Introducing A New Model of Class Identification:
A Mixed Method of Mathematical Modeling andBayesian Statistical Modeling∗
Atsushi Ishida(Osaka University of Economics)

Abstract
This paper aims to introduce a new analytical framework for class identiﬁcation
by applying a mixed method of simple mathematical modeling and Bayesianstatistical modeling. First, I constructed a simple mathematical model which
can explain the middle concentration tendency of class identiﬁcation where
the majority of people tend to regard themselves as middle, assuming thatthe succession of the same Bernoulli m-trials with success probability p determines one’s subjective class identiﬁcation. Second, I estimated parametersof the model from SSM survey data by applying a Bayesian statistical model.
The distribution of latent success probability p and number of trials m wasestimated by the Markov Chain Monte Carlo method. I also analyzed diﬀerences in distributions of p and m among age cohorts and educational levels byhierarchical models.
From the analysis, I found the following point: (1) approximately ﬁve trials ofﬁfty-ﬁfty games with around 0.5 success probability describes well the observed
class identiﬁcation distribution in 2015 data; (2) the Japanese postwar period
can be divided in two based on people’s subjective evaluations—the period of
expanding opportunity (1955 to 1975) and the period of high and constant∗ The study was supported by JSPS KAKENHI Grant Number JP25000001 and 15K13080. The

author thanks the Social Stratification and Social Mobility (SSM) Research Committee of 2015
for the permission to use the SSM 1955-2015 data.

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success probability, but less chance of trials (from 1985); and (3) the diﬀerent
games model on educational levels is always better in terms of goodness of
predictions than the common game model in each survey period. However,these models were closest in terms of goodness of prediction in 1975 and 1985,
possibly indicating that during the “all middle-class society” period, people
evaluated their society almost the same as the common game where all players
have the same opportunities, but dierent luck.
Keywords: class identiﬁcation, random walk, Bayesian statistical modeling

1 Introduction
Social psychological aspects of social inequality and social stratiﬁcation, such as people’s cognitions, attitudes, and emotions in an unequal or stratiﬁed society, can be
regarded as antecedent conditions of people’s rational choices or actions that, if aggregated, would lead to macro (un)change in the society. Thus, it is crucially importantfor micro-macro linkage approaches on social inequality to shed light on the social
psychological aspects as with other aspects of inequality.
Among social psychological features, this paper focuses on class identiﬁcation. Class
identiﬁcation is the extent to which people identify themselves as members of a certainsocial class or stratum, and it has been one of the main subjects in social psychological studies of social stratiﬁcation. In this paper, I will introduce a new analyticalframework for class identiﬁcation by applying a mixed method of simple mathematicalmodeling and Bayesian statistical modeling, rather than the conventional frequentist
statistical analysis that typically applies regression analysis blindly.
First, I construct a simple mathematical model that explains one of the major
tendencies of class identiﬁcation, that is the middle concentration tendency in which
the majority of people tend to regard themselves as middle. This is the life-is-like-arandom-walk model where it is assumed that succession of the same Bernoulli m-trials
with success probability p determines one’s subjective class identiﬁcation.
Second, I will estimate parameters of the model from empirical data by applyinga Bayesian statistical model. The Bayesian modeling enables us to construct more
ﬂexible model which directly reﬂects the mathematical model and is able to explain
generative mechanism of observed distribution. The distribution of latent successprobability p and number of trial m are estimated by MCMC estimation and diﬀer-

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ences in distributions of p and m among diﬀerent social categories are analyzed byhierarchical models.
The new analytical framework is based on an intermediate modeling strategy. In
general, formal modeling of a social phenomenon has three features:(1) a more realistic mechanism description of the phenomenon;(2) more constraints on parameters; and(3) lower ﬁtness of empirical data.
On the contrary, conventional statistical modeling, which tries to summarize empirical
data structures as typically linear equations, has opposite features:(1) less or no realistic mechanism description of the phenomenon;(2) less or no constraints on parameters; and(3) higher ﬁtness of empirical data.
The intermediate modeling strategy, composed of simple mathematical modeling andBayesian statistical modeling, aims to have not the highest but the most reasonable
qualities of all features:(1) a reasonable realistic mechanism description of the phenomenon;(2) reasonable constraints on parameters; and(3) reasonable ﬁtness of empirical data.
A series of Japanese cross-sectional survey (SSM survey) data from 1955 to 2015,
collected every 10 years, will be analyzed using the Bayesian model. The results ofthis analysis will be interpreted and discussed. Section 2 will discuss preceding studies
on class identiﬁcation; section 3 will introduce the life-is-like-a-random-walk model;
section 4 will provide a methodological description of Bayesian modeling; section 5
will oﬀer results of analyses of SSM survey data; and ﬁnally, section 5 will provide aconclusion.

2 Class Identification
Class identiﬁcation (or self-location in the class system or status identiﬁcation) is akind of subjective evaluation of one’s own social status in society. More speciﬁcally,class identiﬁcation is the extent to which people identiﬁes themselves as members of
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a certain social class or stratum.
In a series of studies across many countries, a tendency, referred to as the “middle
concentration tendency,” in which the majority of people tend to regard themselves
as middle regardless of their actual objective status, was common. For example,
Kelley and Evans (1995) and Evans and Kelley (2004) found, using cross-national
survey datasets, that in many countries people tend to select the middle category
for subjective status location. These researchers argued that this tendency existed
because of a comparison mechanism in homogeneous reference groups, that is, anassumption that people tend to assess their social status by comparing themselves withothers whose status are similar with them. Even in Japan, the middle concentration
tendency has consistently been observed since the 1970s when Japan enjoyed aﬄuenteconomics as a result of the postwar economic growth (Kikkawa, 2016).
The formal model by Fararo and Kosaka (2003) further pursued the idea of a mechanism of class identiﬁcation formation via comparison with others. In their model, it
is assumed that actors evaluate their class position by a sequential comparison processwith others in order of importance of several social economic dimensions. Under themodel assumption, the distribution of class identiﬁcation approximates to a normal
distribution under the condition that all dimensions are independent of each other(Ishida, 2012; Hamada, 2012).
Following these preceding studies, in this paper, I introduce an alternative andsimpler mathematical model explaining the middle concentration tendency.

3 Life-Is-like-a-Random-Walk ModelAn ancient Chinese saying states “good luck and bad luck alternate like the strandsof a rope.” This saying tells us that good events and bad events in life are like twosides of a coin ﬂipped by Fate. Many other sayings share the same message relevant
to fortune and life, such as “everything for men is like a Saioh’s horse.” In line withthese wisdoms, I insist here “life is like a random walk.”
The random walk, as well known, is a simple stochastic process in which actors
ﬂip a coin to decide to move left with a probability p or right with 1 − p in every

step in m times sequential steps. As long as the ﬂipping-a-coin trials are independent

from one another, distribution at a suﬃciently large m-th step approximates a normal
distribution. Figure 1 is an example of the random walk process in which the left
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graph shows trajectories of 100 diﬀerent random walks with the probability 0.5 andthe right graph shows the histogram of their positions at 500th step.

40

0.015

20
-20

100

200

300

400

500

-40

0.010
0.005

-60

0.000

-40

-20

0

20

40

60

Fig. 1 An example of the random walk process (p = 0.5, m = 500)

Based on the random walk process, I construct the life-is-like-a-random-walk model(LLRW model) for class identiﬁcation distribution. Here are two assumptions:
Assumption 1 Life is like a series of Bernoulli trials that result in success with aprobability p or failure with 1 − p.
Assumption 2 People evaluate their relative status in society by the identical functionof their success rate (z) of Bernoulli trials at the end of m-th trial.
Based on the assumptions, number of success denoted by s through m trials comesfrom the binomial distribution parameterized by m and p, that is,s ∼ Binomial(m, p).

(1)

And then, according to the central limit theorem, the distribution of success rate(z = s/m) will approximate a normal distribution parameterized by p as mean and√p(1 − p)/m as standard deviation, as m gets suﬃciently larger, that is,z ≈ Normal(p,√p(1 − p)/m).

(2)

Figure 2 shows a simple outline illustration of the life-is-like-a-random-walk model.
From the model, a simple explanation for the middle concentration tendency ofclass identiﬁcation is derived: if the lives of most people are like a series of ﬁfty-ﬁfty
games and they evaluate their status based on their wins, then distribution of class
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Life-is-like-a-random-walk modelcore
mechanism

subjective
evaluation

objective
position

RW process

identity
function

success p
m timesFig. 2 Outline of the life-is-like-a-random-walk model
9

identity eventually shows middle concentration. In addition, two implications withrespect to parameters can be derived immediately from the model.
Implication 1 The higher success probability (p) in society, the higher the social average subjective status.
Implication 2 The larger the number of trials (m) in society, the smaller the varianceof subjective status.
It isnoteworthy that the LLRW model has the same mathematical structure as the
speciﬁc version of Fararo-Kosaka model of class identiﬁcation with the assumption of
two ranks and identical distributions (Fararo and Kosaka, 2003). Hence, it can besaid that the LLRW model and the Fararo-Kosaka model have diﬀerent allegoricalmechanisms, but share the same fundamental mechanism.

4 Parameter Estimation by the Bayesian Statistical Model
I will apply the Bayesian statistical modeling approach for parameter estimation of
the life-is-like-a-random-walk model where there are two parameters to be estimated:
latent success probability p and latent number of trials m.
In contrast to the conventional frequentists approach, the Bayesian modeling approach enables us to construct a ﬂexible model by means of a hierarchical model andto express the generative process of observed distribution. These features help us
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conduct direct estimations of the mathematical model and interpret an outcome ofquantitative analysis based on the concrete theoretical framework.

4.1 Data and Variables
Data to be analyzed comes from the National Survey of Social Stratification and Social Mobility (SSM) surveys, which are national level random sampling cross-sectional
surveys conducted every 10 years from 1955 to 2015. Accordingly, we have 7 wavesof data*1 . SSM surveys focus mainly on the current situation and changes in social stratification structure as well as inter- and intra-generational class mobility inJapan. Moreover, SSM surveys also focus on social psychological aspects of social
stratification, such as people’s cognition and attitude towards stratification, inequality, unfairness, and related policies. By using this longitudinal data, I examine changes
on latent mechanisms of generating class identification distribution in postwar Japan.
It should be noted that since female samples are only available from the 1985 survey,
in this study, I decide to analyze only male samples aged 20 to 69 for examination of
longitudinal trends of class identification*2 .
The outcome variable is relative class identification score z ranging from 1 as thehighest to 0 as the lowest. The variable is recoded from 5-scaled class identification.
In the SSM survey, class identification has almost invariably been asked using thefollowing question*3 : “suppose we were to divide the people living in Japanese society
today into the following five strata (Upper, Upper Middle, Lower Middle, Upper
Lower, Lower Lower), to which group do you think you would belong?” I assign
4/4 = 1 to “Upper,” 3/4 to “Upper Middle,” 2/4 to “Lower Middle,” 1/4 to “Upper
Lower,” 0/4 = 0 to “Lower Lower,” respectively*4 .
*1 The published data and detailed information on surveys from 1955 to 2005 can be foundin the SSJ Data Archive (http://csrda.iss.u-tokyo.ac.jp).
The short summary of2015 survey can be found in the following URL (http://www.l.u-tokyo.ac.jp/2015SSM-PJ/
2015ssmjisshigaiyo.pdf) [in Japanese].
*2 Analyses of female samples are tasks to be completed in the future.
*3 SSM surveys have been simultaneously conducted by two modes of administration: intervieweradministered and self-administered. The question of class identification has been investigatedby interviewer-administered questionnaire in every wave except in 2005.
*4 As for the 2005 data, I employ a 10-scaled class identification variable (from 1 as the highest
to 10 as the lowest) investigated by interviewer-administered questionnaire instead of 5-scaled
variable to avoid bias by mode of administration. According to suggestions by Kobayashi(2008), I recoded the 10-scaled variable to the 5-scaled variable in the following manner: 1 and
2 to “Upper,” 3 and 4 to “Upper Middle,” 5 and 6 to “Lower Middle,” 7 and 8 to “Upper

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I try to examine how a diﬀerence in social category results in a diﬀerence in class
identiﬁcation in the diﬀerent games model, which I will describe in detail in the next
section. In this study, I focus on age cohorts and educational levels as factors thatwould diﬀerentiate social economic opportunities among people in Japan (Kikkawa,2016). I categorize ﬁve age groups according to respondents’ age in each wave: 20–
29, 30–39, 40–49, 50–59, and 60–69. Then, 11 age cohorts are obtained from age
groups in each wave according to year of birth: 1886–1895, 1896–1905, 1906–1915,
1916–1925, 1926–1935, 1946–1955, 1956–1965, 1966–1975, 1976–1985, and 1986–1995.
Three educational levels are categorized by years of schooling: lower-secondary (under
9 years), upper-secondary (11–13 years) and tertiary (over 14 years).

4.2 Comparison of Two Models
In the following analysis, I test two models, the common game model and the diﬀerent
games model. The common game model assumes that every player of the game has
the same opportunities in terms of success probability p and number of trials m;
thus, the dispersion of success rate and identiﬁed class is a matter of probabilistic
luck. On the other hand, the diﬀerent games model assumes that players in diﬀerent
categories play diﬀerent games with diﬀerent opportunities determined by pj and mj ;therefore, the dispersion of success rate depends on diﬀerent opportunities as well asprobabilistic luck.
Figure 3 shows a simple outline illustration of the two models.
The preference of the models in terms of model prediction of new data will be evaluated by the WAIC (the Watanabe-Akaike or widely applicable information criterion)in every wave (Watanabe, 2010; Vehtari and Gelman, 2014).

5 Results of Analyses
5.1 Common Game ModelFirst, I examine the common game model using 2015 data with the assumption that
all members in society have experienced the same kind of trials with common p andLower,” 9 and 10 to “Lower Lower,” and then assigned relative class identification scores inthe same way as the other waves.

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Comparison of two models
Common game model

Different games model

p1, m1

p2, m2p, m
p3, m3Same opportunity, it’s a matter of luck
Different opportunities and luck
Fig. 3 Outline of the two models

Model comparison by WAICm.

15 the life-is-like-a-random-walk model, it is assumed that observed relative
From

status zi is approximately normally distributed with parameters p and m.
zi ∼ Normal(p,√p(1 − p)/m)(3)

Here, p is assumed to be a logistic function of beta for future model extension.
p=

1
1 + exp(−β)(4)

It is assumed that prior distribution of β is a normal distribution with a relatively
larger standard deviation, that is,β ∼ Normal(0, 102 ).

(5)

It is also assumed that m is a continuous variable for simplification of estimation andprior distribution of m is a a truncated normal distribution, such thatm ∼ Normal+ (0, 102 ).

(6)

Figure 4 is the graphical model of the common game model, where grey circle nodes
indicate observed continuous variables (zi in this model), double circle nodes indicate
generative continuous variables (parameter p), and single circle nodes indicate latent
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continuous variables with prior distribution (parameters β and m). Figure 5 is theoutline illustration of the model.
β

p
zi
i

m
Fig. 4 Graphical model of the common game model

Common game model outline (modified)𝒛𝒛𝒛𝒛𝑖𝑖𝑖𝑖 ~iid

𝑁𝑁𝑁𝑁( 𝒑𝒑𝒑𝒑, 𝒑𝒑𝒑𝒑(1 − 𝒑𝒑𝒑𝒑)/𝒎𝒎𝒎𝒎)~

𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑙𝑙𝑙𝑙𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 (𝛽𝛽𝛽𝛽)17

𝑁𝑁𝑁𝑁(0, 102 )𝑁𝑁𝑁𝑁+ (0, 102 )Fig. 5 Outline of the common game model

Posterior distributions for the parameters were estimated by the Markov Chain
Monte Carlo (MCMC) method. In this study, I employed Stan 2.16.0 and RStan 2.16.2(Stan Development Team, 2017a,b) for the MCMC programming*5 . I conducted fourchains of sampling for 5,000 iterations each, which includes 1,000 initial iterations asburn-in samples. The thin interval was set as one, to generate 16,000 sampled pointsof posterior distributions. The sample size (n) used for this analysis was 2895.
Table 1 shows the summary of the MCMC estimation of posterior distributions forthe common game model for SSM 2015 data. The Gelman-Rubin MCMC convergencestatistic (R̂) of each parameter is around 1.000; hence, we can safely conclude that
the MCMC sampling converged (Gelman et al., 2013, 284–286).
*5 I referred Kruschke (2015) and Matsuura (2016) for building Stan codes for the models. Stan

codes are available in Appendix 2.

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Table 1 Result of the MCMC estimation of the common game model in2015 data (n = 2895)

β
p
m

mean

SE

SD

2.50%

25%

50%

75%

97.50%

R̂

0.033
0.508
5.417

0.000
0.000
0.001

0.016
0.004
0.141

0.001
0.500
5.146

0.023
0.506
5.319

0.034
0.508
5.416

0.044
0.511
5.510

0.065
0.516
5.694

1.000
1.000
1.000

SE: standard error of posterior mean, SD: standard deviance of posterior distribution,
x%: x-percentile of posterior distribution, R̂: Gelman-Rubin MCMC convergence statistic.

The success probability of a trial p is estimated around 0.51, meaning that thegame is almost a ﬁfty-ﬁfty game. The number of trials m is estimated to be ﬁvetimes, approximately. Therefore, if everyone played a ﬁfty-ﬁfty game approximately
ﬁve times, then we obtain nearly the same distribution as observed relative status
distribution. Figure 6 shows the histogram of observed relative status overlapped by
predicted normal distribution determined by the medians of posterior distributions ofp and m.

density

1.5

1.0

0.5

0.0
0.0

0.4

z

0.8

Fig. 6 Histogram of observed relative status and predicted distribution in 2015 data

5.2 Diﬀerent Games Model on Age Cohorts
Next, I examine the diﬀerent games model with the assumption that each category
in the society has experienced diﬀerent games with diﬀerent opportunities. I apply
the framework of the Bayesian hierarchical modeling to construct the diﬀerent games
model.
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First, I examine the diﬀerent games model on age cohorts in 2015 data. In thismodel, it is assumed that the parameters p and m diﬀer by age cohort. A person i’s
relative status zi is assumed to be normally distributed, and the distribution to beparameterized by pj(i) and mj(i) , which indicates success probability and number oftrials commonly held in the cohort j to which person i belongs, that is,zi ∼ Normal(pj(i) ,√pj(i) (1 − pj(i) )/mj(i) ).

(7)

pj is assumed to be a logistic function of a liner equation composed of β0 as theintercept and βj as the slope of each category. βj is assumed to be under the zero-sum
constraint and has common normal distribution around zero with σβ as the standard
deviation. β0 and σβ as hyper-parameters have a normal distribution and a uniform
distribution from 0 to ∞ as prior distributions, respectively. These assumptions canbe described in the following notations,
pj =

1
,
1 + exp(−β0 − βj )β0 ∼ Normal(0, 102 ),∑βj ∼ Normal(0, σβ ),βj = 0,σβ ∼ Uniform(0, ∞).

(8)
(9)
(10)
(11)

mj is assumed to have a common truncated normal distribution at zero with hyperparameters µm and σm , that is,
mj ∼ Normal+ (µm , σm ),(12)µm ∼ Uniform(0, ∞),(13)

σm ∼ Uniform(0, ∞).

(14)

Figures 7 and 8 are the graphical model and the outline illustration of the diﬀerent
games model, respectively. The diﬀerent games model can be regarded as a Bayesianversion of analysis of variance (ANOVA) model (Kruschke, 2015, Ch.19).
I would like to discuss the interpretation of diﬀerence of parameters. What are
the actual meanings of parameters p and m? How can we interpret the outcome ofempirical data analysis? p can be regarded as a subjective evaluated advantage in
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σβ

β0

βj

pj
zi
i

mj
j

µm

σm

Fig. 7 Graphical model of the diﬀerent games model

Different game model outline
𝒛𝒛𝒛𝒛𝑖𝑖𝑖𝑖 ~iid
𝑁𝑁𝑁𝑁( 𝒑𝒑𝒑𝒑𝑗𝑗𝑗𝑗(𝑖𝑖𝑖𝑖) , 𝒑𝒑𝒑𝒑𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖 (1 − 𝒑𝒑𝒑𝒑𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖 )/𝒎𝒎𝒎𝒎𝑗𝑗𝑗𝑗(𝑖𝑖𝑖𝑖) )𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 (𝛽𝛽𝛽𝛽0 + 𝛽𝛽𝛽𝛽𝑗𝑗𝑗𝑗(𝑖𝑖𝑖𝑖) )𝑁𝑁𝑁𝑁(0, 𝜎𝜎𝜎𝜎𝛽𝛽𝛽𝛽 )𝑁𝑁𝑁𝑁+ (𝜇𝜇𝜇𝜇𝑚𝑚𝑚𝑚 , 𝜎𝜎𝜎𝜎𝑚𝑚𝑚𝑚 )~𝑁𝑁𝑁𝑁(0, 102 )𝑈𝑈𝑈𝑈 (0, ∞)𝑈𝑈𝑈𝑈 (0, ∞)𝑈𝑈𝑈𝑈 (0, ∞)Fig. 8 Outline of the diﬀerent games modela life event, such as enrollment in a university or getting a job or promotion sharedin a social category. So, a category that has a higher value of p can be assumed to
be a category that achieves higher advantage. Meanwhile, m can be regarded as a
subjective evaluated number of important life events that members of a social category
have experienced. If you played a few games, you could all win from beginner’s luck or
could all lose. But, if you played many games, a winning rate becomes more reflectiveof their actual advantages, so that, the variance within a category becomes smaller.
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However, the value of m should be regarded not as an absolute but rather as a relativevalue, because it partly depends on measuring scales.
Let us examine the result of the analysis. Table 2 shows the summary of the MCMC
estimation of posterior distributions for the diﬀerent games model on age cohorts in
SSM 2015 data. The MCMC setting for parameter estimation is the same as thatin the common game model. MCMC sampling for all parameters can be judged as
being converged from the values of R̂. The index of βj , pj , and mj indicates age
cohorts: 1 1986-95 (20s in 2015), 2 1976-85 (30s), 3 1966-1975 (40s), 1956-1965 (50s),
and 1946-1955 (60s).
Table 2 Result of the MCMC estimation of the diﬀerent games model on age
cohorts in 2015 data (n = 2895)

β0
β1
β2
β3
β4
β5
σβ
µm
σm
p1
p2
p3
p4
p5
m1
m2
m3
m4
m5

mean

se mean

sd

2.50%

25%

50%

75%

97.50%

R̂

0.028−0.030−0.099
0.029
0.096
0.004
0.121
5.427
0.433
0.500
0.482
0.514
0.531
0.508
5.143
5.456
5.611
5.579
5.379

0.000
0.000
0.000
0.000
0.000
0.000
0.002
0.006
0.019
0.000
0.000
0.000
0.000
0.000
0.016
0.003
0.004
0.003
0.004

0.017
0.038
0.033
0.030
0.032
0.027
0.084
0.317
0.454
0.011
0.009
0.008
0.009
0.007
0.349
0.250
0.263
0.262
0.213

−0.005−0.106−0.164−0.0290.032−0.046
0.038
4.784
0.024
0.477
0.464
0.498
0.513
0.495
4.369
4.964
5.155
5.115
4.947

0.017
−0.055−0.1200.009
0.074−0.015
0.071
5.288
0.162
0.492
0.476
0.509
0.525
0.503
4.921
5.293
5.432
5.402
5.241

0.028
−0.030−0.099
0.029
0.096
0.004
0.099
5.438
0.317
0.500
0.483
0.514
0.531
0.508
5.187
5.457
5.583
5.557
5.383

0.040
−0.004−0.076
0.049
0.118
0.022
0.143
5.574
0.552
0.507
0.489
0.520
0.537
0.513
5.403
5.606
5.769
5.736
5.524

0.062
0.039
−0.033
0.089
0.159
0.056
0.339
6.028
1.543
0.521
0.501
0.531
0.548
0.522
5.696
5.968
6.194
6.158
5.789

1.001
1.003
1.000
1.000
1.000
1.003
1.003
1.001
1.005
1.003
1.000
1.000
1.001
1.001
1.007
1.001
1.001
1.000
1.004

The index of βj , pj , mj indicates age cohorts: 1 1986-95, 2 1976-85, 3 1966-1975, 4 1956-1965, 5 19461955.

Figures 9 and 10 show the diﬀerence of parameters pj and mj depending on age
cohorts where each dot indicates the median and the error bar indicates 95% highdensity interval of posterior distribution, respectively. Figure 11 is the path plot of
the medians of pj and mj by age cohorts. Roughly saying, opportunities of games in
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terms of both the success probability and number of trials increases with age up to
respondents in their 50s, except those in their 30s, who, in terms of success probability,
suﬀered the negative eﬀects of severe recession when entering graduate job markets.
Figure 12 shows the histograms of observed relative status overlapped by predicted
normal distributions according to age cohorts.

5.3 Diﬀerent Games Model on Educational Levels
Next, I examine the diﬀerent games model on educational levels. Table 3 shows thesummary of the MCMC estimation of posterior distributions for the diﬀerent games
model on educational levels in SSM 2015 data. The MCMC setting for parameter
estimation is the same as that in the common game model. MCMC sampling for allparameters can be judged as being converged from the values of R̂. The index of βj ,pj , and mj indicates educational levels: 1 lower secondary, 2 upper secondary, and 3
tertiary.
Table 3 Result of the MCMC estimation of the diﬀerent games model on educational levels in 2015 data (n = 2895)

β0
β1
β2
β3
σβ
µm
σm
p1
p2
p3
m1
m2
m3

mean

se mean

sd

2.50%

25%

50%

75%

97.50%

R̂

−0.048−0.280−0.054
0.334
1.345
6.449
4.329
0.419
0.475
0.571
4.687
5.562
6.179

0.000
0.000
0.000
0.000
0.104
0.396
0.690
0.000
0.000
0.000
0.006
0.002
0.003

0.023
0.042
0.026
0.027
3.525
11.55323.783
0.015
0.005
0.006
0.451
0.200
0.260

−0.093−0.364−0.105
0.282
0.192
2.175
0.313
0.389
0.464
0.560
3.833
5.184
5.685

−0.063−0.307−0.071
0.316
0.370
4.983
0.848
0.409
0.471
0.567
4.377
5.427
6.002

−0.048−0.280−0.054
0.334
0.609
5.559
1.509
0.419
0.475
0.571
4.673
5.557
6.172

−0.032−0.252−0.037
0.351
1.146
6.159
3.076
0.429
0.478
0.575
4.988
5.694
6.352

−0.002−0.196−0.001
0.385
6.853
13.16020.581
0.449
0.485
0.582
5.611
5.960
6.700

1.000
1.000
1.000
1.000
1.003
1.007
1.006
1.000
1.000
1.000
1.001
1.000
1.000

The index of βj , pj , mj indicates educational levels: 1 lower secondary, 2 upper secondary, and 3 tertiary.

Figures 13 and 14 show the diﬀerence of parameters pj and mj depending on
educational levels where each dot indicates the median and the error bar indicates
95% high density interval of posterior distribution, respectively. Figure 15 is the path
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0.550
6.0
●

0.525

m

prob

●

●
●

●

0.500

●

5.5

●

●

●

5.0
●

0.475

4.5
1986−95 (20s) 1976−85 (30s) 1966−75 (40s) 1956−65 (50s) 1946−55 (60s)

1986−95 (20s) 1976−85 (30s) 1966−75 (40s) 1956−65 (50s) 1946−55 (60s)Fig. 9 pj by age cohorts (median and
95% HDI)

Fig. 10 mj by age cohorts (median and
95% HDI)agecohort

5.6

agecohort

1966−75 (40s)●

1956−65 (50s)●

5.5

1976−85 (30s)

●

5.4

m

1946−55
(60s)
●

5.3

5.2

1986−95 (20s)●
0.49

0.50

0.51

prob

0.52

0.53

Fig. 11 Path plot of medians of pj and mj by age cohorts
1986−95

1976−85

1956−65

1946−55

1966−75

1.5
1.0

density

0.5
0.0

0.0

0.4

0.8

1.5
1.0
0.5
0.0

0.0

0.4

0.8

0.0

0.4

z

0.8

Fig. 12 Histograms of observed relative status and predicted distributions by age cohorts.

237

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plot of medians of pj and mj by educational levels. We can see from these ﬁgures thatboth the success probability pj and number of trials mj increase as educational level
gets higher, indicating that higher educational levels provide better opportunities interms of both success probability and number of trials in a random walk game. Figure
16 shows the histograms of observed relative status overlapped by predicted normal
distributions according to educational levels.

5.4 Trends of the Parameters of the Common Game Model and the Diﬀerent Games Model on Educational Levels
Finally, I examine the trends of the model parameters in both the common gamemodel and the diﬀerent games model on educational levels in postwar Japan from1955 to 2015*6 . The trends of the parameters of the diﬀerent games model on age
cohorts will be shown in Appendix 1.
Figures 17 and 18 are the trends of parameters of the common game model, andFigure 19 is the path plot of medians of p and m by survey years. As for successprobability p, it increased sharply from 0.33 in 1955 to around 0.5 in 1975 following
the high economic growth in Japan, and then it became almost stable around 0.5
from 1975 to 2015, except for a slight decrease in 2005. As for number of trials m, it
increased from 4.7 in 1955 to 6.5 in 1975 in the same way with success probability,
then it decreased by about 1 point in 1985, thereafter it became almost stable. We
can see clearly from the path plot (Figure 19) that the postwar period can be divided
into two periods in terms of people’s subjective evaluation: the period of expanding
opportunity in terms of both the success probability and possible chance of trials
from 1955 to 1975 and the period of high and constant opportunity, but less chanceof trials from 1985.
Figures 20 and 21 are the trends of parameters of the diﬀerent game model on
educational levels, and Figure 22 is the path plot of the medians of pj and mj byeducational levels by survey years. Both success probability pj and number of trials mj
increase as educational level increases in every survey period. As for pj , the diﬀerenceof success probabilities among educational levels became smaller from 1955 to 1975,*6 The sample size of the data in each survey wave was 1982 in 1955 survey, 1989 in 1965, 2665

in 1975, 2391 in 1985, 2372 in 1995, 2567 in 2005, and 2895 in 2015.

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―255―

●

●

6
●

0.50

m

prob

0.55

●

5

0.45

●

●

4

0.40
lower_secondary

upper_secondary

tertiaryedulevellower_secondary

Fig. 13 pj by educational levels (median and 95% HDI)

upper_secondaryeduleveltertiary

Fig. 14 mj by educational levels (median and 95% HDI)
tertiary●

6.0

upper_secondary
●

m

5.5

5.0

lower_secondary

●

0.45

0.50

0.55

prob

Fig. 15 Path plot of medians of pj and mj by educational levels
lower_secondary

upper_secondary

tertiary

2.0

density

1.5

1.0

0.5

0.0
0.0

0.4

0.8

0.0

0.4

z

0.8

0.0

0.4

0.8

Fig. 16 Histograms of observed relative status and predicted distributions byeducational levels

239

―256―

0.50

●

●

●

●

●
●

6
●

m

prob

0.45

0.40

●
●

●

●

●

5
0.35

●
●

55

65

75

85

95

year

05

15

55

Fig. 17 p by survey years (median and
95% HDI)

65

75

85

year

95

05

15

Fig. 18 m by survey years (median and
95% HDI)
75

6.5

●

6.0

05

●

m

65

●

5.5

95●
●

85

●

15

5.0
●

55
0.35

0.40

prob

0.45

0.50

Fig. 19 Path plot of medians of p and m by survey years

and then become larger after 1975. The shapes of trajectories for each educational
level in the path plot looks mostly homogeneous, but the location moves to the righttop corner as educational level increases, indicating that higher educational levels
provide better opportunities in terms of both the success probability and number oftrials in a random walk game in every survey period.
Figure 23 shows the values of WAIC for model comparison*7 . In terms of WAIC, thediﬀerent games model is adopted as a better model than the common game model inevery survey period. Figure 24 shows diﬀerences of WAIC between the two models*8 .

*7 I employed ‘loo’ R package (Vehtari et al., 2016) for calculation of WAIC.
*8 More precisely, Figure 24 shows diﬀerences of estimated expected log pointwise predictive

density for a new data (eldp), which is related to WAIC as WAIC = −2elpd. See Vehtari andGelman (2014) for more information.

240

―257―

We can see from Figure 24 that with respect to educational level, the common gamemodel and the diﬀerent games model were closest in 1975 and 1985. In those days, it
was commonly discussed in mass media as well as academia that Japan had become
an “all middle-class society” in which almost of all the people regarded themselves as“middle.” From the outcome of our analysis, it can be claimed that in the time of
the “all middle-class society,” people evaluated their society almost the same as thecommon game in which all players have the same opportunities, but diﬀerent luck.

6 Conclusion
Thus far, I have introduced a new analytical model for class identiﬁcation by applying a mixed method of simple mathematical modeling and Bayesian statisticalmodeling. In the ﬁrst part, I introduced the life-is-like-a-random-walk model for class
identiﬁcation distribution, which may be the simplest model for explaining middle
concentration. In the second part, I attempted a parameter estimation of latent success probability and number of trials by the Bayesian (hierarchical) modeling method.
From the analysis, I determined that:(1) assuming approximately ﬁve trials of ﬁfty-ﬁfty games with around 0.5 success probability can well describe the observed class identiﬁcation distribution in 2015 data;(2) the Japanese postwar period can be divided into two periods in terms ofpeople’s subjective evaluation: the period of expanding opportunity from1955 to 1975 and the period of high and constant success probability, butless chance of trials after 1985; and(3) the diﬀerent games model assuming diﬀerences in parameters among educational levels is always better in terms of goodness of prediction than thecommon game model assuming common parameters in the society in every
survey period. However, these models were closest in terms of goodnessof prediction in 1975 and 1985, possibly indicating that in the time of “allmiddle-class society,” people evaluated their society almost the same as thecommon game in which all player has the same opportunities, but diﬀerent
luck.
Thus, although some future tasks, which includes constructing and testing more
241

―258―

edulevel●

lower_secondary

upper_secondaryeduleveltertiary●

lower_secondary

upper_secondary

tertiary

10

0.5

8
●
●
●

●

0.4

m

prob

●

6

●

●

●
●
●

0.3

●

●

●

85

95

4

●

55

65

75

85

95

year

05

15

55

Fig. 20 pj by educational levels by survey years (median and 95% HDI)

65

75

65

7

m

55
●
05

05

●

15●

55

●

0.3

0.4

75

55

65

5

15

a upper_secondary a tertiary
75

8

65

05

Fig. 21 mj by educational levels by survey years (median and 95% HDI)

● lower_secondaryedulevel a

6

year

95● ●

85

75●

95

15

85

85

05

95

15

0.5

prob

Fig. 22 Path plot of medians of pj and mj by educational levels by survey years
model

●

common_game

110

different_games

●

−500

elpd_diff

●

●

●

waic

●

90

●

−750

●

−1000

70

●

●

●

●

95

05

50
●

●

−1250
55

65

75

85

year

95

05

30

15

Fig. 23 Trends of WAIC (error bar:standard error)

55

65

75

●

85

year

15

Fig. 24 Trends of WAIC diﬀerence (error bar: standard error)

242

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empirically realistic and complex models, remain, at least in this study, we could
shed new light on the well-known Japanese class identification trends using a new
analytical framework.

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7 Appendix 1: The Trends of the Parameters of the Diﬀerent
Games Model on Age Cohorts
This appendix shows the trends of the model parameters in the diﬀerent games model
on age cohorts in the postwar Japan from 1955 to 2015, and the results of modelcomparison between the common game model and the diﬀerent games model in eachsurvey wave by WAIC.

245

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246

―263―

prob

0.30

0.35

0.40

0.45

0.50

0.55

55

●
●
●
●
●

●

75

●
●
●
●
●

year

85

●

●
●
●
●

95

●
●
●
●
●

05

●
●
●
●
●

Fig. 25 pj by age cohorts by survey years (median and 95% HDI)

65

●
●
●

●

agecohort

1886−95 ● 1916−25 ● 1946−55 ● 1976−85● 1896−05 ● 1926−35 ● 1956−65 ● 1986−95● 1906−15 ● 1936−45 ● 1966−75
●

15

●
●
●
●
●

247

―264―

4

5

6

7

m

55

●
●●●

●

75

●
●

●
●
●

year

85

●●●●
●

95

●

●

●
●●

05

●
●
●●
●

Fig. 26 mj by age cohorts by survey years (median and 95% HDI)

65

●

●

●●
●

agecohort

1886−95 ● 1916−25 ● 1946−55 ● 1976−85● 1896−05 ● 1926−35 ● 1956−65 ● 1986−95● 1906−15 ● 1936−45 ● 1966−75
●

15

●●
●●
●

248

―265―

6.0

6.5

0.35

55

●

●

65

●

65

0.40

65

●

●

prob

0.45

0.50

05●
●95
05 75●
65●
05● ● ● 95●●
05 95● 15●●
15
05● 85 15
15
85 ● ●● ●●●
85
●
●
85
85
95
15
●
●
95
65

75
●
75●
75
75●

● 1886−95a● 1896−05a● 1906−15a● 1916−25a● 1926−35a● 1936−45a● 1946−55a● 1956−65a● 1966−75a● 1976−85a● 1986−95aagecohortFig. 27 Path plot of medians of pj and mj by age cohorts by survey years

55●●●55●
4.5 5555

5.0

5.5m