# Lesson 5: Expectation of Life at Birth

## 5.0 Overview

Often expectation of life (ex0), and in particular expectation of life at birth e00, is considered to be a summary measure of mortality. Hence, differences in expectation of life (either between gender or other subgroups of population or between two time points) are considered to be indicators of changes in mortality. However, a change in life expectancy does not mean that the age-specific mortality rates change in the same magnitude and direction. Also, the contributions of various causes of death in the changes in expectations of life will not be of the same magnitude and direction. Some age-cause-specific death rates will increase in the two groups compared, thus contributing to a decrease in expectation of life, while others may decrease in some age intervals, thus increasing expectation of life.

Several authors have attempted to disentangle the contributions by age-specific and age-cause-specific death rates to the difference between two expectations of life. Applying these methods we learn to interpret the dynamics of changes in the mortality.

This lesson presents two methods to decompose changes in expectation of life at birth. The method developed by Arriaga (Arriaga, E. 1984. "Measuring and Explaining the Change in Life Expectancies," Demography 21: 83-96) is described first (Section 5.1). The second method was developed by Pollard (Section 5.2) (Pollard, J. H. 1988. "On the Decomposition of Changes in Expectation of Life and Differentials in Life Expectancy," Demography25(2): 265-276). Pollard's method is an exact decomposition developed using a continuous modeling approach of the life table, whereas Arriaga's method is based on the discrete analysis approach. As such, Arriaga's method is an approximate method and is easier to compute. (Refer to the papers for details.)

The final section of this lesson (5.4) explains cause elimination life tables, which yield answers to the hypothetical questions about a cohort's mortality experience if a particular cause (or causes) were eliminated.

Download a zip file containing data for Lesson 5 in Excel and CSV formats.

## 5.1 Arriaga Method

Assume that we would like to compare the life expectancy of two groups (males and females, for example). Designate the two groups as Group 1 and Group 2. The goal is to decompose the difference in the expectation of life at birth to reveal the differences in age-specific death rates (and age-cause-specific death rates). We approach this problem in two steps: In Step 1, we will examine the contributions of the differences in age-specific death rates toward the changes in expectation of life at birth. In Step 2, we will extend this to include age-cause-specific death rates.

Notation Definition e001 ~ expectation of life at birth for group 1 e002 ~ expectation of life at birth for group 2 ex0i ~ expectation of life at age x in group i ( i = 1, 2) lxi ~ number of survivors to age x in group i ( i = 1, 2)Note: As is often done, in the table and calculations that follow, the radix = 1 rather than 100,000. The result is that the terms in the table are expressed as proportions (per person, in other words) rather than as whole numbers (per 100,000 people, in other words.) Either way works fine as long as all the numbers are expressed the same way (all per person or all per 100,000 people). nLxi ~ person years of life contributed in the age group ( x, x + n) in group i ( i = 1, 2)Note: See note above. This term is also expressed as a proportion in the table and calculations below.

#### Arriaga's Formula

Step 1: Age-Specific Decomposition

Arriaga formulates the differences in the expectation of life at birth ( e002e001) due to differences in age-specific death rates as:

 e002 -
 e001

Total effect
of age x =
direct effect of age x + indirect effect of age x (summed over all age intervals)

where

direct effect of age (x , x + n)
(DEx) =
 lx1 [ nLx2lx2 - nLx1lx1 ]

and

indirect effect of age (x , x + n)
(IEx) =
 { lx1 [ lx+n2lx2 ] - lx+n1 } *ex+n02

The direct effect on life expectancy is due to the changes in life years within a particular age interval as a consequence of the mortality change in that age interval.

The indirect effect consists of the number of life years added to a life expectancy because of the changes in the number of survivors at the end of the age interval caused by the change in mortality within the age interval.

For an open-ended interval (x+), the contribution of the indirect effect is considered to be 0. Therefore, the total effect of this age interval is calculated as:

 DEx+ = lx1 [ex+ 02 - ex+ 0 1 ]

Example

The data necessary for decomposing the difference in expectations of life at birth of males and females in Costa Rica in 1960 are given in Table 5.1.1 below.

From that table:

Expectation of life at birth for males (call it e001) = 62.97
Expectation of life at birth for females (call it e002) = 65.44

Difference (e002 - e001) = 65.44 - 62.97 = 2.47

The first goal is to calculate the contribution of the male-female differences in each age interval to a total difference of 3.04 years in expectation of life at birth.

For this purpose, we calculate the direct effect and indirect effect of each age interval using the formulas above.

First, let's look at a specific age interval, 20-24.

From the table:

 l201 (males) = 0.88298
 l202 (females) = 0.89394
 l251 = 0.87504
 l252 = 0.88829
 5L201 = 4.39502
 5L202 = 4.45556

 e2502 = 48.19

Direct effect of age group 20-24:

 l201 [ 5L202l202 - 5L201l201 ]

 = 0.88298[ 4.45556.89394 - 4.39502.88298 ]
= 0.00591
(using rounded values, 0.00589 actual)

Indirect effect of age group 20-24:

 { l201 [ l252l202 ] - l251 } *e2502

 = { 0.88298 [ 0.888290.89394 ] - 0.87504}*48.19
= 0.11369
(using rounded values, 0.11330 actual)

Total effect = direct effect + indirect effect = 0.00589 + 0.11330 = 0.11961 (using rounded values, 0.11920 actual)

Calculations show that the mortality rates in the age interval 20-24 contribute 0.1222 to a total change in expectation of life at birth of 3.04 years. The lower mortality rate for females in this age group is responsible for this positive contribution. The majority of this contribution of 0.11920 comes from the indirect effect: more females surviving to the next age interval.

The contributions of mortality rates in all age intervals are in shown Table 5.1.1. Note that in age intervals 1-4, 25-29, and 30-34 the contributions are negative because of higher death rates for females in those age intervals. The biggest contribution is in the youngest age interval, <1. The gap between male-female mortality widens after age 40.

Males Females Age lx1 nLx1 ex01 lx2 nLx2 ex02 Direct Effect Table 5.1.1: Arriaga MethodAge-Specific Effects on Mortality Differences between 1960 Costa Rican Males and Females 0 1.00000 0.96340 62.97 1.00000 0.96868 65.44 0.00528 0.70915 0.71443 1 0.92770 3.65924 66.84 0.93801 3.69496 68.73 -0.00492 -0.16271 -0.16763 5 0.90204 4.49098 64.69 0.90961 4.52900 66.81 0.00032 0.00804 0.00837 10 0.89436 4.45757 60.22 0.90200 4.49925 62.35 0.00359 0.08264 0.08624 15 0.88867 4.42912 55.59 0.89771 4.47912 57.64 0.00493 0.10402 0.10895 20 0.88298 4.39502 50.93 0.89394 4.45556 52.87 0.00589 0.11330 0.11920 25 0.87504 4.35739 46.37 0.88829 4.42202 48.19 -0.00130 -0.02263 -0.02393 30 0.86792 4.31822 41.73 0.88053 4.37779 43.60 -0.00313 -0.04866 -0.05178 35 0.85938 4.26465 37.12 0.87060 4.32318 39.06 0.00280 0.03855 0.04135 40 0.84651 4.18616 32.65 0.85870 4.26212 34.57 0.01548 0.18493 0.20041 45 0.82802 4.07402 28.32 0.84618 4.18222 30.05 0.01843 0.18768 0.20611 50 0.80173 3.91758 24.17 0.82678 4.06265 25.69 0.02198 0.18664 0.20862 55 0.76558 3.70214 20.19 0.79845 3.88937 21.52 0.02714 0.18666 0.21380 60 0.71583 3.37577 16.43 0.75766 3.61591 17.54 0.04052 0.21940 0.25992 65 0.63605 2.90813 13.18 0.68977 3.21227 14.03 0.05396 0.22101 0.27496 70 0.53040 2.33629 10.32 0.59736 2.66702 10.82 0.03181 0.09369 0.12550 75 0.40933 1.70095 7.67 0.47419 2.01109 8.00 0.03508 0.06547 0.10056 80 0.27931 1.01508 5.14 0.33793 1.23455 5.28 0.00532 0.00531 0.01062 85+ 0.14207 0.42160 2.97 0.17392 0.54971 3.16 0.02746 0.00000 0.02746 Total: 2.46315

Step 2: Age-Cause-Specific Decomposition

In this section we will extend the decomposition of the difference in expectation of life at birth by further decomposing the change by age and causes of death. This decomposition is accomplished as follows:

1. Calculate the proportion of change in the cause-specific mortality rates as a share of the total mortality change in the specific age interval.
2. Distribute the calculated total effect into specific cause contributions according to the proportions calculated in Step a.

Example

From above example for age group 20-24:
Total effect = 0.1222

 Female Death Rates Male Death Rates Difference In Death Rates Proportionate Contrib. Contribution Table 5.1.2: Contribution of Cause-Specific Death Rates for Males and Females 20-24: Arriaga Method Diarrhea 0.21 0.00 0.21 -0.38760 -0.04620 Cancer 0.23 0.12 0.10 -0.19434 -0.02316 CVD 0.15 0.15 0.00 -0.00062 -0.00007 Other causes 0.69 1.54 -0.85 1.58257 0.18864 All causes 1.27 1.81 -0.54 1.00000 0.11920

Table 5.1.2 shows that the difference of -.54 in male-female "all causes" mortality comes from a combined difference of 0.21, 0.10, 0.00, and -0.85 respectively for death due to diarrhea, cancer, CVD, and other causes. The proportionate contribution by each cause is calculated by dividing the contribution of each cause by the change due to all causes. Thus:

Proportionate contribution by diarrhea =
 0.21- 0.54 = - .38760

Proportionate contribution by cancer =
 0.10- 0.54 = - 0.19434

Proportionate contribution by CVD =
 0.00- 0.54 = 0 (using rounded values, 0.00062 actual)

Proportionate contribution by other causes =
 - 0.85- 0.54 = 1.58257

In order to calculate the effect of each cause in the age interval, multiply the total effect for age interval (0.11920) by each proportionate contribution. Thus, the contribution of diarrhea in the age interval 20-24 is:

0.11920 * (-0.38760) = -0.04620

Similarly, the contributions of cancer, CVD, and other causes to the total effect are -0.02316, 0 (rounded, -0.00007 actual), and 0.18864 respectively.

Note that the contributions of each cause may be positive or negative. In the age interval 20-24, females have a lower "all causes" mortality. However, diarrhea and cancer mortality are higher for females in this age group; their contribution to the total change is negative. The contribution of each cause of death for all the age intervals is calculated in this manner and presented in Table 5.1.3.

Note that summing over all age intervals for a specific cause yields the contribution of that cause to the total change in expectation of life at birth. Table 5.1.3 shows the total contributions of diarrhea, cancer, CVD, and other causes are -0.15545, 0.27241, 0.20399, and 2.14214 respectively. Although all contributions are positive, the table shows that in specific age intervals (e.g., 20-24) the contribution may be negative.

Age TotalEffect DiarrheaEffect CancerEffect CVDEffect Other CausesEffect Table 5.1.3: Age-Cause-Specific Contributions to Changes in Expectation of Life at Birth, 1960 Costa Rican Males, Arriaga Method <1 year 0.71443 0.09416 -0.00693 0.01208 0.61511 1-4 -0.16763 -0.06119 0.01811 -0.01165 -0.11291 5-9 0.00837 -0.00810 0.01719 -0.02869 0.02796 10-14 0.08624 -0.12965 0.00747 -0.00455 0.21296 15-19 0.10895 -0.04047 0.01301 -0.00936 0.14577 20-24 0.11920 -0.04620 -0.02316 -0.00007 0.18864 25-29 -0.02393 0.00000 0.00646 -0.03126 0.00088 30-34 -0.05178 0.00538 -0.02690 -0.04506 0.01480 35-39 0.04135 0.01102 -0.03985 -0.00820 0.07838 40-44 0.20041 -0.00555 0.03698 0.01301 0.15597 45-49 0.20611 0.00000 -0.08194 0.02445 0.26360 50-54 0.20862 0.00553 0.02487 0.02912 0.14910 55-59 0.21380 0.00552 0.04359 -0.00616 0.17085 60-64 0.25992 0.01091 0.10454 0.00168 0.14279 65-69 0.27496 0.00018 0.07275 0.06308 0.13895 70-74 0.12550 -0.00002 0.00446 0.13563 -0.01457 75-79 0.10056 -0.00846 0.06610 0.02704 0.01588 80-84 0.01062 0.00352 -0.00054 0.03131 -0.02368 85+ 0.02746 0.00797 0.03625 0.01158 -0.02834 Total 2.46315 -0.15545 0.27247 0.20399 2.14214

## 5.2 Pollard Method

Pollard (1988) proposed a method to decompose the difference in expectation of life at birth to determine the contribution of the difference in age and cause-specific mortality rates.

Unlike the Arriaga method, the Pollard method is based on the exact relationship between expectation of life at birth and age-specific mortality rates (some details are not shown here).

In this section we learn to apply Pollard's method of calculations.

#### Pollard's Formula

Pollard showed the following relationship between the difference in expectation of life at birth and the difference in age-specific death rates:

 e002 -  e001 = ¥óõ0 [m(t,1) - m(t,2)]w(t)dt

where m(t, 1) and m(t, 2) are the age-specific death rates and

 w(t) = 12 [lt1 *et02 + lt2 *et01 ]

For a specific age interval (x, x+n) the contribution is calculated as:

 [ nmx1 - nmx2 ]    *
 n2 [w(x) + w(x + n)]

where nmx1 and nmx2 are age specific death rates in the age interval (x, x+n) and

 w(x) = 12 [lx1 * ex02 + lx2 * ex01 ]

To implement the calculations for each age interval (except for the open-ended interval), one will calculate an average weight as

 Average weight for age interval (x, x+ n) = n2 [w(x) + w(x + n)]

For an open-ended interval (x+) the average weight is calculated as:

 12 [ Tx2mx + 1 + Tx1mx + 2 ]
where mx+1 and mx+2 denote the age-specific death rates of the open-ended age interval x+

The calculations of the average weight for different age intervals are illustrated in Table 5.2.1.

Age lx1 e01 nmx2 e02 w(x) w(x+n) Avg. Weight Table 5.2.1: Average Weight Calculations for 1960 Costa Rican Males and Females, Pollard Method 0 1.00000 62.97 0.0640 65.44 64.20488 63.22912 63.71700 1 0.92770 66.84 0.0077 68.73 63.22912 59.55371 245.56565 5 0.90204 64.69 0.0017 66.81 59.55371 55.04361 286.49330 10 0.89436 60.22 0.0010 62.35 55.04361 50.56409 264.01925 15 0.88867 55.59 0.0008 57.64 50.56409 46.10860 241.68171 20 0.88298 50.93 0.0013 52.87 46.10860 41.68188 219.47619 25 0.87504 46.37 0.0018 48.19 41.68188 37.29248 197.43591 30 0.86792 41.73 0.0023 43.60 37.29248 32.94533 175.59453 35 0.85938 37.12 0.0028 39.06 32.94533 28.65060 153.98982 40 0.84651 32.65 0.0029 34.57 28.65060 24.42252 132.68280 45 0.82802 28.32 0.0046 30.05 24.42252 20.29097 111.78374 50 0.80173 24.17 0.0070 25.69 20.29097 16.29820 91.47294 55 0.76558 20.19 0.0105 21.52 16.29820 12.50084 71.99761 60 0.71583 16.43 0.0188 17.54 12.50084 9.00558 53.76604 65 0.63605 13.18 0.0288 14.03 9.00558 5.95147 37.39261 70 0.53040 10.32 0.0462 10.82 5.95147 3.45550 23.51743 75 0.40933 7.67 0.0678 8.00 3.45550 1.60647 12.65494 80 0.27931 5.14 0.1329 5.28 1.60647 0.48258 5.22263 85+ 0.14207 2.97 0.3164 3.16 0.48258 0.00000 1.48193

Example

For age group 20-24:

Average weight =
 52 [w(20) + w(25)]

where

 w(20) = 12 [l201 *e202 + l202 *e201 ]

and

 w(25) = 12 [l251 *e252 + l252 *e251 ]

 w(20) = 12 [0.88298*52.87 + 0.89394*50.93] = 46.1086

 w(25) = 12 [0.87504*48.19 + 0.88829*46.37] = 41.68188

Thus, average weight for the age interval 20 -24:

 52 [w(20) + w(25)] = 52 [46.1086 + 41.68188] = 219.4762

And average weight for open-ended interval 85+:

 12 [ T852m85 + 1 + T851m85 + 2 ] = 12 [ .54971.33698 + .42157.3164 ] = 1.48193

Contribution of difference in age-specific mortality rates of a specific age interval (x, x + n) to the difference in expectation of life at birth:

nmx1 - nmx2 ] * average weight

For age interval 20-24:

[0.00181-0.0013] * 219.47619 = 0.11778

The age interval 20-24 contributes 0.11778 years to the total difference of 3.04 years in expectation of life at birth between males and females.

Similarly, the contributions for all age intervals are computed and presented in Table 5.2.2. The youngest and the oldest age intervals are the highest contributors to the male-female differences in expectation of life at birth.

Age ASDR(males) ASDR(females) Differencein ASDR AverageWeight Contributionof Age Group Table 5.2.2: Contributions for All Age Intervals, 1960 Costa Ricans, Pollard Method 0 0.07505 0.0640 0.01106 63.71700 0.70477 1 0.00701 0.0077 -0.00068 245.56565 -0.16597 5 0.00171 0.0017 0.00003 286.49330 0.00826 10 0.00128 0.0010 0.00032 264.01925 0.08512 15 0.00129 0.0008 0.00045 241.68171 0.10758 20 0.00181 0.0013 0.00054 219.47619 0.11778 25 0.00163 0.0018 -0.00012 197.43591 -0.02362 30 0.00198 0.0023 -0.00029 175.59453 -0.05094 35 0.00302 0.0028 0.00026 153.98982 0.04054 40 0.00442 0.0029 0.00148 132.68280 0.19652 45 0.00645 0.0046 0.00181 111.78374 0.20277 50 0.00923 0.0070 0.00225 91.47294 0.20612 55 0.01344 0.0105 0.00295 71.99761 0.21247 60 0.02364 0.0188 0.00486 53.76604 0.26126 65 0.03633 0.0288 0.00756 37.39261 0.28263 70 0.05182 0.0462 0.00564 23.51743 0.13262 75 0.07644 0.0678 0.00869 12.65494 0.10994 80 0.13520 0.1329 0.00235 5.22263 0.01227 85+ 0.33698 0.3164 0.02060 1.48193 0.03053 Total: 2.47067

### Contributions by Causes of Death

The contributions by "all causes" of mortality can be decomposed into cause-specific contributions very easily using the Pollard method. Remember that age-specific mortality rate is equal to the sum of age-cause-specific mortality rates over all causes. Therefore, the difference in age-specific mortality also sums to the difference in age-cause-specific mortality rates. Using the symbols above:

 mx1 - mx2 = åd [mxd1 - mxd2 ]

where mxd denotes the age-specific death rates by cause Rd.

Thus, the contribution of a specific cause Rd can now be expressed as:

 * Average weight

Example

Contributions of various causes of death in age interval 20-24:

Table 5.2.3 shows the age-cause-specific death rates for males and females in the age interval 20-24. The contribution of each cause is calculated by multiplying the rate difference by the average weight for the age interval 20-24. The results are in the last column of the table.

Cause ofDeath Cause-SpecificDeath Rates(males) perPerson Cause-SpecificDeath Rates(females) Differencein Death Rates AverageWeight Contributionof Cause ofDeath Table 5.2.3: Contribution of Cause of Death, 1960 Costa Ricans 20-24, Pollard Method Diarrhea 0.00000 0.00021 -0.00021 219.47619 -0.04565 Cancer 0.00012 0.00023 -0.00010 219.47619 -0.02289 CVD 0.00015 0.00015 0.00000 219.47619 -0.00007 Other Causes 0.00154 0.00069 0.00085 219.47619 0.18639 All Causes 0.00181 0.00127 0.00054 219.47619 0.11778

From Table 5.2.3 for age interval 20-24:

• Cancer-specific mortality for males (per person) = 0.00012
• Cancer-specific mortality rate for females = 0.00023
• Difference in rates (male - female) = -0.00011 (-0.00010 with rounding)
• Average weight for age group 20-24 (from Table 5.2.3) = 219.47619
• Contribution of cancer mortality = weight * difference in rates
• = 220.8867 * -0.00010
• = -0.02289

Because of the higher death rates due to cancer for females in this age interval, the contribution is negative.

The calculations for other diseases are also shown in the table.

Similar calculations for all age groups are given in Table 5.2.3.

  Difference in Male-Female Death Rates Due to... Contribution to the Difference in e00 Due to... Age Diarrhea Cancer CVD All Other Causes Diarrhea Cancer Table 5.2.4: Age-Cause-Specific Contributions, 1960 Costa Ricans, Pollard Method 0 0.00146 -0.00011 0.00019 0.00952 0.09289 -0.00684 0.01192 0.60680 0.70477 1 -0.00025 0.00007 -0.00005 -0.00046 -0.06058 0.01793 -0.01153 -0.11179 -0.16597 5 -0.00003 0.00006 -0.00010 0.00010 -0.00800 0.01698 -0.02833 0.02762 0.00826 10 -0.00048 0.00003 -0.00002 0.00080 -0.12798 0.00737 -0.00449 0.21022 0.08512 15 -0.00017 0.00005 -0.00004 0.00060 -0.03997 0.01285 -0.00924 0.14394 0.10758 20 -0.00021 -0.00010 0.00000 0.00085 -0.04565 -0.02289 -0.00007 0.18639 0.11778 25 0.00000 0.00003 -0.00016 0.00000 0.00000 0.00637 -0.03086 0.00086 -0.02362 30 0.00003 -0.00015 -0.00025 0.00008 0.00529 -0.02646 -0.04432 0.01456 -0.05094 35 0.00007 -0.00025 -0.00005 0.00050 0.01081 -0.03907 -0.00804 0.07685 0.04054 40 -0.00004 0.00027 0.00010 0.00115 -0.00544 0.03627 0.01276 0.15294 0.19652 45 0.00000 -0.00072 0.00022 0.00232 0.00000 -0.08062 0.02405 0.25933 0.20277 50 0.00006 0.00027 0.00031 0.00161 0.00547 0.02457 0.02877 0.14731 0.20612 55 0.00008 0.00060 -0.00009 0.00236 0.00548 0.04332 -0.00612 0.16979 0.21247 60 0.00020 0.00195 0.00003 0.00267 0.01096 0.10508 0.00169 0.14352 0.26126 65 0.00000 0.00200 0.00173 0.00382 0.00018 0.07478 0.06484 0.14282 0.28263 70 0.00000 0.00020 0.00609 -0.00065 -0.00003 0.00471 0.14333 -0.01540 0.13262 75 -0.00073 0.00571 0.00234 0.00137 -0.00925 0.07227 0.02956 0.01736 0.10994 80 0.00078 -0.00012 0.00693 -0.00524 0.00407 -0.00062 0.03618 -0.02736 0.01227 85+ 0.00598 0.02721 0.00869 -0.02127 0.00886 0.04032 0.01287 -0.03152 0.03053 Total: -0.15288 0.28633 0.22296 2.11426 2.47067

The sum of the contributions of all ages for a specific cause gives the total contribution of that cause of death to the difference in expectation of life at birth.

Table 5.2.4 shows a major contribution of 2.11426 years from "other causes" of death. The contributions from diarrhea, cancer, and CVD are -0.15288, 0.28633, and 0.22296 respectively.

Although the contributions over all ages are positive for all causes, the table shows negative contributions by specific causes in some age intervals. For example, due to higher death rates among females due to cancer and CVD, the contributions from these causes are negative in age intervals 30-34 and 35-39.

## 5.3 Comparison of Arriaga and Pollard Methods

The Pollard Method is analytically more exact. However, the Arriaga Method is conceptually simple, its components such as direct and indirect effects are easy to interpret, and it is easy to compute. Table 5.3.1 below gives the contribution from "all causes" to the difference in expectation of life at birth as calculated by both Pollard and Arriaga methods.

In this example, the two methods achieve similar results; discrepancies occur from the approximations used in the computations.

Also, Table 5.3.2 shows that the two methods achieve similar results in determining the total contributions for each of the four causes. However, discrepencies can occur when broader age intervals are used because of the discrete approximations used in the computations.

Age PollardMethod ArriagaMethod Table 5.3.1: Total Mortality Contributions 0 0.70477 0.71443 1 -0.16597 -0.16763 5 0.00826 0.00837 10 0.08512 0.08624 15 0.10758 0.10895 20 0.11778 0.11920 25 -0.02362 -0.02393 30 -0.05094 -0.05178 35 0.04054 0.04135 40 0.19652 0.20041 45 0.20277 0.20611 50 0.20612 0.20862 55 0.21247 0.21380 60 0.26126 0.25992 65 0.28263 0.27496 70 0.13262 0.12550 75 0.10994 0.10056 80 0.01227 0.01062 85+ 0.03053 0.02746 Total: 2.47067 2.46315

Method Diarrhea Cancer CVD Other Causes All Causes Table 5.3.2: Contributions of Causes of Death Pollard -0.15288 0.28633 0.22296 2.11426 2.47067 Arriaga -0.15545 0.27247 0.20399 2.14214 2.46315

#### Exercise 14

For this exercise, download the zip file and use the data file with the age-specific and age-cause-specific death rates and life tables for Taiwanese males in 1960 and 1964 (presented in tables below). Let the 1960 data be "Group 1" in your calculations and the 1964 data be "Group 2." Use your spreadsheet software to complete this exercise.

Use these data tables to compute the contributions of each cause of mortality in determining the changes in expectation of life at birth between these periods. Use both the Arriaga Method and the Pollard Method and compare these methods.

Age All Causes Tuberculosis Cancer CVD Other Causes Table 5.3.3: Age-Specific and Age-Cause-Specific Death Rates, 1964 Taiwanese Males 0 0.02920 0.00004 0.00012 0.00037 0.02867 1 0.00401 0.00004 0.00008 0.00005 0.00384 5 0.00083 0.00001 0.00005 0.00003 0.00074 10 0.00071 0.00002 0.00006 0.00005 0.00058 15 0.00126 0.00004 0.00006 0.00010 0.00106 20 0.00292 0.00017 0.00019 0.00018 0.00238 25 0.00208 0.00020 0.00015 0.00015 0.00158 30 0.00262 0.00036 0.00026 0.00022 0.00178 35 0.00352 0.00053 0.00043 0.00040 0.00216 40 0.00500 0.00083 0.00082 0.00075 0.00260 45 0.00746 0.00111 0.00126 0.00145 0.00364 50 0.01186 0.00189 0.00157 0.00300 0.00540 55 0.01802 0.00156 0.00295 0.00557 0.00794 60 0.02951 0.00308 0.00373 0.01027 0.01243 65 0.04600 0.00414 0.00566 0.01624 0.01996 70 0.07068 0.00503 0.00541 0.02575 0.03449 75 0.10439 0.00439 0.00690 0.03377 0.05933 80 0.14863 0.00464 0.00564 0.04616 0.09219 85+ 0.35358 0.00448 0.01023 0.09974 0.23913

Age All Causes Tuberculosis Cancer CVD Other Causes Table 5.3.4: Age-Specific and Age-Cause-Specific Death Rates: 1960 Taiwanese Males 0 0.03860 0.00009 0.00000 0.00072 0.03779 1 0.00619 0.00008 0.00003 0.00011 0.00597 5 0.00119 0.00003 0.00002 0.00006 0.00108 10 0.00089 0.00002 0.00002 0.00080 0.00005 15 0.00154 0.00006 0.00004 0.00014 0.00130 20 0.00272 0.00018 0.00009 0.00022 0.00223 25 0.00244 0.00030 0.00015 0.00024 0.00175 30 0.00302 0.00046 0.00022 0.00034 0.00200 35 0.00376 0.00060 0.00043 0.00048 0.00225 40 0.00563 0.00096 0.00073 0.00090 0.00304 45 0.00819 0.00121 0.00119 0.00188 0.00391 50 0.01305 0.00173 0.00180 0.00360 0.00592 55 0.02069 0.00260 0.00263 0.00596 0.00950 60 0.03081 0.00351 0.00344 0.00988 0.01398 65 0.05047 0.00416 0.00509 0.01681 0.02441 70 0.07540 0.00494 0.00522 0.02269 0.04255 75 0.11552 0.00427 0.00576 0.03259 0.07290 80 0.17400 0.00363 0.00380 0.03867 0.12790 85+ 0.33592 0.00465 0.01009 0.07215 0.24903

Table 5.3.5: Ordinary Life Table 1964 Taiwanese Males

Age
 nmx
 nqx
 lx
 ndx
 nLx
 Tx
 ex0
0 0.02920 0.02878 100,000 2878 98,554 6,452,944 64.53
1 0.00401 0.01591 97,122 1545 385,390 6,354,390 65.43
5 0.00083 0.00414 95,577 396 476,894 5,969,000 62.45
10 0.00071 0.00354 95,181 337 475,061 5,492,106 57.70
15 0.00126 0.00628 94,844 596 472,728 5,017,045 52.90
20 0.00292 0.01449 94,248 1366 467,817 4,544,317 48.22
25 0.00208 0.01035 92,882 961 462,004 4,076,500 43.89
30 0.00262 0.01301 91,921 1196 456,608 3,614,497 39.32
35 0.00352 0.01745 90,725 1583 449,655 3,157,889 34.81
40 0.00500 0.02469 89,142 2201 440,185 2,708,234 30.38
45 0.00746 0.03661 86,941 3183 426,698 2,268,049 26.09
50 0.01186 0.05758 83,758 4822 406,614 1,841,351 21.98
55 0.01802 0.08616 78,935 6801 377,419 1,434,737 18.18
60 0.02951 0.13718 72,134 9895 335,325 1,057,318 14.66
65 0.04600 0.20547 62,239 12788 278,000 721,993 11.60
70 0.07068 0.29770 49,451 14722 208,287 443,993 8.98
75 0.10439 0.40664 34,729 14122 135,283 235,706 6.79
80 0.14863 0.52439 20,607 10806 72,704 100,423 4.87
85+ 0.35358 1.00000 9,801 9801 27,719 27,719 2.8

Table 5.3.6: Ordinary Life Table, 1960 Taiwanese Males

Age
 nmx
 nqx
 lx
 ndx
 nLx
 Tx
 ex0
0 .03860 .03786 100,000 3786 98,095 6,228,352 62.28
1 .00619 .02446 96,214 2353 380,129 6,130,257 63.72
5 .00119 .00593 93,861 557 467,909 5,750,128 61.26
10 .00089 .00444 93,304 414 465,482 5,282,219 56.61
15 .00154 .00767 92,889 713 462,664 4,816,737 51.85
20 .00272 .01351 92,177 1245 457,765 4,354,073 47.24
25 .00244 .01213 90,932 1103 451,897 3,896,308 42.85
30 .00302 .01499 89,829 1346 445,772 3,444,411 38.34
35 .00376 .01862 88,483 1648 438,282 2,998,639 33.89
40 .00563 .02776 86,835 2410 428,121 2,560,357 29.49
45 .00819 .04012 84,425 3387 413,597 2,132,236 25.26
50 .01305 .06317 81,037 5119 392,250 1,718,638 21.21
55 .02069 .09828 75,918 7461 360,618 1,326,388 17.47
60 .03081 014277 68,457 9774 317,225 965,770 14.11
65 .05047 .22303 58,684 13088 259,323 648,545 11.05
70 .07540 .31408 45,596 14321 189,931 389,222 8.54
75 .11552 .43876 31,275 13722 118,784 199,290 6.37
80 .17400 .58105 17,553 10199 58,615 80,506 4.59
85+ .33592 1.0000 7,354 7354 21,891 21,891 2.98

After you finish the exercise, check your work with the answer key below.

## 5.4 Cause Elimination Life Tables and Gains in Expectation of Life

Cause elimination life tables answer the hypothetical questions about a cohort's mortality experience if a particular cause (or causes) were eliminated. The gain in expectation of life after a particular cause of death is eliminated gives a summary measure of the impact of a particular cause of death in the population. When all the causes except a particular cause are eliminated (i.e. all competing causes are eliminated) the resulting cause-eliminated life table gives a life table that can be used to compare the mortality with respect to the selected cause across populations and across time. This comparison is possible because the resulting life table in this situation adjusts for the differences in the intensity of all other competing causes of death among the different populations.

There are a number of methods available to construct life tables eliminating certain causes of death. A number of methods are described in Namboodiri and Suchindran (1987). This session briefly describe one simple method. This method is consistent with the method attributed to Fergany and used in the construction of the life table in Section 3.2.

#### Life Table Construction Eliminating a Specified Cause

Recall that in Section 3.2 under the assumption of constant mortality the relationship between proportion dying in an age interval (nqx) and the age-specific death rate (nmx) is expressed as:

Also recall that the age-specific death rate is related to age-cause-specific death rates ( nmxd ) as:

 nmx = nmx1 + nmx2 + nmx3 + ... + nmxr

In the hypothetical situation when one particular cause Rdis eliminated from the population, the age-specific death rate in the population will change to

 nmx* = nmx - nmxd

The life table eliminating the cause Rdis calculated using the method in Lesson 3.2 with the assumption that the prevailing age-specific mortality in the population is nmx* . Specifically:

 ndx = lx *nqx

 nLx = ndxnmx*

 Tx = Endoftableåh = x nLh

 ex0( - d) = Txlx

where ex0( - d) denotes the expectation of life at age x eliminating cause R.

Example (with 1960 Costa Rican Males)

In the age-specific and age-cause-specific death rates for 1960 Costa Rica males in Table 2.5.2, we note that:

 All cause mortality for the age interval <1 (1m0 ) = 75.05 per 1000 peopleor 0.07505 per person

 Diarrhea-specific mortality for the age interval <1 (1m0Diarrhea) = 18.07 per 1000 peopleor 0.01807 per person

When diarrhea is eliminated as a risk of death in the population, the new risk of death for this age interval ( 1m0*) will be 0.07505 - 0.01807 ( 1m1m0Diarrhea) = 0.05698 (See Table 5.4.1 below.)

Similarly, the new risks of death ( nmx*) when diarrhea is completely eliminated are presented in Table 5.4.1 for all other age groups. These new death rates ( nmx*) are then used to complete the life table in Table 5.4.1.

Remember these formulas for calculating life table entries:

 l0 = 100000

 ndx = lx *nqx lx + n = lx - ndx

 nLx = ndxnmx* Tx = å nLx             ex0 = Txlx

From Table 5.4.1.:

Expectation of life at birth when diarrhea is completely eliminated:

 e00( - diarrhea)= 64.68

Expectation of life at birth when no causes are eliminated (from Table 3.2.1):

 e00 = 62.97

Therefore the gain in expectation of life at birth is:
64.68 - 62.97 = 1.71 years.

Similarly, gain in expectation of life at age 60 by elimination of diarrhea is:

 e600( - diarrhea) - e600 = 16.57 - 16.42 = 0.15 years.

Table 5.4.1: Life Table Eliminating the Risk of Diarrhea

Age
 nmx*
 nqx
 lx
 ndx
 nLx
 Tx
 ex0(-diarrhea)
0 0.05698 0.05539 100,000 5,539 97,204 6,468,621 64.68621
1 0.00551 0.02180 94,461 2,059 373,710 6,371,417 67.45034
5 0.00159 0.00792 92,402 732 460,177 5,997,707 64.90912
10 0.00126 0.00629 91,670 576 456,908 5,537,531 60.40717
15 0.00125 0.00623 91,094 568 454,049 5,080,622 55.77349
20 0.00181 0.00899 90,526 814 450,594 4,626,574 51.10760
25 0.00163 0.00814 89,713 730 446,735 4,175,980 46.54840
30 0.00192 0.00956 88,983 851 442,782 3,729,244 41.90985
35 0.00295 0.01463 88,132 1,289 437,428 3,286,462 37.29033
40 0.00437 0.02163 86,842 1,879 429,498 2,849,035 32.80694
45 0.00645 0.03175 84,964 2,698 418,038 2,419,536 28.47727
50 0.00905 0.04426 82,266 3,641 402,159 2,001,498 24.32960
55 0.01328 0.06426 78,625 5,053 380,353 1,599,339 20.34134
60 0.02323 0.10966 73,572 8,068 347,302 1,218,986 16.56855
65 0.03588 0.16424 65,504 10,759 299,822 871,684 13.30724
70 0.05077 0.22419 54,746 12,273 241,750 571,862 10.44574
75 0.07540 0.31408 42,473 13,340 176,923 330,112 7.77236
80 0.13120 0.48108 29,133 14,015 106,822 153,190 5.25834
85+ 0.32604 1.00000 15,118 15,118 46,367 46,367 3.06711

#### Exercise 15

1. For the 1960 Costa Rican males, construct a life table eliminating CVD as a cause of death and add a final column showing the gain in expectation of life for each age after eliminating CVD as a risk.

Practical Suggestions:

• Start with the spreadsheet you used to construct an ordinary life table for 1960 Costa Rican females in Exercise 7. All of those formulas are needed here. Incorporate a new ASDR for the males based on the elimination of CVD as a risk.
• Table 2.5.2. Note that these rates are "per 1000 people" and need to be converted to a "per person" proportion before continuing.
• For each age group, calculate and enter in your ordinary life table a new death rate (nmx*) that is the CVD-eliminated death rate.
• If you have carried over your formulas properly from the ordinary life table in Exercise 7, the rest of the calculations will be done automatically, ending in the expected life column, which will now mean "expected life when CVD is eliminated as a cause of death."
•  The expectation of life when CVD is eliminated as a cause of death will be 66.23 at birth and 19.66 at age 60. The corresponding expectation of life when no elimination occurs are 62.97 and 16.42. Thus the gains in expectation when CVD is eliminated are respectively 3.26 years at birth and 3.24 years at age 60.

2. Using your spreadsheet, show that the gain in expectation of life at birth when cancer is eliminated as a cause of death is 2.37 years.

When you have finished the exercise, compare your calculations, graphs, and descriptions to the answer key below.

#### Exercise 14

Use these data tables to compute the contributions of each cause of mortality in determining the changes in expectation of life at birth between these periods. Use both the Arriaga Method and the Pollard Method and compare these methods.

Note: Our starting point is the Expectation of Life Improvement for Newborns in Taiwan from 1960 (62.28) to 1964 (64.53) = 2.25

Age ArriagaMethod PollardMethod Exercise 14 Answer Key: Total Mortality Contributions (Showing Age-Specific Improvementsin Mortality for Taiwan from 1960 to 1964) 0 0.5914 0.59911 1 0.5276 0.52994 5 0.1012 0.10120 10 0.0464 0.04634 15 0.0655 0.06550 20 -0.0422 -0.04217 25 0.0676 0.06770 30 0.0661 0.06611 35 0.0343 0.03430 40 0.0761 0.07620 45 0.0726 0.07265 50 0.0940 0.09391 55 0.1595 0.15862 60 0.0551 0.05446 65 0.1229 0.12049 70 0.0746 0.07186 75 0.0854 0.07937 80 0.0729 0.06126 85+ -0.0128 -0.01093 Total 2.2582 2.24592

As you can see, the two methods produced very similar results over all causes. Now we look at this issue on a cause-by-cause basis:

  Pollard Method Arriaga Method All Causes Age Tuberculosis Cancer CVD OtherCauses Tuberculosis Cancer Exercise 14 Answer Key: Age-Cause-Specific Mortality Contributions (Pollard vs. Arriaga Method) 0 0.0032 -0.0076 0.0223 0.5813 0.5991 0.0031 -0.0075 0.0220 0.5737 0.5914 1 0.0097 -0.0122 0.0146 0.5178 0.5299 0.0097 -0.0121 0.0145 0.5155 0.5276 5 0.0056 -0.0084 0.0084 0.0956 0.1012 0.0056 -0.0084 0.0084 0.0956 0.1012 10 0.0003 -0.0103 0.1931 -0.1367 0.0463 0.0003 -0.0103 0.1932 -0.1368 0.0464 15 0.0047 -0.0047 0.0094 0.0561 0.0655 0.0047 -0.0047 0.0094 0.0562 0.0655 20 0.0021 -0.0211 0.0084 -0.0316 -0.0422 0.0021 -0.0211 0.0084 -0.0316 -0.0422 25 0.0188 0.0002 0.0169 0.0318 0.0677 0.0188 0.0002 0.0169 0.0317 0.0676 30 0.0165 -0.0066 0.0198 0.0364 0.0661 0.0165 -0.0066 0.0198 0.0363 0.0661 35 0.0100 0.0001 0.0114 0.0127 0.0343 0.0100 0.0001 0.0114 0.0127 0.0343 40 0.0157 -0.0109 0.0181 0.0532 0.0762 0.0157 -0.0109 0.0181 0.0532 0.0761 45 0.0100 -0.0070 0.0428 0.0269 0.0727 0.0099 -0.0070 0.0428 0.0269 0.0726 50 -0.0126 0.0182 0.0474 0.0410 0.0939 -0.0126 0.0182 0.0474 0.0411 0.0940 55 0.0618 -0.0190 0.0232 0.0927 0.1586 0.0621 -0.0191 0.0233 0.0932 0.1595 60 0.0180 -0.0121 -0.0163 0.0649 0.0545 0.0182 -0.0123 -0.0165 0.0657 0.0551 65 0.0005 -0.0154 0.0154 0.1200 0.1205 0.0006 -0.0157 0.0157 0.1224 0.1229 70 -0.0014 -0.0029 -0.0466 0.1227 0.0719 -0.0014 -0.0030 -0.0484 0.1275 0.0746 75 -0.0009 -0.0081 -0.0084 0.0968 0.0794 -0.0009 -0.0087 -0.0091 0.1041 0.0854 80 -0.0024 -0.0044 -0.0181 0.0862 0.0613 -0.0029 -0.0053 -0.0215 0.1027 0.0729 85+ 0.0001 -0.0001 -0.0171 0.0061 -0.0109 0.0001 -0.0001 -0.0199 0.0071 -0.0128 Total 0.1597 -0.1324 0.3447 1.8738 2.2459 0.1596 -0.1343 0.3359 1.8971 2.2582

The two methods produce very similar results.

This table shows that "other causes" contributed the most to the improvement in expected life for the Taiwanese from 1960 to 1964. Reductions in CVD and tuberculosis mortality rates also contributed to the improvement. Cancer risks actually increased and had a negative effect on the mortality improvement.

#### Exercise 15

1. For the 1960 Costa Rican males, construct a life table eliminating CVD as a cause of death and add a final column showing the gain in expectation of life for each age after eliminating CVD as a risk.

  ASDR ACSDR Cause-Eliminated ASDR   New, Cause-Eliminated Original (All Causes) Gain Age (All Causes) (CVD) nmx* Exercise 15 Answer Key, Part 1: Life Table Eliminating the Risk of CVD: 1960 Costa Rican Males 0 0.07505 0.00057 0.07448 0.07177 100,000 7,177 96,367 6,622,828 66.23 62.97 3.25 1 0.00701 0.00009 0.00693 0.02732 92,823 2,536 366,195 6,526,462 70.31 66.84 3.47 5 0.00171 0.00000 0.00171 0.00851 90,287 769 449,509 6,160,266 68.23 64.69 3.54 10 0.00128 0.00006 0.00122 0.00607 89,518 543 446,230 5,710,758 63.79 60.22 3.57 15 0.00129 0.00007 0.00121 0.00605 88,975 539 443,526 5,264,528 59.17 55.59 3.58 20 0.00181 0.00015 0.00166 0.00827 88,436 731 440,350 4,821,002 54.51 50.93 3.58 25 0.00163 0.00010 0.00154 0.00765 87,705 671 436,845 4,380,652 49.95 46.37 3.58 30 0.00198 0.00017 0.00181 0.00900 87,034 783 433,207 3,943,807 45.31 41.73 3.58 35 0.00302 0.00035 0.00267 0.01325 86,250 1,142 428,389 3,510,600 40.70 37.12 3.58 40 0.00442 0.00056 0.00386 0.01911 85,108 1,627 421,460 3,082,211 36.22 32.65 3.57 45 0.00645 0.00133 0.00512 0.02530 83,481 2,112 412,104 2,660,752 31.87 28.32 3.55 50 0.00923 0.00192 0.00731 0.03590 81,369 2,921 399,499 2,248,648 27.64 24.17 3.47 55 0.01344 0.00336 0.01008 0.04914 78,448 3,855 382,521 1,849,149 23.57 20.19 3.38 60 0.02364 0.00649 0.01714 0.08215 74,593 6,128 357,427 1,466,628 19.66 16.43 3.24 65 0.03633 0.01152 0.02481 0.11666 68,465 7,987 321,947 1,109,200 16.20 13.18 3.02 70 0.05182 0.02001 0.03181 0.14704 60,479 8,893 279,571 787,253 13.02 10.32 2.70 75 0.07644 0.02397 0.05247 0.23075 51,586 11,903 226,870 507,682 9.84 7.67 2.18 80 0.13520 0.04960 0.08560 0.34819 39,682 13,817 161,413 280,812 7.08 5.14 1.93 85+ 0.33698 0.12035 0.21663 1.00000 25,865 25,865 119,399 119,399 4.62 2.97 1.65

As you can see from the table, the gain in expectation of life after eliminating CVD as a risk factor ranges from 3.26 extra years at birth to 1.65 extra years in the 85+ age group.

2. Using your spreadsheet, show that the gain in expectation of life at birth when cancer is eliminated as a cause of death is 2.21 years.

The table below is identical to the table above except cancer has been eliminated instead of CVD. At the top of the far right column, you can see the foreshadowed gain of 2.37 extra years at birth from eliminating cancer as a risk for 1960 Costa Rican Males.

  ASDR ACSDR Cause-Eliminated ASDR   New, Cause-Eliminated Original (All Causes) Gain Age (All Causes) (Cancer) nmx* Exercise 15 Answer Key, Part 2: Life Table Eliminating the Risk of Cancer: 1960 Costa Rican Males 0 0.07505 0.00007 0.07498 0.07224 100,000 7,224 96,343 6,534,446 65.34 62.97 2.37 1 0.00701 0.00014 0.00687 0.02711 92,776 2,515 366,050 6,438,103 69.39 66.84 2.55 5 0.00171 0.00010 0.00161 0.00804 90,260 725 449,487 6,072,053 67.27 64.69 2.59 10 0.00128 0.00007 0.00120 0.00599 89,535 537 446,333 5,622,567 62.80 60.22 2.58 15 0.00129 0.00007 0.00121 0.00605 88,998 539 443,644 5,176,234 58.16 55.59 2.57 20 0.00181 0.00012 0.00168 0.00837 88,460 740 440,444 4,732,591 53.50 50.93 2.57 25 0.00163 0.00017 0.00146 0.00729 87,719 640 436,995 4,292,146 48.93 46.37 2.56 30 0.00198 0.00011 0.00186 0.00928 87,080 808 433,375 3,855,151 44.27 41.73 2.54 35 0.00302 0.00035 0.00267 0.01325 86,271 1,143 428,494 3,421,776 39.66 37.12 2.54 40 0.00442 0.00099 0.00343 0.01701 85,129 1,448 422,014 2,993,282 35.16 32.65 2.51 45 0.00645 0.00095 0.00550 0.02715 83,681 2,272 412,700 2,571,269 30.73 28.32 2.40 50 0.00923 0.00267 0.00656 0.03226 81,409 2,626 400,445 2,158,569 26.52 24.17 2.35 55 0.01344 0.00420 0.00924 0.04514 78,783 3,556 384,957 1,758,124 22.32 20.19 2.12 60 0.02364 0.00751 0.01613 0.07748 75,227 5,829 361,368 1,373,167 18.25 16.43 1.83 65 0.03633 0.00916 0.02717 0.12703 69,398 8,816 324,454 1,011,799 14.58 13.18 1.40 70 0.05182 0.00990 0.04192 0.18910 60,583 11,456 273,274 687,345 11.35 10.32 1.03 75 0.07644 0.01529 0.06115 0.26344 49,127 12,942 211,632 414,071 8.43 7.67 0.76 80 0.13520 0.01840 0.11680 0.44234 36,185 16,006 137,036 202,439 5.59 5.14 0.45 85+ 0.33698 0.02845 0.30853 1.00000 20,179 20,179 65,402 65,402 3.24 2.97 0.27

Lesson 6: Constructing Multiple-Decrement Life Tables