UNDERSTANDING SOCIAL CHANGE

A DECOMPOSITION APPROACH

PARFAIT M. ELOUNDOU-ENYEGUE
SARAH C. GIROUX
MICHEL TENIKUE
 Preface

A story is often told of a drunk man who loses his key at night and only searches for it under a lamppost, because
that is where the light is. Today, the social sciences may be in a similar predicament: Research on social change
(a macro-phenomenon) often relies on (micro) survey data. Compared to macro-level regression, micro-level
research is the ‘methodological lamppost,’ the better-lit area of quantitative social science, where analyses are
deemed statistically most rigorous. Thus, to study how education affects national economies, a researcher might
rely on estimates of how a worker’s education level affects his or her income. In public health, to estimate how
economic growth affects national obesity, our researcher might likewise extrapolate from the individual-level
link between income and obesity. In demography, to capture the economic benefits of declining fertility,
the researcher might explore how the economic fortunes of various family members depend upon the size of
their family.

All these three examples show a discrepancy between the scale at which a problem occurs and the scale at which
it is studied. Social change is a dynamic and macro-process, yet we often approach its study with micro cross-
sectional data. This creates two fallacies. One, ecological, relates to unit of analysis, specifically, studying
individual people rather than the entire society. The second, historical, fallacy amounts to “reading history
sideways” i.e., reading cross-sectional data as indicating a historical trend. Most researchers acknowledge these
fallacies: cross-sectional studies of interpersonal differences may reveal why some people have a higher
propensity to experience a given condition (say, obesity) but they cannot explain why/how the social magnitude
of this phenomenon changes over time. Nonetheless, many studies still directly infer macro-relationships from
micro-results.

If an investigator wishing to avoid these biases turns to macro-level analysis, her peers may see this work as
lacking statistical rigor or obscuring the differences between people in the same country (Rodrik 2005). The
question, therefore, becomes, "How to avoid ecological and historical fallacies while also maintaining some
statistical rigor?"

The challenge is to recognize within-country diversity but still show how the diverse behaviors of people in a
country add up to produce a collective change. For this kind of analysis, decomposition is a useful tool. Even if it
does not reveal ultimate causes, it points to main "sources" of change, i.e., the groups or proximate processes
driving the social change.

Decomposition is widely applied across many disciplines, especially within the social sciences. However, its
different variants remain insufficiently integrated and literature on the topic remains fragmented. Few textbooks
offer a comprehensive introduction highlighting the method’s wide range of possibilities. None to our knowledge
clearly explains how to enrich elementary decompositions let alone combine them with other methods. We fill
a gap with this monograph, which updates and augments an earlier version published in 2010.

Our work on this document was conducted under the auspices of the International Union for the Scientific Study
of Population (IUSSP) with funding from the William and Flora Hewlett Foundation, and material support from
Cornell University (USA), the Institute for Training and Demographic Research (IFoRD, Cameroon), and the
Higher Institute of Population Sciences (ISSP, Burkina Faso). We received feedback from numerous colleagues,
most prominently Françoise Vermeylen, Gervais Béninguissé, Jean-François Kobiané, Silvere Konan Yao, Crispin
Mabika Mabika Gilles Gohi, Nina Trautmann Chaopricha and Latif Dramane as well as from dozens of
participants in the various workshops conducted under this IUSSP project. We tested early versions of this

3
Understanding social change: a decomposition approach

monograph with support from Serge Bahoken, Firmin Zinvi Charles Mouté, Justin Dansou, Sandrine Mbele, and
Habibou Ouedraogo. Several researchers and students — Yédodé Ahokpossi, Jacques Emina, Edzengte Pascal,
Kouam Nadège and Bela Foe Chrystelle. — were early volunteers to apply the method in their own research.
Their experiences informed subsequent revisions, as did the insights of several non-scientists (Pie-Marie Belibi,
Stephanie Hissoak, and Vincent Onguéné) whose comments helped make the document more accessible to a
broader audience. In the end, however, the authors are solely responsible for the remaining shortcomings of
this document.

Ithaca, August 2017

4
 Sommaire

Preface 3

Chapter I Introduction 7
I.1 What is decomposition? 8

I.1.1 Definitions 8

I.1.2 Application areas 8

I.1.3 Mode of explanation 9

I.2 Decomposition versus other methods 9

I.2.1. Advantages and limitations 11

I.2.2 Political relevance and applications 12

I.3 The maintypes of decomposition 12

I.4 Conventional notations 14

I.5 Structure of the monograph 15

Chapter II Demographic decomposition 17

II.1 Basic demographic decomposition 17

II.1.1 Problem type 17

II.1.2 Visual representation 18

II.1.3 Example 18

II.1.4 Mathematical formulation 19

II.1.5 Application 20

II.1.6 Application to the demographic dividend 21

II.2 Derived demographic decompositions 22

II.2.1 Decomposing a difference 22

II.2.2 Decomposing inequality 23

II.2.3 Ordinal decomposition 24

II.2.4 Nested decomposition 25

Chapter III Regression decomposition 27

III.1 Simple regression decomposition 27

III.1.1 Problem type 27

5
Understanding social change: a decomposition approach

III.1.2 Formulation 27

III.1.3 Application 28

III.2 Other regression decompositions 29

III.2.1 Curvilinear regression 29

III.2.2. Multivariate regression 29

III.2.3 Multilevel regression 29

III.3 Application to the démographic dividend 30

Chapter IV Mathematical decomposition 31

IV.1 Simple mathematical decomposition 31

IV.1.1 Problem type 31

IV.1.2 Mathematical formulation 31

IV.1.3 Application 32

IV.2 Derived mathematical decomposition 32

IV.2.1 Extended mathematical chain 32

Chapter V Combination of demographic and regression decompositions 35

V.1 Extension of behavior effect 35

V.1.1 General presentation 35

V.1.2 Illustration 36

V.1.3 Comparison with national transfer accounts 36

V.2 Extension de l’effet de composition 37

V.3 Double extension 38

Chapter VI Combination of demographic and mathematical decompositions 39

Chapter VII Combination of regression and mathematical decompositions 41

Chapter VIII Summary and conclusions 43

References 45

6
 Chapter I
Introduction

Can scientific research be a useful guide for social policy? Whether reducing unemployment, expanding school
enrollment, containing social inequality, or reducing mortality, the constant challenge facing those who plan
and implement policies is to promote desirable social change. To meet this challenge, policymakers must
understand the levers of social change — and, in this context; scientific research has a key role to play.

Social science has made great strides over the last half-century yet these strides were greatest in micro-level
studies. Thanks to advances in computing and communication technology, researchers can now collect, share,
and process statistical data on millions of households and individuals. Researchers can explore the detail and
mix of factors shaping individual behaviors.

However, an analyst of societal change will not be satisfied with this micro-level detail alone. She may find the
detail useful but still need to convert the micro-level information into valid inferences about social change. Over
half a century ago, Robinson (1950) warned against ecological bias, noting that relationships observed
macroscopically need not match those occurring at the individual level and vice versa.1 Thornton (2001)
likewise drew attention to an equally harmful bias, the tendency to “reading history sideways.” In essence, it is
wrong to use cross-country comparisons to draw conclusions about development trajectories. Researchers must
thus be careful in navigating the space between micro-data and macro-issues, building on the detail and
robustness of microlevel statistics as they aggregate them to inform macro-level questions. The decomposition
methods presented here can facilitate this conversion.

1
For example, richer countries may have higher rates of obesity but that does not imply that richer people creases with the level of wealth
of a country does not necessarily imply that within countries, richer people have a greater propensity to become obese than poor people.
Similarly, national rates of divorces increase with national crime rates, but that does not necessarily mean that people from divorced families
are more prone to crime.

7
Understanding social change: a decomposition approach

I.1 What is decomposition?

I.1.1 Definitions
To decompose something is to break it into its elementary components. In biology, decomposition is a process of
organic material decay. In chemistry, it is the process of bursting a complex molecule into simpler molecules or
atoms. In physics, it can describe the trajectory of a projectile, projecting its path into a three-dimensional space
that includes vertical, horizontal, and transversal components. In the social sciences, decomposition can serve to
estimate how several elementary processes (or groups) fuel an aggregate social change. For example, how do
different regions of a country contribute to national wealth, or how do women from different education
backgrounds contribute to change national fertility?

Several variants of decomposition methods exist in the literature but the fundamental idea is the same: how to
partition a functional set into its elementary constituents. To appreciate the uniqueness of decomposition vis-à-
vis other methods, we specify below some situations in which it applies.

I.1.2 Application areas

Within the social sciences, decomposition analysis is particularly useful for studying aggregate social change
considered as any transformation — induced or spontaneous — in the structure, functioning or performance of
a social community. The most relevant changes are quantifiable transformations resulting from an aggregation
of individual behaviors.

Decomposition is thus not applicable in studies where individuals are the unit of analysis. Decomposition is also
of little help when studying societal outcomes that do not result from the aggregation of individual behavior.
These two exceptions aside, the method applies to a wide range of social processes in demography, economics,
political science, and sociology (Kitagawa 1955, Oaxaca 1973; DasGupta 1993; Vaupel and Romo 2003). The
only requirements are to have quantifiable social outcomes that reflect an aggregation of individual outcomes.

Quantification: The outcome under study should be quantifiable, i.e., captured as an absolute number, an
average, a percentage, a ratio, or a measure of inequality. This excludes qualitative processes such as a country’s
political transition from autocratic to democratic rule, or its economic transition from subsistence to market
orientation, or its demographic transition from extended to nuclear family systems. Even in these cases,
dichotomies such as autocracy/democracy, subsistence/market, or extended/nuclear can be expanded into a
continuum. For example, one can replace the subsistence/market dichotomy by a continuous outcome such as
the percentage of people working for an employer other than family. With such reformulation, phenomena that
may a first seem qualitative lend themselves to decomposition analysis.

Aggregation of individual behavior: Sociologists distinguish between social outcomes that an intrinsic feature of
the whole society versus aggregate outcomes. The former have no correspondent at the individual level. An
example might be a country’s laws on abortion. The latter resulting from an aggregation of individual
characteristics. Examples may include the fertility level of a country or its average consumption of tobacco. This
tobacco consumption is an aggregated (not intrinsic) feature of the society because a country does not smoke;
individual citizens do.

Gradual change: Decomposition methods are seldom applicable to rare or sudden, accidental processes (e.g.,
studying the number of casualties in an earthquake or a sudden outbreak of cholera). Rather, they best apply to
studies of processes that change gradually over time.

8
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

III.2
I.1.3 Décompositions
Mode of explanation de régression dérivées
Decomposition analysis is about the sources rather than the ultimate causes of change. It is better at determining
III.2.1 Régression curvilinéaire
the origin of a change rather than why the change occurred. It mechanically accounts for the sources of change,
Pour la plupart
specifically, des phénomènes
how different processes étudiés en sciences
or groups sociales,
contribute un modèle
to generate linéaire
a social et While
change. bivarié,full
tel causal
que présenté
analysisdans
la to
seeks section
revealprécédente,
the ultimateest simpliste.
causes, Dans cet exemple
decomposition qui modélise
merely reveals le revenu
the proximate selon leorniveau
processes groupsd’instruction,
from which un
the effet
changecurvilinéaire
occurred. semblerait plus plausible. Fort heureusement, la décomposition s’étend aisément à des modèles
non-linéaires. Pour un effet quadratique par exemple,
From what? (Proximate processes). In a basic demographic decomposition, a national outcome is cast as the
Y = α + β1X
weighted average of behaviors observed among different + β2X2
sub-populations (III.4)
in the country. The total change is set
as coming
L’analysefrom two proximate
de décomposition sources,
donnera lesnamely (a) compositional
termes suivants : changes such as changes in population
composition like the relative weight of constitutive sub-populations and (b) behavioral changes such as changes
�� � �� � ��� � ��� � � �� ���� � ��� � �� ����� � ��� � � �� ���� � ����
in group behavior or outcomes.2
�� � �� � ��� � ��� � � �� ����� � ��� � � ��� ���� � ������ � � ��� (III.5)
Decomposition can also apply to outcomes resulting from a chain of events where, say, process #1 leads to
Comme
process #2, exemple, l’on#3
then process peut
…retourner sur the
For instance, l’exemple
amounthypothétique des écarts
of locally-produced foodsalariaux entre
available hommesdepends
in markets et femmes,
on amais en spécifiant
sequence of threecette fois-ci une
processes: localrelation curvilinéaire
food production, pour l’effet
allocation de l’instruction
of harvest between sur le revenu.
personal consumption
and market, and food transportation into markets. Any change in amounts of food available in markets will
III.2.2
reflect a mixRégression
of changes inmultivariée
these three elementary processes.

De manière
By whom? similaire,
In addition la décomposition
to revealing d’une
the processes régression
by which changemultivariée inclura un
occurs, researchers mayouseek
destotermes supplémentaires
estimate the relative
contribution of each group to the total change. How for instance how much do people from various regions,variable.
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première age
Commeordans
categories l’exemple
educational précédent,
categories (for le nombrecontribute
instance) de termestosera simplement
the change. plus long. Pourand
The decomposition prendre le cas simple
its identification
de deux facteurs,
of proximate processesl’on
and pourrait
groups canétendre l’analyse
pave the way fordes niveaux
a deeper de salaires
exploration en considérant
of the cette
ultimate causes fois-ci le niveau
of change.
d’instruction et le nombre d’années d’expérience.
I.2 Decomposition versus other
Y = α +methods
β1X1 + β2X2 (III.6)
Imagine
Dans aceresearcher wishing
cas particulier, to explain
les sources a recent decline
du changement dans in mortality
le niveau desinsalaires
her country. In all
incluraient leslikelihood,
composantes
decomposition
suivantes : will not be the first or only tool considered. The researcher has multiple options including
qualitative analysis, trend analysis, and regression.
�� � �� � �� ���� � ��� � � ��
���� � ��� � � �� ���� � ��� �
���� � ��� � � �� (III.7)
She could use a qualitative approach based on key informants, focus group discussions, or archival data to shed
lightIII.2.3 Régression
on historical multi-niveaux
events occurring during the decline. Did the country make new investments or implement
new projects? Did it make recent scientific discoveries or put new drugs on the market? Was there an outbreak
Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
of unusual events, a rise of effective leaders, or an improvement in the water supply and medical services?
l’individu, son groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
Ourexaminer
researcher might also
comment decide to monitor
la performance recentdépend
des élèves trends in
desmortality, to pinpoint
caractéristiques the exact moment
individuelles when
des élèves the 1)
(niveau
decline
ainsibegan and de
que celles to la
review other
société key 2).
(niveau events precedingpour
Les équations this decline. Shetelwill
estimer un ask about
modèle social processes
s’expriment comme suitthat
:
appear to co-vary with mortality. This approach can complement a qualitative reliance on key informants by
Niveau 1 : Yjk = β0k + β1k*xjk + rjk
systematically testing the hunches from the informants with trend data. However, the conclusions remain
Niveau if2 :all she βdid
subjective 0k=�
was
�� �to
���eye-ball
�� � ���and visually compare various trends without attempting a formal
statistical test.
Β1k=��� � ��� �� � ���

En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
2
We will return to these two sources and can already offer one example here. Imagine a country today that experienced war in 1970. One
performance du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
can well-imagine that the views about war –specifically, whether the country ought to embark on another war today — would likely vary
interaction :
by generation. For instance, people born before the 1970s who have lived through a previous war may be less likely to support another war
than the younger generations. Therefore, the results of a poll about a country’s readiness to go to war in 2000 and again in 2025 might
Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
change simply because the percentage of people born before 1970 would have dwindled as this group ages (a compositional effect). It could
also Pour
be thatétudier
the globallepolitical
changement
environment jk dans 2000
de Ybetween le temps, il suffit
and 2025 de in
changes différencier la formule
ways that changes people’sci-dessus et l’incorporer
attitudes, perhaps dans
making them
less supportive
le deuxièmeof war (a behavioral
terme effect). Overall,
de l’équation [1]. the total change in national attitudes will be a combination of these two effects.

9 29
Understanding social change: a decomposition approach

More formally, our researcher could use formal correlation/regression analysis. Unfortunately, her macro-
correlations would obscure the detail of individual responses. On the other hand, regression using individual
data is more detailed and rigorous but it does not address the right level of analysis for someone interested in
social change.

Micro and macroscopic approaches thus complement each other: one being more rigorous and detailed, and the
other more relevant for national policy design. This complementarity permits two possible kinds of integration
(Figure 1). The first looks at how the ‘macro’ (society) affects the ‘micro’ (individual) and is known as multilevel
regression (Luke 2004). The second conversely looks at how micro behaviors aggregate to shape macro-level
outcomes and change therein; this is the essence of decomposition.

Table 1 summarizes the differences between the four approaches discussed above. In particular, they differ in
their presumed drivers of change. The drivers, in demographic decomposition, are either compositional forces
or behavioral factors. In some qualitative analyses, change can come from a single person or a key event in the
study community.

Figure 1. Four modalities of the integration of micro and macro analysis.

Weak points 1. Not statistically representative; 2. Statistical associations are not established;
3. Chronology not established; 4. Competing explanations are not eliminated; 5. Causal relationships
are not clarified.

Table 1. Comparison of four explanation approaches to social change.

10
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

III.2
I.2.1 Décompositions
Advantages de régression dérivées
and limitations
Unlike causal analysis, decomposition does not establish causation. Returning to our earlier example about
III.2.1 Régression curvilinéaire
mortality, decomposition may reveal which sectors, groups, or causes of death accounted for a rise in overall
Pour labut
mortality, plupart des phénomènes
the results étudiés
will not say en sciences
why these sociales,
declines un modèle
occurred. linéaire et bivarié,
This shortcoming tel because
is severe, que présenté
sounddans
la section
policies précédente,
require est simpliste. Dans
a clear understanding cet exemple
of cause quiHowever,
and effect. modélise le
byrevenu selon
revealing theleproximate
niveau d’instruction,
sources of un
effet curvilinéaire
change, decomposition semblerait
analysis plus
can plausible. Fort heureusement,
usefully guide la décomposition
social intervention s’étend
and targeting. aisément it
Moreover, à des
hasmodèles
the
non-linéaires.
advantage of beingPour un effet
simple, quadratique
flexible, par exemple,
easy to interpret, and compatible with other methods.
Y =apply
Simplicity. Decomposition methods are simple to α + β1X
and+ β2X2 (III.4)
interpret. Their application requires no fancy
statistical analysis,
L’analyse complex calculations,
de décomposition donnera lesortermes
advanced software.
suivants : Much of the work is doable with a spreadsheet
program such as Excel, and the input data are often readily available in reports and online tabulations.3 When
�� � �� � ��� � ��� � � �� ���� � ��� � �� ����� � �� � � �� ���� � ����
the input data is publicly available, the analyses are made easier and the� process transparent because other
�� � �� � ��� � ��
researchers can easily check both the input data and � � ������
� � �� � � ���
� the results. �
���
� � ���
�� � � � ��� (III.5)

CommeThe
Flexibility. exemple, l’on peut retourner
decomposition approach sur l’exemple
is quite hypothétique
flexible des écarts
in its application. salariaux entre
A reasonably hommes
creative et femmes,
analyst can
mais
move en basic
from spécifiant
formscette fois-ci une
to complex relation curvilinéaire
combinations, depending pour
on herl’effet
needs.deThe
l’instruction sur and
presentation le revenu.
illustrations in
this monograph seek to highlight this flexibility.
III.2.2 Régression multivariée
Easy interpretation. The interpretation of decomposition results is intuitive and easy. Compared to the results
De regression
from manière similaire,
analysisla (ordinary
décomposition
least d’une régression
squares, logit, multivariée inclura the
and odds ratios), un ou des termes
statistics supplémentaires
generated by a
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première
decomposition analysis are readily expressed in plain language. The results simply indicate the percentage of variable.
Comme
social dans
change l’exemple
coming from précédent, le nombre
a given process de termes sera simplement plus long. Pour prendre le cas simple
or group.
de deux facteurs, l’on pourrait étendre l’analyse des niveaux de salaires en considérant cette fois-ci le niveau
Compatibility. The decomposition method is compatible with many other methods, including micro-regressions,
d’instruction et le nombre d’années d’expérience.
multilevel analysis, simulations, and qualitative analysis. It can help aggregate the results from micro-
Y = αanalysis.
regressions. It can combine fruitfully with multilevel + β1X1 It+ can
β2X2serve as a prelude to a qualitative analysis.(III.6)
4

For all these reasons, decomposition can apply eclectically to a wide range of fields and methodological
Dans ce cas particulier, les sources du changement dans le niveau des salaires incluraient les composantes
traditions. Importantly, it does not replace or compete with other methods but, rather, it complements them to
suivantes :
improve the quality of the findings.
�� � �� � �� ���� � ��� � � ��
���� � ��� � � �� ���� � ��� �
���� � ��� � � �� (III.7)
Transparency. This is a major strength of decomposition. Contrasting with the opacity of most other methods,
the average reader can easily check the internal consistency of results from decomposition analysis. In a
III.2.3 Régression multi-niveaux
multivariate regression, for example, the reader often has few opportunities to check the accuracy of the
Commecoefficients
regression son nom l’indique, la and
presented régression intègre
s/he must trustdes
theanalyses à deux
researcher.5
In ou plusieurs niveaux
a decomposition qui peuvent
analysis, inclure
by contrast,
s/hel’individu,
can checkson groupe etofsafinal
the accuracy société toute
results entière. Dans
by reviewing l’analyse
the initial de la scolarisation
information par within
and following, exemple,
thel’on
tablepeut
examinerthecomment
of findings, la performance
subsequent des élèves
data transformations dépend
and des caractéristiques
the plausibility individuelles des élèves (niveau 1)
of final results.
ainsi que celles de la société (niveau 2). Les équations pour estimer un tel modèle s’expriment comme suit :

Niveau 1 : Yjk = β0k + β1k*xjk + rjk

Niveau 2 : β0k=��� � ��� �� � ���

3
See, for instance, the Demographic and Health Survey (www.statcompiler.com) or the World Bank (http://data.worldbank.org/data-
Β1k=��� � ��� �� � ���
catalog/world-development-indicators).
4
ForEn intégrant
example, in ourles valeurs
previous du niveau
example 2 dansvis-a-vis
about attitudes l’équation
war, de
the niveau 1, l’on
researcher’s obtient l’équation
decomposition mixtethat,
results may show quicontrary
exprimeto la
expectations, much ofdu
performance the groupe
change inànational
la foisattitudes vis-à-visdes
en fonction warcaractéristiques
reflects a true behavioral change rather
du groupe, than a compositional
du contexte national et change.
de leur
In other words, many people in the country actually changed their attitudes about war. Alternatively, she may find that most of the change
interaction :
came from people between the ages of 45 to 60. Armed with this information, she can now go ahead and conduct more investigations into
whyY
these
=�people
�� changed
� � their
� minds
� �during
� � that
� period.
�� � ��� ��� � ��� (III.8)
jk �� �� �� �� � �� � �� ��
5
The situation is improving, as researchers must increasingly make their data publicly available. However, the reader must have access to
Pour étudier
the computing le changement
program de Yjk dans
used by the researchers and beleable
temps, il suffit
to retrace de différencier
all steps, la formule
from coding to the ci-dessus
final modeling, etrarely
which is l’incorporer dans
convenient.
le deuxième terme de l’équation [1].

11 29
Understanding social change: a decomposition approach

I.2.2 Political relevance and applications

Decomposition methods can apply to many of the social changes underway across the globe. In particular, they
can inform the study of several development goals pursued under the United Nations’ Sustainable Development
agenda (SDG), including poverty, health, inequality, gender, or basic education, for instance. Achieving these
ambitious goals requires an efficient use of countries’ scarce resources, which implies a clear understanding of
the drivers of socioeconomic change. The rapid and uneven demographic changes occurring in many countries
create fast-changing and diverse societies that can no longer be fully understood without careful attention to
disaggregated evidence. Decomposition methods can help.

I.3 The main types of decomposition

Decomposition is not a single method but set of related methods. Its variants appear separately in different fields
and they have not been sufficiently integrated. Below, we review some of these variants and their differences
based on four criteria: (a) the type of independent variable, (b) the type dependent variable, (c) the link function,
and (d) the complexity of the analysis.

Criterion 1: The independent variable type

One can distinguish demographic, regression, and temporal decompositions, depending on whether the
independent variable is nominal, interval or ordinal.

• In a demographic decomposition, the independent variable is nominal, e.g., country region, age group,
ethnicity, or marital status. For instance, a national outcome can be viewed as the population-weighted average
of outcomes across all of the countries’ regions. For example, the national support for a given cause or person
(e.g., the president) will be a weighted average of support across all regions. This support will change if the
relative size of the regions happens to change over time or if the views of people within any of the regions
change.
Note that demographic decomposition also applies to an ordinal variable (e.g., socioeconomic status) if it is
treated nominally, i.e., with no explicit attention to the order of categories.
• In a temporal decomposition, the independent variable is ordinal, and the order between categories is explicitly
considered. For instance, if age group is the independent variable, we keep in mind that people aged 15-19
are younger than people aged 20-29 years, who are themselves younger than people aged 30-39 years.
• In a regression decomposition, finally, the independent variable is quantitative. Examples include a person’s
years of education, the number of siblings, or income in dollars. The starting point in regression decomposition
is to have an estimate of the statistical effect of an independent variable (e.g., years of education) on some
outcome (e.g., health). Health outcomes then are expected to change either because the amount of education
changes or because the average effect of education changes. The decomposition in this case seeks to determine
how much of the total change in health reflects the change in the quantity of education versus the effectiveness
of education.

Criterion 2: The dependent variable

For the simplest basic decomposition, the dependent variable is an average or a percentage. In more complex
(derived) decomposition, the dependent variable may be a more complex measure such as inequality.

12
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

Criterion 3: The functional relationship

III.2 Décompositions de régression dérivées
The main criterion here is the type of relationship linking the independent to the dependent variable. The type
III.2.1 Régression curvilinéaire
and complexity of this functional relationship vary (see Figure 2), and we can specify three types:
Pour la plupart des phénomènes étudiés en sciences sociales, un modèle linéaire et bivarié, tel que présenté dans
• A demographic relationship: Basically, the Y value of the entire country is a weighted (by demographic weight)
la section précédente, est simpliste. Dans cet exemple qui modélise le revenu selon le niveau d’instruction, un
average of prevailing values in the various subpopulations of the country (yi);
effet curvilinéaire semblerait plus plausible. Fort heureusement, la décomposition s’étend aisément à des modèles
• A statistical relationship, specifically, a regression relationship between Y and X; and
non-linéaires. Pour un effet quadratique par exemple,
• A mathematical relationship: in this case, the dependent and independent variables are linked by a simple
Y = α +sum,
mathematical relationship (which involves a quotient, + β2X2 or log). Unlike the statistical relationship,
β1X product, (III.4)
it is a true relationship that does not vary across countries or situations. The only change is in the values of
L’analyse de décomposition donnera les termes suivants :
these variables.
�� � �� � ��� � ��� � � �� ����� � ��� � � ��
���� � ��� � �� ���� � ����
����� � ��� � � ���
�� � �� � ��� � ��� � � �� ���� � ��
���� � � ��� (III.5)

Comme exemple, l’on peut retourner sur l’exemple hypothétique des écarts salariaux entre hommes et femmes,
mais en spécifiant cette fois-ci une relation curvilinéaire pour l’effet de l’instruction sur le revenu.

III.2.2 Régression multivariée

De manière similaire, la décomposition d’une régression multivariée inclura un ou des termes supplémentaires
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première variable.
Comme dans l’exemple précédent, le nombre de termes sera simplement plus long. Pour prendre le cas simple
de deux facteurs, l’on pourrait étendre l’analyse des niveaux de salaires en considérant cette fois-ci le niveau
d’instruction et le nombre d’années d’expérience.

Y = α + β1X1 + β2X2 (III.6)

Dans ce cas particulier, les sources du changement dans le niveau des salaires incluraient les composantes
suivantes : Figure 2. Basic types of decomposition.

�� � �� � �� ���� � ��� � � ��
���� � ��� � � �� ���� � ��� �
���� � ��� � � �� (III.7)
Each of these basic functional relationships can become more complex. In regression analysis, for instance, one
can move from a single independent variable and a linear specification to more elaborate forms that feature
III.2.3 Régression multi-niveaux
several independent variables, non-linear forms, or multi-level relations.
Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
Criterion 4: The
l’individu, degreeetofsacomplexity
son groupe société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
examiner comment la performance des élèves dépend des caractéristiques individuelles des élèves (niveau 1)
This criterion distinguishes simple from advanced or mixed decompositions that combine two or more
ainsi que celles de la société (niveau 2). Les équations pour estimer un tel modèle s’expriment comme suit :
elementary forms. The basic forms shown in Figure 2 are thus the building blocks for generating complex forms.
Niveau as
Therefore, Yjkin= Figure
1 :indicated β0k + β3, anjkelementary
1k*x + rjk decomposition may yield a (derived) decomposition when its
dependent variable or functional relationship becomes more complex. We obtain a mixed decomposition by
Niveau 2 : β0k=��� � ��� �� � ���
combining two or three basic forms. This book’s organization follows the typology shown in Figure 2. We begin
with basic and derived =��� �
Β1kforms ���work
and �� � our
��� way up to mixed forms.6

En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
performance du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
interaction :
6
jk =���
WeYshould � here
note ��� �that
�� � ��� �� �
a number ��� ��specialized
of highly ��� � ���decomposition
� ��� ��� �formulas
��� have been developed to study specific questions such(III.8)
as
life expectancy (Vaupel, 2003), job discrimination (Oaxaca, 1973) and poverty or inequality (Shorrocks, 2013). The monograph does not
Pour étudier le changement de Y dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
show these detailed formulations but readersjkwill easily see where these formulations fit into the taxonomy presented here.
le deuxième terme de l’équation [1].

13 29
Understanding social change: a decomposition approach

I.4 Conventional notations

To facilitate the presentation, we use the following conventional notations for variables, indices, and historical
change:

Variables. Following common practice in the social sciences, dependent and independent variables are indicated
by the letters Y and X, respectively. In addition, we use capital letters when referring to national outcomes, and
reserve lowercase letters when referring to individuals or sub-populations. Y thus designates a dependent,
macro-level variable; y designates a dependent variable at the micro or meso-level; X designates an independent
variable at the macro-level; and x designates an independent variable at the micro or meso-level. Therefore, for
instance, if we study the effects of education on mortality, Y is the national mortality rate, while y is mortality
rate in a subpopulation (e.g., for those between the ages of 15 to 19 years). In the study, X is the national
education level, while x is the level of education in a subpopulation, e.g., for the 15-19 year olds.

Weighing. The letter w measuring the weights of different subgroups is mostly used in demographic
decomposition or its close variants. The weights often reflect population size, i.e., the percentage of the national
population in a given category.

Regression parameters. Regression decomposition includes conventional regression parameters such as:

• α, the intercept, which is the value of Y (or y) when X (or x) is 0;

• β, the marginal increase of Y (or y) when X (or x) increases by one unit; the more complex regression analyses
will integrate the case where β is a vector and take multiple values; and
• e, the error term.

Indices. The presentation will also use the following indices:

• j indexes groups; e.g., yj denotes the value of the dependent variable for the j group, while xj indicate the value
of the independent variable for the same group;
• t denotes time; e.g., Yt indicate the value of the dependent variable for a given year t and for the entire
population (e.g., the average mortality in Senegal as of 1990); and
• a indexes age; thus, Ya is the value of the dependent variable for a given age group e.g., the specific fertility
rate for the age group. Occasionally, we use the term + to show all ages above the referenced age; in fertility
analysis Y39+ might be the average fertility for all ages above 39.

Historical change. Δ indicates the historical change. For example, ΔY represents the historical change in
the outcome being studied, specifically, the difference in the Y values observed at two points in time (e.g., Yt1 -Yt,
t1>t …).

Averages. The annotations will distinguish between two types of averages, whether cross-sectional or historical.
A cross-sectional average is the average in the population at a specific time t. It is calculated over several groups
at one point in time. Since the dependent variable in most basic decompositions is an average, these averages
will simply be denoted Y (or y). A historical average is the average for the same group over two periods. We
signal it by adding a bar over the corresponding letter. Thus, ܻത is an average between two periods for the value
of the dependent variable at the national level. Likewise, ‫ݕ‬ത indicates the average value of two years of the
dependent variable for one subpopulation.
ഥ ൌ ሺܻ௧ଵ ൅ ܻ௧ ሻȀʹ
ܻ

14
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

I.5III.2
Structure of the monograph
Décompositions de régression dérivées
Our presentation progresses from simple to complex decompositions. We began by defining and comparing
III.2.1 Régression curvilinéaire
decomposition with other methods. We then offered a typology of decomposition methods, depending on the
typePour
of data and the
la plupart descomplexity
phénomènesofétudiés
the analysis. Decompositions
en sciences are labeled
sociales, un modèle ‘demographic,’
linéaire et bivarié, tel‘temporal,’ anddans
que présenté
‘regression’
la sectiondepending
précédente,on
estwhether
simpliste.the independent
Dans cet exemplevariable is nominal,
qui modélise ordinal,
le revenu selon leorniveau
quantitative. These un
d’instruction,
elementary types are semblerait
effet curvilinéaire then modifiable to yieldFort
plus plausible. nested decompositions.
heureusement, Once again,s’étend
la décomposition elaborations
aisémentmay reflect
à des modèles
greater complexityPour
non-linéaires. in the
untype
effetofquadratique
dependent variable (e.g., a measure of inequality instead of an average) or the
par exemple,
link function between independent and dependent variables (e.g., a curvilinear or multivariate relationships
Y = α + β1X + β2X2 (III.4)
instead of a simple linear relationship).
L’analyse de décomposition donnera les termes suivants :
Finally, one can combine elementary types to produce advanced decompositions (Figure 3). We present a few
examples of such advanced � �� � ��� � ��but
��decompositions, � � �investigators
�� ����� �use
���� � ��� � ��
can ���their ���
� � �� � � ����
own imagination to generate
additional forms. ����� � ��� � � ���
�� � �� � ��� � ��� � � �� ���� � ��
���� � � ��� (III.5)
To facilitate understanding,
Comme exemple, l’on peuttheretourner
documentsurmixes verbal
l’exemple descriptionsdes
hypothétique with mathematical
écarts formulas,
salariaux entre hommesillustrative
et femmes,
examples,
mais enannotated
spécifiant charts, graphs,
cette fois-ci une and figures.
relation This pedagogical
curvilinéaire approach
pour l’effet hopefully
de l’instruction surmakes the material
le revenu.
accessible to a wide pool of readers with different learning styles.
III.2.2 Régression multivariée
De manière similaire, la décomposition d’une régression multivariée inclura un ou des termes supplémentaires
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première variable.
Comme dans l’exemple précédent, le nombre de termes sera simplement plus long. Pour prendre le cas simple
de deux facteurs, l’on pourrait étendre l’analyse des niveaux de salaires en considérant cette fois-ci le niveau
d’instruction et le nombre d’années d’expérience.

Y = α + β1X1 + β2X2 (III.6)

Dans ce cas particulier, les sources du changement dans le niveau des salaires incluraient les composantes
suivantes :

�� � �� � �� ���� � ��� � � ��
���� � ��� � � �� ���� � ��� �
���� � ��� � � �� (III.7)

III.2.3 Régression multi-niveaux

Figure 3. Decomposition types and combinations.

Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
l’individu, son groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
examiner comment la performance des élèves dépend des caractéristiques individuelles des élèves (niveau 1)
ainsi que celles de la société (niveau 2). Les équations pour estimer un tel modèle s’expriment comme suit :

Niveau 1 : Yjk = β0k + β1k*xjk + rjk

Niveau 2 : β0k=��� � ��� �� � ���

Β1k=��� � ��� �� � ���

En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
performance du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
interaction :

Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)

Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
le deuxième terme de l’équation [1].

15 29
 Chapter II
Demographic decomposition

II.1 Basic demographic decomposition

II.1.1 Problem Type
This first type of decomposition applies to national outcomes (Y) that are an aggregated outcome from several
subpopulations (yi), each weighted by its relative size (wj). Formally,

Y = f (yj, wj) [II.1]

For example, the mortality rate of a country is the weighted average of rates in different regions or socioeconomic
groups. In this formulation, Y, the dependent variable, is quantifiable and X, the independent variable, is
measured nominally. The variable X and its categories must meet at least four criteria:

Exhaustiveness: The full set of categories for the independent variable must be such that each member of the
population belongs to one and only one category j. In other words, the set of categories must cover the entire
population, with categories also being mutually exclusive.

Distribution: The number of categories for the classification variable should be neither too small (> 2) nor too
large. With too few categories, the analyses are not detailed enough to be informative. Conversely, having too
many categories spreads the data too thinly. Thus, variables such as sex (with too few categories) or age (too
many categories if measured in singe years) are not ideal as classification variables.

Variability: The size of the individual categories must fluctuate over time. Otherwise, the compositional effect in
the decomposition analysis will remain zero. For this reason, gender is, once again, a poor classification variable
unless the researcher is dealing with a very dynamic population with rapidly-changing sex ratios.

Relevance: A good classification variable should be theoretically relevant or policy-relevant. Variables such as
“region” for instance usually meet the criterion of policy-relevance, whereas education for instance is
theoretically relevant if the outcome being studied is expected to depend on one’s education level.

17
Understanding social change: a decomposition approach

II.1.2 Visual representation

As stated in the introduction, decomposition methods seek to identify the sources of change, whether substantive
(with processes driving the change) or sociological (with groups or people driving the change).

The figure below illustrates a basic decomposition with a study of gender parity in education. The left-hand side
of the diagram shows five squares capturing the trend in parity for the country as a whole, with darker colors
reflecting greater parity. The chart shows a steady progress at the national level from high educational inequality
between boys and girls (the white square on the far left) to parity (the black square on the far right). Obviously,
the country became more gender-equitable but for several reasons, a researcher may wish to understand how
this evolution occurred, specifically, how the country’s various social classes contributed to it.

On the right side of the diagram (Frame B) are two possible, opposite, scenarios of convergence. The first (B1)
shows a horizontal convergence, with the educational gap closing at the same rate for all groups. Conversely,
Frame B2 shows a vertical convergence, with change starting among the top income group before gradually
trickling down. In year 2, the top income group had already achieved parity, while gender inequality remained
prevalent in lower-income groups. If we ask about the groups that led the change, we get different answers from
the two scenarios: in the first case, all groups evolved simultaneously, while in the second case the change came
from above.

Figure 4. Vertical versus horizontal convergence in education.

II.1.3 Example
While the previous figure shows the groups leading the change, Table 2 below shows the leading processes. In
particular, it shows the difference between compositional and behavioral effects. Imagine a country where the
monthly income (in thousands of FCFA7) is initially 142.5. This income is the weighted average of incomes across
all of the economic classes making up the national population, from the richest to the poorest. The table shows,
for years 1 and 2, respectively, the average income and the size of each class. Thus, in year 1, the poorest category
represented 20% of the population and earned an average of 50,000 FCFA.

7
FCFA is the Cameroonian national currency, 1US$ ≈ 500 FCFA.

18
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

III.22, the
In year Décompositions de régression
table shows two possible scenarios of change. Bothdérivées
scenarios lead to the same aggregate result, a
growth in national income from 142.5 to 152.9. Yet they are qualitatively different. Under scenario 1, the
III.2.1
average Régression
incomes curvilinéaire
of different classes did not change; what changed was the relative size of the different classes,
including the poor’s share of the national population, which fell from 20% to 15%. In contrast, scenario 2 does
Pour la plupart des phénomènes étudiés en sciences sociales, un modèle linéaire et bivarié, tel que présenté dans
not involve a compositional change; what changed instead were the average incomes of some groups. The richest
la section précédente, est simpliste. Dans cet exemple qui modélise le revenu selon le niveau d’instruction, un
economic group (350 on average in year 1) became even richer (400), which fully explains the rise in the
effet curvilinéaire semblerait plus plausible. Fort heureusement, la décomposition s’étend aisément à des modèles
country’s average income.
non-linéaires. Pour un effet quadratique par exemple,
The two scenarios in Table 2 show extreme, textbook, illustrations (100% composition versus 100% behavior).
Y = α + β1X + β2X2 (III.4)
In practice, compositional and behavioral change often occur simultaneously. One may thus end up with a
L’analyse
situation de décomposition
where donnera30%
composition explains les termes
of the suivants :
change while behavioral change accounts for the rest. The
value of a demographic decomposition
�� � �� �is���precisely
� �� � �to��
quantify
��� � ��� these
� �������relative
� �� � contributions.
��� � ����
� ��
� � � �

�� � �� � ��� � ��� � � �����

�� ���� � ��
� ��� � � ��� ���� � � ��� (III.5)

Comme exemple, l’on peut retourner sur l’exemple hypothétique des écarts salariaux entre hommes et femmes,
mais en spécifiant cette fois-ci une relation curvilinéaire pour l’effet de l’instruction sur le revenu.

III.2.2 Régression multivariée

De manière similaire, la décomposition d’une régression multivariée inclura un ou des termes supplémentaires
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première variable.
Comme dans l’exemple précédent, le nombre de termes sera simplement plus long. Pour prendre le cas simple
de deux facteurs, l’on pourrait étendre l’analyse des niveaux de salaires en considérant cette fois-ci le niveau
d’instruction et le nombre d’années d’expérience.

Y = α + β1X1 + β2X2 (III.6)

Dans ce cas particulier, les sources du changement dans le niveau des salaires incluraient les composantes
suivantes :

�� � �� � �� ���� � ��� � � ��
���� � ��� � � �� ���� � ��� �
���� � ��� � � �� (III.7)

III.2.3 Régression multi-niveaux

Table 2. Compositional vs behavioral change: An extreme case.

Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
II.1.4 Mathematical formulation
l’individu, son groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
In this first case,
examiner we focus
comment on a nationaldes
la performance average;
élèvesYdépend
is expressed as a weighted average
des caractéristiques (by wdes
individuelles j) ofélèves
the values of 1)
(niveau
individual
ainsi quesubpopulations (yj). (niveau 2). Les équations pour estimer un tel modèle s’expriment comme suit :
celles de la société

Niveau 1 : Yjk = β0k + β1k*xjk + rjk �� � ∑ ��� � ���   [II.2] 

In this formula,
Niveau 2 : a national change
β0k=��� � ���can
�� �be���
broken down into two components:

���� � ∑ �� � ��� � ∑ �
Β1k=��� � ��� �� � �� � � � ��� [II.3]

En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
performance du groupe à la fois en fonction des caractéristiques
Compositional du groupe, du contexte national et de leur
Behavioral
interaction : effect effect

Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)

Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
le deuxième terme de l’équation [1].

19 29
Understanding social change: a decomposition approach

This basic decomposition identifies two sources of change. The first, the compositional effect, measures the
change in the relative representation of the various subpopulations. This change in composition affects the
national average through a mechanical change in the weights and the importance of different subpopulations.

The second source of change, behavior, is less mechanical. It shows a real change of mortality within one or
more groups. If the mortality of a group increases, other things being equal, the national mortality will increase.
What changes here is not the relative size of groups, but rather the mortality levels within some or all
subpopulations.

II.1.5 Application
In practice, one implements a decomposition analysis in four main steps: defining the problem; calculating
national averages and change therein; decomposing the total change; and presenting/ discussing the findings.

Defining the problem: Here, the researcher specifies the nature of the substantive (dependent) variable, the
classification (independent) variable, and the period. In our example below, the substantive variable is child
mortality, the classification variable is socioeconomic status, and the study period is 1991-2001.

Calculating the national averages: The national averages are calculated using formula #1, the year-specific
information about group size (wj), and the value of the dependent variable (yj). These data should be available
for first and the last year of the period at least. The statistical procedures to generate these results are a simple
frequency analysis (for wj) and an equally simple comparison of means (for yj). These basic data can come
directly from published reports or automatic compilations available online (e.g., www.Statcompiler.com).

Doing the actual decomposition: Using the available annual information, one can simply apply the formulation
in equation 3. Given the repetitive nature of the calculations, we recommend the use of spreadsheet software
like Excel but other software or personal programs can be used. Table 3 summarizes the basic data for the
calculations.

Table 3. Decomposing a change in child mortality, Cameroon 1991-2011.

Again, the first four columns include the basic information needed for the analysis, including the child mortality
rate for each of the social classes and the percentage of children in these classes for the years 1991 and 2011.
Using these data, one can easily calculate the national average for each year and the change over the study
period, from 143.6 in 1991 to 117.7 in 2011. The difference here is -25.9.

20
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

TheIII.2 Décompositions
final step demortality.
is to explain this decline in régression
We do so bydérivées
applying formula #3. The results in this case
show that 17% of the decline stemmed from compositional change. Note, for example, how the percentage of
III.2.1
children Régression
in poorer familiescurvilinéaire
dropped by 18 to 15%, while the percentage of children in the richest families rose
from 19 to 21%. The remainder of the change (83%) reflects a behavioral effect. Notice in particular how
Pour la plupart des phénomènes étudiés en sciences sociales, un modèle linéaire et bivarié, tel que présenté dans
mortality rates declined across all social classes, including the poorest.
la section précédente, est simpliste. Dans cet exemple qui modélise le revenu selon le niveau d’instruction, un
effet curvilinéaire semblerait plus plausible. Fort heureusement, la décomposition s’étend aisément à des modèles
Presenting the findings
non-linéaires. Pour un effet quadratique par exemple,
For a scientific audience, the presentation can simply use a table like the one in Section II.2, with the discussion
Y = α + β1X + β2X2 (III.4)
focusing on both the processes and groups driving the change.
L’analyse de décomposition donnera les termes suivants :
Leading processes: The percentages at the bottom of the table indicate the contributions of the two competing
processes (composition and�� behavior)
� �� � ��to� the
� ��total ���� � ���
change
� � � ��
����� � ��mortality.
in�national
�� ��� ����
� � � ��� �In this case, these relative
contributions are 17% for composition
�� � ��and 83% for�behavior.
� � ��
� �� ����� ��� ��� (III.5)
� � �� � ��� � � ���� � ��� � � ���

Leading
Commegroups. In addition
exemple, to clarifying
l’on peut retourner the
surcontributions of various processes,
l’exemple hypothétique des écarts asalariaux
decomposition analysis et
entre hommes reveals
femmes,
which
maisgroups drove the
en spécifiant change.
cette fois-ciIn
uneourrelation
case, the contribution
curvilinéaire of the
pour lower
l’effet income group
de l’instruction surisle-9.69, 37% of the
revenu.
total change. This total contribution reflected effects stemming from changes in group size (-6.92) and behavior
III.2.2
(-2.77). TheRégression multivariée
other income groups contributed 41%, 3%, 16%, and 3%, respectively.

In aDe
decomposition analysis,
manière similaire, the sum of these
la décomposition contributions
d’une régression ismultivariée
always 100%. However,
inclura un ou individual
des termes contributions
supplémentaires
canqui,
be negative (lessseront
cette fois-ci than 0%)
des or greaterindépendantes
variables than 100%. A negative percentage
supplémentaires, indicates
plutôt que leacarré
contribution that goes
de la première in
variable.
the Comme
oppositedans
direction of the précédent,
l’exemple general change. For example,
le nombre a group
de termes making a negative
sera simplement contribution
plus long. to thelenational
Pour prendre cas simple
decline in mortality
de deux facteurs, means that this
l’on pourrait group’s
étendre effect des
l’analyse worked to increase
niveaux mortality,
de salaires i.e., it worked
en considérant against
cette fois-ci the
le niveau
prevailing trend. et
d’instruction In le
contrast,
nombrea d’années
percentage greater than 100% indicates that group or process accounted for more
d’expérience.
than the total change observed at the national level. The national change would have been even greater if it had
Y =offset
depended only on this group, and if it had not been α + β1X1 + β2X2
by the opposite influences of other groups. (III.6)

For Dans ce cas particulier,

a non-scientific lesit sources
audience, might beduuseful
changement dans
to rely on le niveau
graphs ratherdes
thansalaires
a table.incluraient lesone
For instance, composantes
might
suivantes :
use pie charts or 100% (stacked area) histograms. Such diagrams offer nice summaries that clearly identify the
dominant drivers of change.�� � �� � �� ���� � ��� � � ��
���� � ��� � � �� ���� � ��� �
���� � ��� � � �� (III.7)

Policy implications
III.2.3 Régression multi-niveaux
TheComme
policiesson
to recommend will la
nom l’indique, depend on whether
régression intègreades
social change
analyses à reflects
deux ouaplusieurs
compositional or behavioral
niveaux effect.
qui peuvent inclure
If mortality
l’individu, son groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’onofpeut
is driven by a compositional effect, the appropriate response is to target the vulnerable segments
the examiner
population. If, on thelaother
comment hand, thedes
performance change
élèvesreflects
dépend behavioral effects, targeting
des caractéristiques becomesdes
individuelles lessélèves
appropriate,
(niveau 1)
andainsi
one would consider
que celles de la broader-based
société (niveau interventions to all
2). Les équations families
pour andun
estimer their children.
tel modèle Later, we will
s’expriment see how
comme suit to
:
refine policy recommendations with more detailed decompositions.
Niveau 1 : Yjk = β0k + β1k*xjk + rjk

II.1.6 Application
Niveau 2: β0kto the
=��� � �demographic
�� �� � ��� dividend
Β1k=��� � ���can
Basic demographic decomposition �� �apply
��� to the study of demographic dividend. Indeed, it is the logic behind
the En
National Transfer
intégrant Accounts
les valeurs (NTA) methodology
du niveau oftende
2 dans l’équation used in this
niveau 1, field (Masonl’équation
l’on obtient and Lee 2005).
mixte At
quithe heart la
exprime
of this method isdu
performance thegroupe
simpleàobservation that income,
la fois en fonction consumption, and
des caractéristiques savings vary
du groupe, with agenational
du contexte (Figureet5).deInleur
particular, the :income/consumption balance tends to be negative among younger and older populations but
interaction
positive among the middle-aged adult population. Intuitively, therefore, the greater the share of adults in the
Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
national population, the higher the national income.
Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
le deuxième terme de l’équation [1].

21 29
Understanding social change: a decomposition approach

The parallel between NTA and demographic decomposition is obvious. One only needs to note here that ‘age
group’ is the classification variable and income-consumption is the substantive variable (see Table 4 below).
Thus, the composition effect is the demographic dividend, or at least the mechanical component of demographic
dividend. Indeed, the application of decomposition could enhance the standard NTA analysis by (a) considering
cases where consumption/income profiles do change with age, rather than assuming that they remain fixed; (b)
quantifying the dividend both in absolute terms and also in relative terms by comparing it to changes induced
by historical evolution in income and consumption; and (c) exploring the contribution of different age groups
to the dividend.

Figure 5. Income and consumption profiles by age. Hypothetical data.

Table 4. Estimating the demographic dividend with a demographic decomposition approach.

II.2 Derived demographic decompositions

II.2.1 Decomposing a difference
Let us return to our case study on mortality. Imagine a researcher who is not interested in historical change but,
rather, in the mortality difference between two countries or provinces. Fortunately, the same approach works,
as long as the research has data on the size and mortality rate of each social class within each of the two
provinces. The calculations proceed in the same way as previously described.

22
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

III.2
II.2.2 Décompositions
Decomposing inequality de régression dérivées
In our mortality case study, the dependent variable was an average—specifically, the average level of child
III.2.1 Régression curvilinéaire
mortality in the country. There may be cases where a researcher is interested not in the average but in the
Pour lainequality
mortality plupart des phénomènes
across regions,étudiés en sciences
economic classes, sociales,
education unlevels,
modèle orlinéaire et bivarié,
racial groups, fortelinstance.
que présenté
At thedans
la section
global level, a précédente, est simpliste.
researcher may Dans cet
seek to account for exemple qui modélise
the historical le global
change in revenuincome
selon le niveau d’instruction,
inequality, and she can un
use effet
basiccurvilinéaire
demographic semblerait plus plausible.
decomposition FortInheureusement,
to do so. Firebaugh and la décomposition
Goesling (2014), s’étend aisément àformulate
for instance, des modèles
non-linéaires.
economic Pour
inequality as un effet quadratique
a function par exemple,
of differences in national populations (p) and per capita income (i). Taking
the mean log deviation (MLD) as a measure of inequality, they+express
Y = α + β1X β2X2 the global level of inequality as follows:(III.4)
��� � ���: 1���� � 
�� � ∑suivants
L’analyse de décomposition donnera les termes [II.4] 

In this context, the historical��

change
� �� in
� global inequality
��� � �� ��� is: ����� ���
� � � ��� � ��� � �� � ��� � � ��� � ����

����
�� � ���
��∑��̅ � �����
� �����
�� ��� ������∑��
� � �� �� ������
�� � ��
� ����
��� � ���� � � ���
����� � ���
��������� (III.5)
[II.5] 

Comme exemple, l’on peut retourner sur l’exemple hypothétique des écarts salariaux entre hommes et femmes,
mais en spécifiant cette fois-ci une relation curvilinéaire pour l’effet de l’instruction sur le revenu.
Compositional Behavioral
effect effect
III.2.2 Régression multivariée
De manière similaire, la décomposition d’une régression multivariée inclura un ou des termes supplémentaires
We use this approach to analyze the historical change in income inequality between African countries from
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première variable.
1985 to 2005 (Table 5). Between these two years, economic inequality, as measured by the Mean Log Deviation
Comme dans l’exemple précédent, le nombre de termes sera simplement plus long. Pour prendre le cas simple
in GDP per capita, increased from 0.391 to 0.425, or by about 9%. We apply a decomposition analysis to identify
de deux facteurs, l’on pourrait étendre l’analyse des niveaux de salaires en considérant cette fois-ci le niveau
the sources of this economic divergence, looking at the respective contributions of compositional and behavioral
d’instruction et le nombre d’années d’expérience.
forces. The analysis can also show individual country contributions, highlighting the countries that made the
greatest contributions to increasing inequality andY=α + β1X1
those that+instead
β2X2 worked to reduce it. The results of this (III.6)
analysis
Dansare
ce in
casTable 5.
particulier, les sources du changement dans le niveau des salaires incluraient les composantes
Thesuivantes
first four: columns in the table show national data on GDP per capita and total population for 1985 and
2005, respectively. These nominal
�� � �� data ����are
� �� transformed
��� into relative
��� values: raw
��� population numbers, for
(III.7)
� �� � � � ��� � ��� � � ��� � ��� � � ��� � ��� �
instance, are turned into the share of the total African population living in a country as obtained by dividing the
population
III.2.3 of each country's
Régression by the total population of Africa during the indexed of year. Similarly, GDP is
multi-niveaux
converted into relative GDP by dividing the country’s GDP by the weighted average of the African GDP during
thatComme son nomwhere
year. Countries l’indique, la régression
the relative intègre
GDP value des analyses
is greater than 1àare
deux ou plusieurs
richer niveaux
than average; theyqui
arepeuvent inclure
poorer than
l’individu,
average if that son
valuegroupe
is below et 1.saUsing
société toute
these entière.
relative Dans
values, onel’analyse de la inequality,
can calculate scolarisation
andpar
the exemple, l’on peut
values obtained
hereexaminer
are 0.39 comment
and 0.43, la
i.e.,performance des élèves
a nominal increase dépend0.04.
of about des This
caractéristiques individuelles
nominal change des élèves (niveau
is then decomposed into a 1)
ainsi que celles
compositional effectde(-18%)
la société
and(niveau 2). Leseffect
a behavioral équations pour estimer un tel modèle s’expriment comme suit :
(118%).

TheNiveau 1:
compositional Yjk = suggests
effect β0k + β1k*x jk +during
that, rjk the investigated period, African populations grew at different
rates, and this
Niveau 2 : altered βthe relative
0k=� sizes of individual countries. The pattern of these changes contributed to reduce
�� � ��� �� � ���
inequality, suggesting that, perhaps, the countries at economic extremes (the very rich or the very poor) grew
Β1k=��� � ��� �� � ���
more slowly than others. At any rate, this differential population growth helped reduce economic inequality.
En intégrant
However, les valeurs
this influence wasduovershadowed
niveau 2 dansbyl’équation de niveau
the changes 1, l’oncircumstances,
in economic obtient l’équation
withmixte qui exprime
countries’ GDPs la
performance
becoming du groupe
even more unequalà in
la 2005
fois enthan
fonction des caractéristiques
they were in 1985. At thisdu groupe,
stage, du possible
it is not contextetonational
say whyetthese
de leur
interaction
differences in :economic performance or population growth occurred. As we will see later, more detailed
decompositions can
Yjk =��� � ��� ���shed
� ���light
�� �on���
these questions.
�� ��� � ��� � For
��� now,
��� �this
��� first analysis is a good start. (III.8)
It is also useful to examine the individual contributions of each country to the change in inequality as shown in
Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
the le
last column of
deuxième Table
terme de5.l’équation [1].

23 29
Understanding social change: a decomposition approach

Table 5. Decomposition of change in economic inequality between African countries (1985-2005).

II.2.3 Ordinal decomposition

The ordinal decomposition is very similar to the demographic decomposition, except that the classification
variable is ordinal. We illustrate this variant with a study of total fertility rate (TFR). TFR values, roughly the
average number of births per woman in a country, are the sum of all of the fertility rates found in all of the

24
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

III.2age
relevant Décompositions
groups—in this case, the de
ages régression
between 15 and 49dérivées
during which women are assumed to be of
reproductive age:
III.2.1 Régression curvilinéaire Yt = Σ yat (7)
Pour la
Suppose plupart des
a country’s TFRphénomènes
were to fall étudiés en sciences
over time. sociales,
A researcher mayun modèle
wish linéaire etwhy
to understand bivarié,
the tel que présenté
decline occurred,dans
la section
specifically précédente,
whether est simpliste.
the decline occurred Dans cet all
among exemple
groupsqui modélise effect)
(a quantum le revenu selon lewomen
or whether niveau mostly
d’instruction,
altered un
the effet
ages curvilinéaire
at which theysemblerait plus plausible.
bear children, Fort heureusement,
perhaps postponing la décomposition
childbearing s’étend aisément à des modèles
(a tempo effect).
non-linéaires. Pour un effet quadratique par exemple,
Using temporal decomposition, the researcher can express age-specific fertility rates relative to the fertility
observed in the oldest age group . For example, theYfertility
= α + β1X + β2X2
among (III.4)
the 15-19 year olds will be expressed in terms
of the averagedefertility
L’analyse for all women
décomposition aged
donnera les20 ( y15-19 =: r15-19* y20-49);
to 49suivants
termes

�� � �� � ��� �ya��
/ y�a+� �
= r��→
a��� ya = ra�* ��
� � ���
y���� ���� � ����
a+ � � ��� � � ��

The TFR can be re-expressed as�� ����� � ��� � � ���

� �� � ��� � ��� � � ��
follows: ���� � ��
���� � � ��� (III.5)

Comme exemple, l’on peut retourner sur l’exemple at * ya+t

Yt = Σ rhypothétique des écarts salariaux entre hommes et (II.6)
femmes,
mais en spécifiant cette fois-ci une relation curvilinéaire pour l’effet de l’instruction sur le revenu.
This new formulation helps decompose the change in the TFR into two terms that reflect the quantum and tempo,
respectively.
III.2.2 Régression multivariée
�� � � ∑ ��� � ��� � � � ∑ �� � ���� (II.7)
De manière similaire, la décomposition d’une régression multivariée inclura un ou des termes supplémentaires
Illustration
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première variable.
Comme dans l’exemple précédent, le nombre de termes sera simplement plus long. Pour prendre le cas simple
A paper by Ouedraogo (2012) (data not shown here) looked at patterns of change in age-specific fertility curves
de deux facteurs, l’on pourrait étendre l’analyse des niveaux de salaires en considérant cette fois-ci le niveau
in Cameroon from 1991, 1998, and 2004. An examination of these curves shows some parallelism. If the change
d’instruction et le nombre d’années d’expérience.
occurs at the same rate across all age groups, the fertility curves should be perfectly parallel. Otherwise, it
Y = α +between
becomes difficult to succinctly describe the differences β1X1 + β2X2
the various curves. For a researcher wishing (III.6)
to
assess
Danshow
ce much the decline
cas particulier, lesinsources
births du
occurs disproportionately
changement among
dans le niveau desyounger
salairesage groups, the
incluraient temporal
les composantes
decomposition
suivantes : is a good tool.

Calculations here show a decline

�� � ��driven
� �� by�a��
��� quantum ���effect
� � �� � �� (90%):���the
� � �� decline
� �� affected
��� � �� all
� � �� � age groups, but, (III.7)
as
� � � � � � � �
the chart shows, it was larger among younger women, suggesting some postponement of births (10%). Note that
thisIII.2.3
analysisRégression
could be repeated using SES, for example, as a classification variable to study the dispersion of
multi-niveaux
reproduction across social classes.
Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
l’individu,
II.2.4 Nestedsondecomposition
groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
examiner comment la performance des élèves dépend des caractéristiques individuelles des élèves (niveau 1)
Thisainsi
extension offers
que celles demore detail(niveau
la société by expressing demographic
2). Les équations pourchange
estimer(the composition
un tel component)
modèle s’expriment in terms
comme suit of
:
its constitutive elementary processes such as fertility, mortality, and migration. To apply this decomposition, one
Niveau 1 : Y = β + β *xjk + rjk
obviously needs much jkmore0kdata 1k than for a simple decomposition.
Niveau 2 : β0k=��� � ��� �� � ���
In addition, one can also envision an extension based on nesting categories of two classification variables. For
instance, the researcher might
Β1k=��� �be
���interested
�� � ��� in the intersection of age and SES and would group people in terms
of these two variables. Thus, instead of distinguishing only between the poor, middle class, and rich, one would
En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
further distinguish between the poor who are young, middle-aged, or old. Thus, if each of the two classification
performance du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
variables had three categories, the researcher would end up with a categorization of her population including
interaction :
six categories.
Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)

Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
le deuxième terme de l’équation [1].

25 29
 Chapter III
Chapter III
Regression decomposition
Regression decomposition

III.1 Simple regression decomposition

III.1 Simple regression decomposition
III.1.1 Problem type
III.1.1 Problem type
The main difference between demographic decomposition and simple regression decomposition is in the kind of
independent
The variablebetween
main difference involved. It is nominaldecomposition
demographic in a demographic decomposition
and simple regressionbutdecomposition
quantitative here. Moreover,
is in the kind of
the independent variable is linked to the dependent variable through a statistical relationship (derived
independent variable involved. It is nominal in a demographic decomposition but quantitative here. Moreover, from
regression
the analysis)
independent ratheristhan
variable an to
linked arithmetic one. The
the dependent equation
variable has a aregression
through statisticalcoefficient (the
relationship effect offrom
(derived the
independent
regression variablerather
analysis) on Y) than
and an
anintercept.
arithmeticThe error
one. Theterm is omitted
equation has ahere for now.
regression coefficient (the effect of the
independent variable on Y) and an intercept. The error term is omitted here for now.
III.1.2 Formulation
III.1.2 Formulation
The generic form is Y = f (α, β, X) but a more explicit formulation is as follows:
The generic form is Y = f (α, β, X) but a more explicit formulation is as follows:
yt =αt + βtXt [III.1]
yt =αt + βtXt [III.1]
In this case, the decomposition seeks to explain the change in the dependent variable based on the change in the
various
In parameters
this case, of the regression
the decomposition equation.
seeks to explain This changeinisthe
the change expressed as follows:
dependent variable based on the change in the
various parameters of the regression equation. This change is expressed as follows:
�� � � ������ � ������ �̅ ���������� � ��������� �� �� [III.2]
�� � � ������ � ������ �̅ ���������� � ��������� �� �� [III.2]
Change in Change in effect Change in effect
baselinein
Change magnitude
Change in effect of in
Change X effect
baseline of X level
magnitude of X
of X level
Once again, the same procedure can apply to both cross-sectional analysis (the difference between two groups
in
Oncea given year)
again, the and
samelongitudinal analysis
procedure can apply(the change
to both experienced analysis
cross-sectional by one group between years).
(the difference Thetwo
between approach
groups
is
in the same;year)
a given onlyand
the longitudinal
interpretations differ.(the change experienced by one group between years). The approach
analysis
is the same; only the interpretations differ.

27
27
Understanding social change: a decomposition approach
Understanding social change: a decomposition approach

Illustration
Illustration
A classic realm of application for regression decomposition is the study of income/wage differentials (say,
between
A classicmen
realmandofwomen) and the
application for extent to which
regression they reflectisdiscrimination
decomposition the study of in the labor market.
income/wage Differences
differentials (say,
could stem from discrimination (different returns to education for men and women) but it is also possible
between men and women) and the extent to which they reflect discrimination in the labor market. Differences that
men and
could women
stem from enter the workforce
discrimination with different
(different returns toeducation
educationlevels. Oneand
for men must therefore
women) butestimate
it is alsohow the wage
possible that
differentials
men and womenreflect discrimination
enter rather
the workforce withthan differences
different in human
education levels. capital.
One must These two possibilities
therefore estimate howare the
explored
wage
below. The formal
differentials reflectanalysis (III.3) consists
discrimination rather of writing
than the earning
differences equations
in human forThese
capital. malestwo
(h)possibilities
and femalesare
(f) and then
explored
taking the
below. Thedifference between
formal analysis these
(III.3) two equations.
consists of writing the earning equations for males (h) and females (f) and then
taking the difference between these two equations.
�� � � �� � � � �� � ��
��� � � ��� ��� �
������ � ���
�� � � �� �� � � �� � ��
____________________
____________________
�� � ��� � � �̅��� � � ���� [III.3]
�� � ��� � � �̅��� � � ���� [III.3]
III.1.3 Application
III.1.3 Application
Table 6 illustrates with a numerical example showing earnings by level of education. In this case, the average
return
Table 6on a year with
illustrates of education is 4,000
a numerical FCFA
example for men
showing and 5,000
earnings FCFA
by level for women.In The
of education. this base
case, salaries are
the average
20,000 on
return anda25,000
year offoreducation
men and is
women,
4,000 respectively.
FCFA for men With the5,000
and average education
FCFA levels being
for women. 10 and
The base 12 years
salaries are
for menand
20,000 and25,000
women,forrespectively, the average
men and women, salary ofWith
respectively. a man
theisaverage
60,000,education
while the levels
average woman
being would
10 and earn
12 years
85,000
for men FCFA, yielding
and women, an averagethe
respectively, salary difference
average salary ofofa 25,000 betweenwhile
man is 60,000, women and men.
the average A decomposition
woman would earn
analysis FCFA,
85,000 will show that this
yielding pay gap reflects
an average a complex of
salary difference mix of forces:
25,000 between women and men. A decomposition
analysis will show that this pay gap reflects a complex mix of forces:
• 20% of the gap comes from the gender difference in base salaries (Δα),
• 36% levelssalaries
20% of the gap comes from the gender difference in base of education,
(Δα), and
44% of the gap comes from the gender difference in levels
• 36% returnsoftoeducation,
education.and
• 44% of the gap comes from the gender difference in returns to education.
In other words, even if one achieved parity in education, only 36% of the current wage inequality between men
andother
In women would
words, evendisappear. The rest
if one achieved of the
parity inequalityonly
in education, that 36%
comes from
of the discriminatory
current inequalities
wage inequality in basic
between men
salary
and and returns
women wouldtodisappear.
schooling The
would
restremain.
of the inequality that comes from discriminatory inequalities in basic
salary and returns to schooling would remain.

Table 6. Regression decomposition regression for analyzing wage differentials between men and women.
Table 6. Regression decomposition regression for analyzing wage differentials between men and women.
28
28
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique
Understanding social change: a decomposition approach

III.2
III.2 Décompositions
Other de régression dérivées
regression decompositions
III.2 Other regression decompositions
III.2.1
III.2.1 Régressionregression
Curvilinear curvilinéaire
III.2.1
Pour Curvilinear regressionétudiés en sciences sociales, un modèle linéaire et bivarié, tel que présenté dans
la plupart des phénomènes
Most processes of interest to social scientists do not fit the linear bivariate pattern posited in the previous section.
la section précédente,
A curvilinear, est simpliste. Dans cet exemple qui modélise
realistic.lepattern
revenu selon le niveaudecomposition
d’instruction, un
Most processesrather thanto
of interest asocial
linear, specification
scientists do notisfitoften more
the linear bivariate Fortunately, regression
posited in the previous section.
effet curvilinéaire
analysis easily extends semblerait
to these plus plausible.
situations. Fort heureusement,
For instance, earnings la
bedécomposition s’étend aisément à des modèles
A curvilinear, rather than a linear, specification is often more can written
realistic. as a quadratic
Fortunately, (rather
regression than linear)
decomposition
non-linéaires.
functioneasily Pour
of education: un effet quadratique par exemple,
analysis extends to these situations. For instance, earnings can be written as a quadratic (rather than linear)
function of education:
Y =Yα=+αβ+1Xβ1X
+ β2+Xβ2X2
2 (III.4)
(III.4)
Y = α + β1X + :β2X2 (III.4)
TheL’analyse de décomposition
decomposition analysis is asdonnera
follows:les termes suivants
The decomposition analysis is as�follows: � � ��� � ��� ���� � ��� ���� ����� � ��� � ��� ���� � ����
�� ��� �� ��
� ��� �����
� � �� ����
� � � ��� � ��� � �� � ��� � � ��� � ����
� ��
� �� �� ����� ��� ���� ����� � ����� ����� ����� ��������
��� (III.5)
���� � ��
� �� ��
� ��
� �� �����
��� �� ����
� �� �
� ��
� ����
� �� � ��
����� ����
�
�����
� � ����
� ��
�����
� � �������� ��� (III.5)
Comme exemple, one
l’oncan
peut �� � ���sur
retourner l’exemple�����hypothétique ��� � écarts
���� � �salariaux et(III.5)
�� � differences�des entre hommes femmes,
�� � � �� �� � ��� � � ��� �� ���
To give an example, return to our study of wage between men and women, only this time,
we mais en
set the spécifiant cette fois-ci une relation curvilinéaire pour l’effet de l’instruction sur le revenu.
To give aneffects of education
example, on income
one can return to ourtostudy
be curvilinear.
of wage differences between men and women, only this time,
we set the effects of education on income to be curvilinear.
III.2.2
III.2.2 Régressionregression
Multivariate multivariée
III.2.2 Multivariate
De manière similaire, regression
la décomposition d’une régression multivariée inclura un ou des termes supplémentaires
In other cases, the researcher wishes to estimate the contributions of multiple independent variables. This is
qui, cette fois-ci
feasible. seront deswill
variables indépendantes supplémentaires, plutôt quethele carré de la première variable.
In other The decomposition
cases, the researcher simply
wishes to include
estimatemore than a single of
the contributions factor (and
multiple additional
independent factors will
variables. This be
is
Comme
other dans l’exemple
independent variable précédent,
rather thanle the
nombre
squaredeterm
termes
of sera
the simplement
main plus long.
independent Pour prendre
variable). In the le cas simple
simplest of
feasible. The decomposition will simply include more than a single factor (and the additional factors will be
de situations,
these deux facteurs, l’on pourrait
thevariable
researcher étendre
hasthan
twothe l’analyserather
variables, des niveauxade salaires
one.enFor
considérant cettemight
fois-ciwish
le niveau
other independent rather square term ofthan single
the main independent instance,
variable).she
In the simplest to
of
d’instruction
examine et
how income le nombre d’années
is affected has
by both d’expérience.
these situations, the researcher two the level of rather
variables, education
thanand the number
a single of years
one. For of experience:
instance, she might wish to
examine how income is affected by both the levelYof= α
education+and the number of years of experience:
Y = α + β+1β1X1
X1 +β2X2β2X2 (III.6)
(III.6)
Y = α + β1X1dans+β2Xle2 niveau des salaires incluraient les composantes
(III.6)
TheDans ce cas particulier,
decomposition then writeslesassources
follows:du changement
Thesuivantes :
decomposition then writes as follows:
���� � ��� � � ������ � ��� � � �� ���� � ��� �
���� � ��� � � ��
�� � �� � �� (III.7)
�� � ���� ���
� ��� � �� � ���
� ���
�� � �� � � ���
��� � ��� � ���
� ���� � ��� �
�� (III.7)
��� ���
�� � �� � � � ��� � � ��� � ��� � � ��� � ��� � � ��� � ��� �
� � � (III.7)
III.2.3 Multilevel regression
III.2.3
III.2.3 Régression
Multilevel multi-niveaux
regression
This type of regression involves factors at two or more levels, e.g., individual and community levels. Research on
Commemight
schooling son nom l’indique,examine
for instance la régressionthe intègre des analyses à deuxofoustudents
plusieurs niveaux qui peuvent inclure
This type of regression involves factors how at two oracademic performance
more levels, e.g., individual depends
and community on their
levels. individual
Research on
l’individu,
characteristics
schooling son
might(levelgroupe et
1) but also
for instance sa société toute
the characteristics
examine entière. Dans
of the schools
how the academic l’analyse de
(level 2).
performance la scolarisation
The equations
of students depends par exemple,
to estimate
on their this modelpeut
l’on
individual
are examiner
as follows:comment
characteristics
la performance des élèves dépend des caractéristiques individuelles des élèves (niveau 1)
(level 1) but also the characteristics of the schools (level 2). The equations to estimate this model
ainsi que
are as follows: celles de la société (niveau 2). Les équations pour estimer un tel modèle s’expriment comme suit :
Level 1: Yjk = β0k + β1k*xjk + rjk
Niveau 1 : Y = β + β *x + r
Level 1: Yjk = jkβ0k +0kβ1k*x1kjk + jkrjk jk
Level 2: β0k =��� � ��� �� � ���
Niveau 2 : β =� � � � � �
Level 2: β =�0k ������ ��������� ��
Β0k1k=����� � ��� �� � ���
 1k=��� � ��� �� � ���
Β1k=��� � ��� �� � ���
An integration of the values from level 2 into the level 1 equation yields a mixed equation that expresses
En
individualintégrant les
performance valeurs duon
based niveau
thelevel2 dans l’équation
individual de niveau 1, l’onofobtient l’équation mixte qui exprime
and la
An integration of the values from 2 into characteristics and those
the level 1 equation yieldsthe group
a mixed at a particular
equation time
that expresses
performance
their interactions: du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
individual performance based on the individual characteristics and those of the group at a particular time and
theirinteraction :
interactions:
Yjk=��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
Y =� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
To investigate the change ofjkYjk �� over time, one simply needs to differentiate the above formula and incorporate
Pour
it into
To theétudier
second
investigate theleterm
changement
change of the de
of Yequation
over dans
[1]. le
Yjk time, onetemps, il suffit
simply needsdetodifférencier
differentiatelathe
formule
aboveci-dessus
formula andet l’incorporer
incorporatedans
jk
le deuxième terme de l’équation [1].
it into the second term of the equation [1].
29
29 29
Understanding social change: a decomposition approach
Understanding social change: a decomposition approach

III.3 Application to the demographic dividend

III.3 Application to the demographic dividend
The study of demographic dividends is about how changes in national birth rates affects national levels of
schooling.
The study In
of essence, the researcher
demographic must
dividends investigate
is about a relationship
how changes between
in national two rates
birth macro-outcomes. For instance,
affects national levels of
imagine a country
schooling. where
In essence, average family
the researcher mustsize declinesafrom
investigate 6 to 4 children
relationship betweenbetween 1990 and 2010For
two macro-outcomes. at the same
instance,
time as the
imagine percentage
a country of average
where 6-10 year oldssize
family enrolled in school
declines from 6rises from 60 between
to 4 children to 75%. A1990
question one might
and 2010 at the ask
sameis
whether or how much these improvements in education came from the observed reduction in average family
time as the percentage of 6-10 year olds enrolled in school rises from 60 to 75%. A question one might ask is
size. It is or
whether possible that the
how much smaller
these family sizes
improvements played a role,
in education camebut
fromother
thefactors could
observed have also
reduction contributed.
in average A
family
macro-level
size. correlation
It is possible cannot
that the apply
smaller because
family sizeswe are dealing
played a role,with
but aother
single country
factors withhave
could onlyalso
two contributed.
points in time.
A
Furthermore,correlation
macro-level even if a researcher could
cannot apply implement,
because the macro-regression
we are dealing is openwith
with a single country to criticism for lacking
only two points in
in time.
detail and rigor.
Furthermore, evenInstead, one can use
if a researcher decomposition
could to look
implement, the at how the percentages
macro-regression of children
is open to criticism forliving
lackingwith
in
different
detail andsibsizes has changed
rigor. Instead, but use
one can alsodecomposition
how the effects
to of sibsize
look itselfthehas
at how changed. Through
percentages decomposition
of children living with
analysis, one can estimate the relative contributions of several factors, but the researcher will need
different sibsizes has changed but also how the effects of sibsize itself has changed. Through decomposition reliable
information
analysis, oneoncan
how the statistical
estimate relationship
the relative betweenofeducation
contributions and family
several factors, but size
the changed over
researcher time.
will In areliable
need recent
study (Eloundou-Enyegue
information and Giroux
on how the statistical 2012), we
relationship showeducation
between how to combine the size
and family micro relation
changed between
over education
time. In a recent
and fertility
study with information
(Eloundou-Enyegue about fertility
and Giroux 2012), change
we showtohow
estimate the implications
to combine the micro of demographic
relation betweenchange on
education
national
and school
fertility enrollment.
with information about fertility change to estimate the implications of demographic change on
national school enrollment.

30
30
 Chapter IV
Mathematical decomposition

IV.1 Simple mathematical decomposition

We have so far covered demographic and regression decompositions. In demographic decomposition, the
dependent outcome is an average, and the function linking it to the independent variable is simply a weighing
function. In regression decomposition, the dependent variable is a relationship linked to the independent
variables through a statistical relationship, estimated via regression analysis. Next, we want to explore yet
another case, where there is an exact mathematical relationship between the independent variables and the
outcome. For example, the average per-capita income in a country equals the total income divided by the total
population. A change in average income can only result from a change in total income (numerator) or in the
size of the population (denominator).

IV.1.1 Problem type

This type of analysis applies to processes that involve a mathematical relationship between two or more social
variables. GDP per capita is one example. It combines an economic component (GDP) and a demographic
component (number of inhabitants), and it can be broken down into simple terms that show how the GDP per
capita changes as either one of these two components varies. Other similar variables measure individual welfare
by relating available resources to the population served.

IV.1.2 Mathematical formulation

For this example, we will use a slightly more complex example than GDP per capita. Consider the public
education spending per child (r). This expenditure is positively related to the total level of national resources (g)
and the percentage of resources allocated to education (k). On the other hand, it is inversely proportional to the
number of school-aged children in the country (p).
݃݇ൗ
‫ ݎ‬ൌ ‫݌‬ (IV.1)

31
Understanding social change: a decomposition approach

In this case, the historical change in this expense can be broken down as follows:
��������
�� � ����� ��������
⁄�� � � ��� � ��� ��������
⁄�� � ��� � ��� ⁄�� � ��� (IV.2)

Effect of Effect of change Effect of change

change in in national in share of budget
population income to education

IV.1.3 Application
Using World Bank statistics (2014), Eloundou, Tenikue, and Ryu (2014) decompose the changes in public
expenditure per child among Southern Africa countries between 1990 and 2010 and compare the results to the
results of South Korea between 1975 and 1995. The results show that, over their respective study periods, South
Korea’s economy cumulatively grew by 250%, much faster than the growth observed in Southern African nations
(3-94%). Its ratio of youth to adults also fell faster, from 0.65 to 0.23, compared to South Africa’s decline from
0.67 to 0.46. Despite such demographic and economic differences, the relative contributions of demographic
change to the gains in r values were similar (60% for Korea versus 70 % and 61% in South Africa and Swaziland,
respectively). The study also noted an often-overlooked fact: many African countries allocate a larger portion of
their national budget to education (e.g., 10% in Lesotho and almost 6% in Botswana versus 2% in Korea). It is
therefore unsurprising that budget decisions made larger contributions to improving the r values in Lesotho
(32%) and Botswana (24 %) than in South Korea (12%).

Table 7. Mathematical decomposition of trends in public spending per child

(South Korea versus the 'vanguard' countries of Africa).

IV.2 Derived mathematical decomposition

IV.2.1 Extended mathematical chain
Let us return to the mathematical decomposition of GDP per capita as introduced in section IV.1.1. This first
decomposition usefully describes the contributions two components (population and GDP). However, is not very
informative because its two components do not refer to key decision variables but also because these variables
themselves need fuller exploration. With only a light transformation of these initial mathematical expressions,

32
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

III.2
we can getDécompositions debutrégression
a formula that is slightly longer dérivées
conceptually richer. Specifically, the initial formula of per capita
GDP
III.2.1 Régression curvilinéaire
� � ��� was differentiated as
Pour la plupart des phénomènes étudiés
∆� � ��en̅ ∗sciences
∆��⁄���sociales,
� ��� ⁄un
�������modèle
∗ ∆�� linéaire et bivarié, tel que présenté
(IV.3)dans
la section précédente, est simpliste. Dans cet exemple qui modélise le revenu selon le niveau d’instruction, un
which can be rewritten more interestingly as
effet curvilinéaire semblerait plus plausible. Fort heureusement, la décomposition s’étend aisément à des modèles
non-linéaires. Pour un effet quadratique �
par
�exemple,
� � �
� ∗ ��∗� (IV.4)
� � �

Y and
where G and P represent mean the national income = α +the
β1X + β2X2
total (III.4)
national population, respectively. The new term
introduced
L’analyseisde
A,décomposition
the active (adult) population
donnera in the
les termes country.
suivants : In this new formula, G/A (or ) refers to adult
productivity, and it is a conceptually interesting variable. It is frequently cited in economic growth theories and
�� � �� � ��� � ��� � � �� ���� � ��� � �� ����� � ��� � � �� ���� � ����
may be shaped through specific policies to raise productivity through education, research, and technological
�� � �� � �� � � �� � � ������
� � �� � � ���
development. Also, the new term A/P (or α) refers� to the population’s �
���
age � ���
�� � � ���
� structure—specifically,
� the ratio (III.5)
of
adults to theexemple,
Comme total population,
l’on peut aretourner
core variable in demographic
sur l’exemple dividend
hypothétique des theory. Thus, ourentre
écarts salariaux analyst now has
hommes two
et femmes,
verymais
interesting variables
en spécifiant cette( fois-ci
and α),
uneand she can
relation break down
curvilinéaire the changes
pour l’effet de in per capita sur
l’instruction GDPlein terms of these
revenu.
two variables.
III.2.2 Régression multivariée ∆� � ��� ∗ ∆�� � ��� ∗ ∆�� (IV.5)

De manière
Obviously, this similaire, la décomposition
new expression d’une régression
can itself expand further tomultivariée inclura
yield a more un ou
detailed des termes supplémentaires
decomposition, with better
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première
potential to inform policy decision-making. For example, adult productivity can be split into two components: variable.
the Comme dans l’exemple
adult unemployment précédent,
rate le nombre de
and the productivity oftermes sera simplement plus long. Pour prendre le cas simple
adult workers:
de deux facteurs, l’on pourrait étendre l’analyse
� � des
� niveaux
� de salaires en considérant cette fois-ci le niveau
� � � ∗ ∗ ��∗�∗� (IV.6)
d’instruction et le nombre d’années d’expérience.
� � � �

And its historical change can be represented as follows:

Y = α + β1X1 + β2X2 (III.6)
∆� � ��̅�̅ ∗ ∆�� � ��̅ �α ∗ ∆�� � ��̅ α
� ∗ ∆�� (IV.7)
Dans ce cas particulier, les sources du changement dans le niveau des salaires incluraient les composantes
Similarly, the: adult ratio of the total population can be expressed as a function of the youth and the elderly
suivantes
populations, respectively. How far the researcher extends this development depends
���� � �� ��� on how much detail she
�� � �� � �� ���� � ��� � � �� ���
� � � ��� � ��� � � ��� � ��� � (III.7)
needs, and on data availability, the policy relevance and the tractability of the added terms.

III.2.3
Students of Régression multi-niveaux
economic growth can apply the same logic in decomposing popular formulations of economic
performance such as the Cobb-Douglas function, which expresses growth based on physical capital (K), human
Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
capital (h), employment (L), and total factor productivity (A):
l’individu, son groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
examiner comment la performance des élèves �� � ����
� �dépend �
(IV.8) 1)
des caractéristiques individuelles des élèves (niveau
ainsi
From thisque celles de
formula, onelacan
société (niveaueconomic
decompose 2). Les équations
growth pour
as estimer un tel modèle s’expriment comme suit :
Niveau 1 : Yjk = β0k + β∆�
1k*x�
jk +
∆�rjk� � �∆�� � �� � ��∆�� � ∆�� � ∆�� (IV.9)
�

Niveau
where �� is 0k=�capita
2 :the GDPβper �� � ��� �� � ���
(Y/P)
1k=�
k is the stockΒof �� � ���
physical �� � �per
capital �� employee (K/L)

En intégrant les valeurs

h is human capital du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
performance du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
l is the employment rate (L/W)
interaction :
w is the population’s age structure (W/P)
Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
A is the total factor productivity (TFP)
Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
le deuxième terme de l’équation [1].

33 29
Understanding social change: a decomposition approach

Clearly, our analyst has several options to decompose economic growth and education. As long as these
decompositions include the effects of population age structure, then they help assess the magnitude of the
demographic dividend.

34
 Chapter V
Combination of demographic
and regression decompositions

Elementary decompositions have the merit of being simple, but they lack in detail. This problem is solved with
mixed decompositions combining two or more elementary forms of decomposition. One can combine
demographic and regression decompositions to get a more detailed form as described in this chapter. Starting
from a simple demographic decomposition, we will see how to expand each of the main components, starting
with the behavioral effect (Δyj) and then the composition effect (Δwj).

V.1 Extension of behavior effect

V.1.1 General presentation
Let us start with the basic formula (3), which expresses a change in any national outcome in terms of the
composition and behavior of various groups:

� � � ��� �
�� � �∑ �� � ��� � � �∑ �

Compositional effect Behavioral effect

This formula’s extension can express the behavior of any given group (yj) as a function of one or more other
variables. In a simple regression analysis,

yj =α +βxj + μj where,

• α (intercept) is the baseline performance, when x = 0,

• β is the increased mortality associated with a unit increase in X, and
• μj is the error term (the over-performance or under-performance of the group or the residual effect of factors
other than X, that are not considered in the analysis).

35
Understanding social change: a decomposition approach

The change in the value of yj between two periods is thus:

��� � �� � �̅��� � ��� �� � ��� [V.1]

By inserting [V.1] in the basic equation (3), we get

�� � ∑ �� ��� � ∑ �
� � �� � ∑ � � ��� � ∑ �
�� � � � ��� �� � ∑ �
� � ��� [V.2]

A B1 B2 B3 B4

Compositional Effect Behavioral effect

• A (the composition effect) remains unchanged from the previous situation.

• B (the behavioral effect) is now divided into four sub-components that respectively reflect changes in

– baseline performance (B1),

– the level of the independent variable (B2),
– the effect of the independent variable (B3) , and
– the residual effect (B4).

V.1.2 Illustration
To illustrate this mixed decomposition, we can return to our first example in which we used simple demographic
decomposition to study changes in infant mortality between 1990 and 2011 (Table 2). To recall, this simple
decomposition showed a nearly 26-point decline in infant mortality, which reflected a mix of compositional
change (17%) and behavioral change (83%). We can refine this analysis by considering that mortality varies
with one’s education level. We can thus express the mortality within each group as a function of the group’s
average education level. If one can estimate this relationship, and get a reliable estimate for the values of α
(baseline mortality), β (the effect of education on mortality), and μ (the residual term), it becomes possible to
refine our initial decomposition and get more detail about the forces driving change at the national level. This
decomposition has the potential to reveal, in greater detail, the policy areas that were most influential in driving
the change. Thus, in equation V.2,

• B1 (the baseline performance) reflects the improvement in the public sanitation and public health conditions
that raise the minimum health standard of the population, regardless of education level;
• B2 measures the health improvements associated with the gains in the national level of education, assuming
that the payoffs to education remain the same;
• B3 measures the improvement in the educational effects on health; and
• B4 measures the residual effect of other variables not considered.

Although this first example limits itself to a linear bivariate regression model (a single independent variable
modeled linearly), one can easily imagine how this analysis can extend to cases where multiple independent
variables or curvilinear influences are considered. The only difference is that the corresponding equations
become longer and longer!

V.1.3 Comparison with national transfer accounts

The method of National Transfer Accounts is a core method for studying demographic dividends (Mason and
Lee 2005). It builds on the simple idea that economic behavior (income, savings, transfers) varies systematically
with age, even if the exact age profiles of economic behavior vary across countries (Figure 5). One can use a

36
 Comprendre le changement social : apport des méthodes de décomposition
Understanding social change: a decomposition approach
et application à l’étude du dividence démographique

III.2 economic
country’s Décompositions debalance
profile to estimate the régression dérivées
between income and consumption at each age and combine
the data for all age groups to calculate a cumulative balance for the entire country. Assuming a constant age
III.2.1
pattern Régression
of economic curvilinéaire
behavior, any change in a population’s age structure automatically changes the national
balance between income and consumption.
Pour la plupart des phénomènes étudiés en sciences sociales, un modèle linéaire et bivarié, tel que présenté dans
However, it isprécédente,
la section not realistic to assumeDans
est simpliste. a constant age qui
cet exemple profile of economic
modélise le revenubehavior, especially
selon le niveau during a un
d’instruction,
demographic transition,
effet curvilinéaire given theplus
semblerait large changesFort
plausible. during a life coursela(e.g.,
heureusement, age at marriage,
décomposition duration
s’étend of schooling,
aisément à des modèles
andnon-linéaires.
the time spentPour
bearing children
un effet versus par
quadratique participating
exemple, in the labor force). It is therefore useful to consider
situations where a country experiences changes in both the age structure and the consumption profile. Such
Y = α + β1X + β2X2 (III.4)
situations are easily managed in a decomposition framework, and a full NTA approach can be seen as an example
L’analyse
of mixed de décomposition
decomposition donnera
(Eloundou, les termes
Tenikue, suivants
Giroux 2014).: Specifically, the national surplus in a given year is
the weighted average of the ��
specific
� ��surpluses for� each
� ��� � �� ���age group. ����� ���
� � ��� � ��� � �� � ��� � � ��� � ����

��� ���∑��
�� � �� � ��� � �� ����   � � � ���
���������� ���� � ��
���� � � ��� [V.3] 
(III.5)
Change is thus
Comme expressed
exemple, as retourner sur l’exemple hypothétique des écarts salariaux entre hommes et femmes,
l’on peut
mais en spécifiant cette fois-ci une relation
�� �curvilinéaire
� ∑ � �� � ∑ pour
� l’effet de l’instruction sur le revenu.
� �� [V.4]
� � � �

Now, we can additionally consider, as indicated above, that consumption/income profiles can change over time,
III.2.2 Régression multivariée
perhaps even in response to a change in the age structure. It is therefore useful to explore the additional
De manière
possibility that asimilaire,
change lain décomposition d’une
the age structure canrégression
also havemultivariée inclura
an effect on un ouIfdes
behavior. thetermes
analystsupplémentaires
can obtain,
qui, cette
through fois-ci analysis,
regression seront desa reliable
variablesestimate
indépendantes supplémentaires,
of the effect plutôt que
of the age structure le on
of (J) carré
thede la première
economic variable.
behavior
Comme
of each age dans
group l’exemple
[Equationprécédent, le nombre
V.5 below], then thedechange
termesinsera
thesimplement plus long.
national surplus can bePour prendrethrough
estimated le cas simple
a
de deux
mixed facteurs, l’on pourrait étendre l’analyse des niveaux de salaires en considérant cette fois-ci le niveau
decomposition.
d’instruction et le nombre d’années d’expérience.
SJT = ajt + bjtJt +ejt [V.5]
Y = α + β1X1 + β2X2 (III.6)
��� � ��� � �̅��� � � ��� � ��� [V.6]
Dans ce cas particulier, les sources du changement dans le niveau des salaires incluraient les composantes
�� � ∑ �� ��� � ∑ � � � ��� � ∑ � � � �̅��� � ∑ � ��� � ∑ �
� �� � � ��� [V.7]
suivantes :

�� � �� � �� ���� � ��� � � ��
���� � ��� � � �� ���� � ��� �
���� � ��� � � �� (III.7)
m-dividend s-dividend

III.2.3 Régression multi-niveaux

In this last equation, the change in the age structure has both a mechanical effect (the first term) and a
Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
substantive effect on behavior (the penultimate term). The question, of course, is whether one can get a reliable
l’individu, son groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
estimate for the coefficient b—the effect of age structure on economic behavior.
examiner comment la performance des élèves dépend des caractéristiques individuelles des élèves (niveau 1)
ainsi que celles de la société (niveau 2). Les équations pour estimer un tel modèle s’expriment comme suit :
V.2 Extension of composition effect
Niveau 1 : Yjk = β0k + β1k*xjk + rjk
Just as with the behavior effect, the composition effect can also be disaggregated. The disaggregation can focus
Niveau 2 : β0k=��� � ��� �� � ���
on primary demographic groups, on the age structure of the sub-populations, or on the demographic processes
shaping the size of the 1k
Β =�
groups
�� (fertility,
� ��� �� �mortality
��� and migration).

En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
(A)performance
Extensionduaccording
groupe à latofois
primary groups
en fonction des caractéristiques du groupe, du contexte national et de leur
interaction
In this case, the: size of the study group is expressed as a function of the size of a primary group that generates
the Y
members of the group being studied. As one example, the number of children in poor families (wj) might be
jk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
expressed as a function of the number of poor families (nj) and the relative fertility of poor families (f j), i.e., the
Pour étudier le changement de Y dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
fertility of the poor compared to thejknational average.
le deuxième terme de l’équation [1].

37 29
Understanding social change: a decomposition approach

wj = nj * fj [V.8]

The change in the percentage of children from each class will depend on the change in (1) the proportion of
families that belong to that social class and (2) the relative fertility of those families.

��� � � ̅��� � ����� [V.9]

One can thus insert [V.9] in [3], yielding

�� � �∑ �� � ����� � � �∑ �� � � ��� � � �∑ �
� � � ��� � [V.10]

A1 A2 B

Compositional effect Behavioral effect

In this equation, A1 represents the change in the percentage of poor families in a society, and A2 represents the
change in the relative fertility of the various social classes. These two variables are conceptually richer, more
informative, and more policy-relevant. The first (A1) might guide policies to stimulate growth and reduce
poverty. The second might be related to the ability of family planning programs to reduce fertility inequality,
especially if that inequality stems from differential access to family planning.

IV.3 Double extension

Finally, one can merge equations [V10] and [3] and get a detailed system that refines the analysis of both the
compositional and behavioral sides. The result of this combination (below) yields an even more detailed
expression where

• A1 = change in the distribution of mothers by social class

• A2 = change in the relative fertility of mothers
• B1 = change in baseline health
• B2 = change in income levels
• B3 = change in the health benefits of income
• B4 = residual effect of the factors omitted from the regression equation.

�� ��� ��� � � �∑ �
�� ��� � �∑ �
�� � �∑ � �� ��� � � �∑ ��� ��̅ ��� � � �∑ ��� ��� ��� ]
�� ��� ��� � �∑ �

B1 B2 B3 B4 A1 A2

Again, both of the primary components of a basic demographic decomposition (compositional or behavioral)
can be expanded for greater detail. With this more detailed accounting, a planner can get more nuanced insights
into policy priorities.

38
 Chapter VI
Combination of demographic
and mathematical decompositions

This variant also begins with a simple demographic decomposition, but the expansion of the behavioral
component uses an exact mathematical (rather than statistical) expression. To illustrate, let us consider a study
of GDP change in Africa. The region’s GDP per capita is a weighted average across all African countries, and the
decomposition of its change can be expressed as usual.

� � � ��� �
�� � �∑ �� � ��� � � �∑ �

Compositional effect Behavioral effect

Next, we now express the countries’ GDPs as a function of adult productivity (π) and age structure (α). The
change of individual country GDP is as follows:

��� � ���� � ��� � � ���� � ��� � [VI.1]

By inserting VI.1 into the basic equation of the demographic decomposition (II.3), the change in the average GDP
of Africa becomes

� � �� ��� �� � �∑ �
�� � �∑ �� ��� � � �∑ � � � �� ��� � [VI.2]

Effect of Effect of age Effect of adult

population size structure productivity

39
 Comprendre le changement social : apport des méthodes de décomposition
et application à l’étude du dividence démographique

III.2 Décompositions de régression dérivées

Chapter VII
III.2.1 Régression curvilinéaire Combination of regression
and mathematical decompositions
Pour la plupart des phénomènes étudiés en sciences sociales, un modèle linéaire et bivarié, tel que présenté dans
la section précédente, est simpliste. Dans cet exemple qui modélise le revenu selon le niveau d’instruction, un
effet curvilinéaire semblerait plus plausible. Fort heureusement, la décomposition s’étend aisément à des modèles
non-linéaires. Pour un effet quadratique par exemple,

Y = α + β1X + β2X2 (III.4)

L’analyse de décomposition donnera les termes suivants :

�� � �� � ��� � ��� � � �� ����� � ��� � � ��

���� � ��� � �� ���� � ����
����� � ��� � � ���
�� � �� � ��� � ��� � � �� ���� � ��
���� � � ��� (III.5)

Comme exemple, l’on peut retourner sur l’exemple hypothétique des écarts salariaux entre hommes et femmes,
mais en spécifiant cette fois-ci une relation curvilinéaire pour l’effet de l’instruction sur le revenu.

III.2.2 Régression multivariée

De manière similaire, la décomposition d’une régression multivariée inclura un ou des termes supplémentaires
qui, cette fois-ci seront des variables indépendantes supplémentaires, plutôt que le carré de la première variable.
Comme dans l’exemple précédent, le nombre de termes sera simplement plus long. Pour prendre le cas simple
Thede deux facteurs,
combination l’on here
covered pourrait
is lessétendre
common.l’analyse desthe
Consider niveaux de salaires
link between en considérant
money cette
and happiness. fois-cifor
Assume, le niveau
the
saked’instruction
of simplicity,etthat
le nombre d’années d’expérience.
this relationship is linear.

ܻYൌ= ߙ
α൅+ β1X1
ߚ̈́ ൅ +݁ β2X2 (III.6)
[VII.1]

Dans
Next, ce casthat
consider particulier, les sources
people have multipleduincome
changement
sources,dans le niveau
perhaps wagesdes salaires
and incluraient
transfers, and thatlesthey
composantes
derive
suivantes :
different levels of satisfaction from each. If that is true, then the formula VII.1. is simplistic and should ideally
incorporate this differential satisfaction.�� ���� � ��� � � �� ���� � ��� �
�� � �� � ���� � ��� � � �� ���� � ��� � � �� (III.7)
Alternatively, one can begin with a mathematical relationship and incorporate a regression relationship.
III.2.3forRégression
Consider, multi-niveaux
instance, the factors shaping education spending per child. Suppose, as is likely, that the share of a
family budget allocated to its children education depends on parental income , so the researcher can refine the
Comme son nom l’indique, la régression intègre des analyses à deux ou plusieurs niveaux qui peuvent inclure
analysis by integrating details from the regression study.
l’individu, son groupe et sa société toute entière. Dans l’analyse de la scolarisation par exemple, l’on peut
Theexaminer
possibilities dependlaonly
comment on the researcher’s
performance own imagination.
des élèves dépend Still, even
des caractéristiques where further
individuelles expansion
des élèves is 1)
(niveau
possible, the celles
ainsi que researcher must (niveau
de la société compromise
2). Lesbetween detail
équations pourand parsimony.
estimer The combination
un tel modèle opportunities
s’expriment comme suit :
suggested are not ready-made recipes to apply mechanically. Rather, they are tools to be used selectively by
Niveau 1 : Yjk = β0k + β1k*xjk + rjk
researchers in their quest to understand social change.
Niveau 2 : β0k=��� � ��� �� � ���

Β1k=��� � ��� �� � ���

En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
performance du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
interaction :

Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)

Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
le deuxième terme de l’équation [1].

41 29
 Chapter VIII
Summary and conclusions

This monograph offers an introduction to decomposition, a cluster of methods that can enrich the
methodological toolbox of social scientists. The adoption of these methods advances the study of social change
beyond the current limits of micro and macro-regressions. While decomposition does not address causation, it
can reliably locate the social or geographic sources of social change. In doing so, decomposition reduces the
margin of error in understanding societal change and in designing related policies. Within these limitations,
decomposition is a reliable and transparent tool that, alone or with other methods, can usefully guide the
allocation of policy resources. Policymakers faced with several options can use a decomposition approach to
narrow down the list of most promising options.

The flexibility of the decomposition method makes it possible to imagine creative extensions. The few options
presented here are not an exhaustive list, and researchers are encouraged to imagine other possibilities. In
addition, many of the examples used here draw from the fields of population or economics, but the methods
apply to a wide range of social phenomena.

One strength of the decomposition approach is its compatibility with other methods. When carefully combined
with other methods, it can enrich understanding. It is therefore not a substitute but a complement. It
complements other methods by identifying key processes and groups. At the same time, it leaves room for other
methods (say causal analysis or informant interviews) to elucidate questions about causation, processes, key
events, and key actors. Such complementarities facilitate a more complete understanding of social change. In a
sense, and going back to our initial story of the proverbial drunk looking exclusively under the lamp-post, the
contribution of decomposition is twofold: it widens the search area by considering all possible sources of change;
and it can help begin the search closer to where the key—rather than the light—is. Having identified the groups
of processes that lead the change, researchers can launch additional investigations into the reasons why these
groups did change. Ultimately, those who seek to change the world can do so more effectively with a better
understanding of the drivers of social change.

43
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individuelles based(niveau
des élèves on the 1)
Shapley
ainsi cellesJournal
quevalue. of Economic
de la société (niveauInequality 11, 99-126
2). Les équations pour estimer un tel modèle s’expriment comme suit :

Shorrocks,
Niveau 1A.,: & Wan,
YjkG.= β(2005). *xjk + rjkdecomposition of inequality. Journal of Economic Geography, 5(1),
0k + β1kSpatial

59–81.
Niveau 2 : β0k=��� � ��� �� � ���
Smith, H. L., Morgan, S. P., & Koropeckyj-Cox, T. (1996). A decomposition of trends in the nonmarital fertility
Β1k=��� � ��� �� � ���
ratios of blacks and whites in the United States, 1960–1992. Demography, 33(2), 141–151.
En intégrant les valeurs du niveau 2 dans l’équation de niveau 1, l’on obtient l’équation mixte qui exprime la
Solow, R. M. (1957). Technical change and the aggregate production function. The Review of Economics and
performance du groupe à la fois en fonction des caractéristiques du groupe, du contexte national et de leur
Statistics, 312–320.
interaction :
Thornton, A. (2001). The developmental paradigm, reading history sideways, and family change. Demography,
Yjk =��� � ��� ��� � ��� �� � ��� �� ��� � ��� � ��� ��� � ��� (III.8)
38(4), 449–465.
Pour étudier le changement de Yjk dans le temps, il suffit de différencier la formule ci-dessus et l’incorporer dans
UNICEF. (2008). The State of Africa’s Children, 2008: Child Survival. UNICEF.
le deuxième terme de l’équation [1].

47 29
Understanding social change: a decomposition approach

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Nathan Keyfitz’s 90th birthday. Demography, 40(2), 201–216.

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