\times p\\_R$ is defined an analogous way.

  2. 2.

Legislator characteristics

age:

Age at time of being sworn into congress for current session.

gender:

Gender of legislator.

black:

Indicator for membership to the Congressional Black Caucus. The authors of the
data note that to their knowledge, all self-identifying Black members of
congress are members of the caucus.

hispanic:

Indicator for membership to the Congressional Hispanic Caucus. The authors of
the data note that to their knowledge, all self-identifying Hispanic members
of congress are members of the caucus.

numberTerms:

Number of terms served in the House.

numSpon:

Number of bills sponsored by the legislator in the current term.

numCosp:

Number of bills co-sponsored by the legislator in the current term.

numPassH:

Number of bills sponsored by the legislator that were approved by a full House
vote in the current term

  3. 3.

Constituency characteristics (based on data from Census)

recentArrivalPrcnt:

Percentage of the district that recently moved to the district from another
county (note that the census does not track how many people have moved into a
district from within the same county).

prcntForeignBorn:

Percentage of the district that was born in a foreign country.

gini:

Index of economic inequality calculated based on the percentage of the
population in each income bracket.

ses:

Measure of socioeconomic status calculated based on the income and education
level of the district.

prcntUnemp:

Percentage of the district’s population that is unemployed but still in the
labor force.

prcntBA:

Percentage of the district with a bachelor’s degree or higher.

prcntHS:

Percentage of the district with a high school degree or higher.

prcntBlack:

Percentage of the district that is Black, including those who are Black and
Hispanic.

prcntHisp:

Percentage of the district that is Hispanic (both Black and White)

prcntAsian:

Percentage of the district that is Asian.

##  Appendix B Robustness

###  B.1 Other hyperpriors

For the intercepts $\mu_{1},\ldots,\mu_{J}$ we give $\rho_{\mu}$ a standard
Gaussian prior and $\kappa^{2}_{\mu}$ an inverse Gamma prior with shape
parameters 2 and scale parameter 1 (so that it has mean 1 and infinite
variance). This choice is computationally convenient and is consistent with
the logic underlying the logistic scale. It is also consistent with the
recommendations in Bafumi et al. (2005).

While we described the model more generally, in the sequel we work with $d=1$.
There is broad acknowledgement that voting in the U.S. Congress has become
broadly unidimensional since the late 1970s (e.g., see Hare & Poole, 2014 and
McCarty et al., 2016). Furthermore, this choice for the dimension of the basic
space allows us to interpret the model as allowing for issue spaces (Poole,
2007) whose dimension varies with the legislator (see Moser et al., 2021 for
details). With this in mind, we give $\omega_{k}$ a uniform prior and
$\kappa^{2}_{\alpha}$ another inverse Gamma prior with shape parameters 2 and
scale parameter 1. Similarly, we use a standard Gaussian prior for
$\rho_{\beta}$ and an inverse Gamma prior with shape parameters 2 and scale
parameter 1 for $\Sigma_{\beta}^{2}$.

###  B.2 Sensitivity analysis

To investigate the sensitivity of the model to prior choices we consider a few
alternative prior specifications for the key regression parameters $\eta_{0}$
and $\boldsymbol{\eta}_{\boldsymbol{\xi}}$. In particular, we explored
replacing the logistic prior for $\eta_{0}$ with a standard Gaussian prior and
replacing the g-prior on $\boldsymbol{\eta}$ with a mixture of g-priors that
relies on the robust prior developed in Bayarri et al. (2012). The results
were quite robust to these changes, although we do note that the mixture of
g-priors tended to favor slightly lower inclusion probabilities for most of
the variables.

In addition to alternative hyperpriors, we also considered slightly different
choices for the anchor legislators. In particular, we considered using party
whips and subjectively-chosen “extreme” legislators in each House as
alternatives to the party leaders. Our results did not seem to affected by
these changes.