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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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