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Towards a General Methodology of Bridging Ideological
Spaces
Tzu-Ping Liu∗1, Gento Kato†2, and Sam Fuller‡1
1University of California, Davis
2Nazarbayev University
August 15, 2020
Abstract
Bridging ideological spaces is an important, but relatively troubled branch of the
scaling literature. The most common bridging procedure, joint-scaling, ignores struc-
tural differences between groups resulting in uninformative results. Alternatively,
dimensional-mapping addresses this issue by using transformation rather than merg-
ing. However, current implementations cannot bridge multi-dimensional spaces nor
estimate ideal points non-parametrically. Furthermore, these methods require shared
individuals between the two groups. To address these major issues, we introduce a
generalized methodology for dimensional-mapping that enables both non-parametric
and multi-dimensional ideal point estimation using either real or ”synthetic anchors.”
Synthetic anchors remove the stringent anchor assumption and are generated by trans-
ferring a small number of individuals from one group to the other and, when used
appropriately, do not distort the ideological space. We demonstrate the utility of our
methodology on two sets of voter-politician data from the United States and Japan
by comparing its performance with existing approaches. Our results suggest that not
only does our method make less stringent assumptions and is more widely applicable
than existing techniques, but our approach also generates bridged ideal point estimates
comparable to those generated by other methods.
∗Ph.D. candidate, Department of Political Science. E-mail: tpliu@ucdavis.edu.
†Assistant professor, Political Science and International Relations Department. E-mail:
gento.badger@gmail.com.
‡Ph.D. candidate, Department of Political Science. E-mail: sjfuller@ucdavis.edu.
1
Introduction
Ideal point estimation is one of the most commonly used methods for ideological analysis
in the field of political science. Directly derived from the spatial model of voting, it pro-
vides a theoretically and methodologically rigorous way to capture the underlying ideological
preferences of different political actors (Poole 2005; Clinton 2012). In general, ideal point
estimation methods determine the “ideological position” of an individual by exploiting large
sets of either issue-specific preferences or political behavior across a sample of individuals
(e.g. a chamber of Congress). These datasets range from roll-call votes in a parliament,
opinions of judges in court cases, or responses to policy-related questions and feeling ther-
mometers in public opinion polls. Assuming that the latent ideological space actually exists
and matches the model’s pre-specified structure (e.g., one versus two dimensions, different
bases, or quadratic versus Gaussian utility functions), ideal points can be estimated based
on the relative distances between every individual. These distances are derived from the rel-
ative and aggregate patterns in individuals’ responses or behaviors, with those responding or
behaving in similar fashions being grouped close together and those that are more different
being placed further apart.
While conventional methods often make specific parametric assumptions regarding the
structure of the estimated latent dimensions, such as the utility function for all individuals
and the stochastic disturbances (random error) in preferences or behaviors (Poole and Rosen-
thal 1997; Heckman and Snyder 1997; Clinton, Jackman, and Rivers 2004), non-parametric
methods relax many of these assumptions. These less stringent assumptions allow for a more
appropriate and accurate estimation of ideal points among groups that likely have a weaker
ideological structure than political elites, like voters (Poole 2000; Hare, Liu, and Lupton
2018).
A critical issue in ideal point estimation is the methods’ dependency on the data on
which they are applied. Given that ideal points are (most often) generated using relative
Euclidean distances of the individuals in the given group/sample (e.g., voters or politicians),
2
any two sets of ideal points estimated on separate groups (or even subsamples of those
groups) are incomparable in most cases. This problem has been widely discussed in the
literature, especially in the context of comparing the recovered ideal points of voters versus
politicians. Current solutions to this incomparability problem can be divided into two general
categories: joint-scaling and dimensional-mapping. The former addresses this issue by
simply merging the two separate groups into one “pooled” data set prior to ideal point
estimation (Bafumi and Herron 2010; Malhotra and Jessee 2014; Jessee and Malhotra 2013)
while the latter preserves the separate estimation of ideal points and uses some sort of
transformation method (e.g., an ordinary least squares (OLS) regression) to “bridge” the
two ideological spaces.
Dimensional-mapping can further be divided into two separate, distinct approaches:
space-prediction and linear-mapping. The first approach, space-prediction, estimates
ideal points on only one group (e.g., politicians) and then utilizes these estimated (poste-
rior distributions of) parameters to predict the ideal points of the other group (e.g., voters)
(Jessee 2016). The second approach, linear-mapping, requires “anchors” (i.e., common in-
dividuals in both groups): After separate estimations of the two groups’ ideal points, a
transformation model is estimated based on anchors’ positions in two spaces (e.g., Shor,
Berry, and McCarty 2010).
While relatively effective in some situations, all of these approaches suffer from significant
limitations. To start, the key assumption of joint-scaling is that the “ideological space under-
lying the preferences of the two groups is structured in the same way” (Jessee 2016, 1110).
If this assumption is violated, ideal points estimated through joint-scaling are sensitive to
trivial factors such as the relative size of the two groups. Dimensional-mapping does not suf-
fer from this issue, but this method is of limited use when ideal points are multi-dimensional
and/or should be estimated non-parametrically. Space-prediction is simply not a possibility
in these scenarios since non-parametric estimations of ideal points do not generate parame-
ters for predicting the second group’s ideal points. Linear-mapping is a possibility here, but
3
previous methods have only been applied in one-dimension with no explicit generalization
developed for multi-dimensional ideal points.
In this paper, we introduce a new linear-mapping method that accommodates non-
parametric and multi-dimensional ideal point estimation to reduce the limitations present for
current bridging methods. In addition to addressing these issues, our method only requires
two groups (e.g., politicians and voters) to have a common set of votes/questions and does
not necessarily require pre-defined anchors. If pre-defined anchors do not exist, our method
randomly selects a small number of individuals from one group (e.g., voters), merges them
with the other group (e.g., politicians), and treats them as synthetic anchors. Then, after
estimating ideal points for each group, we apply Procrustes transformation to estimate the
transformation matrix based on the anchors’ coordinates in two spaces. This matrix is then
used to transform the rest of ideal points from the space of one group (e.g., voters) to that
of another group (e.g., politicians).
This article proceeds as follows. First, we review existing bridging methods and describe
in detail their limitations. Next, we introduce our new methodology. We then demonstrate
its performance through simulated data experiments. In addition, we apply our method to
two datasets that allow for bridging between politicians and voters: the 2004-2005 Senate
Representation Survey in the United States (Jessee 2009) and 2012 UTokyo-Asahi Survey
(UTAS) in Japan (Imai, Lo, and Olmsted 2016). In both applications, two-dimensional ideal
points are estimated through non-parametric (ordered) optimal classification (OC) (Poole
2000; Hare, Liu, and Lupton 2018) and the performance of our method is compared with
those of joint-scaling and linear-mapping. Lastly, we conduct two Monte-Carlo experiments
for evaluating how well our approach can recover the latent common space for different
sets of ideal points, and for assessing to what extent the inclusion of extra samples impacts
the original distribution of ideal points. We conclude by discussing both the advantages
and limitations of our approach and its implications for future development of ideal point
estimation techniques.
4
Existing Bridging Methodologies and Their Limitations
To overcome the data-driven nature of ideal point estimation, there exist two general types
of bridging methods: joint-scaling and dimensional-mapping. Both methodologies share the
requirement that either there exist common questions/votes between both groups and/or that
there are common individuals shared across both groups. The primary difference between
the two methods is how the two groups are merged or bridged. Joint-scaling combines data
from two (or more) groups with the same or similar set of either roll-call votes or survey
questions into one joint dataset and then uses standard methods to estimate the combined
ideal points. This method is by far the simplest: one merely needs to merge their datasets
and then apply any ideal point estimation method. On the other hand, joint-scaling makes
the strong assumption that the ideological space of each combined group has an identical
structure, i.e., the two sets of ideal points are generated from the same data generating
process (DGP) (Jessee 2016). If this assumption is not satisfied, which is likely for the
case when we compare highly ideologically constrained politicians versus less constrained
voters, the structure of the jointly estimated ideological space can only partially reflect the
true structure of ideological space for any one group. Additionally, since a jointly estimated
ideological space is influenced by the relative sizes of each group, the resulting structure can
be highly skewed towards the larger group if they do not share the same underlying structure
(e.g., a sample of 500 voters can overwhelm a sample of 100 senators).
In contrast, dimensional-mapping methods first estimate separate ideological spaces for
each group and then transform one group’s ideal points onto the other group’s space. This
procedure can thus accommodate the potential differences in the structures of groups’ dif-
ferent ideological spaces. Within dimensional mapping, there exist two popular approaches:
space-prediction and linear-mapping. Space-prediction bridges two groups by simply plug-
ging in one group’s data into the model parameters (estimated on the other group separately)
to generate their ideal points in the same space. For example, to bridge voters and politi-
cians, Jessee (2016) trains a Bayesian probit-link ideal point model (Clinton, Jackman, and
5
Rivers 2004) only on voters (politicians) and then uses the estimated parameters in each
iteration to predict the ideal points of politicians (voters). While intuitive, there are limita-
tions to this approach. One major issue is that it (almost) completely ignores the structure
of ideological space for the transformed group. Since the core idea of bridging is to recognize
the intrinsic differences between and within subspaces, ignoring one subspace makes this
approach at best incomplete as a bridging method. Additionally, space-prediction is simply
unavailable for non-parametric ideal point methodologies, as these methods do not estimate
model parameters.
Finally, linear-mapping first estimates ideal points within each group separately and
then estimates a transformation matrix to merge the two groups’ ideal points. This method
requires “anchors,” or common individuals who exist in both groups. After estimating the
ideal points of two groups separately, a transformation model is estimated based on anchors’
coordinates in the two ideological subspaces. The transformation matrix is then used to
map the rest of the ideal points onto one of the subspaces. This approach overcomes the
limitations in joint-scaling and space-prediction as it uses the structural information from
both groups’ spaces when bridging. The existing methodology of linear-mapping (Shor,
Berry, and McCarty 2010; Shor and McCarty 2011), however, suffers from several issues that
limit its application. Therefore, in the next section, we describe those issues and propose a
new, more generalized methodology of linear-mapping.
A General Methodology of Linear Mapping
While our method can easily be generalized to any dimensional space, we focus on an ap-
plication to a two-dimensional (2-D) ideological space. In a two-group bridging scenario,
our method can be described as merging two separate 2-D subspaces with ideal points. One
obvious question is how we generate one combined space that recovers the complete view of
ideal points. Overall, our method works in two stages to achieve this: First, we estimate
ideal points separately in two 2-D subspaces. In this stage, two subspaces are required to
6
have overlapping data points, i.e., anchors, that are either real or generated by random sam-
pling of cases from one subspace. Second, we use those overlapping data points to re-scale
two subspaces and combine them. The following sections describe each stage in detail.
For the purposes of illustration, suppose that there are two separate, unique survey
datasets of 1,000 respondents sharing an identical set of 30 policy questions. Since our
method is one version of linear-mapping, we estimate ideal points on each dataset separately.
This step not only estimates the coordinates (2-D ideal points) of respondents but also
constructs two separate 2-D Cartesian coordinate systems with independent bases. In other
words, this step remedies the problem that plagues joint-scaling and space-prediction in
which either one space dominates the ideal point estimation of the other (joint-scaling) or
most of the structural information in the second group is lost in merging the two groups
(space-prediction). Simply put, these separate independent coordinate systems retain the
structural information present in both groups. Depending on the data type, ideal points
can be estimated with any method, including non-parametric methods, such as optimal
classification (OC) or ordered optimal classification (OOC), which have the advantage of
unbiased estimation on data generated by an unknown process.
Our method departs from the previous linear-mapping methodologies in the treatment
of “anchors.” In previous linear-mapping methodologies, the datasets are required to share
“real” respondents to enable bridging. Our method relaxes this strong requirement by in-
troducing the use of “synthetic” anchors. They can be generated by taking a few, say 20,
respondents from the first dataset and inserting them into the second dataset. The valid
use of synthetic anchors requires two assumptions. First, those inserted first-survey respon-
dents are assumed to provide the same response if they were indeed answering the second
survey. This assumption is required even when using real, pre-defined anchors. For example,
Shor, Berry, and McCarty (2010) use politicians who served in both state-legislatures and
Congress as anchors to connect state and congressional ideological spaces. In this application,
although these legislators are “real” anchors, it is still assumed that these bridging legislators
7
make similar decisions regardless of whether they are in different institutions while represent
non-identical districts (Gray and Jenkins 2019). Second, given that ideal point estimation
is data dependent, we must also assume that the ideological structure present in the second
survey is not sensitive to the inclusion of synthetic anchors. In a later section, we validate
this assumption using real-world data.
In the next step, we utilize anchor-pairs to estimate a transformation matrix which
transforms the estimated coordinates from one space onto the other. when transforming a
2-D space with homogeneous coordinates, the procedure can be explained, without a loss
of generality, by the equation d1=Td2:
x
y
1
=
h11 h12 h13
h21 h22 h23
h31 h32 h33
x0
y0
1
where d1= (x, y) and d2= (x0, y0) are a pair of corresponding points in the two subspaces
separately, and the matrix with nine parameters, T, is the transformation matrix. When
the depth of each subspace is irrelevant, i.e., when conducting one- or two-dimensional ideal-
point estimation, one usually leaves the three parameters on the last row of Tas constant,
usually h31 = 0, h32 = 0, and h33 = 1, and estimate the remaining six parameters.1This
procedure is called (general) affine transformation.2
The only existing linear-mapping methodology to our knowledge (Shor, Berry, and Mc-
Carty 2010) uses bivariate linear regression to bridge one-dimensional ideological space. The
straightforward extension of this methodology to 2-D space through multiple linear regression
1More generally, for the n-D latent space, one could simply utilize the n-D homogeneous equation, in which
the dimensions of d1and d2are both (n+ 1) ×1 vectors (e.g., (x y z 1)|and (x0y0z01)|in the case of 3-D
latent space), and the transformation matrix Tis a (n+ 1) ×(n+ 1) matrix.
2Affine transformation is a special case of the so-called projective transformation, which refers to when h31,
h32 are not set equal to 0 and have to be estimated.
8
Figure 1: Variations in Forms of Geometric Transformation
can be defined as:
x=αx0+βx1x0+βx2y0
y=αy0+βy1x0+βy2y0
The above formula shows that the use of linear regression in the transformation of a 2-D
ideological space is equivalent to estimating an affine transformation matrix.
When applied to multi-dimensional spaces, affine transformation can cause a problem: It
allows shearing, a form of transformation that uses a direction, or line, on which all points
9
remain fixed, while adjusting all other points parallel to the direction by a distance propor-
tional to the direction (Borg and Lingoes 2012; Solomon and Breckon 2011). To understand
the problem more intuitively, see Figure 1. Four out of five forms of transformation —
rotation,translation,scaling, and reflection— preserve a “shape” or geometric structure of
a space, which is defined by relative distances between data points (Solomon and Breckon
2011). Once the space is sheared, however, the proportional distances between points (the
original shape) is lost in transformation. Simply put, linear-regression-mapping, when ap-
plied to multi-dimensional spaces, cannot preserve the original shape of ideological space
when estimating the transformation matrix.
To preserve the original relationships between data points in the transformed group (i.e.,
the second survey in our illustration), our method adopts Procrustes transformation to esti-
mate the transformation matrix. Given two sets of correspondent point clouds, this method
finds a transformation matrix which fits a configuration to the other as closely as possible
using only scaling (changing the size of a shape without changing its centroid and structure),
rotation (rotating a shape without changing its centroid and structure), translation (moving
the centroid of a shape without changing its structure), reflection (flipping a shape over a line
without changing its size and structure), and not shearing (Borg and Groenen 2005; Cox F.
and Cox A. A. 2000). Unlike affine transformation (or more generally, projective transfor-
mation) which allows shearing, Procrustes transformation preserves the original shape of the
ideological space.
Formally, Procrustes transformation can be described as follows. Given two point clouds
Xand Y, we would like to find a rotation matrix, T, a translation matrix, t, and a scaling
scalar, s, such that Y≈sXT +1t0. Similar to ordinary least squares regression (OLS), we
need to minimize the sum of squared errors to derive unknown parameters and matrices, but
10
Procrustes transformation does so through the Frobenius norm (Borg and Groenen 2005).3
min
||T||2=Itr[Y−(sXT +1t0)]0[Y−(sXT +1t0)] (1)
The primary advantage of Procrustes transformation is that it maintains the projected
points’ original relationships in the new space. This is because Procrustes transformation
does not change the shape of a given configuration, i.e., it preserves the proportion between
edges of a configuration (the shape). Given its characteristics, Procrustes transformation has
a relative advantage over bivariate and multiple regressions: It allows us to examine ideal
points and their relative structure (shape) through the perspective of a (new) common space.
The application of this transformation method is an important and novel contribution to
the bridging literature in that it enables the non-parametric bridging of multi-dimensional
spaces with no loss in the original shape of the respective ideological spaces.
Application
In this section we apply our bridging methodology alongside previous approaches to two
different datasets: The 2004–2005 US Senate Representation Survey and the 2012 Japanese
UTAS Survey. For each application, we bridge voters’ and politicians’ OC or OOC estimated
ideal points. Bridged ideal points are generated on a common space based on a dataset with
(1) voters and politicians pooled together (joint-scaling), (2) only voters (regression and Pro-
crustes transformation), and (3) only politicians (regression and Procrustes transformation).
We demonstrate the utility of our generalized bridging methodology through the derived
results.
3The restriction of ||T||2=Iis necessary or this equation cannot be solved. Furthermore, this restriction
guarantees that the derived rotation matrix, T, is an orthogonal matrix.
11
Case 1: American Voters and Senators
We first examine the data used in Jessee (2016) that surveys voters and politicians on
an identical set of policy questions. The voter survey (N=3599) was conducted between
December 2005 and January 2006 and includes questions corresponding to specific roll-call
votes that occurred in the U.S. Senate between 2004 and 2005 (N=100). Combined with
Senators’ votes, this survey records voters’ and senators’ stances on each of the same set of
votes. Given that the survey was originally fielded to examine voters’ perceptions of their
own senators, the sample includes, at a minimum, 100 respondents from each state.
Figure 2 presents the results. It can be divided into three sets of analyses. We first
present the results of joint scaling in the center panel. Second, in the right-hand panels, we
map Senators’ ideal points on voters’ latent ideal space by using randomly-sampled twenty
voters as synthetic anchors. The top row of the right panel presents separately estimated
ideal points for Senators (with twenty synthetic anchors) and voters. The middle and bottom
rows present Senators’ transformed ideal points mapped onto the voters’ latent space through
Procrustes and regression transformation, respectively. Third, in the left-hand panels, we
map voters’ ideal points on Senators’ latent ideal space by using twenty randomly-sampled
Senators as synthetic anchors. The top row presents separately estimated ideal points for
Senators and voters (with twenty Senator anchors). Again, middle and bottom rows present
Procrustes- and regression-transformed voter ideal points on the Senators’ latent space.
The results presented in Figure 2 demonstrate how ideal point estimation is heavily data-
driven. Comparing untransformed and jointly-scaled ideal points of voters and Senators, one
can see that the shapes of ideological space are not identical between separately and jointly
estimated ideal points. Especially for Senators, the space seems to be shrinking along the
vertical axis (second dimension) when jointly scaled. This result implies that the assumption
required for joint-scaling may not hold: Jointly-scaled ideal points are potentially biased or
distorted. Furthermore, as discussed in Jessee (2016), the joint-scaling result confirms that
the ideological space of the smaller group (i.e., Senator, N=99) is distorted more severely
12
Voter
(Regression)
Voter
(Procrustes)
Senator
(Untransformed)
Voter
(Untransformed)
−.5 0 .5
−.5 0 .5
−.5
0
.5
−.5
0
.5
−.5
0
.5
Transform Voter
Voter
Senator
−.5 0 .5
−.5
0
.5
−.5
0
.5
Democrat
Republican
Joint
X Party Centroid
Senator
(Regression)
Senator
(Procrustes)
Senator
(Untransformed)
Voter
(Untransformed)
−.5 0 .5
−.5 0 .5
−.5
0
.5
−.5
0
.5
−.5
0
.5
Transform Senator
1st dimension
2nd dimension
Figure 2: Comparing Bridging Results for US Senators and Voters
13
than the larger group (i.e., voters, N=3423).
The bottom rows of the left- and right-hand panel demonstrates the potential issue with
using regression transformation. As discussed in the previous section, shearing distorts
the original shape of the ideal point structure. Although ideal points from untransformed
groups are preserved as it is, the transformed ideal points are compressed (especially along
the vertical axis) thus making them relatively uninformative. In contrast, the middle row
of the left- and right-hand panels illustrates that Procrustes transformation preserves the
original shape of the ideological space even after the transformation.
Once two sets of ideal point clouds are transformed on the same latent space, we can
calculate the centroid of each party in each group. For instance, the centroid of Repub-
lican Senators’ ideal points, Crs , is calculated as Crs = ( ¯xrs,¯yrs), where ¯xrs is the mean
x-coordinate and ¯yr s is the mean y-coordinate of all Republican Senators. This information
can be used to determine the degree of polarization between two groups. Specifically, po-
larization can be captured by the Euclidean distance between the groups’ centroids. “X”
symbols in Figure 2 show the location of centroids, and Euclidean distances between Republi-
can and Democratic centroids are presented in Table 1. In the table, group column indicates
a subject group of which a distance is measured, and type and transformation columns
describe bridging methodologies utilized. Both the figure and table show that Procrustes
transformation does a better job than regression transformation in differentiating two parties
and transforms the original centroids from one space onto the other latent space. Regres-
sion transformation, on the other hand, tends to shrink inter-party distances thus making it
difficult to differentiate between Republicans and Democrats after transformation.
Overall, Figure 2 and Table 1 illustrate that Procrustes transformation is an effective
and appropriate strategy to bridge ideological spaces of US Senators and voters. Procrustes-
transformation bridged ideal points (regardless of which group is transformed) show that
the level of polarization is much greater among Senators than ordinary voters, this result is
consistent with a large amount of prior research documenting both polarization and ideolog-
14
Table 1: Distances between Republican and Democratic Party Centroids by Senators and
Voters
Group Type Transformation Distance
Senator Joint 0.907
Transform Voter Untransformed 1.059
Transform Senator Untransformed 1.086
Procrustes 1.086
Regression 0.532
Voter Joint 0.474
Transform Voter Untransformed 0.279
Procrustes 0.279
Regression 0.213
Transform Senator Untransformed 0.293
ical constraint in the mass public versus elites (Lupton, Myers, and Thornton 2015; Hare et
al. 2015). While the similar tendency persists, jointly-scaled and regression-transformation
bridged (especially when Senator is transformed) ideal points potentially underestimate
this Senator-voter gap in polarization by either overestimating inter-party distances among
voters (joint-scaling) or underestimating inter-party distances among Senators (regression-
transformation). These results provide further qualitative evidence that our methodology is
working well in bridging two different groups while addressing and preserving their structural
differences.
Case 2: Japanese Voters and Candidates
In this section we analyze the UTokyo-Asahi Survey (UTAS), fielded in Japan during the
House of Representatives election in 2012.4At each election, UTAS contains two sets of
surveys—voters and candidates—that share an identical set of policy questions. All policy
questions are answered on an ordinal scale and are thus compatible with OOC. The voter
survey randomly samples 3000 respondents from Japan’s list of registered voters’ and re-
sponses are filled out using a mail-in questionnaire. The response rates are 63.3% (N=1900)
4UTAS is conducted by Masaki Taniguchi of the Graduate Schools for Law and Politics, the University
of Tokyo and the Asahi Shimbun. The original dataset is available from the survey’s website (http:
//www.masaki.j.u-tokyo.ac.jp/utas/utasindex.html).
15
in 2012. The candidate survey is sent to all congressional candidates in a given election. The
response rates are 93.4% (N=1404) in 2012.5
Bridging voters’ and politicians’ ideologies is a particularly interesting question in Japan,
for at least three reasons. First, in contrast to the United States, parties’ positions on the
ideological spectrum are not always obvious or clear to voters in Japan (Jou and Endo 2016).
Second, political ideology in Japan is known to be multi-dimensional (Tanaka and Mimura
2006). Third, a large portion of Japanese voters are considered to be “independent” voters
(called muto-ha), and Japanese politicians are also known to frequently switch (or form new)
parties. The uncertain, unstable, and multi-dimensional nature of this ideological space
implies that the improvement made by our bridging methodology would play a particularly
important role in Japan.
Figure 3 maps bridged ideal points for members/supporters of three major parties in the
2012 election – Liberal Democratic Party (LDP), Democratic Party in Japan (DPJ), and
Japan Restoration Party (JRP).6Similar to Figure 2, it demonstrates that the derived ideal
points are highly data-driven, i.e., ideological spaces differ significantly when separately- and
jointly-scaled. Furthermore, the issue with regression transformation persists—the trans-
formed configuration is too compressed and uninformative. As shown in the bottom panels
of the figure, regression transformation tends to project ideal points onto the x-axis and
distorts the information conveyed originally by the y-axis.
In terms of differentiating between parties, Figure 3 and Table 2 calculate and map
party centroids in the same way as in Figure 2 and Table 1. The results suggest that
Procrustes-transformation bridging does a better job at differentiating party centroids than
other methodologies. Also, the Procrustes-transformed ideal points illustrate that DPJ and
LDP/JRP are differentiated on the first dimension while LDP and JRP are differentiated
5Ideal points are not calculated for voters and candidates with insufficient number of responses. The final
result for UTAS 2012 includes 1403 candidates and 1887 voters. In addition, we also analyzed UTAS 2009
and found very similar results. For the sake of brevity, we present these results in the Online Appendix.
6Ideal points for other party members/supporters and independents are not shown in the figure to avoid
confusion.
16
Voter
(Regression)
Voter
(Procrustes)
Candidate
(Untransformed)
Voter
(Untransformed)
−.5 0 .5
−.5 0 .5
−.5
0
.5
−.5
0
.5
−.5
0
.5
Transform Voter
Voter
Candidate
−.5 0 .5
−.5
0
.5
−.5
0
.5
LDP
DPJ
JRP
Joint
X Party Centroid
Candidate
(Regression)
Candidate
(Procrustes)
Candidate
(Untransformed)
Voter
(Untransformed)
−.5 0 .5
−.5 0 .5
−.5
0
.5
−.5
0
.5
−.5
0
.5
Transform Candidate
1st dimension
2nd dimension
Figure 3: Comparing Bridging Results for Japanese Candidates and Voters (UTAS 2012)
17
Table 2: Distances between LDP, DPJ, and JRP Centroids by Candidates and Voters
(UTAS 2012)
Combination Group Type Transformation Distance
LDP and DPJ Candidate Joint 0.409
Transform Voter Untransformed 0.356
Transform Candidate Untransformed 0.384
Procrustes 0.513
Regression 0.404
Voter Joint 0.140
Transform Voter Untransformed 0.073
Procrustes 0.191
Regression 0.089
Transform Candidate Untransformed 0.162
LDP and JRP Candidate Joint 0.563
Transform Voter Untransformed 0.471
Transform Candidate Untransformed 0.158
Procrustes 0.593
Regression 0.078
Voter Joint 0.088
Transform Voter Untransformed 0.061
Procrustes 0.166
Regression 0.069
Transform Candidate Untransformed 0.105
DPJ and JRP Candidate Joint 0.336
Transform Voter Untransformed 0.324
Transform Candidate Untransformed 0.419
Procrustes 0.490
Regression 0.384
Voter Joint 0.106
Transform Voter Untransformed 0.082
Procrustes 0.189
Regression 0.111
Transform Candidate Untransformed 0.188
on the second dimension. This tendency is consistent with the journalistic facts: LDP and
DPJ are traditional right-wing and left-wing parties, while JRP is one of new-right parties
emerged at 2012 election in which many members are previous members of LDP.
Combined Figure 3 and Table 2 illustrate the usefulness of our methodology above
18
and beyond standard methods when ideological spaces are uncertain, unstable and multi-
dimensional. Compared with the 2005–2006 Senate Representation Survey, the results
demonstrate that the level of polarization among Japanese elites is much smaller than those
in the U.S. Furthermore, the derived results indicate that Japanese partisan voters are geo-
metrically overlapping on the ideological space, as demonstrated by short distances between
party centroids. Unlike American voters, Japanese voters’ political preferences have not been
clearly polarized or divided along party lines and, overall, they generally hold moderate views
across all policies. Still, Procrustes-transformed party centroids are mapped on theoretically
consistent relative locations.
Lastly, this is a perfect example, in contrast to the American case, in which two or
more dimensions may be needed to understand latent dimensional spaces. Whereas in the
American situation the first dimension cleanly splits partisans, single dimension is uninfor-
mative in the Japanese case. Overall, not only providing evidence of our method’s utility
and robustness, the Japanese case further supports the importance of being able to bridge
multi-dimensional spaces. Our method makes an important contribution in enabling mul-
tidimensional linear-mapping. In the next sections, we provide further evidence to validate
our assumptions for synthetic anchors and the performance to recover real (true) ideological
space.
Validating the Use of Synthetic Anchors
In our applications, we use synthetic anchors, as oppose to real anchors, to bridge ideo-
logical spaces. This procedure makes an assumption that the inclusion of out-group extra
samples do not distort the estimation of original latent space. In this section, we test this
assumption on UTAS 2012 dataset. The violation is expected to be severer in Japan than the
United States, since latent spaces of politicians and voters tend to be more distinctive from
each other.7We follow the approach taken in Jessee (2016) to conduct Monte-Carlo experi-
7The test on American data is presented in Online Appendix. Its result implies that the violation in American
data is less severe and less responsive to the number of out-samples than in Japanese data.
19
20
50
100
500
1000
X−axis
(1st dimension)
Y−axis
(2nd dimension)
−.5 0 .5 −.5 0 .5 −.5 0 .5 −.5 0 .5 −.5 0 .5
Ideological positions
Density With voters Only candidates (N=1403)
Extra samples (the number of voters)
Figure 4: Density Plots of Candidate’s Positions with Different Amount of Extra Voter
Samples (UTAS 2012)
ment. Specifically, we estimate candidates’ two-dimensional ideal points (N=1403) through
OOC by including zero, twenty, fifty, one-hundred, five-hundred, and one-thousand randomly
sampled voters. We repeat this procedure four-hundred times8and assess if distributions of
candidates’ ideologies change by the inclusion of extra voter samples.
Results are presented in Figure 4. In each panel, a solid line shows the density distribu-
tion of candidates’ ideal points estimated with the inclusion of extra voter samples, averaged
across four-hundred experiments. A dashed line indicates the density distribution of can-
didates’ ideal points without any voter samples. Given that we estimated two-dimensional
ideological space, the top row shows distributions for the first dimension (i.e., x-axis), the
bottom row for the second dimension (i.e., y-axis). Columns indicate numbers of extra voter
8Given that there is no random sampling involved, candidates’ ideal points without voter samples are only
estimated once.
20
samples included in estimation.
We see that in general, the larger the voter samples added, the more distorted the original
distribution of candidates’ ideologies (i.e., distribution estimated without voter samples). For
coordinates on the x-axis shown in the top row, the density is not significantly different than
the original when the extra sample-size is fifty or smaller. As the number of extra samples
grows, however, the original bi-modal density gradually transforms to a single-peaked one.
The distortion caused by synthetic anchors is less severe on the y-axis (the bottom row).
Still, densities with extra samples tend to deviate more from original as the extra voter
sample-size grows.
The results of the experiment provide the intuition regarding the validity of synthetic
anchors. The distortion is tolerable as long as the size of synthetic anchor sample is under a
reasonable level (around fifty or 3.5% in the case of UTAS 2012). While this level may differ
by the sample size of the first group and the severity of differences between ideological struc-
tures of two groups, our result demonstrates that synthetic anchors enable us to accurately
bridge two separate ideological spaces in absence of “real” anchors.
Compare Performances with Simulated Data
While we have demonstrated the utility of our approach, we have yet to address whether
this general approach performs well to reveal a true latent space underlying individuals’
preferences. In this section, we design an experiment to address this concern.
To verify that the general approach can accurately reveal a real latent space, we adopt an
informal Monte-Carlo experiment similar to Hare, Liu, and Lupton’s (2018).9The simulation
uses 2,400 “respondents” and a set of 40 five-point Likert scale policy questions (like a
standard survey). We also, separately, generate each respondent’s ideal point coordinates
in a two-dimensional space by drawing from a bivariate normal distribution centered at
zero with correlations between the two dimensions also drawn from a uniform distribution
9See Hare, Liu, and Lupton (2018) for a detailed explanation for this experiment.
21
between -0.1–0.7. The “policy question normal vectors” are generated by drawing from the
edges of a unit circle in 2-D space, then each of the five possible responses to each issue is
randomly selected and projected onto their respective normal vectors.
To generate each respondent’s response to each question, we first generate each response
level’s location (one to five) along with each normal vector based on the distribution of all
respondents’ locations along that normal vector. We then randomly assign one of three
distributions, the quadratic, normal, and linear distribution (with equal probability and the
same mean and similar variances) to each respondent, and further generate each respondent’s
response based on the probability corresponding to the distance between each level and that
respondent’s location.10 For instance, given a respondent’s distances between each of five
levels, we can derive the probability of each level for that respondent based on his/her
designated distribution, and further randomly pick one level as their actual choice based on
each level’s corresponding probability.
After the data is generated, we randomly split the data into two subsets, one containing
1,400 respondents and the other 1,000. We employ multiple bridging methods to estimate
and merge ideal points on these two datasets, including:
1. Joint Scaling: Merge the two datasets and jointly estimate ideal points.
2. Procrustes Transformation: Our dimensional-mapping approach described above (with
twenty extra anchors).
3. Linear Regression: Existing dimensional-mapping approach described above (with
twenty extra anchors).
4. Concatenation: Estimate ideal points separately on two datasets and then concatenate
them.
10Although the three utility functions are substantively different after the Gaussian’s inflection points, OC
or OOC constrains the boundary of ideal-points to a unit-circle. Under this unit-circle condition, these
three utility functions should be exceedingly similar due to the ideal-points’ close proximity to the mean.
22
Joint
Procrustes
Regression
Concatenated
Real
Joint
Low Med. High Low Med. High Low Med. High Low Med. High
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
Error level
Correlation
Estimated ideal points
Figure 5: Correlation between Estimated and Real Positions
Each approach is repeated four hundred times. After all the results are derived, we further
calculate correlations between the distance matrices between estimated ideal points from
each approach and the real ideal points. The results of this experiment are presented in
Figure 5.
Based on the proportion of incorrect choices (errors), we group results into three cate-
gories: low, medium, and high error. We then calculate two types of correlations by each
error group. The first includes correlations between the real ideal points and the distance
matrix of each method. The second compares the utility of the different methods through
the correlations between the distance matrix of joint scaling and the distance matrices of the
other methods.
As the boxplots in the top panels of Figure 5 demonstrate, first of all, joint scaling,
Procrustes transformation, and regression transformation all perform similarly across the
23
different error levels. Given that all respondents’ ideal points are generated from similar
distributions, the results of joint scaling also show the utility of OOC as demonstrated in
Hare, Liu, and Lupton (2018). Furthermore, the results from Procrustes and regression
transformation both indicate that with appropriate procedures, two different sets of ideal
points can still be projected onto the same latent space.
The results from the separate estimations further validate that the results of the two
transformation approaches are not simply due to the fact that the two sets of ideal points
are generated from the same distribution. As discussed, given that the ideal points are
estimated through the relative distances between individuals, the estimated results are highly
data driven, i.e., different combinations of individuals will generate different latent spaces.
The poor performance of the separate estimations illustrates that even though two sets of
ideal points are generated from the same distribution, the estimated ideal points actually
exist in different latent spaces due the structure of data. Furthermore, the correlations
between the results of joint scaling the the rest of approaches indicate that joint scaling and
the two transformation approaches all derive similar configurations of estimated ideal points.
By and large, the results of Figure 5 provide one important insight into bridging methodologies—
that is, even when two groups meet the assumptions required by joint scaling, applying
bridging methods will generate similar results, but if these assumptions are not met, bridg-
ing methods will yield vastly different and better results. Simply put, when trying to compare
two sets of ideal points without knowing the underlying data generating process (which is
most often the case), one should always use an appropriate bridging method, rather than
joint scaling, to guarantee valid estimates and results.
Overall, the results of current Monte-Carlo experiment addresses important concerns
regarding our method. When the two sets of ideal points are generated by the same data
generating process, our results demonstrate that dimensional-mapping performs no worse
than joint scaling. Furthermore, the results also show that the estimation of ideal points is
highly driven by the combination of respondents, regardless of their distribution.
24
Discussion: Generalizing a Methodology of Bridging
1 Discussion: Generalizing a Methodology of Bridging
In this paper, we develop a generalized approach to ideal-point bridging to address major
issues and limitations present in previous methods. Compared with existing approaches,
this generalized approach has four major advantages: 1) it can employ synthetic anchors
when no real ones exist between the bridged groups; 2) this method can be applied to multi-
dimensional latent spaces, which is often necessary for recovering meaningful dimensions
(e.g., Japan); and 3) it preserves important information regarding the separate ideological
structure of the different groups as much as possible.
The benefits of this method are also fueled in large part due to the significant threat of
bias arising from the highly data-driven structure of ideal-point estimation. Specifically, our
applications and experiments demonstrate that: 1) when two sets of data are not generated
by the same distribution, joint scaling should not be used due to the bias caused by the data-
dependent nature of ideal-point estimation; and 2) when two sets of data are generated by the
same data generating process, dimensional-mapping performs just as well as joint-scaling.
In fact, our method improves upon joint-scaling’s performance when the data generating
processes are heterogeneous and matches it when the DGPs are homogeneous. Given that
prior knowledge of groups’ DGPs is rare, at best, our findings suggest that one should
almost always use linear mapping, generally, and our method, specifically, as demonstrated
in Figure 5. This ensures a high degree of performance and the lowest possible risk of biased
estimation.
Finally, although this method is primarily designed for non-parametric ideal-point esti-
mation, we believe that this general approach can also be applied to parametric methods.
Theoretically, because parametric methods assign specific DGP to the distribution of data,
as compared with non-parametric methods, the post-hoc density of ideal points should be
more stable if applied to parametric methods (i.e., the original density is not significantly
25
distorted when using a reasonable amount of synthetic anchors). In other words, synthetic
anchors should not significantly impact original data’s structure, thus allowing researchers
to apply this general approach to other parametric methods of ideal point estimation, such
as the Bayesian item response model (Clinton 2012) or the blackbox scaling in basic space
(Poole 1998).
26
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