Policy, Planning, and Research
WORKI'4G PAPERS
Population, Health, and Nutrition
Population and Human Resources
Departmnent
The World Bank
December 1989
WPS 337
Projecting Mortality
for All Countries
Rodolfo A. Bulatao and Eduard Bos
with
Patience W. Stephens and My T. Vu
New procedures for projecting mortality in each country mod-
estly change previous mortality projections.
The Policy. Planning, and Research Complex distnbutes PPR Workung Papers to disseminate the findings of work in progress and to
encourage the exchange of idcas among Bank staff and all others interested in development issues These papers carry the names of
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Plicy, Planning, and Research
Population, Health, and Nutrlion
As part of its worldwide population projections, In the short term, the rate of change in life
the Banik annually provides projections of expectancy in a particular country can be pre-
mortality in each country. Bulatao and Bos dicted from its rate of change in the previous
reviewed and updated those procedures. five years and from female secondary enroll-
ment. For the longer temi, alternative logistic
Basically, mortality has been projected by functions are defined to give medium, rapid. and
first projecting male and female life expectancy slow improvements in life expectancy.
according to standard schedules and then
choosing life tables (which give the age pattem The infant mortality rate can also be repre-
of mortality) for successive periods to give the sented by alternative logistic functions that
desired sequences of life expectancy levels. allow the rate to decline to either 6 or 3 per
thousand. In the short term, the trend can be
Bulatao and Bos present new procedures for predicted from the previous tren For the long
projecting short-term (one or two decades) and term, Bulatao and Bos define a medium trend
long-term (one or two centuries) mortality rates. and altemative rapid and slow trends.
These procedures involve calculating rates of
change for and separately projecting male and "Split" life tables can be chosen from thc
female life expectancy and infant mortality and Coale-Demeny models, using the infant morlal-
then selecting appropriate model life tables. ity rate to determine which level to use for
mortality at younger ages, and life expectancy to
Bulatao and Bos derivcd the approaches to determine which level to use for older ages.
projecting life expectancy and infant mortality
from analysis of data for developed and develop- Changes from previous mortality pro'jections
ing countries. resulting from these new procedures are mostly
modest. Projected life expectancies generally
For female life expectancy, altemative stay within a few percentage points of older
maxima of 82.5 and 90 years are used in defin- projections. Infant mortality and crude dealh
ing logistic functions for increase over time. rates vary somewhat more. Projected popula ioni
Male life expectancy is currently 6.7 years lower is affected only slightly; a 2 percent change is
thian female life expectancy in developed close to the maximum effect.
countries, and this differential is assumed to
apply at the maximum.
This paper is a product of the Population, Health, and Nutrition Division, Population
and Human Resources Department. Copies are available free from the World Bank,
1818 H Street NW, Washington DC 20433. Please contact Sonia Ainswonh, room
S6-065, extension 31091 (30 pages with figures and tables).
The PPR Working Paper Series disseminates the findings of work under way in the Bank's Policy, Planning, and Resea;rch
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The findings, interpretations, and conclusions in these papers do not necessarily represent official policy of the Bank.
Produced at the PPR Dissemination Center

CONTENTS
PREVIOUS WORK ....................... 2
Projection methods .................. 2
Limits to life expectancy ....... ........ 3
LIFE EXPECTANCY ...................... 4
Data ......................... 4
General trends ..................... S
Predicting specific trends ....... ....... 9
Projection approach ......... ........ 12
INFANT MORTALITY ............ ....... 14
Data ......................... 14
General trends ............ ........ 14
Predicting specific trends ...... ....... 17
Projection approach ......... ........ 18
AGE PATTERNS OF MORTALIT ..... ..... 18
ILLUSTRATIVE PROJECTIONS ...... ...... 21
Procedures used .......... ......... 21
Results ......................... 22
CONCLUSION ......................... 28
ACKNOWLEDGMENTS .......... ........ 29
REFERENCES ......................... 29

1
Ths cxerdis is an attempt to develop a method for projecting mortality trends in all
countries into the future, both over the short term (for one or twco decades) and over the long term
(for one or two centuries). On the critical assumption that the future wiU resemble what we know
of the recent past, we seek a heuristic model for mortality projections that is universally applicable.
Detailed country by country and disease by disease examination might yield better predictions of
future mortality, but is quite cumbersome if one wishes to project all countries of the world.
Ultimately, of course, future trends in mortality, especiaRy over longer periods, are largely
unknowable. Thus this exercise does not attempt to predict mortality so much as to project it into
the future given reasonable, though inevitably somewhat arbitrary, assumptions.
Two indicators are the focus of this exercise: life expectancy at birth and the infant
mortality rate. The projection of the age pattern of mortality will be considered but no empirical
analysis performed. In this introduction, ihe reasons for focusing on life expectancy and infant
mortality will be discussed and a preview of the major issues wil be provided. Then previous
projection approaches wiU be briefly summarized. The current exercise is meant to update
procedures for projecting mortality for the World Bank, which are among those to be reviewed.
Then analysis will be presented separately for life expectancy and infant mortality, covering
appropriate representations of universal time trends and country-specific trends and how they might
be predicted. Fmatly, iUlustrative projections will be used to show the effects of the derived
procedures, and some conclusions will be drawn.
Life expectancy at birth and the infant mortality rate together provide a much better
description of mortality than the crude death rate, and sufficient time-series data exist on these
two variables for the detection of trends. More precise pictures of mortality could be obtained
with age and cause-specific rates leading to complete life tables, but data on these are more sparse
and generalizations correspondingly much more difficult. Instead, work on sequences of model life
tables (Coale and Demeny 1983; Coale and Guo 1989) can be relied on once these key mortality
parameters have been projected.
Current projections of mortality for multiple countries involve one of two procedures:
incrementing life expectancy according to some schedule and applying appropriate life tables, or
selecting some optimal life table toward which mortality rates gradually converge. The procedure
considered here is of the first type. Essentialy we attempt to refine previous schedules of
incretaents to Ufe expectancy, adjusting them to more closely reflect recent experience, and to take
infant mortality into account in selecting life tables.
Here is a brief preview of critical issues considered below.
* What data on mortality should be used to represent recent trends? *Good data on
life expectancy based on empirically derived life tables will be considered, but will be shown to
give a rather different picture from weaker data. Some amalgam will be necessary.
* Is any mathematical representation of mortality change appropriate? No perfect model
exists, but logistic models will be applied for fife expectancy and infant mortality.
* Should life expectancy be allowed to increase indefinitely, or should it be asymptotic
to some limit? We will assume Umits and, lacking precognition, impose two alternative sets of
Umits, the first set (the limited set) assuming that life expectancy will not progress far beyond
current maximum levels and the second set (the *extended set) assuming that substantial
imprerement will still be made.
* Can current socioeconomic indicators predict future mortality? They will be shown

2
to have relatively Uttle power to predict rates of improvement. Recent mortality trends, on the
other hand, do predct future trends over a decade or so.
* WiU separate projections of life expectancy and infant mortality be consistent? We
will find reasonable consistency, but also some need rules to prevent too great divergence.
* Wil reliance on recent trends radically alter expected mortality in World Bank
projections? Some differences wiUl appear, but they are mostly modest.
PREVIOUS WORK
We consider, first, the methods in use for projecting national mortality levels in
numerous countries simultaneously, and second, calculations of and speculations about ultimate
limits to life expectancy.
Projectioa methods. Although a variety of methods exist for projecting age-speciflc
mortality (e.g., Pollard 1987), current projections of multiple countries generaly rely on only one
or two methods and show many similarities. The United Nations Population Division and the
World Bank both rely on model life tables. Projecting population in Latin American countries only,
the Centro Latinoamericano de Demografla (CELADE) relies on an optimal or ultimate life table
toward which all countries converge. The U.S. Census Bureau relies on model life tables and, in
some cases, on an optimal life table. We consider these procedures briefly here (but do not deal
with the evolution of U.N. procedures, nor with other approaches taken in the past, which are
covered in Freika 1981).
The U.N. Population Division extrapolates life expectancy and applies model life tables
based on this parameter (U.N. 1989:13-19). Life expectancy is extrapolated separately for males and
females, raising it a specific number of years that graduaUy declines as life expectancy rises. For
example, if a country has had 'typical' experience with mortality improvement, and if initial male
or female life expectancy is under 60 years, the U.N. expects it to rise 2.5 years in the next
quinquennium. For another typical country, if initial tife expectancy is between 75 and 77.5, the
U.N. expects the increment for males to be .5 years per quinquennium, and for females to be 1.0
years per quinquennium. Intermediate increments are defined for intermediate levels of life
expectancy. If a country has had unusually fast or unusually slow mortality improvement, schedules
of increments that are slightly higher or slightly lower are applied instead. No definition is given
of what constitutes slow, typical, and fast mortality improvement for purposes of choosing a
schedule of increments. (Presumably comparisons of previous life expectancy gains with the three
schedules of increments can be used.) These increments were obtained by taking means of
quinquennial increments by initial life expectancy levels across low- and moderate-mortality countries
for quinquennia from 1955 to 1985.
Based on the resulting life expectancy estimates, the U.N. chooses appropriate life
tables from among nine models: the four Coale-Demeny (1983) families (North, South, East, and
West) and the five U.N. (1982) models (General, Latin American, Chilean, Far Eastern, and South
Asian). These model families are extended to give, at a maximum, a male life expectancy of 82.5
years and a female life expectancy of 87.5 years. Model life tables with these maximum life
expectancies were devised by adjusting downward the q, values in life tables which combined data
on a few low-mortality countries. Each family of model life tables was then required to converge
to these ultimate life tables.
The World Bank procedures are roughly parallel (Zachariah and Vu 1988&xv-xvi; Vu,
Bos, and Bulatao 1988:2-6). Female life expectancy is incremented according to schedules similar

3
to those used by the U.N., with larger increments at lower levels of life expectar.cy. Two schedules
are provided, one for countries with female primary enrolment under 70 percent and the other for
aU other countries. These schedules were obtained from separate regressions for these two groups
of countries on initial life expectancy in 1965-69 of life expectancy increments in the following
decade. The incremented lfe expectancies are used to select appropriate levels of the Coale-
Demeny life tables. Male life tables are chosen at the same level as the female tables, or with life
expectancy incremented according to similar schedules. At lower life expectancy levels, South or
North family tables are used, at higher levels, West family tables are used. In the long run, females
are allowed to reach a life expectancy of 82.5, and males are limited to a life expectancy of 76.6.
The U.S. Census Bureau (as an account they provided indicates) first estimates logistic
functions by country and sex, using in each case two historical estimates of life expectancy, or a
current estimate and a judgment of probable life expectancy in 2000, or (for developed countries)
a current estimate and life expectancies of 80 for males or 86 for females by 2050. These functions
are assumed to have lower bounds of 25 and upper bounds of 79-81 for males and 86-87 for
females. The annual increment given by these functions is required to be within the range provided
by the U.N. projections. Age patterns for mortality are then obtained in one of three ways:
directly from Coale-Demeny model life tables; by applying to an empirical life table (where one is
available) the relative changes in mortality rates between levels of Coale-Demeny model life tables;
or by generating interpolated life tables between an empirical life table and an optimal Life table
for all countries based on Japanese and Swedish data. Projections of U.S. mortality are done
separately, and involve more complex operations (Long and McMillen 1987:154-155).
The similarities are more notable than the contrasts among these procedures. The
annual increments to life expectancy used by the U.N. and the World Bank are quite similar, as will
be illustrated below, and the Census Bureau requires that its procedures produce increments in the
same range. In addition, the World Bank increments can be approximated with a logistic curve
(Bulatao and Elwan 1985:2W4), the type of curve applied by the Census Bureau. Similar logistic
curves to represent rapid mortality decline have also been defined (Bulatao and Elwan 1985:10-
14).
The CELADE procedure in principle is not that different from any of these.
Survivorship ratios are initiaUy given by life tables selected for each country, and the logits of
these ratios converge linearly to those of optimal life tables for all countries (Po!lard 1987:65). The
U.N., World Bank, and Census Bureau procedures all to some degree also involve such optimal life
tables. However, whether the specific changes in mortaity parameters like life expectancy in the
CELADE projections match those in the other sets of projections is not known.
Limits to life expectancy. These projection methods all make assumptions about
the maximum attainable life expectancy, at least within the projection periods considered. Between
its last two assessments of population prospects, for instance, the U.N. (1986a, 1989) raised assumed
maximum levels of life expectancy between the present and 2025 from 75 to 82.5 years for males
and from 82.5 to 87.5 years for females. Particularly for longer mortality projections of 50 years
or more, maximum levels of life expectancy have significant effect, and we therefore consider what
these maxima are.
Two main approaches have been used to estimate maximum levels; neither between
nor within approaches has there been agreement. One approach has been to extrapolate observed
improvements in mortality and determine some point at which these improvements cease.
Extrapolating rates of increase in life expectancy at different ages until the levels converged, Fries
(1980) obtained limits for the U.S. of 82.4 years for males and 85.6 years for females, and also
determined that these levels would be reached in 2009 and 2018 respectively. Extrapolating life

4
expectancies at birth along exponential curves, which they argued fit the data for advanced
developed countries well, Coale and Guo (1989) obtained limits of 76.1-77.8 for males and 83.25-
84.9 for females. In these two cases and with other extrapolation exercise, the statistical techniques
used affect the results. Even more important, however, are the data used, which represent past
conditions and cannot reflect future breakthroughs in mortality reduction.
The other approach has been to determine the effect on mortaUty of eliminating,
deferring, or reducing the impact of particular causes of death. This approach has a long history,
dating at least to an 1806 study on the effects of smallpox vaccination (see Bourgeois-Pkchat 1978,
Pressat 1974). Among more recent work, Bourgeois-Pichat, with Norwegian data, attempted in 1952
to deternine the effect of eliminating *exogenous causes of death, obtaining maximum life
expectancies of 76.3 for males and 78.2 for females. In 1978, he revised the limit downward for
men to 73.8 and upward for women to 80.3 (barely above current estimates for Norway). Using
the Framingham study and focusing on U.S. white adult males, Manton (1986) showed that
controlling major risk factors could lead to an increase of 12.3 or 12.8 years (his statistica are
ambiguous) in expectation of life at age 30. Even with no changes in mortality rates below 30, this
would effectively raise life expectancy at birth for males above 81.
Some of the maximum estimates various authors have made have already been excoeded.
Life expectancy in Japan, for instance, is now estimated at 75.6 for males and 81.4 for females
(Institute of Population Problems 1989:2). Combining the lowest age-sex specific death rates around
the world gives slightly higher life expectancies, 76.2 for males and 821 for females (Uemura 1989).
In the long run, over 50 years or longer, none of these calculations provide any convincing evidence
of specific limits. Thus the question of ultimate limits to life expectancy is unresolvable at this
time. We will therefore develop two alternative patterns for projection purposes: the first, the
*limited option, will assume that national life expectancies will not rise greatly beyond current
maximum levels; the second, the extended" option, will assume that they will rise by about ten
years.
LIFE EXPECTANCY
First, we discuss the data to be used on expectation of life at birth (e.). Second, we
consider the general trends over time shown in these data. Third, we attempt to determine whether
trends for individual countries can be predicted. Fourth, we suggest an approach to projecting life
expectancy based on the analysis.
Data Data on life expectancy by sex (referred to here as data set A) were drawn
from United Nations Secretariat (1988a:61-64), which provides the estimates from various collections
of life tables, including the input life tables used by Coale and Demeny (1983), a previous U.N.
publication (1952), a U.N. (1986b) database for developing countries, and WHO life tables based
on registered deaths in developed countries. Many of these life tables were constructed with data
for several years. The life expectancy estimates were assumed ., pertain to the midpoint of these
periods. We needed estimates for at least three years for each country, which reduced the countries
considered to 37, with England and Wales, Scotland, and Northern Ireland, as well as Puerto Rico,
considered separately.
The only developing countries in the list, aside from Puerto Rico, were Argentina,
Hong Kong, Sri Lanka, Martinique, Reunion, and Singapore. For some analyses, even these
atypical countries had insufficient data. Therefore, some parallel analysis was run just on
developing countries using additional, if less reliable, data. The life-table based estimates were
augmented with estimates from the U.N. Deinographic Yearbook (various yeats), from which we
excluded, as much as possible, estimates the U.N. obtained by projection. A set of 33 developing

5
countries (set B) was obtained by this process.
Generl tend. We attempted fint to linearize trends over time in these data.
Following previous work, we used a logistic transformation, which captures the slower improvement
in Ufa expecancy at high levels and at very low level (Alternative transformations like
exponentials did not fit as well.) The transL nation of life expectancy at time t (et) was of the
form
logit(e.) - log, [(Ic. + k - et) / (e, - Q)J
where kI s a lower lmit for life expectancy and c + k) an upper limit. After some
experimentation, k1 was fixed at 20.
Alternative values of (Ic + k) were tried: for womn,, these were 82.5 and 90. Logits
using these alternative maxima will be referred to as the limited and "extended transformations.
The first value is one level above the highest model life table in the Coale-Demeny set, but is
covered in later work by Coale and Guo (1989), which also revises the tables at the highest level
of lifec expectancy. The second value allows life expectancy to rise much higher in the long term.
These two maxima are close to the high and low values for long-run female life expectancy-81.5
and 90.1-projected by the U.S. Social Security Administradon (Wade 1988:13).
For men, maximum life expectancy was set at 6.7 years kss than vmen, the current
average gap for developed countries. Both a limited and an Mxtended transformation for men were
thus defined. The gap between men and women has grown over time, but apparently is no longer
growing in some developed countries and may even be narrowing (U.N. Seetariat 1988b). From
experience in higher social classes in developed countries, particululy regarding smoking behavior,
some argue that the gap could eventually decline (Nathanson and Lopez 1987). However, the
generalizability of such trends cannot be assumed, and no firm basis exists so far for assuming either
a larger or a smaller sex differential worldwide in the future Assuming the differential will stay
at current developed-country levels, the resulting limits for men-75.8 and 83.3-will aguin resemble
low and high values projected by the U.S. Social Security Administration-75.2 and 83.6 for long-
run male life expectancy (Wade 1988:13).
Some indication that the logit transformation serves to linearize trends in life
expectancy is given in Table 1, which reports correlations between life expectancy at the beginning
of a quinquennium and the rate of change in life expectancy or its logit in that period. Across the
set A countries, this correlation is generally negative if life expectancy has not been transformed,
indicating more rapid rise in life xpectancy from lower levels. By contrast, rates of change in the
logit tend to be correlated about as often positively as negatively with initial level across these
countries. However, regardleas of transformation, an unusual time trend does emerge, with
countries in set A at lower levels of life expectancy gaining more around 1960 than countries at
higher levels of life expectancy, but the situation being reversed around 1980. The set B data are
more difficult to interpret, the data referring to periods that are less consistent across countries.
For instance, a 19'O.80 rate for one country may be included, in calculating a correlation, with a
1976-79 rate for 'her country. The logit transformation does consistently reduce the link
between initial leve. and rate of change, but does not eliminate iL
Time trends in the logit of life expectancy were examined country by country, and
appeared to be mostly linear, the exceptions being mainly countries with sharp falls in life
expectancy at paticular periods. Linear regression was then used to fit a Une through the logits
for male and female life expectancy in each country. Rz for these regressions was uniformly high,
half the time being above .95 and three-fourths of the time being above .90. Residuals from the

6
Table 1. Correlations between initial life expectancy and subsequent rate
of change in life expectancy and its logit.
Data set
and sub- No. of Males Females
sequent coun- Untrans- Logit, Logit, Untrans- Logit, Logit,
period tries formed limited extended formed limited extended
SetA
1957-62 19 -0.92 0.88 0.91 -0.93 0.82 0.89
1962-67 29 -0.58 0.42 0.52 -0.74 0.44 0.63
1967-72 29 -0.20 -0.07 0.10 -0.12 -0.25 -0.05
1972-77 29 -0.12 -0.25 -0.01 0.02 -0.47 -0.23
19-7-82 29 0.22 -0.54 -0.33 0.23 -0.63 -0.41
Set B
-1957-62 17 -0.41 0.19 0.34 -0.34 0.15 0.27
-1962-67 17 -0.41 0.28 0.36 -0.45 0.31 0.39
-1967-72 20 -0.39 0.32 0.36 -0.12 -0.17 -0.00
-1972-77 20 -0.55 0.14 0.40 -0.69 0.32 0.58
-1977-82 19 -0.36 -0.15 0.15 -0.32 0.00 0.17
Not.: The limited transformtions assus ma5Xma of 75.8 for oen and 82.5 for womn. The extended
trtsformtions assum axima of 83.3 for m-n end 90 for wmen. Set A data, mostly for detvloped
eountries, are froe several sets of life tablas. Set N data, all for developiam countries, add to
the hif table data stimates from the U.N. Demographic Yearbook.
equations showed no consistent trend.
Table 2 provides the means and some percentiles of the estimated slopes from these
regressions for each of four groups of countries: (1) set A countries, using only data for 1950 or
later, (2) only those set A countries with pre-1950 estimates, including both pre-1950 and post-
1950 data; (3) set B countries, using 1950 and later data; and (4) only those set B countries with
pre-1950 estimates, including both pre-1950 and post-1950 data. The set A data give more gradual
rates of change than the (probably less reliable) set B data, and wuntries with pre-1950 data show
more rapid change when such data are included. This seems to imply that change in developing
countries, or from lower levels of life expectancy, may be more rapid. Given the consistency of
these contrasts, reliance solely on the set A data is inappropriate. AU data and al! cases are pooled
to give the last row of each section of the table. On the basis of these pooled estimates, working
estimates of rates of change in life expectancy to represent slow mortality decline, medium mortality
decline, and rapid mortality decline are given at the bottom of Table 2.
What these coefficients mean is demonstrated in Table 3, which shows the annual
increments to life expectancy predicted by using models incorporating the working estimates. If
male life expectancy is now 40, for instance, the medium annual gain (based on the limited
transformation) is .45 years, as opposed to .22 years using the slow estimate and .69 years using
the rapid estimate. If male life expectancy is 70, on the other hand, the medidm annual gain is
only .18 yeams

7
Table 2. Means and percentiles across countries of rates of change in
logit of life expectancies, and working estimates.
Sex, transforma- Percentile
tion, and data set N Mean 90th 75th 50th 25th 10th
Males (limited)
Set A, 1950-85 37 -0.028 -0.006 -0.014 -0.026 -0.039 -0.049
Set A, pre-1950 22 -0.031 -0.023 -0.026 -0.029 -0.034 -0.045
Set B, 1950-85 32 -0.037 -0.011 -0.028 -0.037 -0.046 -C.068
Set B, pre-1950 9 -0.046 -0.034 -0.040 -0.046 -0.052 -0.057
All data, all years 70 -0.035 -0.014 -0.026 -0.034 -0.045 -0.053
Males (extended)
Set A, 1950-85 37 -0.016 -0.003 -0.008 -0.014 -0.021 -0.032
Set A, pre-1950 22 -0.023 -0.017 -0.019 -0.020 -0.026 -0.035
Set B, 1950-85 32 -0.027 -0.009 -0.019 -0.028 -0.033 -0.039
Set B, pre-1950 9 -0.038 -0.029 -0.032 -0.040 -0.044 -0.047
All data, all years 70 -0.025 -0.009 -0.018 -0.023 -0.034 -0.040
Females (limited')
Set A, 1950-85 37 -0.035 -0.017 -0.025 -0.035 -0.045 -0.052
Set A, pre-1950 22 -0.034 -0.025 -0.028 -0.032 -0.038 -0.045
Set B, 1950-85 32 -0.036 -0.013 -0.023 -0.037 -0.049 -0.060
Set B, pre-1950 9 -0.043 -0.033 -0.038 -0.042 -0.047 -0.055
All data, all years 70 -0.036 -0.019 -0.027 -0.036 -0.045 -0.053
Females (extended)
Set A, 1950-85 37 -0.021 -0.010 -0.015 -0.019 -0.026 -0.033
Set A, pre-1950 22 -0.026 -0.017 -0.021 -0.024 -0.029 -0.036
Set B, 1950-85 32 -0.028 -0.010 -0.019 -0.031 -0.037 -0.043
Set B, pre-1950 9 -0.037 -0.026 -0.031 -0.036 -0.043 -0.047
All data, all years 70 -0.027 -0.012 -0.019 -0.025 -0.036 -0.041
Working estimates Si Medium Ra2i
Males (limited) -0.017 -0.035 -0.053
Males (extended) -0.010 -0.025 -0.040
Females (limited) -0.017 -0.035 -0.053
Females (extended) -0.t)10 -0.025 -0.040
Figure 1 compares the estimated annual increments to female life expectancy--from
the slow, medium, and rakjid models, using the limited and extended transformations--with the
increments applied by the U.N. Population Division (1989:16) and those previously used by the
World Bank (Zachariah and Vu 1988xvi). In these comparisons, the previous World Bank
increments for the high-female-education case are treated as medium estimates and those for the
low-female-education case as low estimates. The increments from the medium n.Jdel appear to
agree reawonably well with the previous World Bank pattemn and the U.N. pattern. Although the
U.N. increments are slightly higher at around 65, at higher life expectancies they faU between the

8
FIpure 1
Annual Inrements, in years, to female liWe expectancy from different models,
assuming slow, medium, and rapid improvement
s L o w
0,3
030
037
0.4 -
0. CL~M E -IU
090
0.3
0.1
23 sou r us ai mu 1 0 73 600 U
| osEDIU
0.3
0n,
I~~~~~ A2
0.0 _ l-t-
0.2
am~~~~~~~~~~~~~v
01
23 40 43 30 UU d ou 70 -7 t *-
R PAP ID
030
037
0.2
0.
0. 1
IMIdItUL L.IPN ISCPECrANCY

9
Table 3. Annual increment to life expectancy in years, by sex, with
different equations, frcm varying initial levels.
Sex and
initial Limited Extended
level Slow Medium Rapid Slow Medium Rapid
Maleg
40 0.22 0.45 0.69 0.14 0.34 0.55
50 0.24 0.48 0.73 0.16 0.39 0.63
60 0.19 0.39 0.59 0.15 0.37 0.59
70 0.09 0.18 0.27 0.10 0.26 0.42
80 0.03 0.08 0.12
FeMales
40 0.23 0.48 0.73 0.14 0.36 0.58
50 0.27 0.55 0.83 0.17 0.43 0.69
60 0.24 0.50 0.76 0.17 0.43 0.68
70 0.17 0.35 0.52 0.14 0.36 0.57
80 0.04 0.08 0.12 0.09 0.21 0.34
estimates using the limited and the extended transformadons. However, where the slow and rapid
models are concerned, the working estimates are slower and more rapid, respectively, than the U.N.
patterns, except near the upper mits for life expectancy, where the U.N. continues to allow some
improvements. Comparisons of the patterns for males lead to similar conclusions.
Comparisons are also possible with increments to ife expectancy estimated by the U.N.
from reliable data. For 23 developing countries in the 195&l-60s, the average annual increment was
.61 years and varied little by initial life expectancy. For 33 developing countries in the 1960s-7Us,
the average increments were .62 for countries with initial lIfe expectancies under 50, falling to .24
for countries with initial lfe expectancies of 65 or higher. For 27 developing countries in the
197(s40s, the average increments were .76 for countries with initial life expectancies under 50,
falUng to .37 for countries with initial life expectancies of 65 or higher (U.N. 1988:132). For 24
developed countries, with initial life expectancies anywhere from 65 to 79, annual increments in the
late 197(0 and early 1980s averaged about .2 (U.N. 1988:140-141). Except at vety low levels of life
expectancy, these increments resemble the gains provided by the medium working models and fail
within the range of the slow and rapid models. At very low levels of life expectancy, these
increments are larger than the models provide, possibly because countries with high mortality but
good data are a select group.
Predicting speciflc trends. Trends in life expectancy in specific countries can be
predicted from previous trends and from socioeconomic factors. Table 4 shows that, in the set A
data, the rate of change in one period is correlated with the rate of change in a previous period.
The rate in the immediately preceding quinquennium is the best predictor of the subsequent rate,
and as the periods related move fanher apart, the correlation declnes. From the set B data (Table
5), the same general conclusion cannot be drawn. Only for periods beginning in the early 1980
is the correlation between rates of change in succeeding periods positive; for earlier periods, the
correlation is negative, and occasionally large. Inconsistencies in period definition, or less stability

10
Table 4. Correlations between rate of change in logit of life
expectancy and rate of change in preceding periods, set A countries.
Periods No. of Males Males Females Females
correlated countries (limited) (extended) (limited) (extended)
Previous
p-Wod and:
1962-67 19 0.66 0.76 0.57 0.68
1967-72 29 0.60 0.58 0.58 0.57
1972-77 29 0.62 0.57 0.60 0.53
1977-82 29 0.76 0.70 0.73 0.62
Period twice
removed and:
1967-72 19 0.38 0.46 0.30 0.40
1972-77 29 0.19 0.12 0.14 0.13
1977-82 29 0.54 0.40 0.53 0.40
Period thrice
reoe and*
1972-77 19 0.14 0.09 0.12 0.16
1977-82 29 0.23 0.04 0.24 0.16
Period four times
removed and:
1977-82 19 0.14 0.10 0.24 0.25
Table 5. Correlations between rate of change in logit of life
expectancy and rate of change in previous period, set B countries.
Starting date
of subsequent No. of Males Males Females Females
period countries (limited) (extended) (limited) (extended)
Any year 80 -0.06 -0.14 -0.08 -0.08
Early 1960s 17 -0.68 -0.64 -0.61 -0.59
Late 1960s 14 -0.07 -0.08 0.12 0.08
Early 1970s 20 -0.52 -0.49 -0.09 -0.14
Late 1970s 19 -0.29 -0.34 -0.37 -0.37
Early 1980s 10 0.49 0.48 0.28 0.41

11
in mortality experience, may be responsible for the unstable pattern with set B data.
Regressions were run to predict rate of change from previous rate of change and
various socioeconomic indicators. Given the inconsistencies noted with set B data, only the set
A data were used. The socioeconomic variables initially cousidered were GNP per capita; the
female primary and secondary enrolment ratios; female labor force participation; and percent of
the population urban. Data were obtained from World Bank liles. Initial analysis led to
elimination of three of these variables. GNP per capita was not predictive, neither by itself or in
association with other variables. The primary enrolment ratio did seem to predict more rapid
improvement in lIfe expectancy, but the ratio had limited range, being close to or often above 100
for most of these countries. Female labor force participation did occasionally predict slower
Improvement in life expectancy, but this was difficult to rationalize. Regressions using the
remaining variables are reported In Tables 6 to &
Table 6 uses previous rate of change and the female secondary enrolment ratio to
predict rate of change in life expectancy, Table 7 uses previous rate of change and percent urban.
Table 6. Regressions for rate of change in logit of life expectancy,
with previous rate of change and female enrolment ratio.
Sex, transforma- Previous rate Female secondary
tion, and period of change enrolment ratio
predicted B (t) B (t) Constant R'
Males (limited)
All periods 0.885 8.28 -0.000318 -2.36 0.01159 0.52
1967-72 0.653 3.79 -0.000028 -0.14 -0.00449 0.36
1972-77 0.736 3.76 -0.000315 -1.31 0.00225 0.43
1977-81 1.150 5.86 -0.000361 -1.15 0.02525 0.59
Males (extended}
All periods 0.669 6.74 -0.000132 -2.02 0.00277 0.41
1967-72 0.549 3.57 -0.000011 -0.10 -0.00302 0.34
1972-77 0.614 3.26 -0.000122 -0.99 -0.00223 0.35
1977-81 0.895 4.90 -0.000137 -0.97 0.00795 0.50
Females (limited)
All periods 0.723 6.80 -0.000254 -2.18 0.00379 0.45
1967-72 0.794 3.53 -0.000127 -0.70 0.00159 0.35
1972-77 0.662 3.21 -0.000338 -1.48 0.00039 0.42
1977-81 0.868 5.26 -0.000237 -0.98 0.01241 0.54
Females (extended)
All periods 0.528 5.27 -0.000092 -1.62 -0.00290 0.30
1967-72 0.627 3.47 -0.000064 -0.63 -0.00112 0.33
1972-77 0.515 2.81 -0.000115 -1.03 -0.00550 0.31
1977-81 0.645 3.98 -0.000070 -0.64 0.00010 0.39

12
Table 7. Regressions for rate of change in logit of life expectancy,
with previous rate of change and percent urban.
Sex, transforma- Previous rate Percent urba
tion, and period of change
predicted B (t) B (t) Constant R2
Males (limited)
All periods 0.958 9.49 -0.000339 -2.89 0.01331 0.53
1967-72 0.677 3.73 -0.000061 -0.37 -0.00'.93 0.37
1972-77 0.828 4.62 -0.000364 -1.98 0.00557 0.47
1977-81 1.132 5.88 -0.000407 -1.60 0.02397 0.61
Males (extended)
All periods 0.735 7.67 -0.000144 -2.46 0.00375 0.42
1967-72 0.553 3.35 -0.000007 -0.07 -0.00320 0.34
1972-77 0.704 4.00 -0.000166 -1.72 0.00094 0.39
1977-81 0.884 5.G1 -0.000189 -1.67 0.00987 0.53
Females (limited)
All periods 0.802 8.02 -0.000192 -1.89 0.00106 0.44
1967-72 0.867 3.66 -0.000107 -0.71 0.00270 0.35
1972.77 0.769 4.00 -0.000176 -0.99 -0.00914 0.39
1977-81 0.876 5.48 -0.000284 -1.48 0.01347 0.56
Females (extended)
All periods 0.581 5.87 -0.000058 -1.12 -0.00459 0.29
1967-72 0.661 3.28 -0.000031 -0.35 -0.00242 0.33
1972-77 0.570 3.23 -0.000043 -0.48 -0.01005 0.29
1977-81 0.668 4.23 -0.000113 -1.29 0.00281 0.42
If all three predictors were combined, the effects of the two socioeconomic variables would not be
significanL In these separate regressions, the two socioeconomic variables have reasonably
consistent effects that are sometimes but not always significant. Previous rate of change, on the
other hand, always has a strong and significant effect. Regressions using only previous rate of
change are shown in Table & Using only this predictor, coefficients close to those estimated can
be chosen to ensure that predicted rates from successive applications of the equations conveiL., to
the medium rates provided above. These form the working equations in Table &
Projection approach. One approach to projecting life expectancy that can be
devised from these results involves four steps. First, predict rate of change in lfe expectancy from
the equations in Tables 6 and 7. The equations using pooled data across periods are an appropriate
choice. Tbe equations using secondary enrolment may be slightly preferable to those using percent
urban because the coefficients are slightly more stable across periods, but the other equations could
equally be used. Second, to minimize the effect of unusual recent trends, require that the rate of
change be within certain limits, such as being no more extreme than the slow mortality decline and
rapid mortality decline estimates in Table 1. Third, apply the calculated rate of change for a few
years, perhaps 15. The correlations in Table 4 suggest that, after 15 years, rate of change is

13
Table 8. Regressions for rate of change in logit of life expectancy,
with previous rate of change only.
Sex, transforma-
tion, and period
predicted B (t) Constant
Males (limited)
All periods 0.938 8.94 -0.00976 0.48
1967-72 0.652 3.93 -0.00622 0.34
1972-77 0.766 4.12 -0.01983 0.36
1977-81 1.178 6.02 -0.00263 0.56
Males (extended)
All periods 0.700 7.18 -0.00623 0.37
1967-72 0.549 3.70 -0.00367 0.31
1972-77 0.631 3.57 .0.01083 0.30
1977-81 0.912 5.02 -0.00272 0.46
Females (limited)
All periods 0.792 7.81 -0.01204 0.41
1967-72 0.804 3.69 -0.00583 0.31
1972-77 0.749 3.92 -0.02151 0.34
1977-81 0.893 5.48 -0.00545 0.51
Females (extended)
All periods 0.558 5.75 -0.00885 0.27
1967-72 0.627 3.58 -0.00498 0.30
1972-77 0.552 3.25 -0.01320 0.25
1977-81 0.656 4.11 -0.00525 0.36
Working equations
Limited 0.8 -0.0070
Extended 0.7 -0.0075
predicted poorly at best. Alternatively, apply the calculated rate only for one period, and estimate
from it the rate for the following period, using the working equations in Table 8, which can be used
successvely for the next period. Fourth, apply a standard rate of change to Ufe expectancy beyond
15 years, such as the medium mortality decline pattern in Table 1. The entire procedure could be
carried out using either the Umited or the extended maxima for lfe expectancy.
The medium decUine patterns for males and females are consistent, the rates of change
being identical. However, country-specific trends estimated from equations in Tables 6-8 need not
be consistent, and could, for a specific country, imply an increasingly large gap between the sexes
in life expectanc or even a reversal of the gap. To produce some minimum consistency between
male and female trends, limits can be set on the degree to which rates of change diverge.
We reexamined the set A and set B data for the differences in rates of change in male

14
versus female life expectancies. Differences in the rates of change of the logits of less than -.01 and
of greater than .02 were rare in the set A data. In the set B data they were more common, these
values representing the quartiles of the distribution of differences in rates of change (when the
limited transformation was used) or the 16th and 88th ptrcentiles (when the extended
transformation was used). For projection purposes, these values can be adopted as limits, adjusting
male and female rates of change upward or downward symmetrically when necessary to stay exactly
within these limits.
INFANT MORTALITY
Parallel analysis to that on life expectancy was run on infant mortality. The data win
be described, general trends will be determined, the prediction of specific country trends will be
discussed, and a projection approach will be presented.
Data. Data were drawn from U.N. (1988a) and from World Fertility and Demographic
and Health Surveys. Only those developing countries in U.N. (1988a) were included whose
estimates (in the judgment of Hill and Pebley [19881, working with the author of the U.N. report)
were based on reasonable data rather than informed demographic judgment Infant mortality
estimates greater than 150 per thousand were left out as probably less dependable and largely
irrelevant for projection purposes, since most countries are below this level. The resulting data set
(set A) included at least two quinquennial estimates between 1960 and 1985 for 91 countries.
Estimates will be assumed to apply, for this analysis, to the midpoint of each quinquennium. Again,
these data were supplemented with presumably less reliable estimates for 52 other countries (set
B) for 1975-80 and 1980-8S. The additional data, covering the majority of the developing countries
not already included, were drawn from World Bank files, and closely resemble the data in U.N.
(1988a) not included in set A.
General trends. The first question was whether some simple transformation could
linearize infant mortality trends, which usually show decelerating declines. Roots and logs were
tried, but were not satisfactory. Using the cube root does eliminate any correlation between infant
mortality level and subsequent rate of change (Table 9), but this transformation fits poorly at low
Table 9. Correlations between infant mortality rate at start of period
and rate of change within period in infant mortality, its roots, log, and
logit.
Transformation of infant mortality rate
Untrans- Square Cube Logit Logit
Period formed root root Log (limited) (extended)
1962-67 -0.47 -0.15 0.01 0.44 -0.08 0.04
1967-72 -0.44 -0.11 0.05 0.48 -0.22 -0.07
1972-77 -0.50 -0.08 0.11 0.58 -0.40 -0.19
1977-82 -0.61 -0.19 -0.01 0.48 -0.34 -0.14
Note: TFo LImited traauforU,tion amsume* a mlnL,n infant mortality rate of 6. The extended
tzansfomaetion assume a mizLmin of 3.

15
Table 10. Means and percentiles across countries of rate of change in
logit of infant mortality rate and working estimates.
Transformation No. of Percentile
and data set cases Mean 90th 75th 50th 25th, 10th
Limited
Set: A, 1962-67 73 0.065 0.025 0.043 0.056 0.088 0.111
Set A, 1967-72 81 0.064 0.021 0.038 0.062 0.085 0.104
set it, 1972-77 72 0.085 0.041 0.049 0.076 0.117 0.141
Set A 1977-82 58 0.086 0.033 0.050 0.077 0.120 0.165
Set B, 1977-82 52 0.056 0.000 0.038 0.051 0.085 0.102
Combined 336 O.C71 0.024 0.043 0.061 0.093 0.131
Extended
Set A, 1962-67 73 0.061 0.025 0.039 0.051 0.078 0.102
Set A, 1967-72 81 0.059 0.021 0.035 0.056 0.077 0.090
Set A, 1972-77 72 0.073 0.036 0.048 0.068 0.094 0.112
Set A, 1977-82 58 0.068 0.029 0.040 0.053 0.089 0.125
Set B, 1977-82 52 0.046 0.000 0.037 0.050 0.073 0.093
Combined 336 0.062 0.022 0.041 0.055 0.080 0.104
Working estimates Slo Mediu Raii
Limited 0.024 0.060 0.130
Extended 0.022 0.0z5 0.105
levels of infant mortality, from 20 per thousand or so down--a critical segment for long-run
projections. The natural log, on the other hand, fits well at low levels, but does not eliminate the
correlation between level and rate of change. A logit transformation similar to that for life
expectancy did work better, reducing though not entirely eliminating the correlation between level
and rate of change and fitting the data well at low levels.
The minimum and maximum for the logit were defined in the following way. Two
minima were chosen, a 'limited" alternative of 6 per thousand, roughly corresponding to the lowest
infant mortality level, for men and women combined, in the Coale-Demeny (1983) West family life
tables, and an 'extended' alternative of 3 per thousand, roughly corresponding to the lowest level
infant mortality would attain if the Coale-Demeny tables were extended to allow male and female
life expectancies to reach the "extended" maxima used earlier. A maximum infant mortality rate of
200 per thousand was chosen. This allows infant mortality to fall most rapidly when it has reached
about 100, which roughly corresponds to the level at which life expectancy should rise most rapidly.
Rates of change in the logit between quinquennia were calculated by country, and
averages and percentiles for various periods are shown in Table 10: for set A countries, for each
su'xessive pair of quinquennia; and, for set B countries, for the change between 1975-80 and 1980-
85. Averages vary across periods, appearing to rise over time in the set A data but being much
lower for the set B data for the last period. Arguably, having better data, as the countries in set
A do, is associated with faster reduction in infant mortality. Instead of defining a general pattern
solely on the basis of set A data, therefore, we combined all the data in the last row of each section

16
Figure 2
Annual decrements to infant mortality rate per thousand from different models,
assuming slow, medium, and rapid improvement
-1 .0 SLWETNE
-2.0
-3.0
-4.0
-5.0 PS0-EXTENDED
-6.0 -
-7.0 - . II,I...,........ I II I
160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10
INITIAL INFANT MORTAL ITY (PER rH4USAN)
of Table 10. This forms the basis for working estimates in the same table of slow, medium, and
rapid change.
What these patterns mean is illustrated in Table 11, which shows, for instance, that
the infant mortality rate would indeed drop fastest from around 100, by between 1.2 and 6.3 points
per thousand a year, as opposed to falling only .1 to .7 points a year once it has reached 10.
However, the decline is fastest in percentage terms--over 4 percent by the medium model--from an
infant mortality rate of around 30.
Figure 2 makes comparisons with the U.N. (1988a) and the previous World Bank
(Zachariah and Vu 1988) patterns. Country by country decrements in infant mortality between
1985-90 and 1990-95 in these two sources were approximated with cubic equations, using initial
infant mortality as the predictor, that fit quite well, giving R2 of .86-.87. These equations are used
to provide the curves in Figure 2. The decrements obtained here using the limited and extended
transformations are larger than those in the U.N. or previous World Bank patterns, but never by
more than one per thousand. The slow and rapid models provide a fairly wide interval around the
medium estimates. This interval is not as wide as actually reported declines: developing countries
with reliable data show a range of declines from 0 to more than 10 percent (U.N. 1988:129-131),
whereas the working models allow declines no smaller than .6 percent and no larger than 8.5
percent. Nevertheless, where averages of reliable data have been taken, they fall close to the
medium estimates. For 33 developed countries with initial infant mortality rates of 10 to 40 per

17
Table 11. Annual absolute and percentage changes ln infant mortality
rate, wlth dLfferent equations, from varying initlal levels.
Initial Limited Extended
level Slow Medium Rapid Slow Medium Rapid
Absolute change
150 -0.9 -2.3 -5.0 -0.8 -2.1 -4.0
125 -1.1 -2.8 -6.1 -1.0 -2.6 -4.9
100 -1.2 -2.9 -6.3 -1.1 -2.7 -5.2
75 -1.1 -2.6 -5.7 -1.0 -2.5 -4.7
50 -0.8 -2.0 -4.3 -0.8 -1.9 -3.7
25 -0.4 -1.0 -2.1 -0.4 -1.1 -2.0
10 -0.1 -0.2 -0.5 -0.1 -0.4 -0.7
Percentage decline
150 0.6 1.5 3.3 0.6 1.4 2.7
125 0.9 2.2 4.8 0.8 2.1 3.9
100 1.2 2.9 6.3 1.1 2.7 5.2
75 1.4 3.5 7.6 1.3 3.3 6.3
50 1.6 4.0 8.5 1.6 3.9 7.3
25 1.6 4.0 8.5 1.7 4.2 7.9
10 0.9 2.3 4.8 1.5 3.6 6.8
thousand, the average decline in 1975-80 was 4.9 percent, slightly above the medium estimates. For
34 developed countries with initial rates of 7 to 30, the average decline in 1980484 was 3.6 percent,
essentially identical to the medium estimates.
Predicting specfic trends As with life expectancy, the rate of change in infant
mortality is related to its rate of change in a previous period. Table 12 shows that the correlation
Table 12. Correlations between rate of change in log of lnfant
mortallty rate and rate of change in precedlng periods.
Limited Extended
Preceding period 1967-72 1972-77 1977-82 1967-72 1972-77 1977-82
Previous period 0.52 0.47 0.44 0.54 0.46 0.38
Period twice removed 0.33 0.36 0.32 0.21
Period thrice removed 0.16 0.15

18
Table 13. Regressions for rate of change in logit of infant mortality
rate on previous rate of change in logit.
Transformation and
period predicted B (t) Constant N
Limited
1967-72 0.5367 4.67 0.0298 0.25 61
1972-77 0.6467 4.13 0.0403 0.21 60
1977-82 0.4961 3.49 0.0438 0.18 52
All periods 0.5608 7.36 0.0369 0.23 175
Extended
1967-72 0.5328 4.97 0.0263 0.28 61
1972-77 0.5531 4.03 0.0362 0.20 60
1977-82 0.4355 2.95 0.0379 0.12 54
All periods 0.5050 6.82 0.0331 0.20 177
Working eauations
Limited 0.5 0.03
Extended 0.5 0.0275
is stronger the closer the two periods are, and becomes quite weak after 15 years or so.
Regressions were run to predict the rate of change in the logit of infant mortality
with set A data, using the previous rate of change and the same socioeconomic indicators used
earlier. Six regressions were run in all, for three periods and for both limited and extended
transformelons. The female primary and secondary enrolment ratios and GNP per capita had
significant effects, but only in one regression each. None of the other socioeconomic variables
had a significant e�ectL On the other hand, the rate of change in the logit of life expectancy in
the previous period had a significant effect in each regression. Table 13 shows regressions using
only the previous rate of change as a predictor. Working equations, based on these regressions,
can be defined so that, relative to preceding rates, predicted rates converge toward the previously
defined medium rates when the equations are applied for successive periods.
Projecton apprmach An approach to projecting infant mortality can be devised
from these results First, predict rate of change from the working models in Table 13, with the
rate of change for each period predicted from the rate for the previous period. Apply this process
for 15 years, requiring that the predicted rates fail within the limits described as slow and rapid
decline. Second, for the longer term, either apply the medium decline pattern or derive the infant
mortality rate from model life tables chosen on the basis of life expectancies. As with life
expectancy, either the limited or extended results can be used.
AGE PATIERNS OF MORTALITY
No analsis was done of age patterns of mortality beyond age 1. In order to provide
such patterns for projection purposes, one can rely on previously estimated model life tables. One
approach, using the Coale-Demeny (1983) model life tables, is outlined here, and consistency

19
Table 14. Regressionm for maximum allowable rate of change in logit of
infant mortality rate (standard errors in parentheses).
Predictor (1) (2) (3) (4)
Constant 0.01 0.05 0.10 0.15
(0.01) (0.01) (0.01) (0.00)
Change in eo -2.29 -2.12 -2.37 -2.45
(0.33) (0.28) (0.19) (0.12)
eO (both sexes) -8.6E-04 -1.4E-03 -1.8E-03
(2.3E-04) (1.8E-04) (1.3E-04)
Infanit mortality -2.3E-04 -9.OE-04
(3.8E-05) (l.lE-04)
Infant mortality 3.OE-06
(5.OE-07)
0.64 0.76 0.90 0.96
between projected infant mortality and infant mortality derived from model life tables is considered.
For each projection period, estimated male and female life expectancies can be used
to identify the appropriate levels for males and females of each Coale-Demeny Ufe table family.
The estimated infant mortality rate usually implies different levels. The maximum convergence
between the levels implied by life expectancy and infant mortality can be used to select one of the
four life table families. Alternatively, if good information on child mortality is available, it can be
used in conjunction with infant mortality to choose a life table family. Working within the chosen
family, one can then produce a "split' tife table, combining different levels, to give exactly the
desired infant mortality and life expectancy. First, choose the level that matches the infant mortality
rate, adopting the consequent age pattern of mortality for the childhood years (up to age 9 or 14,
as desired). Second, choose another level to give the age pattern of mortality for older ages such
that life expectancies, given the combined segments of the two life tables, match the projected
values.
If country-specific mortality trends are projected for three periods, split life tables could
be produced for at least that long. Once the country-specific trend is replaced by a universal
trend, it boomes more reasonable to adopt a universal age pattern. One can use projected life
expectancies to choose appropriate levels of the Coale-Demeny West family. the most general of
the four families. Several projection periods should be allowed for a gradual shift from some other
family to the West family.
Whether split or unified life tables are used, the consistency of projected infant
mortality can become a problem. When split life tables are used, projected infant mortality can
be exactly duplicated. In principle, however, it could fall so fast, relative to the rise in life
expectancy, that adult mortality would have to rise. This is seldom desirable, and to prevent it a

20
Figure 3
Infhnt mortaUty medium trend compared with trends when model lIfe tables are
chosen to match medium male or female life expectancy trends
w e s 7 N O R T H
a a a la i a- 1
E A S T S O U T M
e ",a ,a z,
maximum allowable increase for infant mortality can be established. We estimated what rate of
change in the logit of infant mortatity would prevent life expectancy at age ;,5 (els) from declining
at any point over three quinquennia, assuming West model tife tables and given different values of
the cuffent infant mortality rate, the cuffent male and female combined e,), and the current rate of
change in combined co, About 30 estimates of this maximnum were regresse on the other variables,
with results shown in Table 14. (rhis exercise wvas actually performed using a minimum infant
mortality rate, for purposes of estimating the logit, of 4, intermediate between the limited and the
extend-ad minima.) 'Me aUfowable maximum change in infant mortality varied most stfongly with
the ratc of change in IUfe expetancy, but was also affeactd by cuffent levels of life expectancy and
infant mortali, the latter having a nonlinear effect 'Me fina equation incorporating aUl these
effect attained an R2 of .96. Ibis equation does in fact stiUl aUlow a rise in adult mortaliq in
specific circumstances if other lik table families are applied, but ft substantiaRy moderates any such
increase.
When unifie life tables are used, projected infant mortaity is not exactly duplicated
but is approximated. 'Me choice of successive life tables from the life expectan trends imposes
a trend on infant mortality. Figure 3 shows that the imposed trend is in fact quite close to the

21
medium trend for Infant mortality estimated earlier. From an assumed infant mortality rate of 160
in year 0, the cuve described by the medium trend in life expectancy closely parallels the curves
descrbed f life table levels are chosen successively (based either on the male or on the female
medium Ufe expectancy trend) within the West family-though not necessarily if other life table
families are used. Results are shown for the limited transformation; results for the extended
transformation are similar.
ILLUSTRATIVE PROJECTIONS
Some results from applying the projection approaches described will be presented and
contrasted with results from other procedures. First we specify the contrasting procedures, and then
we discuss results for eight countries with current life expectanies varying from 52.5 to 76.8 years:
Zaire, Bolivia, Ghana, Pakistan, Thailand, Poland, Costa Rica, and Norway.
Procedures used. UNeWe projections using essentially the procedures described
above will be contrasted with "old^ projections using the World Bank procedures covered in the
literature review and with results from the latest U.N. (1989) assessment Between the new and
the old projections, base populations, initial estimates of vital rates, and trends for fertility and
migration are identical. Only mortality trends differ. The U.N. projections are potentially different
from the new and the old projections not only in mortality trends but also in other respects.
For the new projections, we used the limited transformations described above, because
the life tables necessary for the extended transformations are not now available. The 1985-90
quinquennium was taken as the base period. Using estimates of 1980.85 Hfe expectancies and
education data drawn from World Bank files, rates of change in male and female life expectancies
were estimated from the 'all periods equations in Table 6. Adjustments to these rates were made
as necessary to keep them within the limits described earlier, in order to prevent too rapid or too
slow change or too great a discordance between male and female rates of change. These rates of
change were applied for one period, and rates for the two following periods were estimated from
the working equations in Table & For subsequent periods, rates of change were assumed to be
equal to the medium working estimates in Table 2. Rates of change in infant mortality were
estimated from the working equations in Table 13, but were required to stay within limits previously
described. Life tables were chosen from the Coale-Demeny '1983) set. For the first three
quinquennia, the life tables were split at age 15 (using a Fortran Irogram called Split), one segment
being chosen to give the estimated infant mortality rate and the oi a r segment being chosen to give
the estimated life expectancy. The life table family chosen %. s the one that minimized the
divergence in levels across the split. For subsequent quinquennki, life expectancies alone were
used to choose unified life tables, and the West family was always applied.
A cohort-component population projection program developed for the World Bank
called ProjPC (Hill n.d.) was used. This program allows the specification of mortality parameters
for arbitrarily selected periods, interpolating linearly to obtain mortality assumptions for other
periods. We specified survivorship ratios for the base quinquennium and the next three
quinquennia, and then for 2025-30, 2050-55, and 2100.2105. This allowed the program to
interpolate and to shift smoothly from whatever life table family was initially chosen to the West
family.
To complement the new projections, which give a medium trend, projections of slow
and rapid improvements in mortality were also made. For rapid mortality decline, rates of increase
in life expectancy were raised by half, and rates of decHne in infant mortality doubled. (Initial rates
that are close to the medium working estimates in Table 2 would by these calculations become
almost equal to the rapid working estimates.) These calculations were done for the first three

22
periods. For subsequent periods, the rates of change in life expectancy were assumed to equal the
rapid working estimates. For slow mortality decline, rates of increase in life expectancy and rates
of decrease in infant mortality are cut in haif for three periods, and the slow working estimates for
life expectancy applied subsequently.
Note that neither the new nor the old projections correspond exactly to the standard
World Bank projections. The new projections incorporate some features not so far used in the
World Bank projections, and the old projections incorporate updates of base-period data and of
projected trends in fertility and migration.
Results. Comparisons of life expectancy, infant mortality, and total population trends
wiU be discussed, and some reference will be made to other population parameters.
Figure 4 shows five alternative projections of life expectancy trends in each of the
eight countries, using the new, old, U.N., rapid, and slow patterns. The new and old patterns are
relatively close: for all of the countries for most periods, the new estimates are within 2 percent
of the old estimates, and are sometimes above and sometimes below them. The biggest differences
are for Ghana and Bolivia about 25 years into the future, when the new estimates are 4 or 5
percent lower because they take into account the slowness of past improvements in life expectancy
in these two countries. The new estimates are also close to the U.N. patterns for half of the
countries, those with higher mortality. For Costa Rica, the U.N.'s initinl estimate is higher but the
U.N. trend is for slower improvement. For Thailand, Poland, and especially Norway, the U.N.
projects faster improvement. One reason for these differences is the use of the limited
transformation, which means that life expectancy in the new projections cannot rise as high as the
U.N. tnodel allows. The rapid and slow patterns generally provide an envelop within which most
of the other projections fall, except at the highest levels of life expectancy. Life expectancy is
shown for both sexes combined; comparisons by sex lead to similar conclusions.
For infant mortality (Figure 5), greater divergence among the patterns appears. This
is because the old Bank procedures used current infant mortality rates derived from model life
tables, rather than attempting to match reported rates. Relative to the old rates for the base
period, the new rates are as much as 40 percent higher or lower. Similar variation exists for future
periods. However, for no country are the new estimates continuously higher or lower than the old
estimates: for Thailand, for instance, the new estimate is 40 percent lower than the old estimate
for 2050 but 30 percent higher than the old estimate for 2100. The new p-attern, because it is
based on the limited transformation, does not allow infant mortality to fall as low vs in the U.N.
estimates.
These variations in mortality assumptions allow the new crude death rate estimates
to vary by 10 percent up or down relative to the old estimates, except for Ghana and Bolivia,
which show greater variation. Variation in probability of dying by age 5 (qs) and expectation of
life at age 10 (e10) were also examined. Comparisons of the former resemble comparisons of infant
mortality trends, and comparisons of the latter resemble comparisons of life expectancy at birth.
The consequences for total population are shown in Figure 6, which expresses the new
estimates as a percentage of the old. The largest changes in population between the two sets of
projections are a reduction of 2.5 percent and an increase of 1.7 percent. These changes may not
be entirely negligible, but they are small. For Ghana, a 5 percent maximum reduction in life
expectancy, relative to the old estimates, translates into a 20 percent maximum increase in the crude
death rate, and eventually, after a lag of some decades, a 2.5 percent maximum decrease in the
population, also relative to the old estimates. Changes in the assumed mortality trend, as measured
by these indicators, translate into delayed and considerably less than proportional changes in total

23
Figure 4
Lifa apency in years projected by different modeb, selected countries,
1985-2100
ZA IRE BOL IV IA
; t XtS all 11Zi A SD5 1S 6 D~ 2165 JS 216S 2S J
G H A N A P A K I S T A N
I. SI~~~~~~~~~~~~~~W
* U~~~~~~~~~~~~~~~~~~~~~~~~3
*;4s5
sa ~ ~ ~ ~ ~ ~ ~ ~ ~~~25233 3
|~~GHN PAl I SEMaAN " " g M3"g

24
Figure 4 (continued)
Life expectancy in years projected by different models, selected countries,
1985-2100
THA I LAND POLAND
71~~~~~~~~~~~~~~~~~~~~~~~~~~7.
C0S T A R I C A NO0R WA Y
u. ..
7- ,,,,,,,.. f .
z Ma sM mM M-

25
Figure 5
Infant mortality rate per thousand projected by different models,
selected countries, 1985.2100
ZA I RE B'JL I V I A
3* U
U U
(3 H A N A P A K I S T A N
4 1
X ~~I 4 "n U"" " Ma 52

26
Figure 5 (continued)
Infant mortality rate per thousand projected by different models,
selected countries, 1985.2100
THA I LAND POLAND
25
* ~~~~~~~~~~~~~~~~~~~~~~~~24
a
45.~~~~~~~~~~~~~~~~~~~~~~~1
a ii''W'6 i~
* mao au nM so ais all Xo a s Mn n oS m
COSTA R ICA NORWAY
IZ 51 l
.1.
S- wB-,-�,w,..,..,..,.~~~~~~~~~~~~~~1
11 M N f E! ls a 11 ax 1 f al

27
Figurc 6
Total population using new mortality pattern as a percentage of population
using old mortality pattern, selected countries, 1985-2100
population. Nor are the age structures of populations greatly different between the new and the
old projections: dependency ratios resemble each other, as population totals do.
More drastic changes in mortality assumptions, such as those represented by the rapid
and slow mortality decline pattems, can produce larger variations in population, particularly for
higher-mortaflty countries. (In the long run, life expectancy in the rapid, medium, and slow patterns
must converge at the predefined upper limit. Therefore, differences among the mortality
assumptions in these projections become greater for a period of 60 or so years at most, after which
they slowly disappear.) Ile situation of Zaire is represented in Figure 7. Relative to the new
medium mortality decfine pattern, the rapid decline pattern allows life expectancy to be up to 6
percent higher vithin the 1985-2100 period, the crude death rate up to 30 percent lower, and the
-.nfant mortality rate up to 50 percent lower. As a result, population is as much as B percent higher
Figure 7
Life expectancy, infant mortaliq, crude death rate, and total population
under rapid and slow mortality decline, as percentages of parallel
estimates under medium mortality decline, Zaire, 1985-2100
n I
__o__a m

28
given rapid mortaUty decHne rather than medium mortality decline. The slow decline pattern allows
life expectancy to be up to 10 percent lower than under medium mortaity decline, the crude death
rate up to 50 percent higher, and the infant mortality rate up to 170 percent higher. Population
wiU be up to 13 percent lower under slow mortality decline than under medium mortality decline.
CONCLUSION
Procedures for short-run and long-run projections of mortality applicable to all
countries of the world have been devised based on analysis of recent mortality trends. These
procedures involve calculating rates of change for and separately projecting male and female life
expectancy and infant mortality and then selecting appropriate model life tables.
The procedures were to have been based on relatively reliable estimates of life
expectancy and infant mortality. However, comparisons of such estimates with other estimates
based on weaker data indicated consistent differences: better estimates of life expectancy showed
slower improvements than other estimates, and better estimates of infant mortality showed faster
improvements than other estimates. Because reasons could be adduced for these differences, the
procedures were ultimately based on all the assembled data rather than solely on the better data.
Attempts to predict change in life expectancy and infant mortality from socioeconomic
variables were minimally successfuL To a large extent, it can be argued, socioeconomic variables
add little preditive power if previous trends in these variables-themselves conceivably dependent
on socioeconomic factors-are taken into account. The procedures finally devised involve predicting
country-specific trends in mortality for 15 years (beyond which current mortality trends have little
influence) and then imposing a uniform mortality trend.
Comparisons of the procedures with previous World Bank procedures and with U.N.
procedures for projecting mortality indicate that life expectancy trends are quite similar. Infant
mortaUty trends differ more. The derived procedures allow infant mortality to fail somewhat faster
overal, and result in greater contrast with previous Bank projections than is the case for life
expectancy. The effect on total population of switching from the previous procedures to these
derived ones is smalL
Assessment of the adequacy of these procedures involves two issues: how well past
trends are represented and how accurately future trends are forecasted. Past trends are represented
by data that is somewhat uncertain for developing countries, and that may undergo revision in the
future. Alternative approaches to representing past trends are also possible, and could be as
satisfactory statistically while implying different future trends. The current representation does
reflect available data and might be taken to provide standards for judging mortality improvement
in different countries, but cannot be considered definitive as both data and approaches will evolve
in the future.
How accurately these procedures forecast future trends will not be known for some
time. The critical assumption in this exercise, that future trends will resemble what we know of
past trends, will undoubtedly not be universaly true. For instance, the spread of Human
Immu4odeficiency Virus infection is not reflected in the data, and lherefore is not factored into the
derived procedures. As this spread should have important impact on mortality in particular
countries, adjustments to the mortality trend may be called for. Without adequate data, we have
not considered thbis issue here. Other uncertainties regarding future mortality trends also exist, such
as uncertainty about the ultimate limits to life expectancy. This exercise cannot resolve such
uncertainty-, instead, it provides a perspective on what the future could be like if it resembled the
more certain past.

29
ACKNOWLEDGMENTS
Kenneth Hill provided some data for this paper. Patience W. Stephens and My T.
Vu assisted In Its preparation.
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