Chapter 16
Multistate Demography
Jacques Ledent
Institut national de la recherche scientifique (INRS), University of Quebec, Canada
Yi Zeng
Duke University and Peking University
Keywords:
Contents
1. Introduction
2. The Multistate Life Table
2.1. A reminder on the ordinary single-state life table
2.2. The multistate life table functions
2.3. Estimating the transition probabilities in practice
2.4. Dealing with the underlying Markov process
2.5. Summary measures
2.6 Additional considerations
3. The Multistate Projection Models
3.1 Multistate projection and stable growth
3.2 Multistate projections with time-varying input
4. Applications of Multistate Demography
4.1. Multistate working life tables
4.2. Active and disabled life expectancies
4.3. Multistate life table analysis on marriage/union formation and dissolution
4.4. Multistate population projections by educational attainment
4.5 Multistate projections of households and living arrangements
4.6 Modeling and estimation of the age and spatial structures of migration flows
5. Bridging the Micro- and Macro-Simulation Models: Recent Development of Multistate
Demography
Summary
This chapter deals with multistate demography, an extension of classic mathematical
demography to the case of individuals grouped in various categories of demographic
1
and/or socioeconomic attribute(s) such as marital/union status, labor activity status,
region of residence or health status. Individuals can move from and to these categories
but—and this the salient feature of multistate demography—they are allowed to return to
a state/status previously occupied; a long standing modeling difficulty which was
eventually overcome by having recourse to a vector/matrix notation. This exposition,
kept as little technical as possible, covers the basic principles of multistate demography,
its two main models—that is, the multistate life table and the multistate projection
model—as well as the generalization of these models to the case of families. It also
alludes to the recent development of multistate demography which tends to bring this
subfield of demography closer to mainstream statistics. Finally, adopting a more
empirical viewpoint, it presents several applications of multistate demography to various
population areas.
Key Words
Multistate demography, multistate life table, multiregional life table, single-state life
table, transition probabilities, Markov process, multistate stable population model,
multistate projection models, multiregional population projections, working life tables,
active and disabled life expectancies, marriage/union statues life table, population
projections by educational attainment, households and living arrangements forecasting,
household momentum, migration model schedules, micro-simulation, macro-simulation,
integrated MicMac approach.
2
1. Introduction
Multistate demography is a topic whose complexity makes exposing and
comprehending it a real challenge. Therefore, this chapter does not attempt in any way to
cover its technical details, which are at the risk of being unpalatable for this
encyclopedia. Rather it is a review limited to presenting the basic principles and
applications of multistate demography—using as simple an exposition as possible—and
offering a quick overview of its extensions and practical uses.
Basically, multistate demography refers to the study of populations stratified by at
least one attribute (other than gender, age or race) that reflects region of residence,
marital/union status, labor activity status, and so forth. Initially concerned with the
generalization of the classical models of mathematical demography—that is, the life table
and the projection model—multistate demography deals with transfers between
alternative categories (i.e., states or statuses) of the population attribute(s), including reentries into these categories. .
Re-entry into a previously occupied category was first tackled as early as a
century ago, in the context of a model involving two states (healthy and disabled)
between which individuals could move back and forth (Du Pasquier 1912/13). However,
despite the relevance of studying changes between multiple categories of population
attribute(s), it was not too long ago that demographers began to seriously investigate such
changes. What posed a problem to them was the complexity in estimating and computing
the reentries into a state or status previously occupied. Such a circumstance continued
until the early 1970s when the advent of mainframe computers made it possible to crunch
the numbers behind the construction of the models initially considered. Since then,
3
multistate demography has grown into a sub-field of its own, tackling various types of
transfers concerning individuals and groups of individuals such as families/households.
Beside significant modeling development, its pioneering contributions include various
applications of the models thus developed to macro population-level (i.e., census and
survey) data on multiregional migration (Rogers 1975; Rogers and Willekens 1986),
marriage and divorce (Schoen and Land 1979; Willekens et al. 1982), labor activity
(Hoem, 1977) as well as extensions to other topics of multi-group population (Land and
Rogers 1982; Schoen 1988; Rogers 1995). Today, multistate demography largely
intersects with the applications of traditional and more recent statistical methods,
including event history analysis, to both macro and micro level data, thus making inroads
into epidemiology, medicine and public health.
In the following sections, we first present the basic principles and the main
constituents of the two fundamental and closely related models of multistate
demography—that is, the multistate life table and the multistate projection model. We
then discuss practical applications of these models in studies concerning populations,
socioeconomics, health, and other related fields. Finally, we present the recent
development of multistate demography aiming at bridging the micro- and macrosimulation models.
2. The Multistate Life Table
Understanding the multistate life table can be made easier by first reviewing the
ordinary single-state life table (a more sophisticated description of which appears in
Section 2 of Chapter 15) and then generalizing from it
4
2.1. A reminder on the ordinary single-state life table
In its ordinary version, the single-state life table is a tool that follows the life
course of a birth cohort (a group of individuals born during a given period of time) until
the death of its last survivor, as it gradually decreases in size under the effect of mortality.
Basically, such a table includes several columns which document the evolution over
successive ages of various functions relating to the life and death of the cohort members.
These ages are usually (but not necessarily) equally spaced n years apart.
Typically, the first two columns are a column containing the number lx of
individuals who, out of the number l 0 of individuals born ( i.e., individuals aged 0), are
alive at age x , and a column containing the numbers L x of person-years lived by the
cohort’s survivors between ages x and x+n. Next are a column containing the cumulated
numbers Tx of person-years lived beyond age x ( Tx   L y ) and a column containing
y x
the numbers e x representing the average remaining lifetime for individuals aged x, also
known as the life expectancy at age x ( e x  Tx / l x ). Usually, there is also a column of the
numbers d x of the cohort members that die between ages x and x+n. Since the cohort is
closed in the sense that i) there is no exit other than through death, and ii) re-entry is not
possible, each number d x of deaths is the difference in the number of survivors at the
beginning and end of the relevant age interval (
.
In practice, a life table reflects a particular mortality pattern embodied in a set of
age-specific mortality rates, with mx being the mortality rate between ages x and x+n
( mx  d x / Lx ). Then, assuming that the number
5
of person-years lived can be
approximated by the arithmetic mean of the number of survivors at ages x and x+n times
the width n of the age interval [
] leads to a simple formula that makes
it possible to derive the column of the successive numbers
where
of survivors [
is the probability to survive (survival probability) from age x to age
x+n]. Note that this simple formula is based on the assumption that the distribution of
deaths occurred between age x and x+n is linear. This is a reasonable approximation
except at infancy (between birth and age one) and extremely advanced ages (e.g. >age
95), which needs some special estimation procedures (Ref. Section 2 of Chapter 15
and/or other standard Demography text book).
Such an ordinary single-state life table is valid only for analysis of non-renewable
events such as death or, in the case of first marriage or first birth. But it can be readily
extended to the multiple decrement life table by recognizing two or more exits
(decrements). These decrements may reflect various causes of death, thus leading to a
cause-of-death life table, the main interest of which resides in its ability to evaluate the impact
of the elimination of a particular cause of death on life expectancy, assuming that the various
causes of death are independent of each other. Alternatively, the decrements may be death and
another non renewable event, such as is the case in a net primo-nuptiality table which
describes the evolution of a never-married cohort of individuals that can exit through not
only death but also marriage.
Both the ordinary single-state life table and the multiple-decrement life table do
not allow re-entering a previously occupied state which is precisely the distintive feature
of multistate life tables.
6
2.2 The multistate life table functions
Basically, a multistate life table, sometimes referred to as a multidimensional life
table, describes the life course of a cohort of individuals born in a given period of time as
they move between the various states considered until the death of the last survivor. It is
typically constructed in the context of marital status (with individuals moving between
the four traditional marital statuses--never-married, married, divorced and widowed--or
more realistically between seven marital and union statuses that take into account
cohabitation), interregional migration (with individuals moving between a set of regions)
or labor activity (with individuals shifting between not being employed and being
employed, the former status being possibly divided further according to whether one
belongs to the labor force or not).
Let us assume for the time being that the cohort under consideration consists of
individuals born in just one state, e.g., the never married state of a multistate life table
focusing on marital status. Drawing on a parallel with the ordinary single-state life table,
it is possible to associate with each state i ( = 1, 2, .., N) .an elementary (state-specific)
table that depicts the mortality and mobility experience in state i of the cohort under
consideration. First, there are the column of the numbers l xi of those alive at age x in
statei i and the column of the numbers Lix of person-years lived in state i between ages x
and x+n. Next, there is the column of the cumulated numbers T xi of person-years lived in
state i beyond age x ( Txi   Liy ) which, it should be noted, are lived by individuals
y x
who at age x were not only in state i but also in all of the other states. As a result, the next
7
column contains the numbers exi of remaining lifetime to be spent in state i beyond age
x, regardless of the state occupied at age x ( e xi 
Txi
l
k
x
).
k
As in the ordinary single-state life table, a column may be included to represent
the numbers d xi of deaths in state i (the δ symbol is added here to stress exits through
death). Moreover, (N-1) additional columns may be included as well to reflect the other
types of exits, the kth column of which contains the numbers of moves d xik to the k-th
state between ages x and x+n. At the same time, of course, these exits constitute entries
into the states to which these moves are directed. As a result, linking the number l xi  n of
survivors in state i at age x+n with the corresponding number l xi at age x requires one to
subtract the deaths d xi in state i and the moves d xik out of state i to all other states but
also to add the moves d xki into state i from all other states:
l xi  n  l xi  d xi   d xik   d xki
k i
(1)
k i
Thus, any given multistate life table consists of N elementary tables (one for each
state) such as the one just described. In practice, it reflects a particular demographic
pattern embodied in a set of age-specific mortality and mobility rates defined for each
state, respectively, by:
m xi 
d xi
Lix
m xik 
d xik
Lix
(2a)
and
for k  i
(2b)
8
From there, drawing on the exposition of the ordinary single-state life table, the
number Lix of person-years lived can be approximated by the arithmetic mean of the
number of survivors at ages x and x+n times the width of the age interval:
l xi  l xi  n
L n
2
i
x
(3)
Combining (1)-(3) gives rise to a system of linear equations involving as many
variables—the numbers of survivors l xi , out of the initial cohort, that reside at age x in
each state i—as there are states considered--that is, N. Such a system, however, is not
tractable using conventional calculus, thus leading one to have recourse to matrix
calculus. Basically, this calls for arranging the age-specific numbers l xi of survivors in a
vertical array or vector
 l 1x 
 2
lx 
x   
 
 .. 
l N 
x 
in which the i-th element is l xi and embedding the age-specific mortality and mobility
rates in a square array or matrix:
m1x   m1xl

l 1
  m12
x

mx  
.

.

  m1xN

 m x21
.
m x2   m x2l
.
l 2
.
.
.
.
. m xN



.


.


  m xNl 
lN

 m xN 1
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in which each kth diagonal element contains the total rate of exit out of state k—that is,
the mortality rate m xk plus the sum
m
l k
kl
x
of the rates of moving to another state and
each k-lth- off-diagonal element contains the rate of moving from state l to state k
preceded by a minus sign. Consequently, the system of linear equations can be written in
a more compact format as (Rogers and Ledent, 1976):
 x n  p x  x
(4)
in which
p x  (I 
n
n
m x ) 1 ( I  m x )
2
2
(5)
is a transition probability matrix which is structurally similar to the single-state
survival probability p x mentioned earlier, except that a matrix of mortality/mobility rates
is substituted for a scalar mortality rate.
Equation (4) enables one to derive the successive vectors  x of survivors and from
there the associated vectors of person-years lived ( Lx , Tx and ex ).
Let us now turn to the situation in which individuals are born in several, if not all
of the states, rather than in just one state as was assumed in the above discussion. In such
a case typical of a multistate life table focusing on migration between a set of regions, or
a multiregional life table, the above framework applies to each state (region) of birth.
Equation (4) can then be written in relation to each state-of-birth-specific cohort  0j of
individuals, thus prompting the addition, in front of  x and  x n , of a j subscript
representing the state of birth:
10
j
 xn  p x j  x
(6)
Then on gathering all of the j  x vectors in a matrix
l x  1  x
2
x
N
x
it follows that the series of the l x can be obtained on the basis of
l x n  p x l x
(7)
Similarly, a region-of-birth-specific index can be assigned to the Lx and Tx vectors and
the ensuing vectors can be gathered in square matrices, respectively, L x and Τ x , which
then allow for the derivation of an alternative type of life expectancies—namely, statuse x  Tx l x
based life expectancies on the basis of:
1
(8)
where ex is a matrix such that its k-lth element expresses the remaining lifetime in region
k to an individual RESIDING in state l at age x. [But such a formula may be applied just
as well in the case of a multistate life table originating from a single-state cohort. Suffices
it to consider along hypothetical cohorts “born” in all of the other states].
2.3 Estimating the transition probability matrices
Much in the same way as the construction of an ordinary single-state life table
relies on setting the age-specific mortality rates m x equal to the corresponding mortality
rates M x in an observed population, the construction of a multistate life table relies on
setting the age-specific mortality/mobility matrices m x to their observed counterparts
.
11
But when one or several mortality/mobility rates take on large enough values,
which may happen in the case of large age-groups (e.g., n = 5), some elements of the p x
matrix estimated from (5) can fall outside the [0,1] range. Fortunately, under such a
circumstance, the p x matrix can be estimated from an alternative formula which is
derived from the continuous specification of the underlying mortality/mobility process.
Let l (t ) denote the matrix of survivors at age t and μ (t ) the matrix of the
mortality/mobility intensities on an infinitesimal interval (t, t+dt) which are built in the
same manner as the l x and m x mentioned above. Then, writing in two ways the net flows
to each state and this, by state of birth, leads to the following differential equation:
l (t )  l (t  dt )  dl (t )  μ(t )l (t )dt
(9)
Assuming that the intensity matrix μ (t ) is constant over the age interval (x, x+n), leads to
the following age-specific probability matrix p x :
p x  exp( nm x )
(10)
in which m x is the constant value of the intensity matrix over the age interval x, x+n.
Formulas (5) and (10) are commonly referred to, respectively, as the linear and
exponential estimators, with the ‘linear’ and ‘exponential’ qualifiers alluding to the
corresponding age variations of the numbers of survivors. Because
it always leads to legitimate probabilities, the exponential estimator is generally the
estimator of choice. Yet, it happens that interregional migration is characterized by low
mobility rates so that the linear approach still enjoys some popularity in the construction
of multiregional life tables, especially in the case of single-year age intervals.
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Sometimes, however, the data that are necessary to estimate the
mortality/mobility rates in an observed population may not be readily available from a
population register or an administrative data bank. In particular, the mobility information
may not come in the form of moves between states but rather in the form of changes in
the state occupied between two points in time. Such is the case when the input
information is obtained from a census or a survey in which respondents are asked for the
state they occupied T years earlier. Appropriate methods, however, have been set forth to
deal with such a situation (see Ledent and Bah, 2005). Finally, if micro (event-history)
rather than macro data are available, then the transition probabilities can be estimated
using the Kaplan-Meier estimator, a non parametric estimator for right-censored data
which has become the standard tool of event history analysis, or rather an extension of it
(Strauss and Shavelle, 1998). But application of this to relatively small panel studies that
typically have survival probabilities exhibiting erratic changes with age due to sampling
and other stochastic variations, is problematic. The problem, however, can be
circumvented by using standard data-smoothing or graduation methods.
2.4 Dealing with the underlying Markov process
It is important to note that underlying any multistate life table is a simple Markov
process which relies on three main assumptions:
-
population homogeneity
-
independence from the past
-
stationary input information
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Fortunately, the numerical impact of the first two assumptions can be reduced in some
ways. First, regarding the homogeneity assumption, one possibility is to break down the
population at hand into more homogeneous subpopulations. For example, in the context
of interregional migration, if the mobility information can be observed separately for the
subpopulations born in each region, a separate and more accurate multiregional life table
can be constructed for the subpopulation born in each region. Another possibility which
arises in the case of event history data is to introduce a transition rate model (e.g., Cox
model) that accounts for personal attributes (covariates) such as sex, race, education,
socioeconomic status etc.. (Hannan, 1984). But, the possibility of constructing group
(covariate)-specific life tables rests on the estimation of transition survival probabilities
on the basis of Coleman-type Markov regressions, which are more stable than those
computed directly from the data, especially over short intervals.
Second, the assumption of independence from the past can also be made less
stringent. For example, in the case that the data come in the form of changes in the state
occupied between two points in time, the transition probability matrices should be
estimated on as long as possible age intervals (Ledent and Rees, 1986 ; Shavelle and
Strauss, 1999). Alternatively, in the case that microdata are available, a more radical
avenue consists of substituting a more realistic semi-Markov or even non-Markov
process for the simple Markov process normally present in a multistate life table
(Rajulton, 2001). And finally, the third and last assumption, the stationary nature of the
input information, can be relaxed by resorting to time-varying transition probabilities (see
Section 3 for details).
2.5 Summary measures
14
As compared to the ordinary single-state life table, one important and unique
feature of the multistate life table is that it decomposes the overall life expectancy (or life
span) into segments of life span by different states, such as life expectancies living in
different regions, different marital/union statuses, health statuses and employment
statuses. Besides these, the multistate life table models offer some other informative
summary measures, which are particularly useful in the context of life course studies. For
the purpose of a clear illustration, the following discussions focus on the summary
measures of the multistate marital/union status life tables, but they are readily applicable
to other kinds of multistate life tables dealing with inter-regional migration, employment,
health statuses, etc.
The percentage of the life span spent in different marital/union statuses is defined
by the person-years spent in union status i among all members of the life table cohort
divided by their total number of person-years regardless of union status. The other major
summary measures of the multistate marital/union status life tables are the propensities of
marital/union status transitions, average life-time numbers of cohabitations (including
cohabitations before first marriage and after marriage dissolution), marriages (including
first marriage and remarriages), and divorces. The propensities of marital/union status
transitions are defined as the total number of transitions from union status i to j divided
by the total number of transitions into union status i, in the context of a hypothetical
cohort for period analysis or a real cohort for cohort analysis. The propensities of union
status transitions are clearly defined and easily understood demographic summary
measures, and represent the overall average intensity of occurrence of the union status
transitions among the at-risk population. For example, if the female propensity of
15
divorce in a period is 0.4, then this means that 40 percent of all marriages would
eventually end in divorce, given that a hypothetical cohort experienced the observed
female occurrence/exposure (o/e) rates of union formation and dissolution in the period.
The average number of cohabitating unions, marriages, and divorces over a person’s life
time is defined as the total number of events of cohabitations, marriages, and divorces
occurred to all members of the life table cohort during their lives divided by the initial
size of the life table cohort. The initial size of the life table cohort is called radix; it is
usually a round number such as 1 or 1,000 or 100,000.
In summation, as compared to the ordinary single-state life table, a key property
of multistate life tables is that, within an age interval, they permit individuals exit from
the state they occupied at the beginning of the age interval to another state and then to
return to the original state (Land et al. 2005). In the multistate life table models of
demographic processes, individuals may make multiple transitions within age intervals,
such as from non-married to married and back to non-married again. This is an
indispensable property and much better than the ordinary single-state life table, which
does not allow re-entry to the previous status. Multi-state life tables condense the
unwieldy sex-age-specific o/e rates of status transitions into several interpretable
summary measures. These life table summary measures are much less affected by
distortions inherent in the conventional period crude rates based on the total numbers and
total rates of union formations and dissolutions based on the age-specific frequencies,
which are not capable of distinguishing between at-risk and non-risk populations.
2.6 Additional considerations
16
Given the fact that an event may be considered as a transition from one state to
another, multistate demography provides a useful modeling framework for event history
data analysis, which has been widely applied in recent years. The statistical model
specifications via statuses transition intensities and likelihood inference are introduced in
the framework of multistate models. Examples of applying the basic principles and
methods of multistate demography for event history analysis include the multistate
models for survival analysis, the competing risks and illness-death models, and models
for bone marrow transplantation, etc. Andersen and Keiding (2002) presented an
overview on applications of multistate models for event history analysis in the fields of
medical research, and as a concrete illustration, they discussed the empirical analysis
concerning mortality and bleeding episodes in a liver cirrhosis trial.
3. The Multistate Projection Models
Recall that the discrete projection model of single-state demography enables one
to carry forward a population on the basis of a predetermined demographic regime
embodied in a set of age-specific fertility and mortality rates. It is a rather straightforward
matter to extend it to a multistate setting as shown by Rogers (1975, 1995) in the context
of interregional migration.
3.1 Multistate projection and stable growth
Just, like the ordinary projection model, it relies over each projection interval on a
matrix operation in which a population, here set out as a vector of age-specific
individuals in various states of a given attribute (region of residence, marital status,
17
etc…), is multiplied by a growth matrix—in fact, a generalized version of the Leslie
matrix of the single-state projection model—which reflects the demographic regime at
hand embodied in a set of age-specific fertility, mortality and mobility rates pertaining to
each state. Basically, such a multiplication results in a vector of the state- and agespecific survivors which also includes the new births that survive to the end of the time
interval, with the process being repeated on as many intervals as needed.
In particular, if the process is repeated an infinite number of times while keeping
the demographic regime unchanged, the multistate population being projected will
ultimately reach a stable state—that is, it will increase at a constant rate of growth and
achieve a fixed state- and age-specific composition. Such an outcome can also be
obtained using a continuous rather than a discrete formulation of the projection process,
thus leading to a multistate extension of i) the Lotka model and ii) the stable population
theory associated to this model (see Rogers, 1975 for such an extension in the context of
multiregional population growth).
In practice, however, the demographic regime considered need not be invariant
and thus it can change from a projection interval to the next to reflect extrapolation of
past and current trends with the idea of identifying alternative demographic futures and
anticipating the problems and needs associated to such futures.
3.2 Multistate population projections with time-varying input
The earliest application of the multistate projection model was the multiregional
projections of population by age, sex and place of residence by Andrei Rogers, Frans
Willekens and Jacques Ledent in the 1970s and 1980s at the Population Program of
18
International Institute for Applied Systems Analysis (IIASA). A remarkable extension of
the multistate projection methodology and applications at IIASA in the past decade has
been the “Multistate Population Projections by Education Attainment (MPPEA).” As
shown in Figure 1 cited from Lutz and Goujon (2001: 325), MPPEA explicitly considers
the important fact that people with different education levels tend to have different
fertility and mortality levels. Projections of educational attainment levels are based on
two different forces: for those who are still in the process of educational attainment, the
model projects the transitions from one educational level to the next higher level by age
and sex. For those who have already completed their highest level of education, the
projection simply increases their age as time passes by and exposes them to educationspecific mortality, fertility and migration rates. The MPPEA projection model explicitly
and clearly demonstrates the process of translation from schooling outcomes to human
capital, measured by the average educational attainment of the entire work force. The
projections clearly illustrate the long-term implications of near-term investments in
education or, on the negative side, the consequences of the lack of such investments.
19
Figure 1. Structure of multi-stage population projection model by level of education
Source: Wolfgang Lutz and Anne Goujon (2001). The world’s changing human capital
stock: multistate population projection by educational attainment. Population and
Development Review, 27(2): 325.
Starting in the early 1990s, the multistate population projection model was
extended to family/household simulation and forecasting, such as the LIPRO model
developed by Van Imhoff and Nico Keilman, which was applied in the Netherlands and a
couple of other Western European countries. Most multistate projection models for
family/household simulation and forecasting use the household as the basic unit of
analysis and require data on transition probabilities among household-type statuses—data
that have to be collected in special surveys because they are not available in vital
statistics, censuses, or ordinary surveys (Van Imhoff and Keilman 1992). As stated by
Van Imhoff and Keilman (1992), such a stringent data demand is an important factor in
the slow development and infrequent application of these models. Due to the aggregated
20
nature of the household statuses identified in these models, it is impossible to make
inferences about groups other than those based on the predefined household-type state
space. The distribution of households by size is impossible to project unless the size of
the household is explicitly incorporated into the state space. Incorporation of household
size into the household-type state space would greatly increase the size of the transition
probabilities matrices at each age for males and females, which is likely not feasible in
practical applications. Furthermore, the household-type status-transition model cannot
directly link changes in household structure with demographic rates. Thus, it is difficult
for such a model to identify the impacts of demographic factors on changes in household
structure.
Benefiting from methodological advances in multistate demography, Bongaarts
(1987) developed a nuclear-family-status life table model using the individuals as the unit
of analysis. Perhaps the most significant innovation of Bongaarts’ family household
model using individuals as the unit of analysis is that it substantially relaxed the stringent
data demand of the household projection models which use household as the unit of
analysis and require usually unavailable data on age-sex-specific transition probabilities
among different household types. In the family household models using individuals as the
unit of analysis and conventional demographic rates at input, it is theoretically possible
for an individual to have any realistic combination of the statuses identified in the model.
We may call the combination the composite state (e.g. a composite state of currently
married, with parity three, living with two children and one parent). Let li(x,t) denote the
number of persons of age x with composite state i (i=1, 2 ... N) in year t. Let Pij(x,t)
denote the probability that a person of age x with composite state i in year t will survive
21
N
and be in composite state j at age x+1 in year t+1. Thus, lj(x+1,t+1)=

li(x,t) Pij(x,t)
i 1
If Pij(x,t), which are elements of a N x N matrix, were properly estimated, the
calculation of lj(x+1,t+1) would be straightforward. Unfortunately, the estimation of
Pij(x,t) is usually not practical when the total number of composite states N is large,
which is the case in the family household projection models. For instance, if seven
marital/union statuses, three statuses of co-residence with parents, six parity and six
statuses of number of surviving and co-residing children are distinguished, as in the case
of households projection for the United States (Zeng, Land, Wang, and Gu, 2006), the
total number of composite statuses distinguished at each age for each sex in this
illustrative application is:
5
N=7 x 3 x

(p+1) = 4411. The total number of cells in the transition matrix is thus 441
p 0
x 441 = 194,481. There would be one such large matrix for males and females,
respectively, at each single age. Although there are many zero cells in the matrices, the
number of non-zero cells to be estimated is still much too large and the estimation of such
large transition matrices is not practical. Bongaarts overcame the difficulty by assuming
that particular events take place at particular points in time between age x and x+1
(Bongaarts, 1987: 209-211). In particular: 1) Births occur throughout the first half and
the second half of the year; the birth probabilities used refer to the corresponding half
year. They depend on the status at the beginning and middle of the year, respectively. 2)
1
Note that although we have distinguished 6 (0 to 5) parity statuses and 6 (0 to 5) statuses
5
of number of co-residing children, we use

(p+1) instead of 6 x 6 to compute the total
p 0
number of composite statuses of parity and number of co-residing children, because the
number of co-residing children is always less than or equal to parity in the model.
22
Deaths, migration, changes in status of co-residence with parents, marital status
transition, and changes in number of surviving and co-residing children due to children's
death or leaving or returning home occur at the middle of the year. These probabilities of
occurrence of the events refer to the whole year and depend on the status at beginning of
the year. One calculates probabilities only for those transitions triggered by the specific
demographic events included in the projection model – births, deaths, marriage/union
formation and dissolution, remarriage and reunion, leaving and returning to parental
home, migration, and so on. As shown mathematically and numerically by Zeng (1991:
61-63, 81-84), the strategy of assuming births occur throughout the first and the second
half of the year and other status transitions at the middle of the year leads to accurate
numerical estimates.
Based on Bongaarts’s nuclear-family-status life table model and Zeng’s generalfamily-status life table model including both nuclear and extended families, Zeng, Vaupel,
and Wang (1997) developed a two-sex dynamic macro projection model known as
ProFamy that permits demographic schedules to change over time and requires only
conventional data that are available from ordinary surveys, vital statistics and censuses.
Zeng et al. (2006) extended the ProFamy model by adding cohabitation and race
dimensions to all computation and estimation procedures. In addition to statuses defined
by the number of co-residing children and parents and parity, the extended ProFamy
family household projection model includes seven marital/union statuses: (1) Nevermarried & not-cohabiting, (2) Married, (3) Widowed & not-cohabiting, (4) Divorced &
not cohabiting, (5) Never-married & cohabiting, (6) Widowed & cohabiting, and (7)
Divorced & cohabiting. Validation tests of household forecasts from 1990 to 2000 for the
23
United States as a whole, and the 50 states, the District of Columbia, Orange & Chatham
Counties and the Town of Chapel Hill in North Carolina, respectively, show that the
ProFamy multistate model/software for household forecasting at national, state and substate levels works reasonably well.
In addition to the projections of residential regions, educational attainments and
family households summarized above, the multistate projection models can also be
readily applied to other interesting and useful sub-fields as well. The applications include
multistate projections of working and unemployment groups, health statuses and care
costs of elderly, income categories of the population, “birth,” survival/growth, and
decline/”death” of companies, and sub-populations with different political attitudes and
ages, etc.
4. Applications of Multistate Demography
The earliest milestone research on multistate demography include work by Andrei
Rogers (1975) on multiregional mathematical demography, and by Schoen (1975),
Schoen and Land (1979), and Willekens et al. (1982) on nuptiality. Due to space
limitations, we will summarize here only a few illustrative examples with some
subsequent and useful refinements in techniques used for constructing multistate life
tables and carrying out multistate projections
4.1. Multistate working life tables
Hoem (1977) introduced an application to multistate working life tables that used
large-sample survey data rather than population-level data. His study demonstrated the
24
proportion of the adult life time spent in being employed and unemployed, as well as the
average number of lost jobs in the lifetime, given the observed age-specific rates of labor
force participation and unemployment. They discovered that even with relatively large
surveys, sub-sample sizes for the estimation of single-year-age and sex specific transition
probabilities may become too small. The result is more age-to-age variability in the
estimated transition probabilities. Accordingly, Hoem introduced a method of smoothing
the estimates of age-specific transition probabilities by moving averages across the age
intervals. Moving average and related methods of graduation or smoothing estimates of
transition probabilities across age intervals have subsequently been used for the
estimation of labor force participation life tables by Schoen and Woodrow (1980), voting
in U.S. Presidential elections by Land, Hough, and McMillan (1986), and tables of school
life by Land and Hough (1989).
4.2. Active and disabled life expectancies
As reviewed by Land et al (2005), a particularly active area of methodological
development and empirical applications of multistate life tables has been the estimation
of years of active/disabled (or healthy/unhealthy) life among the elderly population. The
initial definition of active life expectancy in terms of an absence of limitations in
activities of daily living was given by Katz et al. (1983). The standard life table method
for estimating active life expectancy used by these authors is a prevalence-rate based life
table method of decomposing the nLx column of a population life table into years lived in
an active state and years lived in an inactive or disabled state, a method presented by
Sullivan (for an exposition, see Molla, Wagener, and Madans 2001).
25
Due to its minimal data requirements (the existence of a life table of the
population under study and a set of age-specific prevalence rates or proportions of
persons identified as either in the active or inactive states), the Sullivan method has been
widely applied and remains useful today (see Crimmins, Saito, and Ingegneri 1997;
Cambois, Robine, and Hayward 2001).
However, Rogers, Rogers, and Branch (1989) noted that one critical limitation of
the Sullivan prevalence-rate-based life table method is its static nature -- it does not
estimate the dynamic transitions between active and inactive states and it could not reveal
the large difference in active/disabled life expectancies between those who are active and
those who are disabled at the initial age. Using data on age-specific transition rates
among the active and inactive statuses from longitudinal panel surveys of the elderly,
they applied multistate life table methods to the estimation of active life expectancy.
Subsequent research (Crimmins, Saito, and Hayward 1993) indicates that the Sullivan
method for estimating the overall active life expectancy (disregarding the status at initial
age) generally yields estimates that do not differ greatly from those obtained by
multistate models. Nonetheless, when data on transitions among the states are available,
the methodological and empirical research accumulated over the past two decades
suggests that multistate life table models should be applied.
The multistate life table model has been applied to analyze the active/disabled or
healthy/unhealthy life expectancy across sub-populations with different covariates in
addition to the traditional age and sex dimensions (Land et al. 1994). For example,
Minicuci and Noale (2005) estimated disability-free life expectancy among older Italians
by different levels of education. They found that men and women with more than four
26
years of education at age 65 were expected to live three more years of their remaining life
without disability than those persons with low education. They also found that the
positive effect of a medium/high educational level increased with increase in ages and
degree of severity of disability, and the effect was more pronounced among men than
women. Kaneda et al. (2005) estimated expected years of active life by socioeconomic
status (SES) for older adults in Beijing. They defined active life as a state without any
assistance in eating, dressing, getting on and off a bed, bathing, walking 300 meters, and
walking up and down a flight of stairs. They found that the active life expectancy of the
old men of higher SES was 30-60 percent higher than those with lower SES. The active
life expectancy of the female elderly of higher social status (measured by education and
occupation) was significantly higher than their counterparts with lower social status, but
the difference by income was not statistically significant. Their results further revealed
that the disparities in active life expectancies by SES generally increased with increase in
ages.
Research on improving the methodology concerning the measurements and
estimation procedures of multistate active/disabled life table models have also received
increasing attention in recent years. Lynch et al. (2003) developed a Bayesian approach
to estimating confidence intervals for active/disabled life expectancies to account for
parametric uncertainty and sampling error within the context of multistate life table
models. They applied and verified their new approach through empirical analysis using
the data on community-dwelling older adults from the U.S. 1989 and 1994 National Long
Term Care Survey. They found that altering the threshold for measuring disability has
relatively little effect on active life expectancy estimates, especially at older ages.
27
Research by Zeng, Gu, and Land (2004) demonstrates that the disabled life expectancies
based on conventional multistate life table methods are significantly underestimated due
to an assumption of no functional status changes between age x and the time of death
occurred between ages x and x+n. They presented a new method to correct the bias and
applied it to 1998 and 2000 longitudinal survey data of about 9,000 oldest old Chinese
aged 80-105. The results show that estimates of active life expectancy can be improved
if data on the disability status of the elderly in the month or so prior to death are
available. These data permit a more accurate estimation of individuals’ functional status
in the time period between a last wave of panel interviews and death.
4.3. Multistate life table analysis on marriage/union formation and dissolution
Robert Schoen and his colleagues have published a series of important and
influential articles in 1985, 1987, 1993, 2001 and 2006 on multistate life table analysis of
U.S. marriage, divorce, and remarriage, based on vital statistics and other types of data.
In their 2001 article, Schoen and Standish analyzed the marriage formation and
dissolution trend from 1970 to 1995 for all races combined. They found that the life table
percentage ever marrying among those men surviving to age 15 declined from 96 percent
in 1970 to 89.2 percent in 1980, and 83.1 percent in 1995, while the corresponding
figures for women were 96.5, 90.8, and 88.7 percent, respectively. The age-specific rates
observed in 1970, 1980, and 1995 implied that 37.3, 44.4, and 43.7 percent of the male
marriages and 35.7, 42.9, and 42.5 percent of female marriages would eventually end in
divorce – the U.S. divorce propensity reached a peak in 1980 and remained at a high level
with slight decline thereafter.
28
While Schoen’s pioneering research employing the multistate life table model has
made groundbreaking contributions to a better understanding of the dynamics of
marriage, divorce, and remarriage from a life course perspective, it was somewhat limited
by data. As the vital statistics data have serious problems concerning consistency of
race/ethnicity classifications in the numerators and denominators of the marital status
transition rates, Schoen’s analysis did not include the racial differentials (Schoen and
Standish, 2001). Moreover, cohabitation was excluded in Schoen’s analysis, because
cohabitation data were not available in vital statistics or in the Current Population Survey
used by Schoen’s studies. A most recent study by Zeng, Morgan and colleagues (2008)
estimates trends and racial differentials in marriage and cohabitation union formation and
dissolution in the 1970s, 1980s, 1990s, and 2000-2002 in the United States, based on
multistate life table synthetic cohort analysis and pooled survey datasets with a large
sample size of 97,787 men and 304,526 women 2 . They document both change and
stability, such as huge increases in the propensity of cohabitation and in the average
number of cohabiting unions over a person’s life time, significant decreases in the
propensity of marriage, and stability in both the likelihood of divorce and in a number of
important racial differentials. For example, based on the multistate marital/union status
life table synthetic cohort analysis, Zeng, Morgan and colleagues estimated that the
lifetime propensities of cohabitation of American women before their first marriage were
2
The pooled datasets include the event history data of marriage/union status changes
from the following national surveys: (a) National Survey of Families and Households
(NSFH) conducted in 1987-1988, 1992-1994, and 2002. (b) National Survey of Family
Growth (NSFG) conducted in 1983, 1988, 1995, and 2002. (c) Current Population
Surveys (CPS) conducted in 1980, 1985, 1990, and 1995. (d) Survey of Income and
Program Participation (SIPP) conducted in 1996. Please refer to Zeng, Morgan and
colleagues (2008) for justifications of pooling the data from the four surveys.
29
25, 46, 60, and 61 percent based on the rates observed in the 1970s, 1980s, 1990s, and
2000-2002, respectively3. The corresponding figures for American men were 33, 55, 66,
and 70 percent4. The numbers of events of cohabitations over lifetime per woman and per
man in 2000-2002 were estimated as 1.15 and 1.36, which are about 2.5 times as large as
those in the 1970s. They found that in contrast to the large difference between Blacks and
the other three race groups, the racial differentials in union formation and dissolution
between Whites, Hispanics and Others are generally modest, except that widowed and
divorced Whites had a distinctly higher propensity of transition to cohabitation and
remarriage.
4.4. Multistate population projections by educational attainment
The program led by Wolfgang Lutz at IIASA and the Vienna Institute of
Demography has so far carried out multistate population projections by educational
attainment for 120 countries according to four scenarios: Constant absolute enrollment
scenario; constant enrollment rate scenario; global trend scenario; and fast-track scenario
(http://www.iiasa.ac.at/Research/POP/humancapital.html). Education is considered to be
the best means of improving the quality of human resources, thereby promoting
economic development. Also, education is a crucial variable in explaining differences in
fertility behavior and differential patterns of mortality and migration. Therefore,
applications in multistate population projections by level of education are an important
3
Ref. to Section 2.7 for definition of the propensities of marital/union status transitions.
As explained in Appendix A of Zeng and Morgan, et al. (2008), the lifetime propensity
of cohabitation before first marriage is NOT equal to proportion of ever-cohabiting
before 1st marriage, and the lifetime propensity of cohabitation is likely somewhat higher
than the proportion of ever-cohabiting.
4
30
step forward in improving population forecasting and its practical usefulness. Adding
education to age and sex as an explicitly included demographic dimension in population
forecasting affects the demographic output parameters themselves due to the fact that
fertility, mortality and migration largely depend on education levels. More importantly,
however, due to the overriding substantive importance of education, forecasting the
future educational composition of the population is of interest in its own right as it shapes
the human capital component of population.
4.5 Multistate projections of households and living arrangements
Using data from national surveys and vital statistics, census micro files, and the
“ProFamy” multistate family household projection method, Zeng et al. (2006) published
the multistate projections of U.S. family households and population by four race groups
(White & Non-Hispanic, Black & Non-Hispanic, Hispanic, Asian and Others & NonHispanic) from 2000 to 2050. Medium projections, smaller and larger family scenarios
with corresponding combinations of assumptions of marriage/union formation and
dissolution, fertility, mortality, and international migration, were performed to analyze
future trends of U.S. households and their possible higher and lower bounds, as well as
the racial differentials. In addition, the ProFamy multistate model for projecting family
households and population has been used to generate household automobile consumption
forecasting in different regions in the United States by Wang et al., household automobile
consumption forecasting in Austria by Prskawetz, Jiang, and Brian, German households
and living arrangement projections by Hullen, and family household projections by
various scholars in several other countries.
31
One of the interesting findings of the multistate family household projections by
Zeng et al. (2006) is that it has provided empirical evidence for the first time to
numerically illustrate the concept of family household momentum. In the multistate
family household projections, although the marriage/union formation and dissolution
rates are assumed to remain constant during the entire period of 2000-2050, the
distributions of households and elderly living arrangements in the first two decades of the
50-year projection period would change considerably, because the older cohorts, who had
more traditional family patterns, will be replaced by the younger cohorts with modern
family patterns. The new concept of family household momentum is similar to Keyfitz’s
original concept of population momentum, in which population size could continue to
increase after the fertility is equal to or even below the replacement level.
4.6 Modeling and estimation of the age and spatial structures of migration flows
Finally, before closing this section on the applications of multistate demography, let us
mention some work of a different nature which has been carried out in the last two
decades in the context of interregional migration. Basically, this work has been devoted
to the derivation of good estimates of the data, otherwise missing or poor, that are needed
for constructing multiregional life tables or carrying out multiregional projections (see
Rogers, 2008). In a nutshell, it has relied on the use of models to structurally infer the
parameters that are needed to obtain the good estimates sought. Two models are here
particularly relevant. One model describes age patterns of migration by means of model
migration schedules. The other describes spatial interaction patterns and has proved to
lead to a formal approach to the estimation of migration.
32
5. Bridging the Micro- and Macro-Simulation Models -- Recent Development of
Multistate Demography
Demographers use micro- and macro- models for simulations and projections of
the population as well as its disaggregation and decomposition. Such micro- and macro
models are usually referred to as micro-simulation and macro-simulation models. The
micro-simulation model simulates the life course events and keeps detailed records of
demographic status transitions for each individual of the sample population. Following a
micro-simulation approach, the SOCSIM model developed by Hammel and Wachter, the
KINSIM model developed by Wolf, the MOMSIM model developed by Ruggles, and other
efforts of this type have made important contributions in studying kinship patterns and
family support for elderly persons.
As compared with the macro-simulation approach, micro-simulation offers three
major advantages: it can handle a large state space with many covariates, the relation of
individuals can be explicitly retained, and it provides richer output including the
probabilistic (and stochastic) distribution of outcomes. It is particularly powerful in
complex kinship simulations and projections, which the macro-simulation methods
cannot do.
As always, however, these advantages are bought at a cost. Three kinds of random
variations in micro-simulation have been discussed in detail in the literature (see, for
example, Van Imhoff and Post, 1998). The first is due to the nature of Monte Carlo random
experiments: different runs of the same model and input produce different sets of outcomes.
This inherent stochasticity can be reduced by increasing the sample size or by taking an
33
average over a large number of runs, but it cannot be wholly eliminated. In the second
variation, the starting population of the micro model is a sample from the total population.
Thus, it is subject to classic sampling errors, especially for small sub-groups such as oldest
old persons and very young teenage mothers. The third variation is that micro-simulation,
which includes many explanatory variables and complex relationships among individuals,
increases the stochasticity and measurement error to which the model outcomes are subject.
Some scholars call this kind of bias “specification randomness.” A correctly specified
micro-simulation model, with many explanatory variables and complex relationships, may
significantly increase the specification randomness (Van Imhoff and Post, 1998: 111).
The unit of macro-simulation models is the group, for instance a group of
persons of the same age and parity and/or marital status, for which the calculations
proceed iteratively, group by group and time period by time period, generally by means
of status transition probabilities. Although not as flexible as micro-simulation models in
analyzing variability and probability distributions and simulating relationships among
individuals, macro-simulation models do not have the problems of the inherent random
variations of Monte Carlo experiments and sampling errors in the starting population. It
can fully and effectively use the data from a census or a large survey as a starting point.
The specification randomness is also present in a macro model, but is likely to be less
serious than that in a micro-simulation model, since it is much less tempting to introduce
additional complications (Van Imhoff and Post, 1998: 111). Some scholars suggest that
adding more covariates to a projection model may improve forecasts, but others think that a
very complicated projection model may not perform as good as a simpler one does.
Furthermore, it is much easier to develop a macro-simulation model into a user-friendly
34
demographic tool of computer software, for use by researchers and policy analysts, who lack
sophisticated mathematical and programming expertise. Planners and policy analysts can
conduct macro-simulation projections relatively easily if an user-friendly computer
software is provided.
While both micro- and macro-models include multiple states and their dynamic
transitions, the choice between a micro- or macro-model obviously depends on the
complexity of the user’s task. For detailed analyses of behavioral patterns and complex
family kinship relationships, a micro-simulation approach may be preferable. For relatively
straightforward demographic and household consumption projection based on commonly
available data for the purposes of policy analyses, market trends studies, and socioeconomic
planning, a macro-simulation approach may well be satisfactory.
Although almost all of the earlier authors in the field did not consider the microsimulation models as part of multistate demography, its applications in demography do
deal with the study of populations at the individual level stratified by more than one
attribute and the transitions between the multiple states including exits, entries and reentries to the states. Such fundamental similarity to the multistate macro-models and the
complementary strengths between the micro-and macro- simulation models as discussed
above have led to the recent development of the multistate model for biographic analysis
and projection, which integrates both micro- and macro- simulation models into one
framework of modeling (Willekens, 2005). A directly related piece of good news is that
after years of discussion (and sometimes even debates), the mainstream in the fields of
multistate demography and micro-simulations are now very much in favor of and committed
to bridging the micro- and macro- simulation models for further developing the multistate
35
demography. The new European multi-institutional MicMac project led by Frans Willekens
combines the micro- and maco- simulation/projection approaches to produce forecasts of the
population in multidimensional states (defined by age, sex, health, and labor force
participation, etc.) as well as the time spent in those states. The macro-simulation involves
the construction of the cohorts biographies at the population level using the census and other
large data sets as a base, while the micro-simulation based on the sampling data produces
more detailed synthetic individual biographies which may not be produced by the macro
models. The results from the micro-simulation at the individual level are then aggregated to
determine the population characteristics. The bridge between the cohort biographies (macrosimulation) and individual biographies (micro-simulation) is the common use of transition
rates (occurrence/exposure rates) estimated from the survey data while controlling for the
confounding covariates. Thus, the dynamics at the macro- and micro levels are internally
consistent. The age-status-specific transition rates are estimated by regressions of the event
history analysis or by direct demographic computing if the sample sizes of the sub-groups
are sufficient. The summary measures of the future trends of transitions are estimated or
forecasted either by time-series regression or expert-opinion evaluations. Thus, structural
modeling is being incorporated into demographic forecasting. Clearly, the MicMac
approach is complementary and pragmatic: it makes use of causality where possible, and
otherwise of statistical association (Booth, 2006). Further complementarities of the microand macro- models are likely to be found in the estimation of uncertainty and stochastic
confidence intervals of the multistate demographic forecasting.
Glossary
36
AGE-SPECIFIC RATE. A rate calculated to express the incidence of a demographic
process at a given age or within a specified age group. The size of the age category used
varies according to the phenomenon studied, but five-year age groups are most common.
COHORT. A group of persons who experienced the same kind of event during a given
period of time. For example, a birth cohort is a group of persons born during a given
period of time
EVENT-HISTORY. A detailed account of a person’s experience of one or more
demographic processes. The term life-history is used synonymously. These data are often
collected in retrospective surveys and may refer to the lifetime up to the time of the
survey. Complete information on all aspects of a person’s life is rarely collected, most
histories being limited to date of a single or several aspects of the life.
HYPOTHETICAL COHORT. A theoretical concept by which measures can be
calculated for a particular period analogous to those calculable for a true cohort. The
resulting construct is said to refer to a hypothetical cohort. Terms fictitious, fictive or
synthetic cohort are also used with the same meaning.
LIFE EXPECTANCY AT AGE X. The average number of additional years a person
would live beyond age x, if the observed age-specific mortality rates were applied to a
hypothetical cohort. Life expectancy at birth (i.e. age 0) is very widely used as a overall
indicator of mortality conditions of a population.
LIFE SPAN. A characteristic of life history, which refers to the duration of an
organism’s entire life. Application of the concept is straightforward at both individual
and cohort levels. At the individual level, it is the period between birth and death; at the
cohort level (including both real and synthetic cohorts), it is the average length of life or
life expectancy at birth.
MACRO-SIMULATION MODEL. The unit of analysis for macro-simulation models is
a group, for instance groups of persons of the same age and parity and/or marital status,
for which the calculations proceed iteratively, group by group and time period by time
period, generally by means of status transition probabilities.
MICRO-SIMULATION MODEL. The micro-simulation model simulates life course
events and keeps detailed records of demographic status transitions for each individual of
the sample population.
SINGLE-STATE LIFE TABLE. A tool that follows the life course of a hypothetical or
real birth cohort until the death of its last survivor, as it gradually decreases in size under
the effect of mortality. Basically, such a table includes several columns which document
how various functions relating to the life and death of the cohort evolve over successive
ages, usually (but not necessarily) equally spaced n years apart. The concept and the
methods for constructing the single-state life table of a birth cohort are also valid for
37
other non-renewable events such as first marriage, first birth, and first participation in
labor force, etc.
MULTISTATE LIFE TABLE. Life table that allows for re-entry into a previously
occupied state, and it is concerned with not only exits (decrements) but also entries
(increments). Initially known as increment-decrement life tables, they are today more
commonly referred to as multistate or sometimes multidimensional life tables.
MULTISTATE DEMOGRAPHY. Multistate demography studies populations stratified
by more than one attribute. It deals with transfers between alternative categories (i.e.
states or statuses) of population attributes, including entries and re-entries to the states.
The states can be regions of residence, marital/union statuses or employment status, and
so forth.
NUCLEAR FAMILY. The unit consisting of husband (father), wife (mother), their
children, and siblings if any. The relationships involved are those basic to kinship
reckoning: husband-wife, parent-child and inter-sibling. The terms elementary family,
immediate family and conjugal family are also used to refer to this basic grouping of kin
relations.
PERSON-YEARS. The sum, expressed in years, of the time spent in a given category by
the individuals comprising the population. Person-years can be used as the average
population in a given category or exposed to a certain risk.
THE PROPENSITY OF MARITAL/UNION STATUS TRANSITION. The total
number of events of transition from union status i to j divided by the total number of
events that lead to entering union status i, in the context of a hypothetical cohort for
period analysis or a real cohort for cohort analysis.
Bibliographies
Andersen, P. K. and Keiding N. (2002). “Multi-state models for event history analysis”,
Statistical Methods in Medical Research, 11: 91-115 [provide an overview of how
multistate models often provide a relevant modeling framework for event history
analysis].
Bongaarts, J. (1987). ‘The projection of family composition over the life course with
family status life tables’, Pp. 189-212, in: J. Bongaarts, T. Burch and K.W. Wachter,
(Eds.), Family Demography: Methods and Applications. Clarendon Press, Oxford.
[Presenting a way to construct and apply the nuclear family status life table]
Booth, H. (2006). ‘Demographic forecasting: 1980 to 2005 in review’, International
Journal of Forecasting, 22: 547-581 [Reviewing and assessing approaches and
developments in demographic and population forecasting since 1980] .
Cambois, E., Robine, J.M. and Hayward, M.D. (2001). ‘Social inequalities in disabilityfree life expectancy in the French male population, 1980-1991’, Demography 38(4):513-
38
524 [Calculating aggregate indicators of population health for occupational groups to
gauge changes in health disparities during the 1980-1991 period, based on the
experiences of French adult men in three major occupational classes: managers, manual
workers, and an intermediary occupational group]
Crimmins, E.M., Saito, Y. and Hayward, M.D. (1993). ‘Sullivan and multistate methods
of estimating active life expectancy: two methods, two answers’, Pp 155-160 in: J.M.
Robine, C.D. Mathers, M.R. Bones and I. Romieu (Eds.) Calculation of Health
Expectancies: Harmonization, Consensus Achieved and Future Perspectives. Libbey
Eurotext, Paris [Discussing the difference of the Sullivan and multistate approaches in
estimating the active life expectancy]
Crimmins, E.M., Saito, Y. and Ingegneri, D. (1997) ‘Trends in disability-free life
expectancy in the United States, 1970-90’, Population and Development Review, 23(3):
555-572 [Readdressing the trend of disability-free life expectancy in the United States by
adding information of 1980-1990 to the existing 1970-1980 estimates].
Du Pasquier, L. G. (1912/13). “Mathematische theorie der invaliditatsversicherung” Mitt.
Verein. Schweiz. Versich.-math. 7-7:1-7 and 8:1-153.
Hannan, M. (1984). ‘Multistate demography and event-history analysis’. Pp. xx-xx in A.
Diekmann and P. Mitter. Stochastic Modeling of Social Processes. Academic Press,
Orlando, Florida.
Hoem, J.M. (1977). A Markov chain model of working life tables. Scandinavian
Actuarial Journal, 1: 1-20.
Kaneda, T., Zimmer, Z. and Tang, Z. (2005). ‘Socioeconomic status differentials in life
and active life expectancy among older adults in Beijing’, Disability and Rehabilitation,
27(5): 241-251 [Comparing active life expectancy estimates across indicators of
socioeconomic status for a cohort of older adults in the Beijing municipality and
confirming the existence of inequality].
Katz, S., Branch, L.G., Branson, M.H., Papsidero, J.A., Beck, J.C. and Greer ,D.S. (1983).
‘Active life expectance’, New England Journal of Medicine, 309:1218-1223 [Using lifetable techniques, this paper analyzes the expected remaining years of functional wellbeing, in terms of the activities of daily living, for noninstitutionalized elderly people
living in Massachusetts in 1974].
Land, K.C., Hough, G.C., McMillen, M.M. (1986). ‘Voting status life tables for the
United States, 1968-1980’, Demography, 23(3): 381-402 [Applying multistate methods to
construct voting status life table and analyzing the resulting tables for differentials by sex,
race, and election period]
Land, K.C., Guralnik, J.M. and Blazer, D.G. (1994). ‘Estimating increment-decrement
life tables with multiple covariates from panel data: the case of active life expectancy’,
Demography, 31(2): 297-319 [To estimate tables from data on small longitudinal panels
in the presence of multiple covariates, this paper presents an approach based on an
isomorphism between the structure of the stochastic model underlying a conventional
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specification of the increment-decrement life table and that of Markov panel regression
models for simple state spaces].
Land, K.C. and Hough, G.C. (1989). ‘New methods for tables of school life, with
applications to U. S. data from recent school years’, Journal of the American Statistical
Association, 84(405): 63-75 [Specifying new methods for the estimation of school-life
tables].
Land, K.C. and Rogers, A. (1982). Multidimensional Mathematical Demography.
Academic Press, New York [Presenting recent research advances and successful
substantive applications in the field of multidimensional mathematical demography].
Land, K.C., Yang, Y., and Zeng, Y., 2005. ‘Mathematical demography’, Pp.659-717 in
D. L. Poston, Jr. and M. Micklin (Eds). Handbook of Population. Springer Publishers,
New York [Introducing the mathematical demography].
Ledent, J. and Bah, S. (2005) ‘Applying the multiregional projection model using census
migration data: A theoretical basis’ Pp. 327-353 in P. Kok, D. Gelderblom, J. Oucho and
J. van Zyl (Eds) Patterns and Causes of Migration in South and Southern Africa. Human
Sciences Research Council, Cape Town, South Africa.
Ledent, J. and Rees, P. (1986). ‘Life tables’ Pp. 385-418 in A. Rogers and F. Willekens
(Eds). Migration and Settlement: A Multiregional Comparative Study, D. Reidel,
Dordrecht and Boston..
Lutz, W. and Goujon, A. (2001). ‘The world’s changing human capital stock: multistate
population projection by educational attainment’, Population and Development Review,
27(2): 325 [Presenting the first global population projections by educational attainment
using methods of multi-state population projection].
Lynch, S. M., Brown, S. J. and Harmsen, K. G. (2003). ‘The effect of altering ADL
thresholds on active life expectancy estimates for older persons’, Journal of Gerontology:
SOCIAL SCIENCES, 58(3): S171-S178 [Demonstrating that disability measurement,
including altering the definition of being disabled and possibly expanding the state space
of a model, may not affect population-based estimates of active life expectancy].
Minicuci, N. and Noale, M. (2005). ‘Influence of level of education on disability free life
expectancy by sex: the ILSA study’, Experimental Gerontology, 40: 997-1003 [Assessing
the effect of education on Disability Free Life Expectancy among older Italians].
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21: 1-11. [Presenting a comprehensive discussion of the methods for calculation and
methodological issues related to the interpretation of healthy life expectancy].
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Rajulton, F. (2001). ‘Analysis of life histories: a state-space approach’, Special Issue of
Longitudinal Methodology-Canadian Studies in Population, 28(2):341-359 [Presenting
the LIFEHIST, a computer program to analyze life histories through a state-space
approach].
Rogers, A. (1975). Introduction to Multiregional Mathematical Demography. Wiley,
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Rogers, A. (1995). Multiregional Mathematical Demography: Principles, Methods, and
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of multiregional population systems with many numerical illustrations based on empirical
data collected in Belgium, India, Mexico, the former Soviet Union, Sweden and the
USA] .
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population: A multirefional approach’, Geographical Analysis, 40:276-296 [Focuses on
the development and evolution of migration and population redistribution modeling
within the context of multiregional demography].
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Demography 13:287-290. [Restates in a matrix format the linear estimator of the
transition probability matrices in a multistate life table]
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expectancy’, Public Health Reports, 104(3): 222–226. [Introducing the multistate life
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Rogers, A. and Willekens, F.J. (1986). Migration and Settlement: A Multiregional
Comparative Study, D. Reidel, Dordrecht and Boston [Providing a comprehensive view
of the multi-regional demography with reporting fundamental findings of the Migration
and Settlement Study].Schoen, R. and Land, K.C. (1979). ‘A general algorithm for
estimating a Markov-generated increment-decrement life table, with application to
marital status patterns’, Journal of the American Statistical Association, 74:761-776
[Developing a general algorithm for estimating a Markov-generated increment-decrement
life table, with application to marital status patterns].
Schoen, R. and Woodrow, K. (1980). ‘Labor force status life tables for the United States,
1972’, Demography, 17(3): 297-322 [Using data from the January 1972 and January
1973 Current Population Surveys, two types of labor force status life tables were
calculated for the United States in 1972].
Schoen, R. (1985) “Constructing increment-decrement life tables”, Demography
12(2):313-324 [Presents one of the first general expositions of the multistate life table
model]
Schoen, R. (1988). Modeling Multigroup Population. Plenum Press, New York [dealing
with models that can capture the behavior of individuals and groups over time, this book
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considers a broad range of multistate population models, showing that such models can
be calculable, clearly interpretable, and analytically valuable for studying many kinds of
social behavior].
Schoen, R. and Standish, N. (2001). ‘The retrenchment of marriage: results from marital
status life tables for the United States, 1995’, Population and Development Review,
27(3):553-563 [Marital status life tables were calculated using 1995 US rates of marriage,
divorce, and mortality with many interesting findings].
Shavelle, R. and Strauss, D. (1999). “The semi-longitudinal multistate life table”,
Mathematical Population Studies 7:161-177.
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projection’, New Methodological Approaches in the Social Sciences, Population: An
English Selection, 10(1): 97-138. [Presents a brief survey of the microsimulation models
which exist in demography, and a number of the essential characteristics of
microsimulation are illustrated using the KINSIM model for projecting the future size
and structure of kinship networks].
Willekens, F.J., Shah, I., Shah, J.M. and Ramachandran, P. (1982). ‘Multistate analysis of
marital status life tables: theory and application’, Population Studies, 36(1): 129-144
[The mathematical theory of multi-state life table construction is reviewed with an
application on the marital status of Belgian women].
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population forecasting’, New Zealand Population Review, 31(1): 77-124 [Outlines a new
modeling approach for demographic projections by detailed population categories that are
required in the development of sustainable health care and pension systems, employing
the complementary strengths of both micro and macro multistate models].
Zeng, Y. (1991). Family Dynamics in China: A Life Table Analysis. The University of
Wisconsin Press, Wisconsin [Presenting the methodology and application (to China) of
the multistate family status life table model which included both nuclear and threegeneration family households].
Zeng, Y., Gu, D. and Land, K. C. (2004). ‘A new method for correcting underestimation
of disabled life expectancy and application to Chinese oldest-old’, Demography, 41(2):
335-361 [Demonstrating that disabled life expectancies that are based on conventional
multistate life-table methods are significantly underestimated because of the assumption
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of no changes in functional status between age x and death occurred between ages x and
x+n, and presenting a new method to correct the bias and apply it to data from a
longitudinal survey of about 9,000 oldest-old Chinese aged 80-105 collected in 1998 and
2000].
Zeng, Y., Land, K.C., Wang Z., and Gu, D. (2006). ‘U.S. family household momentum
and dynamics -- extension of ProFamy method and application’, Population Research
and Policy Review, 25(1): 1-41 [Applying the ProFamy multistate household projection
model to make the projections of US households and elderly living arrangements from
2000 to 2050].
Zeng, Y., Morgan, P., Wang, Z., Gu, D. and Yang, C. ‘Union regimes in the United
States: trends and racial Differentials, 1970-2002’, Paper presented at the 2008 Annual
Meeting of the Population Association of America, April 17-19, New Orleans, LA.
[Constructing the race-sex-period-specific multistate life tables to analyze the marriage
and cohabitation union formations and dissolutions in the United States].
Zeng, Y., Vaupel, J.W. and Wang, Z. (1997). ‘A multidimensional model for projecting
family households -- with an illustrative numerical application’, Mathematical
Population Studies, 6: 187-216 [Developing a multistate model for projecting family
households, living arrangements and population].
Biographical Sketches
Active in the initial development of multistate demography at a time when he was
affiliated with the International Institute for Applied Analysis (IIASA) located near
Vienna (Austria), Jacques Ledent is Research Professor at the Institut national de la
recherche scientifique (INRS) which is part of the University of Quebec. After a long
standing focus on internal migration, his research is now concerned with international
migration and, more specifically, the integration of immigrants in Montreal, Quebec and
Canada.
Yi Zeng is a Professor at the Center for the Study of Aging and Human Development and
Geriatric Division / Dept of Medicine of Medical School, and Institute of Population
Research and Dept. of Sociology, Duke University. He is also a Professor at the China
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Center for Economic Research, National School of Development at Peking University in
China, and Distinguished Research Scholar of the Max Planck Institute for Demographic
Research (MPIDR) in Germany. He received his doctoral degree from Brussels Free
University in May 1986, and conducted post-doctoral study at Princeton University in
1986-87. Up to July 15, 2009, he has had 96 professional articles written in English
published in academic journals or as book chapters in the United States and Europe;
among them, 60 articles were published in anonymously peer-reviewed academic
journals. He has had 89 professional articles written in Chinese and published in China;
among them, 61 articles were published in national Chinese academic journals. He has
published nineteen books, including 6 sole-author books, 3 co-author books, 7 chiefeditor books and 3 co-editor books. Eight of Yi Zeng’s published books were written in
English (including three by Springer Publisher and one by the University of Wisconsin
Press), one was written in both Chinese and English, and the remainders were written in
Chinese. Yi Zeng has been awarded eleven national academic prizes and three
international academic prizes, such as the Dorothy Thomas Prize of the Population
Association of America, the Harold D. Lasswell Prize in Policy Science awarded by the
international journal Policy Sciences and Kluwer Academic Publishers, the national
prizes for advancement of science and technology awarded by the State Sciences and
Technology Commission of China and the State Education Commission, the highest
academic honor of Peking University: "Prize for Outstanding Contributions in Sciences,"
and the “Chinese Population Prize (Science and Technology)”, jointly awarded by nine
ministries and seven national non-governmental associations in China.
44