CHANGES IN MORTALITY AND LIFE EXPECTANCY: SOME METHODOLOGICAL ISSUES

CHANGES IN MORTALITY AND LIFE EXPECTANCY:

SOME METHODOLOGICAL ISSUES

*

Marcia Caldas de Castro

Office of Population Research Princeton University Wallace Hall, room 229

Princeton, NJ – 08544 Tel.: (609) 258-4943 Fax: (609) 258-1039 mcaldas@opr.princeton.edu

Measuring and explaining the effects of mortality changes on life expectancy has been discussed for the past three decades. Different approaches have been proposed using discrete or continuous methods. Two basic ideas underlie these approaches. The first compares two different mortality schedules and quantifies the contribution of each age group to the increase in life expectancy. The second analyzes how the progress in the mortality schedule translates into progress in life expectancy. This paper discusses and compares the approaches proposed by the United Nations 1982), Arriaga (1984), Pollard (1982 and 1988), and Vaupel (1986), identifying their problems, advantages, and the types of situations where each one can best be applied.

INTRODUCTION

Consider two populations that had life expectancies equal to 55 years at the same point in time. If 10 years later both populations have increased their life expectancies to 62, does that mean that mortality rates have changed exactly the same throughout all age groups? If, instead, one population had increased its life expectancy to only 60, would that population have experienced lower improvements in mortality at all ages? The answer to both questions is no. But to understand this whole process

*

I thank the valuable comments and suggestions of Noreen Goldman, German Rodriguez, Burt Singer and James Trussell. This is a modified version of the paper Changes in life expectancy in Brazil: 1940/1990: methodological issues, which was awarded the Charles Westoff Prize in Demography in 2000.

perfectly, it is necessary to understand how changes in mortality affect life expectancy, and to devise a method that allows the decomposition of changes in life expectancy by age groups. Although some methods have been proposed, it is necessary to compare them and identify the types of situations where each one is best recommended.

In 1982, three different authors dealt with this issue. A United Nations (1982) study proposed a method to decompose the change in life expectancy at birth in three major age ranges (0 to 29 years, 30 to 64 years and 65 years and over), based on Kitagawa’s (1955) technique to compute the differences between two sets of specific rates. Arriaga (1982) introduced the notion of temporary life expectancy, and suggested an index to measure, by each age group, the annual relative change in the years to be lived. Finally, Pollard (1982) proposed a continuous method that compares the difference between two different mortality schedules, and quantifies the contribution of each age group to the increase in life expectancy at birth.

Two years later, Arriaga (1984) developed a technique to measure and explain the changes in life expectancy, considering the changes in the age-specific mortality rates. It was a discrete method constructed on the basis of life table functions.

Later, Vaupel (1986) extended Keyfitz’s work (Keyfitz and Golini, 1975 and Keyfitz, 1977) and proposed a continuous method to analyze how the progress in the mortality schedule would translate into progress in life expectancy at birth.

Finally, Pollard (1988) revisited the method he first proposed in 1982, showing its similarity with Arriaga’s discrete approach.

The purpose of this paper is to compare the different methods proposed, using Brazilian data for the period between 1940 and 1990, pointing out the strengths,

weakness and limitations of each method, the most feasible situation when each one should be used, and the gaps still to be pursued in this kind of analysis.

EFFECTS OF MORTALITY CHANGE ON LIFE EXPECTANCY

The simple comparison of mortality rates at two different points in time shows that mortality differs in all age groups by different magnitudes. Changes in mortality in a particular age group affect life expectancy in direct and indirect ways. However, contributions to life expectancy increase cannot be measured only in terms of changes in mortality in each age group, since the overall change has associated with it the notion of interaction effects.

Formally, two basic relationships show this problem. The life expectancy at birth is given by da a p e =

∫

ω 0 0 ( ) , (1)

where ω is the oldest age considered (beyond which no one survives), and p(a) is the probability of surviving from birth to age a

   − =

∫

a u du a p 0 ( ) exp ) ( µ , (2)

where µ(u) is the force of mortality at age u. From (1) and (2), it is easy to see that

da du u e =

∫

ω −

∫

aµ  0 0 0 exp ( ) . (3)

So, when µ(u) changes at a particular age group x to x+i, both the probability of surviving and the average time lived at those ages will change. This is called the direct effect. Moreover, this change in µ(u) affects the number of survivors that will move on

to the next age group, and hence the time lived at ages x+i to ω. This is called the indirect effect. But, in reality, µ(u) changes are observed at all ages, and not only at a particular age group. This means that the new survivors at age x+i will spend more time at ages x+i to ω than under the old mortality regime. This is called the interaction effect. Figure 1 explains this problem graphically. As a whole, changes in the mortality schedule can be represented by a movement on the survivors curve (lx curve), and each

of these curves has a different life expectancy at birth (Figure 1.a). However, the contribution of each age group to the change in life expectancy is not the same.

<Figure 1>

Figure 1.b shows the effects of changes in mortality that occur in the particular age group x to x+i. Two main effects are observed on the number of survivors and, consequently, on the life expectancy at birth. The first one, highlighted as the direct effect in Figure 1.b, is due to the change in years of life within that particular age group as a consequence of its observed change in mortality. The second one, the indirect effect, is due to the change in life expectancy because the mortality change within the particular age group x to x+i affects the number of survivors at the end of the age group. It is called indirect because the changes occur after age x+i.

As was formally introduced before, both direct and indirect effects (shown in Figure 1.b) consider that mortality changed only within the age group x to x+i, independent of the mortality change in other age groups. But, as said before, mortality changes simultaneously in all age groups. As a consequence, part of the change in life expectancy comes from the fact that the new survivors at the end of the age group x to x+i (responsible for the indirect effect) won’t necessarily experience an unchanged

mortality. This fact is shown in Figure 1.b as the interaction effect, which is produced by additional survivors from one age being subject to lower mortality at later ages.

The total effect on life expectancy given a mortality change is the sum of the three effects described above. However, adding up direct, indirect and interaction effects individually needs some caution. The total effects are additive by age, since they are just the amount by which the life expectancy changed. Nevertheless, each of the three effects doesn’t hold this property individually (Arriaga, 1984). For example, to get the total effect of a mortality change between ages 50 to 60 it is correct to add up the total effects of the age groups 50-55 and 55-60. However, adding the direct effects of these two age groups would produce a smaller effect than the one for the age group 50-60 (part of the effect produced at older ages would be missing). The opposite would be observed for indirect and interaction effects (the effect of the older groups would be counted twice).

MEASURING THE CHANGE IN LIFE EXPECTANCY

The methods developed to measure the changes in life expectancy, given a change in mortality rates, can be classified according to two different approaches. The first one is based on the comparison of two mortality schedules. Main and interaction effects are calculated from the basic relations among functions of the life table. A decompositional technique was developed by the United Nations (1982), a discrete method was proposed by Arriaga (1984), and a more general and continuous one by Pollard (1982 and 1988).

The second approach, proposed by Vaupel (1986) uses the notion of entropy, first introduced by Demetrius1 (1974 and 1979), and later extended by Keyfitz and Golini (1975) and Keyfitz (1977), to analyze how the progress in the mortality schedule can be translated into progress in life expectancy.

United Nations’ Method

The UN proposed a simple method to decompose the gains in life expectancy in three major age ranges. Based on the techniques described in Kitagawa (1955) to decompose the difference in two or more sets of specific rates, Eq. (1) was written as the sum of three different integrals. Then, the contribution of mortality change between times 1 and 2 to the change in life expectancy was expressed by the sum of the contribution below age 30, the contribution between ages 30 and 65, and the contribution above age 65.

There is no clear discussion regarding the different effects on life expectancy, caused by a mortality change, and no discussion about potential heterogeneity in the mortality schedule.

Compared to the other methods proposed, this is the one with the least power of explanation and interpretation. The use of only three age groups hide important contributions of specific ages. For example, increases in mortality at particular age groups would not be captured (as was, in fact, observed when applying the method to male life tables of the Southeast region of Brazil during the 70’s and 80’s). Analogously, although the contributions below 5 years of age is usually significantly

1

The concept of entropy was developed on the basis of the ergodic theory and statistical mechanics, as a measure to describe the dispersion of the net maternity function and the convexity of the survivorship curve.

higher than for the other age groups (specially for populations with low life expectancy at birth), the aggregation used can make the contribution of the 0-30 and 30-65 age ranges very close to each other. Even though the method could be easily applied to a five-year age group perspective, the interpretation of the results is not very powerful.

The great contribution of this method, at the time it was proposed, was to provide a simple way to decompose the gains in life expectancy, when data were not available for single or five-year age groups, which was the case for some developing countries. Considering that the UN publishes worldwide indicators, this method was a great step in providing a way to compare changes in life expectancy among different countries. However, the techniques proposed by Arriaga, Pollard and Vaupel can be used with any age range, and are able to provide a better understanding of the observed changes. Based on that, a detailed analysis will be focused on these three methods.

Arriaga’s Method

Considering the different effects described before, Arriaga (1984) proposed a method to measure the changes in life expectancy, given a change in the age-specific mortality rates. Using a discrete perspective of basic functions of the life table, the author presents the following relations for each of the effects, given a change in mortality at ages x to x+i, observed between the periods t and t+n:

Direct effect ⇒ interval, ended -open for the , and ,     − =     − ₋ − = + + + + + + + t x t x n t x n t x t a t x x t x t i x t x n t x n t i x n t x t a t x x i l T l T l l DE l T T l T T l l DE ω (4)

Indirect effect ⇒     − = ₊ + + + + ₁ n t x t i x n t i x t x t a t i x x i l l l l l T IE . (5) Interaction effect ⇒ _{t n i x} i x t i x n t x t x t a n t i x x i IE l l l l l T I −     − = ₊ + + + + + _{. (6)}

The open-ended interval has only direct effects, as is clear from the definition of each type of effect. Eq. (4) to (6) measure the effects on the change of life expectancy at age a (most commonly at birth, a=0). Similar formulas could be written for the effects on temporary life expectancies (life expectancies calculated between any two chosen ages), when the open-ended interval does not have reliable data. So, for example, instead of working with life expectancy for the age range 0 toω, one could work with a different age range, believed to have good and reliable information. In this case, the temporary life expectancy for the age range would be

x i x x x i l T T e = − + , (7)

where x is the lower limit and i the upper limit of the age range considered (x ≥ 0 and i < ω).

The same idea could be applied to measure racial and sex differentials in life expectancy. Since this is a discrete technique, there are no approximations on the computation of the effects, and their sum is exactly the difference between the life expectancies of the two periods analyzed. Also, since it just considers basic relations among life table functions, there is no assumption about heterogeneity in the mortality schedule.

Although it is easy to analyze and interpret the results for the main effects (direct and indirect), the interaction effects are very hard to explain. In the examples found in Arriaga (1984) the interaction effects were often small, but this is not always the case (as the examples with the Brazilian data will later show).

Pollard’s Method

This method describes the changes in life expectancy using only the elementary notions of life tables (written in continuous form), and the knowledge of the types of effects that a change in mortality can cause in the years of life to be lived (main and interaction effects described before).

Consider the basic relations between mortality and life expectancy implicit in a life table and given by Eq. (1) and (2). Based on (1), changes of life expectancy between times 1 and 2 can be written as

[

p a p a

]

dt e e − =

∫

ω − 0 1 2 1 0 2 0 ( ) ( ) , (8)

which after some algebraic manipulation is equivalent to

[

a a

]

{

[

u u

]

du

}

p a e da e e t 1 _a1 0 0 2 1 2 1 1 0 2 0 − =

∫

( ) − ( ) exp

∫

( ) − ( ) ( ) ω µ µ µ µ . (9)

Expanding the exponential term in (9) in terms of the power of the integral, it is possible to get the main and interaction effects of the mortality change in life expectancy. However, Pollard (1988) assumes that the interaction effects are small, except when life expectancy at birth is low (below 55 years). Using this assumption, and considering that interaction terms are hard to compute and not easy to interpret, the alternative is to get an exact solution for (9), which combines both main and interaction effects. The proposed solution is

[

a a

]

p a e da e e 2 _a1 0 2 1 1 0 2 0 − =

∫

( ) − ( ) ( ) ω µ µ . (10)

This difference result can also be written as

[

a a

]

p a e da e e 1 _a2 0 2 1 1 0 2 0 − =

∫

( ) − ( ) ( ) ω µ µ . (11)

Expressions (10) and (11) can be interpreted as weighted averages of the mortality improvements between times 1 and 2. The weights

(

p(a)1e_a2 and p(a)2e_a1

)

are of comparable magnitude and approximate a straight line. There is no theoretical reason to prefer the weights in (10) or those in (11), so Pollard (1982) suggests the use of their arithmetic mean. The final relation between changes in mortality and life expectancy at birth can be expressed as

[

x x

]

w dx e e − =

∫

ω µ −µ _x 0 2 1 1 0 2 0 ( ) ( ) , (12)

where wx are the final weights given bywx =

[

p(x)2e1x+ p(x)1ex2

]

2.

For numerical evaluation of (12), the following approximation is used

(

−

) (

+ −

) (

+ −

)

+

(

−

)

+. ≅ − 12.5 2 10 5 1 10 5 5 . 7 2 5 5 1 5 5 2 2 1 4 1 1 4 0 2 0 1 1 0 1 1 0 2 0 e Q Q w Q Q w Q Q w Q Q w e , (13) where_iQ_x =

∫

i _x+tdt=−ln

(

l_x+i l_x

)

0µ .

For the first two age groups, considering that deaths are concentrated at younger ages, the weights are not calculated for the mid-age of the interval, but for ages 0 and 2.

Pollard (1988) argues that the continuous method given by expression (12) is similar to the discrete approach proposed by Arriaga (1984), and if very small age groups were used, the main effects would converge, in the limit, to the first term of the

expansion of the exponential term in (9)2, while the interaction effects would converge to the difference between (12) and that first term of main effects. A comparison between this method and Arriaga’s approach can give an idea of how accurate is the assumption that the interaction effects are captured by (12).

The same method could be applied to the study of the contribution of different causes of death, since the force of mortality at each age is the sum of the forces of mortality for the various causes. Furthermore, this method could be applied to examine sex differentials in the life expectancy (Pollard, 1983)3.

Similarly to Arriaga’s method, the analysis does not take into account any heterogeneity concerns.

Vaupel’s Method

Supposing an equal proportional change in mortality at all ages

( ) ( )(

µ δ

)

µ∗ = +

1 x

x Keyfitz (1975 and 1977) has shown that the rate or intensity of progress in mortality over time, ρ(a)=

[

∂µ

( )

a ∂t

] ( )

µ a , and the rate of progress in life expectancy over time, π =

[

∂e₀ ∂t

]

e₀, can be related as π =ρH , where H is the measure known as entropy, which can be expressed as

( ) ( )

( )

∫

∫

− = _ω ω 0 0 ln da a p da a p a p H . (14) 2

According to Pollard (1982) the first term of the expansion is the main effect, and can be written as

[

a a

]

p a eada 1 1 0 2 1 ) ( ) ( ) (

∫

ω µ −µ 3

Although Arriaga’s method can also be applied to measure sex differentials, the two approaches cannot be compared (and, hence, will not be analyzed in this paper). While Arriaga would measure the contribution of each age group to the difference between a male and a female life expectancy for a given period, Pollard is measuring the effects of a change in mortality rates between two periods in the sex differentials.

Goldman and Lord (1986) and Vaupel (1986) presented an alternative way to express the entropy, allowing a better understanding of this expression. After some calculation, H can be written as

( ) ( )

0 0 0 0 e dx e d e dx e x p x H =

∫

x =

∫

x x ω ω µ , (15)

where dx is the number of deaths, and the numerator can be interpreted as the average

years of future life that are lost by observed deaths.

Vaupel (1986) proposed a way to decompose H. Writing

( ) ( ) ( )

x µ x p x ex e0

η = ₍₁₆₎

one can then write H as H =

∫

ωη

( )

x dx

0 . This decomposition allows the evaluation of

the rate of progress in life expectancy when progress in mortality is observed at an instantaneous age a as

( ) ( )

a η a . ρ

π = ₍₁₇₎

So η(a) can be defined as the potential increase in life expectancy given that mortality was reduced at age a. As a rule of thumb, Vaupel (1986) proposes that for populations with life expectancy higher or equal to 65 years, the age group with the highest potential for saving years of life will usually be near the life expectancy of birth.

The same idea can be extended for an age interval, and generalized for the entire age distribution

( ) ( )

.

0 η a ρ a da

To compute the average potential for saving years of life over the age interval x to x+i, Vaupel (1986) proposes

. where , 2 e0 x i a e e i d_{x x x i} i + + = = + η (19)

This expression comes directly from the definition of η(a), and is very similar to the simple average of η(a) over the length of the interval. The only difference is that the proposed expression considers an average of the life expectancies at the beginning and end of the interval.

To compute the average progress in mortality over a period t, the proposed formula is

(

)

(

)

(

(

)

)

(

− −_iq_x − − −_iq_x

)

t = ln ln1 ' ln ln1 ρ , (20)

where iq’x is the earlier mortality. This expression comes directly from the initial

formulation of ρ(a), and from the basic relations between functions of the life table (µ(x), lx and qx).

Since in practice there is no available information of the progress of mortality for each instantaneous age a, the method assumes that improvements in mortality are uniform within the age group. Although actual mortality changes do not occur this way, the assumption is more reasonable if one works with single year age groups, instead of five-year age groups. However, complete life tables are not frequently available.

Analogously to the other methods, it is assumed that people who die at some age, if saved, would have the same life expectancy of those who stay alive. That means that no heterogeneity concerns are incorporated into the model.

Similarly to Pollard’s method, the results are an approximation of the changes in life expectancy implicit on the life tables used.

Finally, the results are based on a stationary population (life table), and not on the actual age distribution of the population (that would probably have more young people than the life table, for example). Vaupel (1986) makes an exercise for 1979 US data and shows that 50% of the increase in life expectancy would be produced by an improvement in mortality after 65 years old, while using the actual age distribution and the observed number of deaths this number would be only 36%.

METHODOLOGICAL COMPARISON OF THE METHODS

To analyze and compare the methods described here, male and female Brazilian life tables for the years 1940, 1950, 1960, 1970, 1980 and 1990, for the whole country and for the Northeast and Southeast regions were used. These 36 life tables were constructed from Census and Civil Register data, using indirect methods of mortality estimation (Simões, 1997), and provide a very rich example, since they gather different patterns of mortality change, both in magnitude and in direction.

All the results are based on the life expectancy at birth. Arriaga (1984) suggests the use of temporary life expectancy from birth to the last age that has reliable information, when the quality of the data for older ages is questionable. However, as Table 1 shows, the results for the contribution of each group to the gains in life expectancy at birth and to the temporary life expectancy from birth to age 69 are very similar, especially for the Southeast region, which is the area with one of the best register coverages in the country.

The different methods, in general, lead to the same conclusions concerning the contribution of different age groups to the improvement in life expectancy at birth through time. However, issues related to interaction effects of mortality change and increases in mortality rates bring up significant differences between them. Table 2 shows the percent contribution of each age group to the gains in life expectancy for 3 different scenarios. Each one has interaction terms of different magnitudes, allowing a better comparison of its effects on the results of each method.

The first comparison to be made is between the results given by the techniques proposed by Arriaga and Pollard, in order to check if the interaction terms are actually being well captured by the latter one. If they are not, one could expect to find the difference between the two methods very close to the interaction effects measured by Arriaga’s approach.

The magnitude of the interaction effects measured by Arriaga’s method can be analyzed in Table 1. The first scenario, females from the Southeast region during 1940/50, has very little interaction effects (only 1.4% of the total change), although the life expectancy at birth in 1940 for this population was only 50 years. Despite this low life expectancy, major changes in mortality were not observed during the decade, resulting in very small interaction effects.

A similar situation is observed in the second scenario, males from the Southeast region during 1970/80. However, although the interaction effects are small (only 1.5% of the total change), they are twice as big as the direct effects. The life expectancy at birth in 1970 was 56 years, and mortality rates varied in a very different way than in the previous decades (decreases in younger ages, and increases between ages 50 and 65).

The most peculiar situation is observed for men in the Northeast region during 1970/80, which had a life expectancy at birth of 44 years. Mortality rates changed substantially during the decade, and the interaction effects represent 13.4% of the total change in life expectancy.

<Table 2>

Despite the interaction effects, however, the results of the methods proposed by Arriaga and Pollard are very close, even when the interaction terms are big (Table 2 and Figure 2). The differences between observed and calculated improvements in life expectancy, by Pollard’s method, are small (lower than 3%), and the differences between the two methods, actually, are not a result of interaction effects alone, but a combination of interaction effects and approximation issues. Comparing them with the interaction effects measured by Arriaga’s method would lead one to the conclusion that when interaction effects are bigger, they are better captured by Pollard’s method. In reality, since the differences are small, they are more likely to have the same magnitude of small interaction effects than of bigger ones. Furthermore, most of the difference is driven by the open-ended age group, which has an insignificant contribution according to Pollard’s method4. So, it is reasonable to say that the discrete and the continuous approaches give similar and comparable results.

<Figure 2>

4

The open-ended interval, for all combinations of region, period and sex, has an insignificant contribution for the increase in life expectancy. This is a peculiarity of the method when the last age group is open. An alternative would be to work only with a particular age range, as proposed by Arriaga (1984), discarding the last age group (or the older ages groups that are not reliable). This issue suggests that the contribution of the open-ended interval may not be perfectly accounted for by Pollard’s method.

Although Pollard (1988) assumes that a life expectancy at birth below 55 years is the threshold for significant interaction effects, this is not a sufficient condition. In many of the Brazilian life tables analyzed, the combinations of sex, region and period that had life expectancy at birth below that threshold presented interaction terms ranging from 1.4% to 3%, leading to similar results despite the method used. Though, it is also important to consider how mortality rates changed during the period.

It is also worth mentioning that since the life tables used are for five-year intervals, the discrete formulas for the main effects include the interaction effects inside the interval. The smaller the interval, the better the interaction effects would be measured. This fact, alone, represents a clear advantage of the continuous approach over the discrete one.

The last scenario in Table 2, Northeast males for the period 1970/80 is the one associated with interaction terms with the highest contribution to the overall effects (13.4%). As is shown in Figure 2.c, these are exactly the cases where the most significant differences between Vaupel’s method and the other two techniques are observed. In this case, the differences between observed and calculated rates of progress of life expectancy were 26% of the observed rate.

With an interaction term around 5% of the total change in life expectancy at birth (second scenario in Table 2), Vaupel’s approach also shows some differences in comparison with the other two methods (Figure 2.b). On the other hand, Figure 2.a shows that when the interaction term is very small (only 1.4% of the total change in life expectancy at birth), all the three methods lead to the same conclusions, and the differences between them are insignificant (first scenario in Table 2).

Since the interaction effects are higher for the younger age groups (those that contributed the most to the change in life expectancy at birth), Vaupel’s method underestimates the percent contribution of these age groups (Figure 2.b and 2.c).

These three examples make it clear that the interaction effects do play an important role in the analysis. For a comparison between the methods proposed by Arriaga and Pollard, the interaction terms are not the major concern, since they basically end up giving similar results. The concept behind both methods is the same: use of life table functions to measure the different effects of a change in the mortality rates. However, Vaupel’s method is based on a different concept. It does not use the idea of main and interaction effects, but it has a unique power of explanation.

Table 3 shows the potential for saving years of life given the mortality schedule implicit in the life tables for Brazilian women and for men in the Northeast region, calculated using Vaupel’s method. A general pattern is that as life expectancy at birth increases, the potential for saving years of life decreases for earlier ages and increases for older ones. For Brazilian females, a 1% reduction of mortality at 65 years or more would increase life expectancy by only 16.5% in 1940, but in 1990 this increase would be 51.5%. On the other hand, the same mortality reduction below 5 years old would increase life expectancy by 40% in 1940, but by only 22% in 1990. This pattern is observed for any combination of sex, period and region. The Northeast region has the lowest life expectancies, and consequently is the region where the potential for saving years of life, given reductions in mortality rates at older ages, is smaller.

As life expectancy increases we can also see that entropy (H) falls. So, as life expectancy increases, the progress in reducing mortality will be translated in less further increases in life expectancy (that is the general idea behind the use of entropy as a measure of convexity of the survival curve).

So, for example, if we consider Northeast men in 1940, which had a life expectancy of only 35 years, a 1% reduction in the force of mortality at all ages would increase life expectancy by 70%. But if, instead, we consider Brazilian women in 1990, which had a life expectancy of 70.9 years, a reduction in the force of mortality by 1% at all ages would increase life expectancy by only 25%.

The analysis of the age group with higher potential for saving years of life is consistent with Vaupel’s rule of thumb. So, usually in the cases where life expectancy was below 65 years, infant ages have the greatest potential for saving years of life, while in the cases that life expectancy has transposed this threshold the greatest potential is associated with older ages. To compare how this potential for saving years of life was translated into actual gains in life expectancy, Table 4 shows the cumulative percentage of the potential and actual improvement in life expectancy of three different scenarios, which allow a better understanding of Vaupel’s method.

<Table 4>

In the first one, women for the whole country, mortality decreased in all age groups from 1960 to 1970. The results show that 39% of the potential for saving years of life occurs after age 50, but only 14% of the actual improvement in life expectancy is due to progress in reducing mortality after this age. Comparing younger and older ages, almost half of the actual gains occur before age 20, but only 2% after age 65. This

pattern is observed as a whole for the different scenarios where only decreases in mortality were observed, and reflect a general trend of very significant progress in reducing mortality at the early years of life, but a modest progress at older ages.

The second scenario in Table 4 is for Southeast men between 1970 and 1980. In this case, mortality did not decrease in all age groups. The last four age groups (boldfaced entries in the table) experienced an increase in mortality during the decade. Since the table has the cumulative percentage, some of the numbers are higher than 100%, which has no logical meaning.

A similar problem happens with the third scenario in Table 4, Southeast men during 1980 and 1990, in which mortality also increased. In this case the increase occurred between ages 15 and 405 (age groups boldfaced in Table 4), and the cumulative percentages, consequently, decline between this age range.

These scenarios make clear that Vaupel’s method cannot be used when mortality changes in different directions across age groups, since the basic assumption of the method is that mortality is changing by a proportional difference at all ages,

( ) ( )(

µ δ

)

µ∗ _x = _{x 1}+ _.

This problem would not happen if one uses any of the other two approaches (Arriaga or Pollard). Table 5 shows that an increase in mortality in a particular age group would be associated to negative contributions to gains in life expectancy at birth, and the interpretation is very simple. For example, while the age group 1 to 4 years

5

Studies that analyzed mortality by causes of death for this same group found that the mortality increase was driven by an increase in the number of deaths by violent causes (Simões, 1997). The Southeast concentrates two of the major cities in the country (Rio de Janeiro and São Paulo), which account for most of these deaths. In numbers, the homicide rate (per 100,000) jumped from 13.4 in 1980 to 23.6 in 1990. Considering only males between 15 and 29 years the national average was around 44 in 1990.

contributed 25.8% to the gains in life expectancy at birth, the contribution of the age group 20 to 24 years was to reduce life expectancy by 1.9%. When mortality change in different directions, there is no inconsistency or theoretical violation of the assumptions implicit in these two models.

<Table 5>

The comparison of all these results indicates that when changes in mortality and interaction effects are small and only in one direction for all age groups, the three methods give similar results, and the choice of which one to use depends on the desired type of interpretation (the richer analysis is obtained by using Vaupel’s method). If, instead, mortality is increasing for some age groups and decreasing for others, Pollard and Arriaga are the only methods that can be applied. Finally, if changes in mortality are significant, and the interaction terms are large, Vaupel’s method should be avoided and both Arriaga and Pollard approaches could be chosen.

Two issues apply to the three methods compared here. First, the results are based on life table measures (a stationary population). As Vaupel (1986) showed, considering the actual age distribution and observed deaths, the numbers are lower than those given by the life table. Second, it must be kept in mind that the results did not take into account heterogeneity, and the contribution of the different age groups can be lower or even higher. Vaupel (1986) introduces some discussion about the effects of heterogeneity on the life expectancy of those saved from death. Vaupel and Yashin (1986) developed a model, called the second-chance model, which can be used to evaluate the effects of heterogeneity in life expectancy. Although there is no accurate way to measure heterogeneity, its consideration in the analysis is very important, in

order to understand the complexity of the issue, and to show how different would be the effects of policies designed to reduce mortality.

CONCLUSIONS

This paper reviewed different approaches available to measure and decompose the gains in life expectancy by age groups, given changes in mortality rates. The comparisons were restricted to the methods proposed by Arriaga, Pollard and Vaupel, since the technique developed by the UN offers a limited power of analysis and interpretation of the changes in life expectancy that can be attributed to different age groups.

Using 36 Brazilian life tables for the decades between 1940 and 1990 as an example, each of the three methods was applied and the comparison of their results pointed out their major strengths and limitations.

All methods are expected to give better results when applied to smaller age groups, since the techniques proposed by Vaupel and Pollard are approximations, and the approach developed by Arriaga would be able to better separate main and interaction effects.

Specifically concerning the strengths and limitations of each method, it is wrong to say that one method is better than the other, but they have different purposes and apply to different situations. When mortality didn’t change a lot and, consequently, the interaction effects are not significant, any of the three methods could be used. The choice depends on the desired type of interpretation: based on main and interaction effects (Arriaga and Pollard) or on a more analytical view considering potential and actual gains in life expectancy (Vaupel).

On the other hand, if mortality changes were very significant, the methods proposed by Arriaga and Pollard should be preferred, since Vaupel’s technique will underestimate the effects for younger ages. The choice between Arriaga and Pollard won’t imply in major differences on the results, since the error in Pollard’s estimations won’t probably be higher than 5%.

However, if in any of the two previous situations mortality rates didn’t change in the same direction in all age groups, then Vaupel’s method has to be avoided, and the choice, again, is restricted to Arriaga and Pollard.

Regarding the interpretation of the results, the method proposed by Vaupel is the most powerful, since it offers a comparison of what could be achieved just based on the mortality schedule, and what was actually observed in terms of gains in life expectancy. It is a great indicator of possible failures and successes of social policies implemented in the period studied.

Finally, heterogeneity is not considered in any method, and this is the main problem. However, this is a hard issue to take into account. Individuals are different, behave differently and live in distinct environments, and there is no accurate way to capture all these differences. Although one could argue that exercises considering different functions to evaluate heterogeneity would be merely exploratory, since the “true” heterogeneity cannot be measured, it is important to emphasize that the impacts on life expectancy can be rather different from the results shown in before. More studies are yet to be done in this area.

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Figure 1 – Survival curves, direct, indirect and interaction effects of a change in mortality a) Female survival curves – Brazil, 1940 and 1980

b) Different effects of a change in mortality in the age group x to x+i

Table 1 – Effects of changes in the mortality schedule on life expectancy at birth and on temporary life expectancy from birth to age 69, for selected periods and regions of Brazil, based on Arriaga’s

method 0 20000 40000 60000 80000 100000 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 Ages lx Brazil - females - 1980 e0 = 66.63 Brazil - females - 1940 e0 = 48.25 x x+i Direct effect Indirect effect Interaction effect lx in 1940

lx in 1980 given that mortality changed

in the age group x to x+i

lx in 1980 given that mortality changed in

the age group x to x+i , but remained the same for ages after x+i

Age range Effects of mortality change on life expectancy used Direct Indirect Interaction Total Southeast Females - 1940/50 0-70+ 0.1038 1.4337 0.0225 1.5601 6.7% 91.9% 1.4% 0-69 0.0856 1.2431 0.0169 1.3456 6.4% 92.4% 1.3% Males - 1970/80 0-70+ 0.0226 3.0397 0.0457 3.1081 0.7% 97.8% 1.5% 0-69 0.0246 3.0526 0.0540 3.1313 0.8% 97.5% 1.7% Northeast Females - 1970/80 0-70+ 0.8030 8.2759 1.0960 10.1749 7.9% 81.3% 10.8% 0-69 0.6667 6.8290 0.7289 8.2247 8.1% 83.0% 8.9% Males - 1970/80 0-70+ 0.8361 7.6838 1.3136 9.8335 8.5% 78.1% 13.4% 0-69 0.7205 6.5046 0.8551 8.0802 8.9% 80.5% 10.6% Region/Sex/Period

Table 2 – Percent contribution to the gains in life expectancy at birth, given by three different techniques, by age groups, for selected periods and regions of Brazil

% contribution of each age group to the change in life expectancy at birth

Southeast females - 1940/50 Brazil females - 1960/70 Northeast males - 1970/80 Ages Interaction represents 1.4% Interaction represents 4.6% Interaction represents 13.4%

of the total change in e0 of the total change in e0 of the total change in e0

Arriaga Pollard Vaupel Arriaga Pollard Vaupel Arriaga Pollard Vaupel 0 26.52 27.05 25.21 24.58 25.52 23.39 17.53 18.50 16.85 1 31.71 31.55 29.62 19.56 19.60 17.50 20.41 20.03 15.39 5 6.04 6.14 6.20 3.04 3.08 2.96 3.45 3.49 3.05 10 4.59 4.67 4.73 2.02 2.05 1.95 2.20 2.23 1.98 15 3.61 3.67 3.81 4.00 4.05 3.72 3.53 3.57 3.23 20 3.23 3.29 3.44 6.05 6.13 5.50 6.02 6.09 5.22 25 3.14 3.20 3.37 6.10 6.17 5.77 6.09 6.15 5.56 30 3.51 3.58 3.78 5.67 5.73 5.59 6.09 6.15 5.87 35 3.48 3.55 3.78 5.23 5.28 5.39 5.96 6.02 6.06 40 2.96 3.02 3.25 4.64 4.69 5.03 5.91 5.97 6.38 45 2.21 2.25 2.47 4.27 4.31 4.79 5.58 5.62 6.45 50 2.20 2.25 2.49 3.80 3.82 4.43 4.96 5.00 6.27 55 1.91 1.96 2.20 3.43 3.45 4.18 4.34 4.37 6.02 60 2.14 2.19 2.49 3.19 3.20 4.02 3.84 3.87 5.84 65 1.60 1.65 1.90 2.92 2.93 3.85 2.91 2.94 5.00 70+ 1.17 0.00 1.26 1.50 0.00 1.92 1.18 0.00 0.82 Total 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00

Figure 2 – Comparison of the three methods for measuring the contribution of each age group to the gains in life expectancy at birth, for selected periods and regions of Brazil

Interaction effects as % of total change in e0: a) 1.4% b) 4.6% c) 13.4% a) Southeast female – 1940/50

b) Brazil female – 1960/70

c) Northeast male – 1970/80

Table 3 – Life expectancy at birth, entropy, and potential for saving years of life given mortality changes in particular age ranges, based on Vaupel’s method – Brazil and Northeast region –

1940/90 0 5 10 15 20 25 30 35 0 10 20 30 40 50 60 70 Ages % Arriaga Pollard Vaupel 0 5 10 15 20 25 30 0 10 20 30 40 50 60 70 Ages % Arriaga Pollard Vaupel 0 5 10 15 20 25 0 10 20 30 40 50 60 70 Ages % Arriaga Pollard Vaupel

Potential for saving years of life Age group with 0-5 0-10 60-ω 65-ω great.potential BRAZIL - WOMEN 1940 0.5381 48.25 0.4006 0.4414 0.2060 0.1654 0-1 1950 0.4883 51.63 0.4146 0.4286 0.2395 0.1951 0-1 1960 0.4394 55.14 0.3914 0.4040 0.2784 0.2306 0-1 1970 0.3779 59.80 0.3625 0.3728 0.3417 0.2892 0-1 1980 0.2928 66.63 0.2957 0.3022 0.4691 0.4098 70+ 1990 0.2457 70.89 0.2223 0.2275 0.5731 0.5151 70+ NORTHEAST - MEN 1940 0.7027 35.21 0.4484 0.5024 0.0976 0.0670 0-1 1950 0.6679 37.29 0.4379 0.4892 0.1109 0.0773 0-1 1960 0.6385 38.91 0.4232 0.4733 0.1207 0.0847 0-1 1970 0.5602 44.15 0.4013 0.4442 0.1626 0.1186 0-1 1980 0.4377 53.99 0.3664 0.3960 0.2663 0.2111 0-1 1990 0.3723 58.53 0.3264 0.3493 0.3316 0.2662 0-1 Period H e0

Table 4 – Cumulative percentage of the potential and actual improvement in life expectancy, based on Vaupel’s method - Brazil and Southeast region – selected periods

Cumulative % of the improvement in e0

Brazil - Women Southeast - Men

1960/70 1970/80 1980/90

Potential Actual Potential Actual Potential Actual

0 25.37 23.39 16.65 65.25 11.11 38.51 1 34.54 40.89 23.25 86.26 12.87 54.65 5 36.25 43.86 25.68 91.81 13.64 60.89 10 37.28 45.81 27.41 95.36 14.49 65.43 15 38.90 49.53 30.02 99.79 17.85 63.16 20 41.02 55.03 33.41 105.01 22.22 60.14 25 43.52 60.80 36.64 109.68 26.41 57.20 30 46.24 66.39 39.92 113.94 30.63 54.59 35 49.32 71.78 43.49 117.66 35.15 52.43 40 52.88 76.81 47.66 120.49 39.99 52.98 45 56.74 81.60 52.63 121.49 45.64 54.91 50 60.97 86.03 58.59 120.12 52.04 59.98 55 65.83 90.21 65.49 115.25 59.18 68.51 60 71.08 94.24 73.04 107.25 66.50 82.14 65 76.52 98.08 80.61 96.92 73.86 97.96 70+ 100.00 100.00 100.00 100.00 100.00 100.00 Age

Table 5 – Percent contribution to the gains in life expectancy at birth, based on Arriaga’s and Pollard’s method

Brazilian southeast region, males – 1980/90

% contribution of each age group to the Age change in e0 - Southeast males 1980/90

Arriaga Pollard 0 36.92 37.38 1 25.80 25.84 5 8.37 8.67 10 4.68 4.87 15 -1.40 -1.46 20 -1.86 -1.93 25 -1.81 -1.87 30 -1.61 -1.65 35 -1.34 -1.37 40 0.36 0.36 45 1.26 1.27 50 3.40 3.41 55 5.77 5.81 60 9.32 9.56 65 10.27 11.11 70+ 1.87 0.00 Total 100.00 100.00