DECOMPOSITION ANALYSIS OF SEGREGATION
Kazuo Yamaguchi
University of Chicago

Email: kyamagu@uchicago.edu

ある。
ABSTRACT

Although substantive studies on segregation, such as residential or school segregation by raceand occupational segregation by gender, are many in sociology, the analytical methodology isalmost exclusively focused on measurement issues. This paper introduces a set of two statisticalmodels for the decomposition analysis of segregation. These models can be regarded as a tool toanalyze whether one dimension of racial or gender inequality is related to another dimension of
inequality, because they can assess, for example, the extent to which gender differences inhuman capital are related to gender segregation in occupation. One of the new models is a simple
extension of the DiNardo-Fortin-Lemieux decomposition method of inequality, which implicitly
assumes a supply-driven determination of positional status attainment, and another modifies it toincorporate demand-based macrosocial size constraints on positional status attainment, but bothmodels rely on Rubin’s conception of modeling counterfactual outcomes and inverseprobability-of-treatment weighting based on propensity score. An application focuses on gendersegregation in occupation in Japan and will lead to a paradoxical result: equalizing human capitaland labor supply characteristics between men and women increases, rather than decreases,
gender segregation in occupation. Although the underlying behavioral mechanism for genderdifferences in occupational choice remains to be investigated, the analysis clarifies at least
demographically why segregation increases under the counterfactual situation.
1

DECOMPOSITION ANALYSIS OF SEGREGATION
I. INTRODUCTION
Segregation analysis is a central topic in sociological research. Substantive research on
segregation includes numerous studies of residential segregation and school segregation by raceand ethnicity, and occupational segregation by gender. This paper introduces two alternative
mutually complementary models for the decomposition of the extent of segregation into
“explained” and “unexplained” components. Generally, this paper is concerned with themethodology of estimating the unique effect of the group variable X on segregation when thecovariates of the group variable also affect segregation and, therefore, the observed effect of X
on segregation is confounded by the effect of the association of X with covariates on segregation.
The “explained” component of the segregation represents the extent of segregation that is
explained by differences in the covariate distribution between the group with X = 1 and the groupwith X = 0, and the “unexplained” component represents the remaining extent of segregation.
Substantive examples may include (1) a case where race and household income affect the choiceof residential district, and we are concerned with the hypothetical extent of residential racial
segregation after eliminating the indirect combined effect of (a) the association between race andhousehold income and (b) the effects of household income on the choice of residential district;and (2) a case where gender is associated with the characteristics of educational attainment, suchas years of education and college majors, and both gender and educational characteristics affectthe choice of occupation, and we are concerned with the hypothetical extent of gendersegregation in occupation after eliminating the indirect combined effect of (a) the association ofgender and educational characteristics and (b) the effects of education on the choice ofoccupation. In both cases, the models described in this paper eliminate the indirect effect by2

removing factor (a) by attaining statistical independence between the group variable andcovariates.
The two models introduced in this paper both rely on propensity-score weighting (Rubin
1985; Robins 1998), employed in Rubin’s causal model (RCM). One of them is a simple
extension of the DiNardo-Fortin-Lemieux (hereafter DFL) method (DiNardo et al. 1997) for
decomposition analysis, extended for a polytomous outcome variable and applied to decompose
the index of dissimilarity. The other model, which is newly introduced in this paper, is what I
refer to as the matching model.1 Like the RCM, the DFL method makes the SUTVA (stable unit
treatment assignment) assumption, and this assumption leads to the fact that the distribution ofoutcomes, such as occupational distribution in the analysis of gender segregation in occupation
and residential distribution in the analysis of residential racial segregation, depends only on the
supply-side characteristics, namely, characteristics of people who attain those occupational orresidential positions. As a result, when we consider a counterfactual situation, such as that which
would be realized if women came to have the same human capital characteristics as men’soccupational distribution changes according to changes in the supply-side labor characteristics.
On the other hand, the matching model assumes that the outcome distribution is determined onlyby the demand-side characteristics, so that job vacancies or residential vacancies will be filled tothe extent to which demand for the occupants of those positions exists, and, therefore, changes inthe supply-side characteristics conceived under a counterfactual situation do not affect theoutcome distribution of positions but alter only the matching between people and positions. The
matching model does not satisfy the SUTVA assumption, because it introduces macrosocial

The term “matching” here has nothing to do with the matching method of the RCM. As described below,
matching here implies matching between people and positions.
13

constraints on the outcome distribution (Yamaguchi 2011). However, as shown in this paper, thematching model retains all other characteristics of the DFL method.
If in reality the attainment of social positions will neither be completely supply driven nor
completely demand driven, but will depend on both the supply and the demand for thosepositions, the results from both the model based on the DFL method and the matching model
identify a range of outcomes in which the counterfactual outcome will be realized.
Not unlike the DFL method, the two models for the decomposition of segregation are notmethods for causal analysis, because I do not intend to claim that the covariates included exhaust
all determinants of the outcome other than the group variable X, and, therefore, omitted-variablebias certainly exists. In addition, the issue of controlling endogenous intervening variables also
exists for the DFL method in making a causal inference (Yamaguchi 2015). Instead, the purposeof the two models introduced in this paper is to separate the observed extent of segregation into acomponent explained by the association of the group variable X with a given set of covariates V
that also affect the outcome and the remaining unexplained component—without claiming thatthe unexplained effect is a causal effect.
Methodological interest in segregation analysis has been centered on which measure has
the most desirable characteristics (e.g., Lieberson 1976; Winship 1977; Blackburn and Marsh
1991; Reardon and Firebaugh 2002; Reardon and O’Sullivan 2004), given the fact that the index
of dissimilarity, which is still a representative measure of segregation, may be better modified toreflect certain additional characteristics such as distance among the units of segregation formeasuring residential segregation. This is an important methodological issue but is largely
independent of the topic of this paper, because the decomposition methods introduced in thispaper can be combined with different segregation indices of the researcher’s choice as long as
4they are modifications of the index of dissimilarity based on the measurement of differences
between groups in the categorical distribution of segregation units. Charles and Grusky (1995,
2004) introduced loglinear and log-multiplicative models for segregation focusing on gendersegregation in occupation. Their method measures segregation based on characteristics of theodds ratio and requires a distinct methodological approach to the decomposition of segregation.
This paper, however, focuses on the decomposition of segregation based on the index ofdissimilarity. Below, I introduce the decomposition of the index of dissimilarity by an extensionof the DFL method and by the matching model and apply them to an analysis of gendersegregation of occupation in Japan.
II. METHODS
II_1 A brief review of the DiNardo-Fortin-Lemieux method
Unlike the Blinder-Oaxaca decomposition method (Blinder 1973; Oaxaca 1973),
which relies on a pair of regression equations for the outcome, the DFL method does not assume
any parametric model for the outcome. As Barskey et al. (2002) point out, a major limitation of
the Blinder-Oaxaca method is that it requires a linear relationship between the dependent
variable and its covariates. On the other hand, the DFL method, which does not assume such a
relationship, is considered (Fortin et al. 2011) to be a method that generates a more robust
estimate for decomposition than the methods that rely on regression models.
The pair of equations assumed for men and women can be expressed in the DFL method

as y M   (V M , θ M )   for men and yW   ( VW , θW )   for women, where  is an unspecified
function, V M and VW are covariates of the outcome, and θ M and θW are parameters that indicate

5the covariate effects of the outcome, for men and women, respectively. Under the assumed
independence of the error term from the covariates, we may obtain the following decomposition:y  y   (VW , θW )   (V M , θW ))    (V M , θW )   ( V M , θ M ))  . (1)W

M

The first component—the “explained” component—of equation (1) reflects inequality in themean of Y that would be eliminated if women had men’s covariate distribution, and the second
component—the “unexplained” component—reflects inequality that would remain because
women having the same covariate distribution as men’s are treated differently in the society.

Since the estimates of  ( V M , θ M ) and  ( VW , θW ) are simply sample means, we only need an
estimate of  ( V M , θW ) , which can be obtained from the following equation for the genderdummy variable X that takes a value of 0 for men and 1 for women: (V M , θW )    ( v, θW ) p ( v | X  0)dv   E (Y W | v) p ( v | X  0)dv
   ( v ) E (Y | v ) p ( v | X  1)dv  E (Y W ),
v

v

W

v

(2)

where p ( v | X ) indicates the conditional probability density of V, and E indicates the weighted
mean with weights:

 ( v) p ( v | X  0) p ( X  0 | v ) p ( v ) / p( X  0) p ( X  1) p( X  0 | v)


. (3)
p( v | X  1)
p( X  1| v) p ( v ) / p( X  1) p ( X  0) p ( X  1| v )Unlike the Blinder-Oaxaca method, the DFL method can be applied to the decompositionof difference in proportion, as in the decomposition analysis of gender difference in the
proportion of managers, because it does not assume any outcome regression model. The method
introduced in this paper takes this advantage of the DFL method further by extending its
application to the case of a polytomous dependent variable.

6II_2. The DFL model—where the marginal distribution of outcomes is endogenously
determined under counterfactual situations
I introduce below a decomposition method, which I call the DFL model, for the index ofdissimilarity (hereafter the ID), using an example of gender segregation in occupation, byextending the DFL method. We are interested in the decomposition of PjM  PjW , for a set ofoccupations with j = 1, … , J, where PjM is the proportion, among men who have jobs, of thosein the j-th occupation, and PjW is the proportion, among women who have jobs, of those in thej-th occupation, into the component attributable to gender differences in covariates V, such as
educational attainment, age, marital status, and hours of work, and the unexplained component.
If this decomposition is attained, then we can decompose the ID, defined as(1/ 2)  j | PjM  PjW | , into the unexplained component, defined as (1/ 2)  j | ( PjM  PjW ) unexplained |—where ( PjM  PjW ) unexplained is the unexplained gender difference in the proportion estimated by
the method described below—and the explained component, which is the difference between theobserved ID value and the unexplained ID value.
Generally, we assume that the marginal distribution of the group variable X, such as
gender or race, is fixed under any counterfactual situation. On the other hand, the marginal
distribution of covariates V summed over the states of X depends on a particular counterfactual
situation that we assume but is assumed nonetheless to be exogenously determined
independently of the association between X and V. First, in this section we assume further amodel that implicitly makes the SUTVA assumption (Morgan and Winship 2007) for potential
outcomes. With this assumption, we can define the saturated linear probability model for the7

outcome with categorical covariates for the probability of having outcome j as a function of thecovariates of individuals and outcome-specific parameters as follows:
Pr(Yi  j )  Pij   j ( v i ; θ 0 j )   j ( v i ; θ1 j ) X i , for j  1,..., J  1 ,(4)

where Yi is a categorical outcome variable for person i, Pij is the probability for person i withcovariates vi , where vi is the value of V observed for person i, to attain the j-th occupation, j ( v i ; θ 0 j ) represents the “main effects” of covariates V with the outcome-specific parameters
θ 0 j , X i is a gender dummy variable that takes a value of 0 for women and a value of 1 for men

for person i , and  j ( v i ; θ1 j ) X i represents heterogeneous effects of gender, with heterogeneity
varying with covariates V , on the outcome with the outcome-specific parameters θ1 j . Note that
although it may be thought that a linear probability model will generate inconsistent parameter
estimates, we assume here a saturated model with categorical covariates where the estimatedprobabilities are equal to sample proportions for each combined state of covariates, and for any
saturated model no inconsistency issue exists. Note also that the marginal distribution of the j-thoccupation is determined by the effects of X and V on Pij characterized by parameters j ( v i ; θ 0 j ) and  j ( v i ; θ1 j ) . In other words, the size of each occupation here is implicitly

assumed to be endogenously determined by supply-side characteristics of the labor market, thatis, the joint distribution of ( X , V ) for a given set of parameters {θ 0 j , θ1 j } .

From equation (1), we obtain Pr(Y  j | X  0)  PjW  E  j ( v ) | X  0  andPr(Y  j | X  1)  PjM  E  j ( v ) | X  1  E   j ( v ) | X  1 , where PjW and PjM are the meansof Pij for men and women, respectively, with each outcome j, and those means for men andwomen are equal to their sample proportions for saturated models. It follows that
8





PjM  PjW  E  j ( v ) | X  1  E  j ( v ) | X  0   E   j ( v ) | X  1 .

(5)Equation (5) indicates that the observed gender difference in the proportion of having the j-th job
is the sum of the gender difference in the main average covariate effects and the average gendereffect among men.
Suppose that we realize a counterfactual situation in sample data where the covariate
distribution becomes statistically independent of X by multiplying weights ω(v) to sample









observations. Then, since E  j ( v) | X  1  E  j ( v ) | X  0 holds, where E indicates theweighted mean, under V  X , we obtainE (Y  j | X  1)  E (Y  j | X  0)  E   j ( v) | X  1  E   j ( v)  ,(6)and, therefore, E (Y  j | X  1)  E (Y j  j | X  0) reflects the weighted average of the gendereffect. Note that the conditioning by X = 1 for  can be omitted, because the weighted averageof  depends only on the weighted distribution of V, which is made equal between X = 1 and X= 0.
Regarding the distribution of V, which is to be made equal between the group with X = 1
and the group with X = 0 for the weighted data, we can consider three meaningful alternative
situations:(1) a counterfactual situation where both men and women come to have the averagecovariate distribution P ( v ) ,(2) a counterfactual situation where women come to have men’s covariate distribution

P ( v | X  1) , and(3) a counterfactual situation where men come to have women’s covariate distribution

P ( v | X  0) .

9

As shown below, these three counterfactual situations are similar to those considered for theestimation of the average treatment effect, the average treatment effect for the treated, and theaverage treatment effect for the untreated (Morgan and Winship 2007)—although we do notmake a causal interpretation of those “treatment effects,” for reasons explained above. In
addition, the distinction between X = 1 and X = 0 is not a distinction between the treatment groupand the control group here, and we will therefore refer to the effect of X under the threealternative situations for the covariate distribution as the average group effect (AGE), theaverage group effect for group 1 (AGE1) and the average group effect for group 0 (AGE0).
As initially shown by Rubin (1985; see also Robins [1998], and described by Morgan andWinship [2007]), the AGE can be estimated by using the following sets of inverse-probabilityof-treatment (IPT) weights:

1 ( v) P( v)P( X  1)
P( v | X  1) P( X  1| v)(7)

for men, for whom X = 1 holds, and0 ( v) P( v)P( X  0)
P( v | X  0) P( X  0 | v)

(8)

for women, for whom X = 0 holds.
By those weightings, the covariate distributions for both men and women become nearly

equal when the propensity score, P( X  1| v) , is adequately estimated. Note that whenP ( X  1| v ) can be estimated nonparametrically because we have observations for both X = 1

and X = 0 for each combined state of the covariates, the conversion of P( v | X  1) andP ( v | X  0) to P ( v ) by using their nonparametric estimates for the IPT weighting will be

perfect. For a parametric estimate of P ( v | x ) , we expect that the weighting, when the propensity
score is adequately estimated, will attain statistical independence between X and V. However, we
10need to diagnose this expected property of the IPT weights, since this property may not be
attained either when we have a model of P ( v | x ) that does not fit the data or when observations
that lie outside the “common area of support” for the propensity score generate nonnegligible
bias in attaining statistical independence between X and V. This issue will be addressed in theapplication.

It is also noteworthy that in order to retain the marginal distribution of X, weights 1 ( v)and 0 ( v) need to be replaced by the adjusted weights 1 * ( v ) and 0 *( v) for the ratio

estimation so that the sum of the weights becomes equal to the sample size in each group of X.

1 * ( v i ) N11 ( v i ) (vk )
k 1| X 1 1
N1

and 0 *( v i ) N 00 ( v i ) (vk )
k 1| X  0 0
N0

,

(9)

where N1 and N0 indicate the number of sample observations with X = 1 and X = 0, respectively.
Then the unexplained difference in the proportion in unit j between men and women is given asE1* (Y  j | X  1)  E0 * (Y  j | X  0),(10)

where E1 * (Y  j | x ) indicate the weighted average probability of being in unit j with adjusted
weights of equation (9). This is the estimate for the AGE.
When we consider an alternative counterfactual situation where women have thecovariate distribution of men, the weights for women and their adjustment for the ratio
estimation are given, as in equation (3), as

01 ( v) P ( v | X  1) P ( X  0) P( X  1| v ), andP( v | X  0) P ( X  1) P ( X  0 | v )01 * ( v i ) N 001 ( v i ) (vk )
k 1| X  0 01
N0

.

(11)
(12)

Then the unexplained difference in the proportion in unit j between men and women is given as
11E (Y  j | X  1)  E01 * (Y  j | X  0) .

(13)

This is the estimate for the AGE1. Similarly, the estimate for the AGE0 is obtained for thedifference in the proportion in the counterfactual situation where men have the covariate
distribution of women by switching men and women in formulas (11) through (13).
II_3. The matching model—where the marginal distribution of outcomes is exogenously
determined under counterfactual situations
In the model introduced in the previous section, it is implicitly assumed that the size of theoutcome categories such as the proportion of people in each occupation, is endogenously
determined and can change depending on supply-side characteristics, characterized by the joint
distribution of the group variable X and covariates V. Let us demonstrate this first by using asimplified case of dichotomous occupational distinction, such as managers and other employees,
and dichotomized covariate states, which we denote by V. We assume that the probability
distribution of the cross-classification of gender and covariates is as given in Table 1.
(Table 1 about Here)
Then the saturated linear probability of being a manager at the individual level, its average, andthe difference in the proportion of managers between men and women are given, respectively, asPi    1 X i   2Vi  3 X iVi ,(14)

P    ( 1   2   3 )Q  1 R   2 S

   1 (Q  R )   2 (Q  S )   3Q , andP P MW

 (Q  R)  ( 1   2  3 )Q  1R  ( S  T )   2 S
QR



S T

 Q 
 Q
S 
 1  3   2 .
QR Q  R S T 

12

(15)

(16)

We retain the proportions of men and women, Q+R and S+T, for any of thecounterfactual situations considered here. The proportion of V = 1, Q+S, is retained only for the
average group effect (AGE) but not for the average group effect for group 0 (AGE0) or that forgroup 1 (AGE1). Then equation (15) shows that neither the AGE0 nor the AGE1 preserves theaverage proportion of managers under each counterfactual situation, because Q+S changes foreach counterfactual situation. On the other hand, the AGE preserves the average proportion ofmanagers if the effects of X and V are additive, but this does not hold when an interaction effectof X and V exists, because then P changes with Q , which does not remain constant under any of
the three counterfactual situations. This holds in the general case where the covariate V has many
more categories, because equation (15) holds for the dichotomous distinction between eachcategory and the rest of the categories combined. Equation (16) simply demonstrates the fact thatthe confounding covariate effect will be eliminated when X becomes statistically independent of
V and, therefore, Q/(Q+R) = S/(S+T) holds.
The model of equation (4) assumes that the sizes of the outcome categories vary freely
with the supply-side characteristics of labor under the counterfactual situation. This assumption
may be unrealistic, however. In the case of residential segregation, change in the association
between race and household income may not change the number of people living in eachresidential district. Similarly, for the case of occupational segregation by gender, change in theassociation between gender and education may not change the number of people in each
occupation but change only the matching between occupation and people. Below, we consider analternative model, the matching model, where the sizes of outcome categories are invariant undera counterfactual situation where the group variable and covariates become independent. In the13analysis of occupational segregation, this leads to a model where demand by employers, rather
than supply-side characteristics, determines the number of occupants of each occupation.
For the matching model, we need to modify the model of equation (4) as follows:

Pr(Yi  j )  Pij   j (V, x)  j ( vi ; θ0 j )   j ( vi ; θ1 j ) X i  , for j  1,..., J  1 ,(17)

where  j is an outcome-category-specific parameter and is a function of the joint distribution ofV  { v i } and x  { x i } . Parameters  j are constant and take a value of 1 for observed data andtake values under a given counterfactual situation to make the average probability of attaining
each outcome status remain the same as the proportion of people that is observed in the unit.

Note that since parameters {  j } depend on the macrosocial joint distribution of X and V, thematching model does not satisfy the SUTVA assumption.
The model of equation (17) might be thought to be underidentified, because we assume

that  j ( v i ; θ 0 j )   j ( v i ; θ1 j ) X i is the saturated model for observed data with zero degrees of

freedom. In fact, equation (17), as well as equation (4), is a model for predicting outcomes for acounterfactual situation. Hence, if conditions to be met for a particular counterfactual situation
are given, it can be identified. For the model of equation (17), we assume that a counterfactual
situation satisfies the following five conditions four of which, (a) through (d), are shared by the
DFL model, and one of which, condition (e), is unique to the matching model:(a) Parameters  j ( v i ; θ 0 j ) and  j ( v i ; θ1 j ) are fixed for each given vi.

(b) The marginal distribution of the group variable X is fixed and is equal to the observed
distribution.

14

(c) The marginal distribution of the covariates is exogenously imposed on one of thefollowing depending on whether we estimate the AGE, AGE0, or AGE1:
P ( v ), P ( v | x  1), or P ( v | x  0) .

(d) The group variable and covariates become statistically independent.

(e) The marginal distribution of the outcome categories Pr(Y  j ) is fixed and isequal to the observed distribution, and this is attained by the adjustment of parameter  j
for each counterfactual joint distribution of X and V.
Given the model of equation (17), we obtain the following equalities:

E (Y  j | X  0)  PjW   j E  j ( v ) | X  0  , andE (Y  j | X  1)  PjM   j E  j ( v ) | X  1   j E   j ( v ) | X  1 .

(18)
(19)

Hence, under the counterfactual situation where X becomes independent of covariates V

by the use of weights M ( v) , which differ from the weights for the DFL method (hereafter, “the









DFL weights”) as explained below, EM  j ( v) | X  1  EM  j ( v) | X  0 holds.
Hence, we obtain;EM (Y  j | X  1)  EM (Y  j | X  0)   j EM   j ( v)  .

(20)

The remaining issue is how to obtain estimates of the set of { j } that satisfies conditions (a)through (e) above. Note that  j is not simply equal to E (Y  j ) / EDFL (Y  j ) , where DFL
denotes the DFL weights, because the adjustment by E (Y  j ) / EDFL (Y  j ) does not preserve
the statistical independence between X and V.
Let us denote by f ( x, y , v ) the observed joint frequency distribution, and denote by
F1 ( x, y , v ) the initial estimate of the counterfactual joint frequency distribution by the DFL
15method. F1 ( x, y, v) is an estimate that, when the propensity score is estimated nonparametrically,satisfies conditions (a) through (d) but may not satisfy condition (e). When the propensity scoreis estimated parametrically using logit or probit regression, we need to diagnose and make surethat condition (d), regarding statistical independence between X and V, is satisfied in the sample
with the DFL weights. When F1 ( x, y, v) satisfies conditions (a) through (d), we then need toestimate the expected joint distribution F ( x, y , v ) that satisfies both F ( x, , v )  F1 ( x , , v ) , inconsequence of which F ( x, y , v ) will satisfy conditions (b) through (d), andF ( , y , )  f ( , y ,  ) , in consequence of which F ( x, y , v ) will satisfy condition (e). Here the+ sign indicates the sum across categories. Condition (a) is also satisfied by the use ofmultiplicative iterative adjustments of frequencies described below. In the case of the AGE that
preserves the marginal distribution of covariates, the above-described model is in fact a special
case of the marginal model (Bergsma and Rudas 2002), for whichF ( x,  , v )  f ( x , ,  ) f ( ,  , v ) and F ( , y , )  f ( , y ,  ) hold.

It might be objected that while the marginal model is a loglinear model, we have amodification of the additive model here. Note, however, that the additive component is saturated
for the observed data, and we fix it for any counterfactual situation and, therefore, do not need toestimate those parameters. On the other hand, the model of equation (17) is multiplicative, or
loglinear, regarding the set of  parameters that we need to estimate and, therefore, we can
employ the method of model fitting and parameter estimation for the loglinear model.
We can employ the iterative proportional adjustment, across rounds of iterative
estimation starting with t = 1, to obtain the final estimate of F *( x, y, v) that should be attained
when the adjustments lead to a convergence.
16

(1) Calculate F2 t 1 (  , y ,  )   x  v F2 t 1 ( x , y , v ) .
(2) Calculate F2 t ( x, y, v )  F2t 1 ( x, y, v)

f ( , y ,  ).
F2t 1 (, y,  )(3) Calculate F2 t ( x ,  , v )   y F2 t ( x , y , v ) .

(4) For the case where F1 ( x, , v ) > 0, calculate F2t 1 ( x, y, v )  F2 t ( x, y, v )F1 ( x, , v ).
F2t ( x, , v )Then, the ratio of the final estimate to the initial estimate can be expressed as
F * ( v, j, x ) / F1 ( v, j , x )   j ( v, x ) ,(21)

where  j is the product of iterative adjustments made in step (2) of the iterative procedure and is
the estimate of parameter  j in equation (17), and  ( v, x ) is the product of iterative adjustments
made in step (4) of the iterative procedure and the modifier of the DFL weights. Note thatweights DFL ( v, x ) ( v, x ) preserve the statistical independence between X and V in F *( x, , v)
because F * ( x, , v )  F1 ( x, , v ) , and they are the weights for the matching model, that is,M ( v, x)  DFL ( v, x) ( v, x) .

III. APPLICATION
III_1. Data and the main descriptive characteristics of occupational segregation
As an illustrative analysis, I focus on gender segregation in occupation among employeesof ages 23–59 in Japan based on the 2005 survey of Social Stratification and Mobility (SSM).
The SSM survey collects data from a national representative sample of men and women of ages
20–69. Those under age 23 and those over 60 are excluded in this analysis in order to reduce
selection bias caused by gender-specific college attendance rates and gender-specific retirement17

processes. This survey is chosen for this analysis because of its detailed collection of information
on occupation and education, including high school types and college majors for the latter.
Gender segregation in occupation in Japan exists for several reasons. Among nonmanual
workers, women are overrepresented in clerical occupations and the “middle-and-lower-status”
human service professions, which we will refer to as “type-II professionals,” distinguished fromother professionals, which we will refer to as “type-I professionals.” By human services, I meanoccupations related to medicine, education, human care, and welfare. By the term “middle-andlower-status,” I imply the exclusion of the high-status professionals in medicine and education,
namely, physicians, surgeons, and dentists in medicine, and post-secondary-school teachers ineducation. Among manual workers, women are overrepresented in service work. Table 2
presents the composition of occupation by gender and the ID, and, for comparison, it alsopresents numbers for the 1995 SSM survey for Japan and numbers for the United States based on
the 2010 population census. The results in Japan are weighted by sampling weights provided by
the researchers who conducted the survey. Information on the correspondence of the eightoccupational categories to the 2010 USA census classification codes for occupations is available
from the author on request.
Table 2 shows that a great deal of gender segregation in occupation exists in both Japanand the United States, and the extent is greater in Japan than in the United States. However, amore important fact is that while there are commonalities and differences between the twocountries in the pattern of gender segregation in occupation, those differences always indicate
greater disadvantages of women in Japan than in the United States, as explained below.
(Table 2 about here)18

Commonalities between Japan and the United States are that women are significantly
overrepresented in type-II professional work and clerical work. The differences are that in Japan
women are strongly underrepresented in type-I professional and managerial positions andstrongly overrepresented in service work. While gender differences exist for those occupations in
the United States as well, the extent of gender difference is much smaller. Thosedifferences between Japan and the United States indicate relative disadvantages of women in
Japan in comparison to women in the United States. On the other hand, men are much more
overrepresented in nonservice manual occupations in the United States than in Japan. Thisdifference between Japan and the United States indicates a relative disadvantage of men in the
United States in comparison to men in Japan, given that average earnings are lower for manual
workers than for nonmanual workers. Hence, all differences between Japan and the United Statesin gender segregation in occupation indicate stronger disadvantages of women in Japan than in
the United States.
A comparison of the 1995 and 2005 results in Japan also shows that the extent of gendersegregation in occupation increased overtime in Japan. This change occurred because both theextent of women’s overrepresentation in type-II professional occupations and the extent of
women’s underrepresentation in nonservice manual work became greater in 2005 than in 1995
despite an increase in gender equality of educational attainment over the decade.
Table 3 also shows that Japanese women are disadvantaged in other respects, that is, inpostsecondary education and in employment status. Women are severely underrepresented inreceiving a four-year college education in Japan and are overrepresented in irregular
employment. The overrepresentation of women in irregular employment comes mainly from a
combination of the facts that the majority of women work either part-time or full-time without
19

overtime work and that the majority of men work full-time with overtime work, as shown in
Table 3, and there is a very strong association between hours of work and employment status in
Japan, such that among those who work less than 35 hours per week, in which women areheavily overrepresented, 91.1% are irregularly employed, among those who work 35-40 hours
per week, in which women are somewhat overrepresented, 24.5% are irregularly employed, andamong those who work 41 hours or more per week, in which women are strongly
underrepresented, 8.7% are irregularly employed. Gender inequality in educational attainment
also varies strongly with age. We will examine whether gender inequality in education, age, andlabor supply explains part of gender segregation in occupation.
(Tables 3 about here)
III_2. Decomposition analysis of occupational segregation by gender
Table 4 presents the main results of the decomposition analysis of occupationalsegregation based on the data from the 2005 SSM survey for the counterfactual situation where
women’s covariate distributions become equal to men’s (AGE1). They are based on the use ofdata weighted by sampling weights. Model 1 includes age and education, including the categoryby-category interaction effects of education and age, which are found to be significant, as
covariates. Model 2 adds to model 1 marital status (married versus single) and number ofchildren (0, 1, 2, and 3 or more) as covariates,2 including the category-by-category interaction
In the preliminary analysis, I tested the use of a three-category distinction (never married, married,
divorced/widowed) for marital status. However, since very few samples with children existed among the nevermarried in Japan, a full cross-classification of this marital-status variable with the number of children could not be
made. Hence, I tested a nine-category distinction of combined marital-and-child states by treating “nevermarried” as a single category and cross-classifying “married” and “divorced/widowed” with the number ofchildren. The use of this variable did not improve the fit of the model for predicting the propensity score comparedwith the use of the combination of number of children with the dichotomous marital status employed in the final
analysis.
220

effects of age and marital status, and the category-by-category interaction effects of marital
status and number of children, which are also found to be significant. Model 3 adds to model 2
the six-category distinction of hours of work per week (less than 35 hours, 35-40 hours, 41-49
hours, 50-59 hours, 60 hours or more, and “missing”) and the two interaction effects that arefound to be significant in predicting propensity score and are necessary to attain statisticalindependence between gender and the hours of work in the IP-weighted sample. Thoseinteraction effects are (1) the interaction effect of being married and working less than 35 hours
per work and (2) the interaction effect of being a college graduate and working 60 hours or more
per week, both of which imply that women are more likely to have those combinations beyond
the additive effects of the two variables.
A surprising finding in the results of Table 4 is that, contrary to our expectation, theequalization of those achieved statuses and attributes between men and women increases, rather
than decreases, the extent of occupational segregation. The results of Table 4 indicate that thegreater the range of achieved statuses in which women obtain status distributions equal to men’s,
the greater the extent of segregation, and the tendency is even stronger in the supply-driven DFL
model than in the matching model.
(Table 4 about here)
Table 5 presents the predicted occupational compositions for models 2 and 3. Since themen’s occupational composition remains the same as the sample composition for the DFL
estimates, the results for the DFL estimates are presented only for women.
(Table 5 about here)
The results in Table 5 indicate that the increase in gender segregation in occupation when
women come to have the same distribution of age, education, marital status, and number of21

children as men in the results of model 2 and the further increase in segregation when womencome to have the same distribution of work hours as men’s in the results of model 3 come fromthe fact that the gender gaps in the proportion of type-II professionals and in the proportion of
nonservice manual workers both increase when those individual characteristics are equalized.
This outcome occurs because when women’s educational attainment and work hours, the meansof which are lower than men’s means, as shown in table 3, are equalized with men’s, (a) many
more women attain type-II professional occupations, in which women are alreadyoverrepresented, than type-I professional occupations, in which women are underrepresented,
and (b) many fewer women obtain nonservice manual work, in which women are
underrepresented, than manual service occupations, in which women are overrepresented.
These two expected changes under the counterfactual situation are very similar to actual changes
that took place over a decade from 1995 to 2005 as shown in Table 2.
The increase in the extent of segregation predicted by the matching model is less, because,in the matching model, an increase in type-II professional occupations and a decrease in
nonservice manual work are both smaller because constraints on the size of those occupations
make changes in men’s occupational attainment partly offset the changes in women’soccupational attainment predicted by the DFL model.
Table 6 is concerned with a diagnosis of the adequacy of constructed propensity scores.
Since model 1 employs a saturated nonparametric model for predicting propensity scores, inwhich the statistical independence of gender and the covariates is perfect, the diagnosis wasmade only for models 2 and 3.
(Table 6 about here)

22

Table 6 presents a test of statistical independence between the gender dummy variable
and each covariate before and after the IPT weighting for model 3. Similar results for model 2,
where all the P values for covariates after the IPT weighting become greater than 0.89, areomitted in Table 6. For the test of independence after the IPT weighting, I employed Clogg andEliason’s method (Clogg and Eliason 1987), which uses the ratio of unweighted to weighted
frequencies as “cell weights” for cross-classified data to calculate the chi-square statistics basedon unweighted sample counts while testing the independence of the weighted frequencies.
However, since cell weights are treated as given by the Clogg-Eliason method rather than as
random variables that vary within cells, standard errors are likely to be underestimated and chisquare statistics are likely to be overestimated (Skinner and Vallet 2010). Hence, the CloggEliason method provides a conservative test for confirming statistical independence between
gender and covariates, which is shown in Table 6 to hold for model 3 after the IPT weighting.
The remaining analysis is concerned with the effect of gender differences in high school
types and college majors on gender segregation in occupation. Junior college graduates areclassified by the type of high school attended, and those with less than a high school education
are treated as a separate single category. Hence, the classification of education in Table 7
distinguishes only three levels of educational attainment. Table 7 presents unweighted sample
counts and relative frequency for a classification of educational attainment combined with highschool types and college majors by gender based on the data of the 2005 SSM survey.
(Table 7 about here)
Table 7 shows that for certain categories we cannot simply assume that women’s relative
frequency will become the same as men’s for realizing the counterfactual situation for two
distinct reasons: men’s extreme underrepresentation in two categories, and women’s extreme
23underrepresentation in two other categories. First, there are no men in the sample majoring inhome economics or nutrition in college. Making the women’s sample proportion equal to themen’s is possible, but it leads to a complete ignoring of women in this category. Even though thesample size is not zero, a similar situation exists for graduation from high schools specializing inhome economics or nursing. Making the women’s relative distribution equal to the men’s implies
that we are virtually ignoring those people, because the weighted sample size for women will bejust three. I decided to omit samples of these two categories from the analysis entirely. A distinct
situation exists for engineering high schools, and science and engineering majors in college, inwhich women are severely underrepresented and women’s sample sizes are very small. There are
two problems involved in including these categories of people in the analysis. Since the women’ssample size is too small, there is a clear lack of corresponding men when the category is
combined with age and other covariates, leading to a nonnegligible lack of “common area of
support” in propensity score between men and women. Accordingly, I cannot construct a set ofpropensity scores that attains statistical independence between educational classification andgender when I include those two categories. In addition, their inclusion causes a very small
number of women to represent a fairly large number of men in the weighted sample for thosecategories, thereby leading to unstable results. Hence, I decided to drop samples of those twocategories as well from the following analysis to assess the impact of differences in high school
type and college major on segregation.
Table 8 presents the results of decomposition analysis for the samples that are retained. In
addition to the sample results, the results from two models are presented. Model 3R is a
modification of model 3 and includes age, education, marital status, number of children, andwork hours and the significant interaction effects included in model 3. Unlike model 3, however,24

model 3R employs a three-category distinction of educational attainment by combining juniorcollege graduates with high school graduates. This three-category variable for educational
attainment is employed for the purpose of comparison with the results of model 4, which
employs a further distinction of educational attainment by using the classification of Table 7. In
the estimation of propensity scores, the three-category distinction of educational attainment wasused for both model 3R and model 4 regarding the category-by-category interaction effects ofage and education. The main effects of education differ, however, between the two models
because model 4 employs the educational classification of Table 7 while model 3R employs thethree-category distinction of educational attainment.
(Table 8 about here)
Table 8 shows for the sample results that the elimination of the four categories of people
described above reduces the extent of occupational segregation by 9.6% (0.096 =(.427–.386)/.427). Hence, gender segregation in science and engineering, in which men arestrongly overrepresented, and in home economics and nursing, in which women are strongly
overrepresented, explains about 10% of gender segregation in occupation. Similar to the resultsof model 3 in Table 6, results from model 3R in Table 10 indicate that making women’s
educational attainment, marital status, number of children, and hours of work equal to those ofmen tends to increase gender segregation in occupation.
The results from model 4 show that making women’s high school types and college
majors equal to those of men yields a somewhat ambiguous overall effect on the extent ofsegregation. According to the prediction from the DFL method, it has little effect on segregation(a change from 0.463 to 0.466), but the outcome of the matching model indicates a small
increase in segregation, from 0.456 to 0.464. On the other hand, if we compare the change in the25

distribution of occupation from model 3R to model 4, we find outcomes that are consistent
between the predictions from the DFL model and the matching model. The further equalizationof high school types and college majors reduces the gender gap in the proportion of type-II
professionals but increases the gender gap in the proportion of clerical workers, and those twochanges largely offset each other in yielding the overall extent of segregation. The major
remaining gap in high school types and college majors comes from a larger proportion of highschool graduates from general high schools and a smaller proportion of college graduates withsocial science majors among women than among men. Somewhat unexpectedly, women withsocial science majors in college have a much higher probability of becoming clerical workers(62%) than women who graduated from general high schools (33%), while the probability of
becoming type-II professionals is slightly lower for the former (18%) than for the latter (20%).
Hence, the results in model 4 reflect female college graduates’ strong tendency to become
clerical workers and the consequent increase in clerical occupation among women when thewomen’s proportion of college graduates with social science majors is equated with men’s, and aconsequent reduction in the proportion of women in the type-II profession.
IV. CONCLUSION
This paper introduced a new method for the decomposition analysis of segregation. The
juxtaposition of the model based on the supply-driven DFL method and the matching model,
where the occupational size is constrained by demand, will be useful in predicting the range ofpossible outcomes under a given counterfactual situation. The semiparametric approach
employed for the method will also generate robust estimates for the decomposition.

26

The illustrative analysis of the decomposition of occupational segregation by gender inJapan showed a paradoxical result that the gender equalization of human capital and labor supply
characteristics such as educational attainment and hours of work will increase the extent ofgender segregation in occupation. It is clear that the role that gender differences in human capitaland labor supply play in generating the gender gap in occupational attainment is quite different
from the role that gender differences in human capital and labor supply play in generating thegender gap in income, because we usually observe that equalizing the gender gap in humancapital and work hours leads to a smaller gender gap in income. It is clear we need further
research on the role that gender plays in occupational opportunities and attainment, and in theconsequent gender segregation in occupation.

27

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Yamaguchi, Kazuo. 2011. “A Macrocounterfactual Analysis of Group Differences: An
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Yamaguchi, Kazuo. 2015. “Decomposition of Gender or Racial Inequality with Endogenous
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30

Table 1. Cross-Classified Proportions of Gender and a Covariate
Men: X = 1 Women: X = 0

V=1 Q
V=0 R

Q+R+S+T=1

S

T

31

Table 2. Gender Segregation in Occupation

Japan, 2005
Japan,1995
U.S.A., 2010
Men Women1 Men Women1 Men
Women
Sample size
1,261 1,187
1,468 1,084
81,323,085 72,714,395
Composition
composition
Composition
1. Professional, type I
0.116 0.018*
0.094 0.019*
0.156
0.127
2. Professional, type II
0.041 0.196*
0.048 0.142*
0.043
0.208
3. Managerial/administrative 0.100 0.007*
0.101 0.012*
0.108
0.075
4. Clerical
0.167 0.329*
0.218 0.345*
0.070
0.219
5. Sales0.131 0.104*
0.093 0.113
0.106
0.120
6. Manual excluding service2 0.305 0.159*
0.294 0.197*
0.255
0.047
7. Manual service
0.026 0.136*
0.031 0.132*
0.117
0.158
8. Other3
0.114 0.050*
0.121 0.040*
0.145
0.046
Index of dissimilarity
0.427
0.343
0.369
1The asterisk indicates, for Japanese data, significant gender difference.

2This category includes production workers, construction workers, and skilled manual workers.
3This category includes protective service workers, agricultural workers, and occupations notclassifiable.

32

Table 3. Gender Inequality in Education and Employment Status1(2005 SSM)

Men Women2
Sample size
1,261 1,187
Composition
1. Proportion who are 4-year college graduates 0.377 0.148*
2. Proportion who are irregular workers
0.078 0.520*
3. Composition by hours of work per week
Less than 35 hours
0.024 0.333*
35-40 hours
0.323 0.398*
41-49 hours
0.235 0.161*
50-59 hours
0.192 0.062*
60 or more hours
0.211 0.032*
Missing
0.015 0.014
Total
1.000 1.000
1The results of the 2005 SSM survey.
2The asterisk indicates a significant gender difference.

33

Table 4. Main Results for the Decomposition of Segregation
Occupationalsegregation:
Sample ID=0.427
DFL Matching
Model 1 0.443 0.430
Model 2 0.450 0.440
Model 3 0.493 0.481

Model 1: age × education.
Model 2: age × education; age × marital status; marital status × number of children.
Model 3: Variables in model 2 plus hours of work per week, marital status × less than 35 hours
of work, and college graduation × sixty or more hours of work.

34

Table 5. Detailed Decomposition Results for Models 2 and 3
SampleMen
1. Professional,
type I
2. Professional,
type II
3. Managerial/
Administrative
4. Clerical
5. Sales
6. Manual,excluding service

7. Manual
service
8. OtherIndex ofdissimilarity

Women

Composition.

DFL

Model 2
Matching

Women

Men

Composition

Women

DFL

Model 3

Women

Matching
Men

Composition

Wome
n

0.116

0.018

0.041

0.097

0.030

0.024

0.110

0.025

0.100

0.007

0.008

0.100

0.008

0.011

0.097

0.010

0.041
0.167
0.131
0.305
0.026

0.196
0.330
0.104
0.159
0.136

0.114 0.050
0.427

0.238
0.320
0.083
0.139
0.126

0.045
0.450

0.033
0.167
0.143
0.315
0.027

0.204
0.329
0.091
0.148
0.134

0.117
0.047
0.440

35

0.291
0.308
0.085
0.126
0.129

0.022
0.493

0.017
0.164
0.135
0.318
0.026

0.221
0.333
0.099
0.144
0.135

0.131
0.032
0.481

Table 6. Diagnostic Tests of Statistical Independence between
Gender and Covariates after the IPT Weighting
Covariates

Before the IPT
Weighting
L2
df P
Age
14.0 6 0.030
Education
282.2 3 0.000
Marital Status
6.3 1 0.012
Number of Children
14.7 3 0.002
Hours of work/week 686.8 5 0.000

After the IPT
weighting: Model 3
L2
df P
4.17
6 0.6540.50
3 0.9190.07
1 0.786
4.02
3 0.259
4.86
5 0.433

36

Table 7. Gender Differences in Specialization in Educational Attainment1
Educational
Attainment
Middle school
High school

CollegeHigh school typeor college major
N.A.
GeneralEngineering
Commerce
Agricultural
Home economics &Nursing
Other
Education
Humanities
Social sciences
Sciences & engineering
Medicine & healthHome economics & nutrition
Other

Total
Index of Dissimilarity
1
The results from the 2005 SSM survey.

Men
N

%

Women
N
%

102
291
221
85
62
3

8.1
23.1
17.5
6.7
4.9
0.2

61
629
13
191
18
87

5.1
53.0
1.1
16.1
1.5
7.3

22
1.7
12
1.0
24
1.9
29
2.4
36
2.9
52
4.4
215
17.0
36
3.0
146
11.6
10
0.8
18
1.4
15
1.3
0
0.0
20
1.7
37
2.9
14
1.2
1,262 100.0 1,187 100.0
0.501

37

Table 8. Results with a Detailed Classification of Educational Attainment
N = 890 for men, N = 1,049 for womenSampleMen
1. Professional, type I
2. Professional, type II
3. Managerial/administrative
4. Clerical
5. Sales
6. Manual, excluding service
7. Manual service
8. OtherIndex of dissimilarity

Women

Composition.

0.066
0.049
0.101

0.015
0.185
0.008

0.197
0.340
0.160
0.107
0.286
0.156
0.029
0.136
0.112
0.052
0.386

Model 3R
DFL
Matching
Women Men
Women
Composition

0.020
0.297
0.013

0.313
0.079
0.121
0.127
0.030
0.463

38

0.060
0.016
0.095

0.020
0.213
0.012

0.192 0.345
0.171 0.097
0.305 0.140
0.030 0.135
0.131 0.037
0.456

Model 4DFL
Matching
Women Men
Women

0.016
0.227
0.007

0.377
0.079
0.126
0.137
0.031
0.466

Composition

0.064
0.040
0.102

0.016
0.192
0.007

0.166
0.367
0.172
0.097
0.299
0.146
0.027
0.138
0.131
0.036
0.464