Contrastive Multiple Correspondence Analysis (cMCA): Using Contrastive Learning to Identify Latent Subgroups in Political Parties - arXiv
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Contrastive Multiple Correspondence Analysis
(cMCA): Using Contrastive Learning to Identify
Latent Subgroups in Political Parties
Takanori Fujiwara* 1 and Tzu-Ping Liu2
1 Department of Computer Science, University of California at Davis, Davis, CA 95616, USA. Email:
tpliu@ucdavis.edu
1 Department of Political Science, University of California at Davis, Davis, CA 95616, USA. Email:
tpliu@ucdavis.edu
arXiv:2007.04540v2 [cs.SI] 15 Jan 2021
Abstract
Scaling methods have long been utilized to simplify and cluster high-dimensional data. However, the latent
spaces derived from these methods are sometimes uninformative or unable to identify significant differences
in the data. To tackle this common issue, we adopt an emerging analysis approach called contrastive learn-
ing. We contribute to this emerging field by extending its ideas to multiple correspondence analysis (MCA) in
order to enable an analysis of data often encountered by social scientists—namely binary, ordinal, and nom-
inal variables. We demonstrate the utility of contrastive MCA (cMCA) by analyzing three different surveys of
voters in Europe, Japan, and the United States. Our results suggest that, first, cMCA can identify substantively
important dimensions and divisions among (sub)groups that are overlooked by traditional methods; second,
for certain cases, cMCA can still derive latent traits that generalize across and apply to multiple groups in the
dataset; finally, when data is high-dimensional and unstructured, cMCA provides objective heuristics, above
and beyond the standard results, enabling more complex subgroup analysis.
1 Introduction
Scaling has been a prolific and popular topic for research throughout political science. Scholars
have developed and utilized a diverse set of methods to identify political actors’ “positions” on the
left-right ideological space. This has been accomplished by utilizing one of two general ideas: “op-
timal scoring”(Jacoby 1999) and “spatial voting”(Enelow and Hinich 1984). While optimal scoring
methods include principal component analysis (PCA), multiple correspondence analysis (MCA),
and factor analysis(e.g., Brady 1990; Davis, Dowley, and Silver 1999), spatial voting methods en-
compass Aldrich-McKelvey scaling, the NOMINATE class of models, two-variable item response
theory, and optimal classification(e.g., Aldrich and McKelvey 1977; Clinton, Jackman, and Rivers
2004; Hix, Noury, and Roland 2006; Martin and Quinn 2002; Poole 2000; Poole and Rosenthal 1991).
These methods have most often been applied to roll-call votes in the United States Congress, or
other political bodies; however, they have been increasingly used to analyze a range of new data,
including surveys and text (Hare, Liu, and Lupton 2018; Imai, Lo, and Olmsted 2016; Monroe, Co-
laresi, and Quinn 2008; Poole 1998; Wilkerson and Casas 2017).
In general, these methods uncover the similarities/differences between observations in a given
dataset. These similarities are then used to estimate a small number of (underlying) dimensions
that reflect the similarities as a set of spatial distances between all of the estimated points. These
lower-dimensional representations can both help us to understand the general structure of a given
dataset, such as its distribution, and identify underlying patterns, like clusters and outliers (Fuji-
wara, Kwon, and Ma 2020; Wenskovitch et al. 2018). More precisely, these methods are often ex-
* Equal contribution and listed alphabetically.
1
plicitly used to (1) identify clusters; (2) derive influential factors (dimensions); and (3) compare
the derived clusters and/or factors with predefined classes or groups (e.g., Armstrong et al. 2014;
Brady 1990; Davis, Dowley, and Silver 1999; Grimmer and King 2011; Hare et al. 2015; McCarty,
Poole, and Rosenthal 2001; Rosenthal and Voeten 2004).
In addition to deriving lower-dimensional spaces, correctly interpreting these recovered di-
mensions is just as, if not more, important. Since the work of Coombs (1964), the latent left-right
dimension of ideology is often recovered as a principal component/direction (PC) in political data.
Indeed, almost all of the methods discussed previously recover ideology as their first PC. Never-
theless, these methods can face situations when the left-right scale is uninformative, e.g., when
respondents are highly moderate or the data contains large amounts of missing values. Because of
this clustering, it is difficult to delineate between voters of some given group or category, such as
those that belong to an established political party. While this does not happen particularly often,
cases of strong moderation are found throughout political science (such as in Japan’s electorate,
see below).
Although derived scaling results may not be very interesting or informative, it does not neces-
sarily mean that supporters from different parties are politically similar. Rather, supporters from
different parties may be divided by specific issues or directions that are not captured by the left-
right scale. Thus, to analyze potential differences above and beyond simple ideological divides
Abid et al. (2018) derive a contrastive learning variant of PCA, named contrastive PCA (cPCA). This
method, instead of analyzing the data as a whole, first splits data into different groups, usually
by predefined classes (e.g., party ID), and then compares the data structure of the target group
against the background group to find PC(s) on which the target group varies maximally and the
background group varies minimally. This method can uncover within-group differences, as shown
below, such as euroskepticism among British voters within both major parties, and xenophobia
within the major conservative party in Japan.
We contribute to the field of contrastive learning by extending this idea from PCA to MCA so as
to enable the derivation of contrastive PC(s) using different types of features/data that social sci-
entists encounter frequently: binary, ordinal, and nominal data. With contrastive MCA (cMCA), we
find that even when voters are highly moderate and homogeneous across party lines, we can still
explore how specific issue(s) effectively identify subgroups within the parties themselves. Similar
to cPCA, cMCA first splits the data by predefined groups, such as partisanship or treatment and
control groups; second, as is the case with MCA, cMCA applies a one-hot encoder to convert a cat-
egorical dataset into a binary format matrix, called a disjunctive matrix; finally, cMCA takes one of
the groups as the background/pivot and then compares this group with the target group to derive
PCs on which the positions of members’ ideal points from target groups vary the most but those
from the background group vary the least.
To demonstrate the utility of cMCA, we analyze two surveys, the 2018 European Social Survey
(ESS 2018, the U.K. module only) and the 2012 Asahi-Todai Elite Survey (UTAS 2012). The results
show that when voters are highly moderate and the derived latent spaces are uninformative, as
shown by the results from the ESS 2018 and UTAS 2012, cMCA objectively detects subgroups along
with certain directions which are overlooked by traditional approaches, such as pro- and anti-EU
attitudes among supporters from each of the U.K.’s Conservative and Labour parties; or the anti-
Korean and Chinese sentiments among supporters of the Liberal Democratic Party in Japan.
This work makes an important contribution to scaling methods and unsupervised learning by
enabling researchers to identify latent dimensions that effectively classify and divide observations
(voters) even when the left-right scale cannot effectively discriminate between classes/groups (par-
tisanship). This method, as evidenced by its application to voter surveys, objectively and meaning-
fully identifies latent dimensions that would be, and are, overlooked by existing methods. Simply
put, cMCA’s results provide important information for researchers to better understand and make
Fujiwara and Liu 2
meaningful distinctions between underlying groups in their data.
Moreover, compared with existing scaling methods such as the (Bayesian) Aldrich-McKelvey
Scaling method (Aldrich and McKelvey 1977; Hare et al. 2015), the ordered optimal classification
(Hare, Liu, and Lupton 2018), the basic-space procedure (Poole 1998), and the Bayesian mixed
factor analysis model (Martin, Quinn, and Park 2011; Quinn 2004), all of which tend to remedy the
missing data issue through imputation or deletion; cMCA, on the contrary, allows researchers to
analyze regular and missing data simultaneously by treating missing data as one of the levels—this
is important given that sometimes the presence of missing data may be theoretically meaningful
and interesting (Greenacre and Pardo 2006).
Finally, to date, almost all of the methods for subgroup analysis, such as class specific MCA
(CSA) (Hjellbrekke and Korsnes 1993; Le Roux and Rouanet 2010) and subgroup MCA (sMCA) (Greenacre
and Pardo 2006) (see Section 4 for detailed discussion), requires researchers to already know how
data should be “subgrouped” in the latent dimensions, and therefore, without such prior knowl-
edge, these methods are almost infeasible, especially when data is high-dimensional and unstruc-
tured. By objectively revealing the overlooked influential factors, cMCA indeed provides a spring-
board for further subgroup analyses when information about how to determine those subgroups
is unavailable. Overall, we believe that cMCA provides an important contribution to data visual-
ization, subgroup analysis, and unsupervised learning. The paper proceeds with a description of
cMCA, its application to our two voter surveys, and then concludes with general thoughts about,
and rules of thumb for using, cMCA, i.e., its comparison with current subgroup analysis methods,
and its application to substantive topics.
2 Contrastive Learning and Contrastive MCA
Contrastive learning (CL) is an emerging machine learning approach which analyzes high-dimensional
data to capture “patterns that are specific to, or enriched in, one dataset relative to another” (Abid
et al. 2018). Unlike ordinary scaling approaches, such as PCA and MCA, that usually aim to capture
the characteristics of one entire dataset, CL compares subsets of that dataset relative to one an-
other (target versus background group). The logic behind CL is to explore unique components or
directions that contain more salient latent patterns in one dataset (the target group) than the other
(background group). So far, this approach has been applied to several machine learning meth-
ods, including PCA (Abid et al. 2018), latent Dirichlet allocation (Zou et al. 2013), hidden Markov
models (Zou et al. 2013), regressions (Ge and Zou 2016), and variational autoencoders (Sever-
son, Ghosh, and Ng 2019). We utilize this contrastive learning approach by applying it to MCA,
an enhanced version of PCA for nominal and ordinal data analysis (Greenacre 2017; Le Roux and
Rouanet 2010).
2.1 Multiple Correspondence Analysis (MCA)
When applying PCA to non-continuous data, the method considers each level or category in the
data as a real numerical value. As a result, PCA necessarily ranks the categories and assumes that
the intervals are all equal and represent the differences between categories (i.e., it treats it as inter-
val data). To overcome these issues, MCA first converts an input dataset X ∈ Òp×d (p: the number
of data points, d : the number of variables) of non-continuous data into what is called a disjunc-
tive matrix G ∈ Òp×K (K : the total number of categories) by applying one-hot encoding to each
of d categorical dimensions (Harris and Harris 2010). For illustration, assume that X consists of
two columns/variables: color and shape, and each variable has three (red, green, blue) and
two (circle, rectangle) levels, respectively. In this case, the disjunctive matrix G will contain
five categories: red, green, blue, circle, and rectangle. For instance, a piece of blue rectangle
paper will be recorded as [0, 0, 1, 0, 1] in G.
Fujiwara and Liu 3
We could further derive a probability/correspondence matrix Z through dividing each value of
G by the grand total of G (i.e., Z = N −1 G where N is the grand total of G). This probability matrix
can now be treated as a typical dataset for use in analyses. Similar to PCA, we first normalize Z and
def
then obtain what is called a Burt matrix, B, with B = Z⊤ Z (B ∈ ÒK ×K ). This Burt matrix B under
MCA corresponds to a covariance matrix under PCA. Thus, as in PCA, to derive principal directions,
MCA performs eigenvalue decomposition (EVD) to B to preserve the variance of G.1
2.2 Extending MCA to cMCA
To apply CL to MCA, we first split the original dataset X ∈ Òp×d into a target dataset XT ∈ Òn×d and
a background dataset XB ∈ Òm×d (where p = n + m) by a certain predefined boundary, such as
individuals’ partisanship or whether they were assigned to treatment or control groups. We then
further derive two Burt matrices from the target and background datasets, BT and BB , respectively.
Importantly, because the CL procedure utilizes matrix differences, the Burt matrices must have the
same dimensions, i.e., BT ∈ ÒK ×K and BB ∈ ÒK ×K .
Let u be any unit vector with K dimensions. Similar to cPCA (Abid et al. 2018), we can derive
def
the variances of the target and background datasets along with u by using σT2 (u) = u⊤ BT u and
def
σB2 (u) = u⊤ BB u. To find the direction u∗ on which a target dataset’s probability matrix, ZT , has
large variances and a background dataset’s probability matrix, ZB , has small variances, one needs
to solve the optimization problem:
u∗ = arg max σT2 (u) − α σB2 (u) = arg max u⊤ (BT − α BB )u (1)
u u
where α is a hyperparameter of cMCA, called the contrast parameter. From Equation 1, we can
see that u∗ corresponds to the first eigenvector of the matrix (BT − α BB ). Note that all of the
eigenvectors of (BT − α BB ) can be derived through performing EVD over (BT − α BB ).2 We call
these derived eigenvectors contrastive principal components (cPCs).
2.3 Selection of the Contrast Parameter
The contrast parameter α in Equation 1 controls the trade-off between having high target variance
versus low background variance. When α = 0, the resulting cPCs only maximize the variance of a
target dataset producing results that are equivalent to applying standard MCA to only the target
dataset. As α increases, cPCs place greater emphasis on directions that reduce the variance of
a background dataset. As proved by Abid and Zou (2019), by selecting arbitrary α values within
0 ≤ α ≤ ∞, CL methods allows the researchers to produce different latent spaces with each
latent space showing a different ratio of high target variance to low background variance—rather
than representing a best solution to Equation 1, the contrastive parameter, α , instead represents
a “value set.” Thus, researchers can manually apply different α values as long as the derived latent
spaces demonstrate “interesting” or meaningful patterns.
In addition to manual selection, we extend the method of Fujiwara, Zhao, Chen, Yu, et al. (2020)
to develop a technique for automatically selecting a special value of α , from which researchers can
derive the latent space where a target dataset has the highest variance relative to a background
dataset’s variance. This specific α can be found by solving the following ratio problem:
tr(U⊤ BT U)
max (2)
⊤
U U=I tr(U⊤ BB U)
1. Similar to PCA, without computing B, one could directly apply singular value decomposition (SVD) to Z to obtain the
same principal directions.
2. Unlike MCA, SVD cannot be applied to compute cPCs for cMCA due to the difference between target and background
datasets’ Burt matrices (i.e., BT − αBB ) may generate negative numbers.
Fujiwara and Liu 4
′ ′
where U ∈ ÒK ×K is the top-K eigenvectors obtained by EVD. Following Dinkelbach (1967), we
employ an iterative algorithm to solve Equation 2 which is usually difficult to solve. This algorithm
consists of two steps: given eigenvectors Ut at iteration step t (t ≥ 0 and t ∈ Ú), we perform
tr(U⊤
t BT Ut )
Step1. αt ← (3)
tr(U⊤
t BB Ut )
tr U⊤ (BT − αt BB )U
Step2. Ut +1 ← arg max
⊤
(4)
U U=I
At t = 0, because the computed U0 does not exist, we self-define α0 = 0 as the default solution to
Step 1. As demonstrated, αt in Equation 3 is an objective value of Equation 2, which is computed
with the current Ut . The second step (Equation 4) is to derive the eigenvectors, Ut +1 , for the next
iteration. This step just solves the original cMCA problem based on the current contrastive parame-
ter, αt . With this iterative algorithm, αt monotonically increases to the maximum value and usually
converges quickly (i.e., in less than 10 iterations).
Note that when BB is nearly singular, αt approaches infinity. One potential solution to avoid
this issue is adding a small constant value, ε (e.g., ε = 10−3 ), to each diagonal element of BB . How-
ever, given that ε does not have a clear connection to the optimization problem in Equation 2, and
consequently, it is hard for researchers to control α ’s search space. Therefore, we add ε tr(UT BT U)
tr(U⊤
t BT Ut )
to the denominator of Equation 3, i.e., αt ← tr(U⊤ BB Ut )+ε tr(U⊤ BT Ut )
. Through this way, we now can
t t
control the search space so that αt reaches at most 1/ε (e.g., when ε = 10−3 , αt may increase up
to 1,000).
2.4 Data-Point Coordinates, Category Coordinates, and Category Loadings
As in ordinary MCA, in cMCA, we provide three essential tools for helping researchers relate data
points and categories/variables to the contrastive PC space: (1) data-point coordinates (also known
as coordinates of rows (Greenacre 2017), or clouds of individuals (Le Roux and Rouanet 2010)), (2)
category coordinates (also known as coordinates of columns (Greenacre 2017), or clouds of cat-
egories (Le Roux and Rouanet 2010)), and (3) category loadings. These three tools each provide
unique and useful auxiliary information for understanding the structure of the data.
Data-point coordinates provide a lower-dimensional representation of the data, which is stan-
dard in most dimensional reduction techniques. Furthermore, category coordinates and loadings
provide essential information on how to interpret these data-point coordinates. Similar to data-
point coordinates, category coordinates present the position of each category/level in a lower-
dimensional space. Given that the coordinates of each category are on the same contrastive la-
tent space with data-point coordinates (see Equation 7), by comparing these two sets of coordi-
nates, we can better understand the associations and relationships between the data points and
categories. Through comparing the positions of data-point and category coordinates, a given cate-
gory’s position implies that respondents who are close to that position are highly likely to withhold
that specific attribute (Greenacre 2017; Le Roux and Rouanet 2010; Pagès 2014). On the other hand,
category loadings indicate how strongly each category relates to each derived contrastive PC.3
′ ′
Similar to MCA, for a target dataset XT , cMCA’s data-point coordinates YTrow ∈ Òn×K (K < K )
can be derived as:
YTrow = ZT U (5)
′ ′
where U ∈ ÒK ×K is the top-K eigenvectors obtained by EVD. Similarly, we can obtain data-point
3. We provide a detailed discussion of the category selection procedure in Section 3.
Fujiwara and Liu 5
coordinates, YBrow , of a background dataset XB onto the same low-dimensional space of YTrow through:
YBrow = ZB U (6)
As discussed in Fujiwara, Zhao, Chen, and Ma (2020), visualizing low-dimensional representations
of both target and background datasets can help researchers determine whether or not a target
dataset has certain specific patterns relative to a background dataset. When the scatteredness/shape
of plotted data points of YTrow is larger than YBrow , we can conclude that there exists a set of unique
patterns within YTrow .
In MCA, the most common way of deriving a dataset X’s category coordinates, Ycol , is through
Ycol = DW, where D is a diagonal matrix with the sum of each column of Z on the diagonal and W
′
is a matrix whose column vectors are the top-K eigenvectors obtained with EVD on B. However,
because cMCA applies EVD to the matrix, (BT −α BB ), the result is necessarily affected by XB as well.
As a consequence, we cannot simply compute category coordinates using the methods described
above. To overcome this issue, instead, we utilize MCA’s translation formula derived from data-
point coordinates to calculate category coordinates (Greenacre 2017). More precisely, the category
′
coordinates of the target dataset, YTcol ∈ ÒK ×K , can be derived through:
YTcol = DT−1 ZT⊤ YTrow diag(λ) −1/2 (7)
where DT ∈ ÒK ×K is a diagonal matrix with the sum of each column of ZT on the diagonal and
′ ′
λ ∈ ÒK is a vector of the top-K eigenvalues derived from Equation 1.
Finally, since MCA’s derivation procedure is highly similar to that of PCA, we utilize the concept
of principal component loadings in PCA to calculate category loadings in cMCA. More precisely,
′
category loadings, LT ∈ ÒK ×K , under cMCA can be derived through:
LT = U diag(λ) 1/2 (8)
3 Application
To demonstrate a), the utility of cMCA with high-dimensional and unstructured data, and b), the
derived results based on the manual- or auto-selection techniques, we analyze two surveys— the
ESS 2018 (the U.K. module) and the UTAS 2012. As suggested by Le Roux and Rouanet (2010), we
first recode some variables, before applying cMCA, to deal with the potential issue of infrequent
categories of active variables, such as the most extreme categories or missing values, that these
infrequent categories could be overly influential for the determination of latent axes. In general,
we keep all missing values and code them as 99; furthermore, when the number of levels of an
active variable is greater than five, we pool the adjacent two or three categories as a new category.4
For instance, the left-right ideological scale in ESS 2018 is originally an eleven-point scale from
zero through ten, we transfer this variable as a new five-point scale—respondents who originally
responded as 0 or 1 are recoded as 1, those who originally responded as 2 or 3 are recoded as 2,
those who originally responded as 4, 5, or 6 are recoded as 3, those who originally responded as 7
or 8 are recoded as 4, and those who originally responded as 9 or 10 are recoded as 5. Given that the
home countries of these surveys, the U.K. and Japan, have very different political environments,
these surveys provide a wide-range of political situations for testing. Furthermore, given that the
general political systems and cultures of each country are unique, we also test the consistency of
the derived cMCA results with the observed, qualitative political realities in each country.
4. For a more detailed recoding scheme, please refer to Appendix B.
Fujiwara and Liu 6OOC Result of ESS 2018
MCA Result of ESS 2018 (All British Parties)
1.0
1
0.5
0
1
PC2
PC2
0.0
Con
2 Lab
−0.5
LD
SNP
3 Green
UKIP
−1.0 Other
4
−1.0 −0.5 0.0 0.5 1.0 2 1 0 1 2 3
PC1
PC1
(a) OOC (b) MCA
Ordinal IRT Result of ESS 2018
BlackBox Result of ESS 2018
4
1.0
2
0.5
PC2
PC2
0.0 0
−0.5 −2
−1.0 −4
−1.0 −0.5 0.0 0.5 1.0 −4 −2 0 2 4
PC1 PC1
(c) BS (d) BMFA
Figure 1. OOC, MCA, BS, and BMFA Results of ESS 2018
Fujiwara and Liu 7cMCA (tg: Con, bg: Lab, : 998.5) cMCA (tg: Lab, bg: Con, : 995.9)
2.5
3.0
2.0
2.5
1.5
1.0 2.0
0.5 1.5
cPC2
cPC2
0.0 1.0
0.5 0.5
1.0
0.0
1.5
Con 0.5 Lab
Lab Con
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
cPC1 cPC1
(a) Target: Con, Background: Lab (b) Target: Lab, Background: Con
Figure 2. cMCA Results of ESS 2018 (Con versus Lab)
3.1 Case One: ESS 2018 (the U.K. Module)—the Brexit Cleavage in the Labor and Con-
servative Parties
The first survey we examine is the British module of the ESS 2018. We first manually select twenty
six questions which are generally related to political issues and recode this data through the afore-
mentioned procedure. As shown in Figure 1, we present the estimated results of all respondents
through four commonly used scaling methods for ordinal data in social science: the ordered op-
timal classification (OOC), MCA, the Basic-Space (BS) procedure, and the Bayesian mixed factor
analysis model (BMFA).5 According to Figure 1, all of the estimated results indicate that there are
no clear party lines under the ordinary PC space. This suggests that the status of political polar-
ization does not clearly exist among the mass public in the U.K. Important for the validity of these
scaling results, this conclusion is consistent with the current academic understanding that the
U.K. has undergone a period of political depolarization since the second wave of ideological con-
vergence between the elites of the two major parties, the Conservative and Labour parties (Adams,
Green, and Milazzo 2012a, 2012b; Leach 2015).
In addition, this survey was conducted from August 10th, 2018 through February 22nd, 2019,
which was between the two snap elections triggered around the issue of the UK’s departure from
the European Union (EU), commonly known as Brexit (Cutts et al. 2020; Heath and Goodwin 2017).
Theoretically, given how salient and polarized the Brexit issue is (Hobolt, Leeper, and Tilley 2018),
we should expect that such high salience should be reflected in voters’ responses and be revealed
on the derived PC space. However, as we see in Figure 1, the ESS results from traditional methods
do not demonstrate any clear evidence that voters are separated by preferences related to Brexit.
This suggests a clear failing of traditional methods in examining issue divisions in the U.K.’s elec-
torate.
In contrast to the traditional methods, we apply cMCA with the automatic selection of α to ESS
2018 and present the results in Figure 2 and Figure 3. As shown in Figure 2, the first pair we examine
includes the two major parties, the Conservatives and Labour. However, the results are relatively
uninformative. Figure 2a and Figure 2b together show that the two main parties fail to enrich each
other in a meaningful latent direction. In both figures, given that the clusters of Conservative and
5. Given that the purpose is only to demonstrate how well each derived result is able to cluster data, none of the sub-
figures through this paper calibrate the structure of the latent space to align the first PC with the left-right scale.
Fujiwara and Liu 8cMCA (tg: Con, bg: UKIP, : 999.9) cMCA (tg: Con, bg: UKIP, : 999.9)
3 Con 3 C
UKIP
cPC2 (European Union Related Attitude)
cPC2 (European Union Related Attitude)
UKIP
2 2
1 1
0 0
1 1
4 3 2 1 0 4 3 2 1 0
cPC1 cPC1
(a) Target: Con, Background: UKIP (b) Target: Con, Background: UKIP
(Con is color-coded by their opinions on EU)
c KIP, : 999.7 !" #$%& '()* +,- .KIP, : 999./0
3 3
cPC2 (European Union Related Attitude)
cPC2 (European Union Related Attitude)
2 2
1 1
0 0
1 1
2 2 1234568
L 9:;?
UKIP UKIP
3 3
5 4 3 2 1 0 5 4 3 2 1 0
cPC1 cPC1
(c) Target: Lab, Background: UKIP (d) Target: Lab, Background: UKIP
(Lab is color-coded by their opinions on EU)
Figure 3. cMCA Results of ESS 2018 (Con versus UKIP, Lab versus UKIP)
Fujiwara and Liu 9the Labour supporters almost perfectly stack onto one another (except for a few outliers), regard-
less of which one is the target or background group, we conclude that there are no hidden patterns
discernible by cMCA between these two main parties. This conclusion is consistent to Figure 1 and
academic wisdom of British politics (i.e., Adams, Green, and Milazzo 2012a, 2012b; Leach 2015)
Nevertheless, the sub-figures of Figure 3 tell a very different story. Instead of comparing the
two main parties themselves, we separate the parties and compare them as target groups with the
UK Independence Party (UKIP) selected as the background group. Through Figure 3a, we observe
that when UKIP supporters (the yellow points in Figure 3a) are the anchor, it splits Conservatives
(the blue points in Figure 3a) into two adjacent sub-groups along with the second contrastive PC
(cPC2) when α is automatically set to around 1000.
When using cMCA we derive two types of auxiliary information—the category coordinates and
the total category loadings—on each latent direction as heuristics for determining which categories
and variables are the most influential.6 To calculate total category loadings, we first calculate
each category’s loading through Equation 8 and further sum each category’s normalized absolute
loading (over the whole dataset) for any given variable to derive that variable’s overall loading
along with each latent direction. This means that differences in responses to the highest loading
variable(s) lead to the largest division or distance between respondents’ positions along the esti-
mated latent component. The nine variables with the highest absolute loadings (i.e., the top nine
most influential variables) and their categories in each latent direction are colored in Figure 6a and
Figure 6b—the order of the colors in the legend refers to the ranking of each variable’s loading.
The second auxiliary information we rely on is category coordinates. Through Equation 7, we
derive each variable’s coordinates and present them with the same top nine most influential vari-
ables by each latent direction in Figure 6c and Figure 6d separately. As discussed in Section 2, we
use this information to explore potential categories or attributes that respondents from different
subgroups may hold.
Although Figure 6a illustrates that the distribution of the Conservatives along the first cPC
is largely impacted by the missing values of the variables tradition (imptrad), environmental
protection (impenv), strong government (ipstrgv)), understanding (ipudrst), and equality
(ipeqopt) (in order by total category loadings), the first cPC is rather uninformative. Together with
Figure 3a and Figure 6, we see that only a handful of Conservatives are separated from the main
group by voters from UKIP along cPC1.7 Nevertheless, Figure 3a also demonstrates that Conser-
vatives can be divided by the anchor, UKIP voters, along with the second cPC, i.e., the second cPC
represents a certain trait, revealed by UKIP voters, that divides the Conservatives into two sub-
groups.
Furthermore, Figure 6b reports that the top nine most influential variables in order of dividing
Conservative voters are: religiosity (rlgdgr), immigration to living quality (imwbcnt), environ-
mental protection (impenv), tradition (imptrad), strong government (ipstrgv), emotionally
attached to Europe (atcherp), immigration to culture (imueclt), understanding (ipudrst),
and equality (ipeqopt). Interestingly, according to Figure 6b, the response of 5 to the immigra-
tion to living quality question (immigration makes the U.K. an extremely better place to live) is the
most influential category which “pulls” some Conservatives away from their co-partisans along
cPC2. The second most influential category is also the response of 5 to the immigration to cul-
ture question (the respondent’s life is extremely enriched by immigrants), which also significantly
divides Conservative voters, specifically pulling these respondents in the positive direction along
cPC2. Other influential categories which divide Conservatives along cPC2 include the response
of 1 to religiosity (extremely nonreligious), the response of 1 to the environmental protection
6. Please refer to Appendix A for figures of this auxiliary information.
7. For the combinations of new variable names and original abbreviations in the data, please refer to online Appendix B.
Fujiwara and Liu 10question (it is extremely important to care for nature and environment), and so on. In contrast, one
can also find several categories that significantly push some Conservative toward the negative end
of cPC2, such as the response of 2 to immigration’s effect on living quality, the response of 2 to
religiosity, and the responses of 2 and 3 to the environmental protection question.
The results of Figure 6b more or less reveal that Conservative voters can be divided into two
subgroups—the extremely liberal and the rest. We further utilize the coordinates of categories in
Figure 6d as another heuristic. After comparing respondents’ positions in Figure 3a, one can see
that among the top nine most influential variables, the Conservatives located on the positive side
of cPC2 are likely to hace responded with 5 to the immigration to living quality, immigration to
culture, and emotionally attached to Europe (extremely emotionally attached to Europe) ques-
tions.
More interestingly, although not all of these are top three or five loading variables for cPC2 in
Figure 3a, we find that Conservative supporters who are on the top of cPC2 in Figure 3a are as-
sociated with extremely open views over at least one of the three EU- or Brexit-related variables:
immigration to living quality, immigration to culture, and emotionally attached
to Europe.8 As shown in Figure 3b, we further color this extremely EU-supportive sub-group
(Con_Pro) in teal and color the rest of the Conservative supporters (Con_Oth) in pink.
Analogously, we also find subgroups within Labour party voters by using UKIP supporters as
the background group. Similar to the previous results, we find that when α is automatically set to
around 1000, Labour supporters’ (the orange points in Figure 3c) are split by UKIP supporters (the
yellow points in Figure 3c) along cPC2. In fact, from Figure 3c we see that only a handful of Labour
voters are separated from the main group, and thus we do not consider the first cPC to represent
any meaningful traits or issue dimensions. Nevertheless, the results in Figure 7b and Figure 7d
indicate that Labour voters have a similar distribution as the Conservatives. The top nine most
influential variables which divide respondents along the cPC2, in order, are: immigration to living
quality (imwbcnt), emotionally attached to Europe (atcherp), religiosity (rlgdgr), tradition
(imptrad), environmental protection (impenv), understanding (ipudrst), strong government
(ipstrgv), equality (ipeqopt), and immigration to culture (imueclt).
According to Figure 7b and Figure 7d, we see that Labour party voters located on the positive
side of cPC2 are highly associated with the response of 5 to the immigration to living quality,
emotionally attached to Europe, and the immigration to culture questions. Similar to the di-
mension obtained for Conservative voters, we speculate that cPC2 represents the same latent trait
which also divides Labour voters into two subgroups—those who are extremely liberal on EU re-
lated issues and the rest. We further divide Labour supporters into two sub-groups with different
colors and present them in Figure 3d: those who hold extremely open opinions over at least one of
the three variables, i.e., immigration to living quality, immigration to culture, or emotionally
attached to Europe, are in green (Lab_Pro) and the rest colored red (Lab_Oth).
Overall, given that recent research has linked negative immigration attitudes to anti-EU sen-
timents (Colantone and Stanig 2016), we conclude that both Conservative and Labour party sup-
porters are internally divided by Brexit attitudes. These results further demonstrate that cMCA
can identify latent traits often ignored by traditional scaling methods, that generalize across large
pre-defined groups. As we may expect, although the two parties are divided by Brexit attitudes,
the size of the pro-EU sub-group is significantly larger among Labour supporters, as compared to
Conservative ones.
The ESS 2018 analysis emphasizes an important component of cMCA—that is, the selection of
the background group is highly influential in deriving either meaningful or meaningless results.
8. All three variables are five-point scales where 1 refers to the feeling of extreme anti-EU or anti-immigrant and 5 to the
feeling of extreme pro-EU or pro-immigrant. Please refer to online Appendix B for more details.
Fujiwara and Liu 11While this is intuitive, given the basic conceit of contrastive learning methods, it is very impor-
tant to consider when using cMCA: if the selected target and background groups are highly similar,
cMCA may not be able to capture any informative patterns, as illustrated by the results in Figure 2a
and Figure 2b. Nevertheless, this “uninformative” situation can actually be informative when ex-
amined from a different perspective—researchers can apply the contrastive learning method to
any two groups to examine their level of similarity. From this perspective, the results of Figure 2a
and Figure 2b actually validate previous findings about the wave of ideological convergence be-
tween the two major parties in British politics.
BlackBox Result of UTAS 2012
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BlackBox Result of UTAS 2012 Ordinal IRT Result of UTAS 2012
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Figure 4. OOC, MCA, BS, and BMFA Results of UTAS 2012
3.2 Case Two: UTAS 2012—the Hidden Xenophobic Wing of the LDP
Next, we turn our attention to the 2012 UTokyo-Asahi Survey, conducted in Japan during their
House of Representatives election in 2012.9 For each election, UTAS surveys both candidates and
voters, and thus contains three sets of survey questions: candidate-specific, voter-specific, and
9. UTAS is conducted by Masaki Taniguchi of the Graduate Schools for Law and Politics, the University of Tokyo and the
Asahi News. The original data can be retrieved from http://www.masaki.j.u-tokyo.ac.jp/utas/utasindex.html.
Fujiwara and Liu 12shared questions. We focus on the voter-specific and shared questions only—we first select forty
questions which are also presented in UTAS 09 and recode this data through the aforementioned
procedure. With OOC, MCA, BS, and BMFA, we present the positions of all respondents who sup-
port either the Liberal Democratic Party (LDP), the Democratic Party of Japan (DPJ), or the Japan
Renewal Party (JRP) in Figure 4.
Similar to Figure 1, the UTAS results highlight how Japanese voters are highly mixed and over-
lapping along these dimensions—not one of the sub-figures of Figure 4 reveal any obvious patterns
that represent clear and distinct party lines. Indeed, these results are consistent with previous re-
search finding a downward trend in the ideological distinctions between Japanese parties, elites,
and voters (Endo and Jou 2014; Jou and Endo 2016; Miwa 2015). In short, while simple, left-right
ideology may still be the heuristic that Japanese voters rely on for determining their vote-choice,
the political culture of Japan has become decidedly more centrist. Given this high degree of mod-
eration, we apply cMCA with the manual-selection method to the UTAS 2012 to explore hidden
patterns within the Japanese electorate.
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cPC1 cPC1 (Feelings to South Korea/China)
(a) Target: LDP, Background: DPJ (b) Target: LDP (color-coded by the attitudes to South
Korea/China), Background: DPJ
Figure 5. cMCA Results of UTAS 2012 (LDP versus DPJ)
As Figure 5 shows, when we assign LDP supporters to the target group and DPJ supporters
to the background group, cMCA identifies two subgroups, with minor overlap, within LDP sup-
porters. We start by manually setting α equal to 1 and find that LDP supporters can be divided
by DPJ supporters in the contrastive space; we then increase the value of α gradually (by 0.1) at
each iteration to adjust the position of every data-point, and finally, when α is set to 1.6, cMCA
shows that the LDP supporters (in blue color) are well divided by the DPJ supporters (in orange
color) along cPC1, as demonstrated in Figure 5a. Furthermore, based on the auxiliary informa-
tion provided in Figure 8a, we find that the top nine most influential variables in order of dividing
LDP supporters along cPC1 are: feeling close to South Korea (policy55), feeling close to Rus-
sia (policy55), feeling close to China (policy54), campaign contribution (policy50), party
control (policy52), diet (policy51), family form (policy43), US-Asia priority (policy47), and
electoral system (policy01). Upon closer investigation, we find that the first cPC is associated
with the LDP voters’ attitudes towards four foreign countries: South Korea, China, Russia, and the
U.S. Indeed, as demonstrated in Figure 8a, almost half of the top nine most influential variables
which disperse the LDP supporters along with cPC1 are related to these attitudes—feeling close
to South Korea, feeling close to Russia, feeling close to China (top three most influential vari-
Fujiwara and Liu 13ables), and US-Asia priority (the eighth most influential variable).
Furthermore, through category coordinates presented in Figure 8c, we find that the LDP sup-
porters who are located on the negative side of cPC1, in general, respond with 1 to the campaign
contribution, 5 to the feeling close to South Korea, 5 to the feeling close to China, and 1 to
the US-Asia priority questions.10 We speculate that one of the main latent traits to which cPC1
corresponds are attitudes towards Japan’s Asian neighbors: LDP supporters who are located on
the left of Figure 5a feel extremely distant to South Korea and China.11
We further divide LDP supporters into two subgroups and present them in Figure 5b: those
who feel extremely distant to South Korea or China are colored in green (LDP_Anti) and the rest
are colored in purple (LDP_Oth). We consider these feelings of extreme distance as anti-South
Korea and anti-China attitudes, respectively. Importantly, this xenophobia identified within the
LDP is consistent with the reality of current Japanese politics.12
Indeed, scholars have consistently identified a wave of xenophobic sentiments in Japan against
Korean, Chinese, and other peoples since the late 2000s—coined the “Action Conservative Move-
ment” (ACM) (Smith 2018). These new right-wing groups are considered to be quite different to
the traditional right-wing politicians and parties of post-war Japan. While these traditional par-
ties, which are the predecessors of the modern LDP and the Shinzo Abe administration, focused
on anti-communism and the Emperor-centered view of history and the state of nature, these new
parties are much more preoccupied with nativistic and xenophobic issues and policies (Gill 2018;
Smith 2018; Yamaguchi 2018). Interestingly, although these new right-wing groups in the LDP have
been identified and discussed by the news media and scholarly research, they are rarely identified
through surveys and traditional scaling methods. The use of cMCA on this data perfectly illustrates
the utility of this method in capturing influential features or categories that are substantively im-
portant, and which are often left unidentified by traditional methods.
4 Discussion: Comparing cMCA with Other Subgroup-Analysis Methods
Scaling has been widely used for both pattern recognition and latent-space derivation. Neverthe-
less, due to the structure of different datasets and the methodology behind these ordinary scaling
methods, the derived results are sometimes uninformative. In this article, we contribute to both
the scaling and contrastive learning literature by extending contrastive learning to MCA enabling
researchers to derive contrasted dimensions using categorical data.
Through the analytical results from two surveys, the ESS 2018 and UTAS 2012, we illustrate the
two main advantages of cMCA, relative to traditional scaling methods. First, cMCA can identify sub-
stantively important dimensions and divisions among (sub)groups that are missed by traditional
methods. Second, while cMCA primarily explores principal directions for individual groups, it can
still derive dimensions that can be applied to multiple groups. As demonstrated in Figure 3, when
Conservative and Labour supporters are distributed almost identically, cMCA is able to identify the
Brexit division that exists in both parties.
It is also important to note that cMCA, and contrastive learning more generally, is different from
ordinary subgroup-analysis methods: almost all standard methods for subgroup-analysis require
researchers to have prior knowledge about how data should be “subgrouped.” For example, as
10. According to both Figure 8a and Figure 8b, the level of 99 of feeling close to South Korea seems to be the most
“powerful” category which pulls LDP supporters in the negative direction of cPC1. However, based on both Figure 8c and
Figure 8d, one can find that voters whose positions are extremely positive on cPC2 are more likely to not respond (99) to
feeling close to South Korea; and on the contrary, voters who are extremely negative on cPC1 are more likely to respond
with 5 to feeling close to South Korea.
11. Both variables are five-point Likert scales where 1 refers to feeling extremely close to South Korea/China and 5 refers
to feeling extremely not close to South Korea/China. Please refer to online Appendix B for more details.
12. Similar to the ESS 2018, cPC2 in Figure 5 represents mostly non-responses/missing values in certain variables so the
discussion of that dimension is ignored in this section.
Fujiwara and Liu 14compared with two of the most popular of these methods, class-specific MCA (CSA) and subgroup
MCA (sMCA), cMCA allows researchers to agnostically explore all possible latent traits from differ-
ent perspectives in the space. In other words, while conducting cMCA, without prior knowledge,
researchers need only apply different values of α , either with manual- or auto-selection, to derive
latent dimensions or traits which could subgroup data in meaningful ways.
In contrast, both CSA and sMCA requires researchers to subjectively subset/subgroup the origi-
nal data first and then compare the derived patterns with the reference group, usually the original
complete data. The ideas behind CSA and sMCA are similar: CSA is used to study whether a pre-
defined subset of data-points has a different latent pattern or not; and sMCA is used to explore
whether data-points distribute differently after excluding certain subgroup(s) from original data
(Greenacre and Pardo 2006; Le Roux and Rouanet 2010). Therefore, without any prior knowledge,
it is almost impossible for researchers to effectively apply CSA and sMCA to objectively explore
meaningful subgroups, especially when data is high-dimensional and unstructured, such as in our
two examples. This is because both datasets have a considerable amount of missing values and ac-
tive variables/categories. Given that both CSA and sMCA require researchers to manually compare
the latent distribution of every possible subset of data, as one can imagine, this issue becomes
much worse when the number of dimensions and missing values of data increases.
This comparison by no means indicates that cMCA and contrastive learning are superior to CSA,
sMCA, or any existing subgroup-analysis methods. Rather, it shows how cMCA is an additional tool
for researchers to agnostically explore latent traits. In that vein, we believe that this new approach
can complement the existing subgroup-analysis methods through providing preliminary heuris-
tics regarding hidden patterns within data. There are several use-cases for cMCA and contrastive
learning, in general. First, cMCA can be used for analyzing covariate-balance between treatment
and control groups in experiments; as demonstrated by Figure 2, cMCA instead can be utilized to
explore the level of similarity of two sets of identical active variables, and therefore, if there are
any categories or variables specific to either group, these results indicate potential issues regard-
ing the procedure of randomization.
Second, cMCA can be applied to substantive research as well. For instance, given that partisan
lines among British voters are not as clear as among American voters, a derived “issue cleavage”
from cMCA can be a good source of studying affective polarization (see Iyengar et al. 2019) by po-
litical issues instead of by party ID. In that vein, it has been noted that traditional scaling methods
usually classify the U.S. Congressperson group known as the “Squad” as “moderate Democrats”
based on their voting patterns. While clearly not moderate, this is due to how they sometimes
vote against with their Democratic colleagues’ more moderate policies (Lewis et al. 2019). Through
cMCA and contrastive learning, researchers could identify the specific kinds of bills that the “Squad”
typically disagree with other Democrats on so as to better understand the cleavage between the
progressive and moderate wings of the Democratic Party.
Supplementary Material
For coding scheme, please visit online Appendix B
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