# Lesson 4: Construction of Multiple-Decrement Life Tables

## 4.0 Overview

In this lesson, we learn to construct a multiple-decrement life table and learn to interpret the summary measures from the table. Because the examples described here pertain to causes of death, the table constructed here may also be called a causes of death life table. The method of constructing the table also applies to other situations, such as marital disruptions, etc.

Download a zip file containing data for Lesson 4 in Excel and CSV formats.

#### Causes of Death Life Tables

The data required for constructing causes of death life tables are age-specific and age-cause-specific death rates. This data could be in the form of age-specific death rates and age-cause-specific death ratios. (See Lesson 2 for details.)

The following steps will be used in constructing the table:

Step 1

Construct an ordinary life table with the data on age-specific death rates. (See Lesson 3 for details.)

Step 2

Distribute the deaths in each age interval in the life table to various causes using age-cause-specific death ratios.

Step 3

Use this information to compute cause-specific death probabilities.

Step 4

Cumulate deaths by cause in the life table to compute eventual probabilities of death by cause.

## 4.1 Distribution of Life Table Deaths by Cause

This section will explain Step 2, distribution of deaths to various causes using age-cause-specific death ratios.

In an ordinary life table the total number of deaths in a specific age interval is given by the ndx values in the table.

Denote the cause-specific death ratio for a specific cause Rd in the age interval (x, x + n) asnCxd.

Then number of deaths due to a specific cause R(denoted as ndxd) in the age interval (x, x + n) is calculated as:

ndxd = ndx * nCxd   (total deaths * cause-specific death ratio)

Example

From the 1960 Costa Rican Males Life Table in Table 3.2.1 of Lesson 3.2:

Number of deaths in the age interval 0 - 1 (1d0) is 7230.

From Table 2.5.3 of Lesson 2.5, the cause-specific death ratios (expressed here as proportions) for the same age group:

Cause of Death Cause-Specific Death Ratio Cause-Specific Deaths (age <1) Table 4.1.1: Calculation of Cause-Specific Deaths Diarrhea 0.2407 7230*.2408 = 1740 Cancer 0.0009 7230*.0009 = 7 CVD 0.0076 7230*.0076 = 55 Other causes 0.7507 7230*.7507= 5428

The distribution of deaths by cause for all age intervals is given below in Table 4.1.2

Deaths Age-Cause-Specific Death Ratio Deaths by Cause Age ndx Diarrhea Cancer CVD Other ndxDiarr Table 4.1.2 Life Table Deaths by Cause: 1960 Costa Rican Males <1 year 7,230 0.24072 0.00089 0.00761 0.75078 1740 6 55 5428 1-4 2,566 0.21407 0.01988 0.01223 0.75382 549 51 31 1934 5-9 768 0.07042 0.05634 0.00000 0.87324 54 43 0 671 10-14 569 0.01149 0.05747 0.04598 0.88506 7 33 26 503 15-19 570 0.02778 0.05556 0.05556 0.86111 16 32 32 490 20-24 793 0.00000 0.06897 0.08046 0.85057 0 55 64 675 25-29 712 0.00000 0.10448 0.05970 0.83582 0 74 43 595 30-34 854 0.02857 0.05714 0.08571 0.82857 24 49 73 708 35-39 1,287 0.02326 0.11628 0.11628 0.74419 30 150 150 958 40-44 1,849 0.00971 0.22330 0.12621 0.64078 18 413 233 1185 45-49 2,629 0.00000 0.14706 0.20588 0.64706 0 387 541 1701 50-54 3,615 0.01887 0.28931 0.20755 0.48428 68 1046 750 1751 55-59 4,975 0.01136 0.31250 0.25000 0.42614 57 1555 1244 2120 60-64 7,979 0.01717 0.31760 0.27468 0.39056 137 2534 2192 3116 65-69 10,564 0.01220 0.25203 0.31707 0.41870 129 2662 3350 4423 70-74 12,107 0.02033 0.19106 0.38618 0.40244 246 2313 4676 4872 75-79 13,002 0.01364 0.20000 0.31364 0.47273 177 2600 4078 6147 80-84 13,724 0.02959 0.13609 0.36686 0.46746 406 1868 5035 6415 85+ 14,207 0.03247 0.08442 0.35714 0.52597 461 1199 5074 7473

Exercise 8

This exercise will use data on 1960 Costa Rican females that you have downloaded and worked with in previous exercises.

1. From the life table for 1960 Costa Rican females constructed in Exercise 7 obtain nqxlx, and ndxcolumns. (No new calculations are needed here).
2. Use the data in Exercise 5 to obtain age-cause-specific death ratioss for diarrhea, cancer, and CVD for 1960 Costa Rican Females. (No new calculations are needed here.)
3. Use the life table deaths (ndx) in Part 1 above and the age-cause-specific death ratios in Part 2 to obtain the number of deaths by cause in every age interval.

When you have finished your work, compare it to the answer key below.

## 4.2 Computation of Cause-Specific Death Probabilities

Step 3 uses the information from Step 2 to compute cause-specific death probabilities.

Recall that in an ordinary life table the proportion dying in a specific age interval (x, x + n), or nqx, may be interpeted as the probability that a person of age x will die in that age interval. This quantity is computed as:

 nqx = ndxlx

Similarly, in the multiple-decrement life table the proportion of individuals dying in the specific age interval due to a specific cause Ris considered to be the probability that a person of age x will die in that age interval (x, x + n) due to cause Rd (in the presence of all other causes). Denote:

 nQxd

~ Probability that a person of age x will die in the
age interval (x, x + n) due to a specific cause Rd when
all other competing causes are present in the population.

Because the probability nQxd is computed as death probability under the assumption of presence of all other possible causes of death, it is often termed as the crude probability of death due to a specific cause. This probability is computed from the life table deaths due to a specific cause as:

 nQxd =
 ndxdlx

Example

The life table in Table 3.2.1 and Table 4.1.2 give the following data for age interval <1 :

Data for Age Interval <1:
 l0 = 100,0001d0 = 72301d0Diarrhea = 17401d0Cancer = 61d0CVD = 551d0Other = 5428

Table 4.2.1 Crude Probabilities in Age Interval <1

Diarrhea
 1740100,000 = .01740
Cancer
 6100,000 = .00006
CVD
 55100,000 = .00055
Other
 5428100,000 = .05428

Note that the sum of the crude probabilities is equal to 1q0, or .07230.

The table below (Table 4.2.2) reproduces data presented in previous tables to illustrate the computation of crude probabilities of deaths for specific causes of death. Table 4.2.3 shows the crude probabilities calculated from Table 4.2.2 by dividing number of deaths (ndxd) by the number alive at the beginning of the age interval (lx).

Age Interval lx Total Deaths Diarrhea Deaths Cancer Deaths CVD Deaths Other Causes Table 4.2.2 Number of Life Table Deaths by Cause: 1960 Costa Rican Males <1 year 100,000 7,230 1740 6 55 5,428 1-4 92,770 2,566 549 51 31 1,934 5-9 90,204 768 54 43 0 671 10-14 89,436 569 7 33 26 503 15-19 88,867 570 16 32 32 490 20-24 88,298 793 0 55 64 675 25-29 87,504 712 0 74 43 595 30-34 86,792 854 24 49 73 708 35-39 85,938 1,287 30 150 150 958 40-44 84,651 1,849 18 413 233 1,185 45-49 82,802 2,629 0 387 541 1,701 50-54 80,173 3,615 68 1,046 750 1,751 55-59 76,558 4,975 57 1,555 1,244 2,120 60-64 71,583 7,979 137 2,534 2,192 3,116 65-69 63,605 10,564 129 2,662 3,350 4,423 70-74 53,040 12,107 246 2,313 4,676 4,872 75-79 40,933 13,002 177 2,600 4,078 6,147 80-84 27,931 13,724 406 1,868 5,035 6,415 85+ 14,207 14,207 461 1,199 5,074 7,473

Table 4.2.3 Crude Probabilities of Death for Specific Causes of Death: 1960 Costa Rican Males

Age Interval
 nqx
 nQxDiarrhea
 nQxCancer
 nQxCVD
 nQxOther
<1 year 0.07230 0.01740 0.00006 0.00055 0.05428
1-4 0.02765 0.00592 0.00055 0.00034 0.02085
5-9 0.00851 0.00060 0.00048 0.00000 0.00743
10-14 0.00636 0.00007 0.00037 0.00029 0.00563
15-19 0.00641 0.00018 0.00036 0.00036 0.00552
20-24 0.00899 0.00000 0.00062 0.00072 0.00764
25-29 0.00814 0.00000 0.00085 0.00049 0.00680
30-34 0.00984 0.00028 0.00056 0.00084 0.00815
35-39 0.01497 0.00035 0.00174 0.00174 0.01114
40-44 0.02184 0.00021 0.00488 0.00276 0.01400
45-49 0.03175 0.00000 0.00467 0.00654 0.02055
50-54 0.04509 0.00085 0.01304 0.00936 0.02184
55-59 0.06498 0.00074 0.02031 0.01624 0.02769
60-64 0.11146 0.00191 0.03540 0.03062 0.04353
65-69 0.16609 0.00203 0.04186 0.05266 0.06954
70-74 0.22826 0.00464 0.04361 0.08815 0.09186
75-79 0.31765 0.00433 0.06353 0.09963 0.15016
80-84 0.49135 0.01454 0.06687 0.18026 0.22969
85+ 1.00000 0.03247 0.08442 0.35714 0.52597

Comment:

Occasionally we will be able to compute the crude probabilities directly from data (See Lesson 5). In such situations the interest will be in computing the number of deaths by specific diseases when the number alive at the beginning of the age interval is known. We will use the following relationships to calculate the number of deaths by cause:

 ndxd = lx * nQxd

#### Exercise 9

Use your spreadsheet software for this exercise and continue to build on the work you've done in previous exercises.

Calculate cause-specific crude probabilities of death for 1960 Costa Rican females using the lx values and the number of cause-specific deaths from Exercise 8.

When you are done, check the answer key below.

## 4.3 Computation of Cumulative Numbers and Eventual Probabilities of Cause-Specific Deaths

In step 4 we will compute the cumulative number of deaths by cause in order to compute eventual probabilities of death by cause.

From Table 4.1.2 one can compute the number of persons dying due to a specific cause after a specific age. For example, the number of deaths after age 65 due to diarrhea is:

 5d65Diarrhea + 5d70Diarrhea + 5d75Diarrhea + 5d80Diarrhea + d85+Diarrhea

=129 + 246 + 177 + 406 + 461
= 1420 (with rounding)

Similar sums give the cumulative number of deaths after age 65 due to cancer, CVD, and other causes. These sums are shown in Table 4.3.1

Cause of Death Number of Deaths after Age 65 Table 4.3.1 Number of Life Table Deaths after Age 65 by Cause: 1960 Costa Rican Males Diarrhea 1420 Cancer 10,643 CVD 22,212 Other causes 29,330 Total 63,605

Note that the total number of deaths after age 65 from all causes in Table 4.3.1 is 63,605 and this number is also the l65 value in the life table (Table 3.2.1). This is consistent with the interpretation that lx values in a life table can be interpreted as the cumulative number of persons dying after age x in the life table. Thus:

 lx = End of tableåh = x ndh

Similarly, let's introduce the following notation:

 lxd = End of tableåh = x ndhd

The number lxd is interpreted as the cumulative number of deaths after age x due to cause Rd.

From Table 4.3.1, l65Diarrhea = 1420, cumulative number of diarrhea deaths after age 65.

Similarly, one can compute cumulative number of deaths by cause for all ages. Table 4.3.2 gives the cumulative number of deaths by cause for 1960 Costa Rican males.

Table 4.3.2 1960 Cumulative Number of Deaths by Cause in the Life Table: Costa Rican Males

Age Interval
 lx
 lxDiarrhea
 lxCancer
 lxCVD
 lxOther Causes
<1 year 100,000 4,120 17,070 27,645 51,165
1-4 92,770 2,379 17,063 27,590 45,737
5-9 90,204 1,830 17,012 27,559 43,803
10-14 89,436 1,776 16,969 27,559 43,132
15-19 88,867 1,769 16,936 27,533 42,629
20-24 88,298 1,754 16,905 27,501 42,138
25-29 87,504 1,754 16,850 27,437 41,463
30-34 86,792 1,754 16,776 27,395 40,868
35-39 85,938 1,729 16,727 27,322 40,160
40-44 84,651 1,699 16,577 27,172 39,203
45-49 82,802 1,681 16,164 26,939 38,018
50-54 80,173 1,681 15,778 26,397 36,317
55-59 76,558 1,613 14,732 25,647 34,566
60-64 71,583 1,556 13,177 24,403 32,446
65-69 63,605 1,420 10,643 22,212 29,330
70-74 53,040 1,291 7,981 18,862 24,907
75-79 40,933 1,045 5,668 14,187 20,034
80-84 27,931 867 3,067 10,109 13,888
85+ 14,207 461 1,199 5,074 7,473

#### Notes on Table 4.3.2

The first row of the table gives the cumulative number of deaths in the life table population at age 0. This means that, among the 100,000 newborn children who eventually die,

• 4120 will die due to diarrhea,
• 17,070 will die due to cancer,
• 27,645 will die due to CVD,
• and 51,165 will die due to other causes.

Therefore, in this population the probability that a newborn will eventually die

• due to diarrhea is 0.04120 (4,120/100,000),
• due to cancer is 0.17070 (17,070/100,000),
• due to CVD is 0.27645 (27,645/100,000)
• and due to other causes is 0.51165 (51,165/100,000).

Later in this lesson, we will examine the death probabilities of individuals surviving to various ages. We will also examine the ages at which people die of specific causes.

#### Exercise 10

Compute the cumulative number of life table deaths after a specific age by cause for 1960 Costa Rican females using the data generated in Exercise 8, Part 3.

## 4.4 Eventual Death Probabilities

Table 4.3.2 from the previous page shows how many persons die due to specific diseases after a specific age. From the table we also calculated the eventual death probabilities by cause for newborn babies (i.e., eventual death probability at age zero). We will extend this concept to other ages. We now ask the question: What is the probability that a person surviving to age x will eventually die due to a specific cause Rd? We will denote this probability as Pd. This probability is calculated as:

Eventual death probability at age x by cause Rd:
 Pd = lxdlx

Example

Data from Table 4.3.2:

 l65 = 63,605l65Diarrhea = 1,420l65Cancer = 10,643l65CVD = 22,212

Calculating eventual death probabilities:

 P65Diarrhea = 142063605 = 0.02233
 P65Cancer = 1064363605 = 0.16733
 P65Cvd = 2221263605 = 0.34922

The results show that the probabilities that a 65-year-old will eventually die due to diarrhea, cancer, or CVD are respectively 0.02202, 0.16736, and 0.34939.

Table 4.4.1 shows the eventual death probabilities for all age intervals for 1960 Costa Rican males.

Table 4.4.1 Eventual Death Probabilities by Cause of Death:
1960 Costa Rican Males

Age Interval
 PxDiarrhea
 PxCancer
 PxCVD
 PxOther Causes
<1 year 0.04120 0.17070 0.27645 0.51165
1-4 0.02565 0.18393 0.29741 0.49301
5-9 0.02029 0.18860 0.30552 0.48560
10-14 0.01986 0.18973 0.30814 0.48227
15-19 0.01991 0.19058 0.30982 0.47969
20-24 0.01986 0.19145 0.31146 0.47723
25-29 0.02004 0.19256 0.31356 0.47384
30-34 0.02020 0.19328 0.31564 0.47087
35-39 0.02012 0.19464 0.31792 0.46732
40-44 0.02007 0.19583 0.32099 0.46311
45-49 0.02030 0.19522 0.32534 0.45914
50-54 0.02097 0.19679 0.32926 0.45298
55-59 0.02107 0.19243 0.33500 0.45150
60-64 0.02174 0.18408 0.34091 0.45326
65-69 0.02232 0.16733 0.34922 0.46113
70-74 0.02433 0.15046 0.35562 0.46958
75-79 0.02552 0.13846 0.34658 0.48944
80-84 0.03105 0.10981 0.36192 0.49722
85+ 0.03247 0.08442 0.35714 0.52597

Exercise 11

Use the data generated in Exercise 10 to compute the eventual probabilities of death for each cause. Compare with table 4.4.1 to describe the male-female differences in the eventual death probabilities at age 0 in Costa Rica in 1960.

## 4.5 Cumulative Probability of Death Due to a Specific Cause before a Specific Age

In the previous section we looked at the eventual death probabilities by cause. Similarly, one can also examine the probability of dying due to a specific cause before a specific age x. The steps involved in calculating this quantity are as follows:

Step 1: Compute the cumulative number of deaths from the beginning of the life table by cause before age x in the life table.

Step 2: Divide the cumulative number obtained in Step 1 by the radix of the life table. This quantity will give the probability that a newborn will die due to a specific cause before reaching a specific age.

 Example Table 4.5.1 gives the cumulative number of deaths due to specific causes by age. This cumulative number is obtained by adding the number of deaths by cause in Table 4.2.2. The table shows that out of 100,000 newborns (radix of the life table) 2350 died due to diarrhea by age 15 (Column 2). Thus the cumulative probability of death due to diarrhea before age 15 is calculated as 2348/100,000 = 0.02350 (Column 6). Similarly, one can look at cumulative probability of deaths due to a specific cause by certain age in Table 4.5.1. For example, cumulative number of deaths by age 65 among the life table population due to cancer and CVD respectively are 6427 and 5445. With these numbers one calculates the cumulative probabilities of death due to cancer and CVD before age 65 respectively as 0.06427 and 0.05445.

#### Exercise 12

1. Use the data generated in Exercise 10 to compute the cumulative number of deaths that occur before a specific age by cause.
2. Use these cumulative numbers of deaths to calculate the cumulative probabilities of death for each cause.
3. Compare with Table 4.5.1 to describe the male-female differences in the cumulative probabilities of death at age 65 in Costa Rica in 1960.

By Age Cumulative Deaths by Cause Cumulative Probability of Death Diarrhea Deaths Cancer Deaths CVD Deaths Other Causes Diarrhea Cancer Table 4.5.1: Cumulative Probabilities of Death by Cause: 1960 Costa Rican Males 1 1740 6 55 5428 0.01740 0.00006 0.00055 0.05428 5 2290 57 86 7362 0.02290 0.00057 0.00086 0.07362 10 2344 101 86 8033 0.02344 0.00101 0.00086 0.08033 15 2350 133 113 8536 0.02350 0.00133 0.00113 0.08536 20 2366 165 144 9027 0.02366 0.00165 0.00144 0.09027 25 2366 220 208 9702 0.02366 0.00220 0.00208 0.09702 30 2366 294 251 10297 0.02366 0.00294 0.00251 0.10297 35 2391 343 324 11005 0.02391 0.00343 0.00324 0.11005 40 2420 493 473 11962 0.02420 0.00493 0.00473 0.11962 45 2438 906 707 13147 0.02438 0.00906 0.00707 0.13147 50 2438 1292 1248 14848 0.02438 0.01292 0.01248 0.14848 55 2507 2338 1998 16599 0.02507 0.02338 0.01998 0.16599 60 2563 3893 3242 18719 0.02563 0.03893 0.03242 0.18719 65 2700 6427 5434 21835 0.02700 0.06427 0.05434 0.21835 70 2829 9089 8783 26258 0.02829 0.09089 0.08783 0.26258 75 3075 11402 13459 31131 0.03075 0.11402 0.13459 0.31131 80 3252 14003 17537 37277 0.03252 0.14003 0.17537 0.37277 85 3658 15871 22572 43693 0.03658 0.15871 0.22572 0.43693 85+ 4120 17070 27645 51165 0.04120 0.17070 0.27645 0.51165

## 4.6 Descriptive Measures of Age Distribution of Deaths by Specific Causes

This section covers several measures to describe the age pattern of death due to specific causes. In Section 4.1, we calculated the number of deaths due to a specific cause and denoted it as ndxd. Recall the formulas used to calculate ndxd  as:

ndxd = ndx * nCxd   (total deaths * cause-specific death ratio)

or

ndxd = lx * nQxd    (survivors at age x * crude probability of death)

For the 1960 Costa Rican males, the number of deaths by various causes were presented in Table 4.1.2; these numbers are repeated here in Table 4.6.1 for easy reference:

Age Interval Diarrhea Deaths Cancer Deaths CVD Deaths Other Causes Table 4.6.1: Age Distribution of Deaths by Cause: 1960 Costa Rican Males <1 year 1,740 6 55 5,428 1-4 549 51 31 1,934 5-9 54 43 0 671 10-14 7 33 26 503 15-19 16 32 32 490 20-24 0 55 64 675 25-29 0 74 43 595 30-34 24 49 73 708 35-39 30 150 150 958 40-44 18 413 233 1,185 45-49 0 387 541 1,701 50-54 68 1,046 750 1,751 55-59 57 1,555 1,244 2,120 60-64 137 2,534 2,192 3,116 65-69 129 2,662 3,350 4,423 70-74 246 2,313 4,676 4,872 75-79 177 2,600 4,078 6,147 80-84 406 1,868 5,035 6,415 85+ 461 1,199 5,074 7,473 Total 4,120 17,070 27,645 51,165

For each cause, the number of deaths shown in Table 4.6.1 gives the frequency of deaths by age. Several measures can be calculated to describe these frequency tables: proportional age distribution, graphic representation of age distribution, cumulative distribution of cause-specific deaths, and median age at cause-specific deaths.

#### Proportional age distribution

The proportional age distribution describes the proportion of deaths due to a specific cause in a specific age interval. Table 4.6.2 shows the proportional age distribution of life table deaths due to specific causes. These proportions are calculated by dividing each frequency in a specified age group by the total number of deaths by that cause. Specifically, the proportions for an age group are calculated as:

Proportion of deaths due to Rin age interval (x, x+ n) =
ndxd

 åall age groups ndxd

Age Interval Diarrhea Cancer CVD Other Causes Table 4.6.2: Proportion of Cause-Specific Deaths by Age:1960 Costa Rican Males <1 year 0.42248 0.00038 0.00199 0.10610 1-4 0.13331 0.00299 0.00114 0.03780 5-9 0.01313 0.00253 0.00000 0.01311 10-14 0.00159 0.00191 0.00095 0.00984 15-19 0.00384 0.00185 0.00114 0.00959 20-24 0.00000 0.00321 0.00231 0.01319 25-29 0.00000 0.00436 0.00154 0.01163 30-34 0.00592 0.00286 0.00265 0.01383 35-39 0.00726 0.00877 0.00541 0.01872 40-44 0.00436 0.02419 0.00844 0.02316 45-49 0.00000 0.02265 0.01958 0.03325 50-54 0.01656 0.06127 0.02714 0.03422 55-59 0.01372 0.09107 0.04499 0.04143 60-64 0.03325 0.14845 0.07928 0.06090 65-69 0.03127 0.15598 0.12116 0.08645 70-74 0.05973 0.13551 0.16912 0.09523 75-79 0.04304 0.15234 0.14751 0.12013 80-84 0.09856 0.10942 0.18212 0.12538 85+ 0.11197 0.07026 0.18354 0.14605 Total 1.00000 1.00000 1.00000 1.00000

#### Graphic representation of age distribution

The proportional age distribution in Table 4.6.2 can be represented graphically to examine the age patterns of deaths by cause. Because of unequal class intervals in Table 4.6.2 (the first age interval is one year long; the second is four years long; the remaining except the last are five years long; the last interval is open-ended), the graphic representation is preferred.

In order to calculate the graph, divide the proportion of deaths in each age interval by its length. The open-ended interval will be closed at an arbitrarily chosen upper age. (In the example here this upper age is set at 110). The graphical representation of Table 4.6.2 is given in Figure 4.6.1.

Figure 4.6.1: Graphic Representation of Age Distribution

1960 Costa Rican Males

The figure clearly shows that the diarrhea deaths are concentrated in the very young ages. Both CVD and cancer deaths are concentrated after age 50.

#### Cumulative distribution of cause-specific deaths

The proportion of deaths by cause in each age interval shown in Table 4.6.2 can be converted into cumulative proportion of deaths by cumulating the proportions in each age interval up to specific ages. These cumulative proportions are given in Table 4.6.3 and the graphic representation of these cumulative proportions is given immediately below in Figure 4.6.2.

Age Diarrhea Cancer CVD Other Causes Table 4.6.3: Cumulative Proportion of Deaths Due to Specific Causes by Age: 1960 Costa Rican Males 0 0.00000 0.00000 0.00000 0.00000 1 0.42248 0.00038 0.00199 0.10610 5 0.55579 0.00337 0.00312 0.14389 10 0.56892 0.00590 0.00312 0.15700 15 0.57051 0.00782 0.00407 0.16684 20 0.57435 0.00967 0.00521 0.17643 25 0.57435 0.01288 0.00752 0.18962 30 0.57435 0.01723 0.00906 0.20125 35 0.58027 0.02009 0.01171 0.21508 40 0.58754 0.02886 0.01712 0.23380 45 0.59190 0.05305 0.02556 0.25696 50 0.59190 0.07570 0.04514 0.29021 55 0.60845 0.13697 0.07228 0.32442 60 0.62218 0.22804 0.11727 0.36585 65 0.65542 0.37649 0.19655 0.42676 70 0.68670 0.53247 0.31771 0.51321 75 0.74643 0.66798 0.48683 0.60844 80 0.78947 0.82032 0.63434 0.72857 85 0.88803 0.92974 0.81646 0.85395

Figure 4.6.2:

The cumulative distribution in Table 4.6.3 and Figure 4.6.2 clearly show a concentration of deaths due to diarrhea at very young ages. The graph also shows the concentration of cancer and CVD deaths at the older ages.

#### Median age at cause-specific deaths

The median age of death due to a specific cause is the age at which 50% of the deaths occur.

Examples

Median age of cause-specific deaths among males, Costa Rica 1960

Cause of Death - Diarrhea

• From Table 4.6.1 the total number of deaths due to diarrhea = 4099
• 50% of total deaths due to diarrhea = 4099 * .50 = 2049.5
• Cumulative number of deaths due to diarrhea up to age 1 = 1741
• Number of diarrhea deaths in the age interval 1-4 = 549
• Length of age interval 1-4 = 4
• Therefore, median age at death for diarrhea deaths:
 1 + 2049.5 - 1741549 *4 = 3.24

Cause of Death - Cancer

• From Table 4.6.1 total number of cancer deaths = 17071
• 50% of total cancer deaths = 170171 * .50 = 8535.5
• Cumulative number of cancer deaths by age 65 = 6427
• Number of cancer deaths in age interval 65-70 = 3362
• Therefore, median age for cancer deaths:
 65 + 8535.5 - 64273362 *5 = 68.1

Cause of Death - CVD

• From Table 4.6.1 total number of CVD deaths = 27666
• 50% of total CVD deaths = 27,666 * .50 = 13833
• Cumulative number of CVD deaths up to age 75 = 13481
• Number of CVD deaths in age interval 75-79 = 4077
• Therefore, median age for CVD deaths:
 75 + 13833 - 134814077 *5 = 75.4

#### Exercise 13

Use the age distribution of life table deaths generated in Exercise 8, Part 3 for 1960 Costa Rican females to compute the following summary measures:

1. Proportional age distribution of deaths by cause. (Give a graphical representation of the proportional age distribution.)
2. Cumulative proportions of deaths by cause. (Draw a graph.)
3. Median age at death for specific causes.
4. Use these summary measures to describe the age pattern of various causes of death.

#### Exercise 8

1. From the life table for 1960 Costa Rican females constructed in Exercise 7 obtain nqxlx, and ndxcolumns. (No new calculations are needed here.)
2. Use the data in Exercise 5 to obtain age-cause-specific death ratios for diarrhea, cancer, and CVD for 1960 Costa Rican Females. (No new calculations are needed here.)
3. Use the life table deaths (ndx) in Part 1 above and the age-cause-specific death ratios in Part 2 to obtain the number of deaths by cause in every age interval.
Age Interval nqx lx ndx Age-Cause-Specific Death Ratios per Person Deaths by Cause Diarrhea Cancer CVD ndxDiarrhea Exercise 8 Answer Key: Calculation of Deaths by Cause: 1960 Costa Rican Females <1 year 0.0620 100,000 6,199 0.25954 0.00273 0.00600 1,609 17 37 1-4 0.0303 93,801 2,840 0.22734 0.00863 0.01727 646 25 49 5-9 0.0084 90,961 761 0.08824 0.02206 0.05882 67 17 45 10-14 0.0048 90,200 429 0.52381 0.04762 0.07937 225 20 34 15-19 0.0042 89,771 377 0.23913 0.02174 0.13043 90 8 49 20-24 0.0063 89,394 565 0.16393 0.18033 0.11475 93 102 65 25-29 0.0087 88,829 776 0.00000 0.07895 0.14474 0 61 112 30-34 0.0113 88,053 993 0.01163 0.11628 0.18605 12 115 185 35-39 0.0137 87,060 1,191 0.00000 0.21951 0.14634 0 261 174 40-44 0.0146 85,870 1,251 0.02857 0.24286 0.15714 36 304 197 45-49 0.0229 84,618 1,940 0.00000 0.36000 0.24000 0 699 466 50-54 0.0343 82,678 2,833 0.01639 0.34426 0.22951 46 975 650 55-59 0.0511 79,845 4,078 0.00730 0.34307 0.32847 30 1,399 1,340 60-64 0.0896 75,766 6,789 0.01075 0.29570 0.34409 73 2,008 2,336 65-69 0.1340 68,977 9,241 0.01523 0.24873 0.34010 141 2,299 3,143 70-74 0.2062 59,736 12,317 0.02283 0.21005 0.30137 281 2,587 3,712 75-79 0.2874 47,419 13,626 0.02618 0.14136 0.31937 357 1,926 4,352 80-84 0.4853 33,793 16,401 0.02424 0.13939 0.32121 398 2,286 5,268 85+ 1.0000 17,392 17,392 0.01569 0.00392 0.35294 273 68 6,138

#### Exercise 9

Calculate cause-specific crude probabilities of death for 1960 Costa Rican females using the lx values and the number of cause-specific deaths in Exercise 8.

Age Interval lx Deaths by Cause Crude Probabilities     Diarrhea Cancer Exercise 9 Answer Key: Calculation of Crude Probabilities: 1960 Costa Rican Females <1 year 100000 1609 17 37 0.0161 0.0002 0.0004 1-4 93801 646 25 49 0.0069 0.0003 0.0005 5-9 90961 67 17 45 0.0007 0.0002 0.0005 10-14 90200 225 20 34 0.0025 0.0002 0.0004 15-19 89771 90 8 49 0.0010 0.0001 0.0005 20-24 89394 93 102 65 0.0010 0.0011 0.0007 25-29 88829 0 61 112 0.0000 0.0007 0.0013 30-34 88053 12 115 185 0.0001 0.0013 0.0021 35-39 87060 0 261 174 0.0000 0.0030 0.0020 40-44 85870 36 304 197 0.0004 0.0035 0.0023 45-49 84618 0 699 466 0.0000 0.0083 0.0055 50-54 82678 46 975 650 0.0006 0.0118 0.0079 55-59 79845 30 1399 1340 0.0004 0.0175 0.0168 60-64 75766 73 2008 2336 0.0010 0.0265 0.0308 65-69 68977 141 2299 3143 0.0020 0.0333 0.0456 70-74 59736 281 2587 3712 0.0047 0.0433 0.0621 75-79 47419 357 1926 4352 0.0075 0.0406 0.0918 80-84 33793 398 2286 5268 0.0118 0.0677 0.1559 85+ 17392 273 68 6138 0.0157 0.0039 0.3529

#### Exercise 10

Compute the cumulative number of life table deaths after a specific age by cause for 1960 Costa Rican females using the data generated in Exercise 8, Part 3.

Age Interval Deaths by Cause Cumulative Deaths by Cause   Diarrhea Cancer CVD Exercise 10 Answer Key: Calculation of Cumulative Deaths by Cause: 1960 Costa Rican Females <1 year 1609 17 37 4375 15178 28352 1-4 646 25 49 2766 15161 28315 5-9 67 17 45 2120 15136 28265 10-14 225 20 34 2053 15119 28221 15-19 90 8 49 1828 15099 28187 20-24 93 102 65 1738 15091 28138 25-29 0 61 112 1646 14989 28073 30-34 12 115 185 1646 14928 27960 35-39 0 261 174 1634 14812 27776 40-44 36 304 197 1634 14551 27601 45-49 0 699 466 1598 14247 27405 50-54 46 975 650 1598 13548 26939 55-59 30 1399 1340 1552 12573 26289 60-64 73 2008 2336 1522 11174 24949 65-69 141 2299 3143 1449 9166 22613 70-74 281 2587 3712 1308 6868 19470 75-79 357 1926 4352 1027 4281 15758 80-84 398 2286 5268 670 2354 11406 85+ 273 68 6138 273 68 6138

#### Exercise 11

Use the data generated in Exercise 10 to compute the eventual probabilities of death for each cause. Compare with table 4.4.1 to describe the male-female differences in the eventual death probabilities at age 0 in Costa Rica in 1960.

Compared to the eventual death probabilities for the males at age 0 presented in Table 4.4.1, the probabilities for the females presented above show that the risks are very similar for both sexes:

• Diarrhea: .0412 males, .0437 females
• Cancer: .1707 males, .1518 females
• CVD: .2765 males, .2835 females
Age Interval lx Cumulative Deaths by Cause Eventual Death Probabilities     Diarrhea Cancer Exercise 11 Answer Key: Eventual Probabilities of Death by Cause: 1960 Costa Rican Females <1 year 100000 4375 15178 28352 0.0437 0.1518 0.2835 1-4 93801 2766 15161 28315 0.0295 0.1616 0.3019 5-9 90961 2120 15136 28265 0.0233 0.1664 0.3107 10-14 90200 2053 15119 28221 0.0228 0.1676 0.3129 15-19 89771 1828 15099 28187 0.0204 0.1682 0.3140 20-24 89394 1738 15091 28138 0.0194 0.1688 0.3148 25-29 88829 1646 14989 28073 0.0185 0.1687 0.3160 30-34 88053 1646 14928 27960 0.0187 0.1695 0.3175 35-39 87060 1634 14812 27776 0.0188 0.1701 0.3190 40-44 85870 1634 14551 27601 0.0190 0.1695 0.3214 45-49 84618 1598 14247 27405 0.0189 0.1684 0.3239 50-54 82678 1598 13548 26939 0.0193 0.1639 0.3258 55-59 79845 1552 12573 26289 0.0194 0.1575 0.3293 60-64 75766 1522 11174 24949 0.0201 0.1475 0.3293 65-69 68977 1449 9166 22613 0.0210 0.1329 0.3278 70-74 59736 1308 6868 19470 0.0219 0.1150 0.3259 75-79 47419 1027 4281 15758 0.0217 0.0903 0.3323 80-84 33793 670 2354 11406 0.0198 0.0697 0.3375 85+ 17392 273 68 6138 0.0157 0.0039 0.3529

#### Exercise 12

1. Use the data generated in Exercise 10 to compute the cumulative number of deaths that occur before a specific age by cause.

2. Use these cumulative numbers of deaths to calculate the cumulative probabilities of death for each cause.

By Age Cumulative Deaths by Cause Cumulative Deaths by Cause (reversed) Cumulative Probabilities of Death   Diarrhea Cancer CVD Diarrhea Cancer Exercise 12 Answer Key: Cumulative Number and Probabilities of Death by Cause: 1960 Costa Rican Females 1 4375 15178 28352 1609 17 37 0.0161 0.0002 0.0004 5 2766 15161 28315 2255 42 87 0.0225 0.0004 0.0009 10 2120 15136 28265 2322 59 131 0.0232 0.0006 0.0013 15 2053 15119 28221 2547 79 165 0.0255 0.0008 0.0017 20 1828 15099 28187 2637 87 214 0.0264 0.0009 0.0021 25 1738 15091 28138 2729 189 279 0.0273 0.0019 0.0028 30 1646 14989 28073 2729 250 392 0.0273 0.0025 0.0039 35 1646 14928 27960 2741 366 576 0.0274 0.0037 0.0058 40 1634 14812 27776 2741 627 751 0.0274 0.0063 0.0075 45 1634 14551 27601 2777 931 947 0.0278 0.0093 0.0095 50 1598 14247 27405 2777 1630 1413 0.0278 0.0163 0.0141 55 1598 13548 26939 2823 2605 2063 0.0282 0.0260 0.0206 60 1552 12573 26289 2853 4004 3403 0.0285 0.0400 0.0340 65 1522 11174 24949 2926 6012 5739 0.0293 0.0601 0.0574 70 1449 9166 22613 3067 8310 8882 0.0307 0.0831 0.0888 75 1308 6868 19470 3348 10897 12594 0.0335 0.1090 0.1259 80 1027 4281 15758 3705 12824 16946 0.0370 0.1282 0.1695 85 670 2354 11406 4102 15110 22214 0.0410 0.1511 0.2221 85+ 273 68 6138 4375 15178 28352 0.0438 0.1518 0.2835

3. Compare with Table 4.5.1 to describe the male-female differences in the cumulative probabilities of death at age 65 in Costa Rica in 1960.

The chances of dying before reaching age 65 by cause are very similar for the females presented above and the males presented in Table 4.5.1:

• Diarrhea: .0270 males, .0293 females
• Cancer: .0643 males, .0601 females
• CVD: .0543 males, .0574 females

#### Exercise 13

Use the age distribution of life table deaths generated in Exercise 8, Part 3 for 1960 Costa Rican females to compute the following summary measures:

1. Proportional age distribution of deaths by cause. (Give a graphic representation of the proportional age distribution.)

2. Cumulative proportions of deaths by cause. (Draw a graph.)

(The two graphs follow the table.)

      Deaths by Cause Proportion of Cause-Specific Deaths(for the interval) Proportion (adj. for unequal age intervals)(per year during the interval)  Cumulative Proportion by Age Start Age Mid-point for Plots Age Interval ndxDiarrhea ndxCancer ndxCVD Diarrhea Cancer Exercise 13 Answer Key: Proportions: 1960 Costa Rican Females 0 0.5 <1 1609 17 37 0.3677 0.0011 0.0013 0.3677 0.0011 0.0013 0 0 0 1 3 1-4 646 25 49 0.1476 0.0016 0.0017 0.0369 0.0004 0.0004 0.3677 0.0011 0.0013 5 7.5 5-9 67 17 45 0.0154 0.0011 0.0016 0.0031 0.0002 0.0003 0.5153 0.0027 0.0030 10 12.5 10-14 225 20 34 0.0514 0.0013 0.0012 0.0103 0.0003 0.0002 0.5307 0.0038 0.0046 15 17.5 15-19 90 8 49 0.0206 0.0005 0.0017 0.0041 0.0001 0.0003 0.5820 0.0052 0.0058 20 22.5 20-24 93 102 65 0.0212 0.0067 0.0023 0.0042 0.0013 0.0005 0.6026 0.0057 0.0076 25 27.5 25-29 0 61 112 0.0000 0.0040 0.0040 0.0000 0.0008 0.0008 0.6238 0.0124 0.0098 30 32.5 30-34 12 115 185 0.0026 0.0076 0.0065 0.0005 0.0015 0.0013 0.6238 0.0165 0.0138 35 37.5 35-39 0 261 174 0.0000 0.0172 0.0061 0.0000 0.0034 0.0012 0.6264 0.0241 0.0203 40 42.5 40-44 36 304 197 0.0082 0.0200 0.0069 0.0016 0.0040 0.0014 0.6264 0.0413 0.0265 45 47.5 45-49 0 699 466 0.0000 0.0460 0.0164 0.0000 0.0092 0.0033 0.6346 0.0613 0.0334 50 52.5 50-54 46 975 650 0.0106 0.0643 0.0229 0.0021 0.0129 0.0046 0.6346 0.1073 0.0498 55 57.5 55-59 30 1399 1340 0.0068 0.0922 0.0472 0.0014 0.0184 0.0094 0.6452 0.1716 0.0728 60 62.5 60-64 73 2008 2336 0.0167 0.1323 0.0824 0.0033 0.0265 0.0165 0.6520 0.2638 0.1200 65 67.5 65-69 141 2299 3143 0.0322 0.1514 0.1109 0.0064 0.0303 0.0222 0.6687 0.3961 0.2024 70 72.5 70-74 281 2587 3712 0.0643 0.1705 0.1309 0.0129 0.0341 0.0262 0.7009 0.5475 0.3133 75 77.5 75-79 357 1926 4352 0.0815 0.1269 0.1535 0.0163 0.0254 0.0307 0.7652 0.7180 0.4442 80 82.5 80-84 398 2286 5268 0.0909 0.1506 0.1858 0.0182 0.0301 0.0372 0.8467 0.8449 0.5977 85 92.5 85+ 273 68 6138 0.0624 0.0045 0.2165 0.0042 0.0003 0.0144 0.9376 0.9955 0.7835 TOTALS 4375 15178 28352 0.9999 1.0000 1.0000

3. Median age at death for specific causes.

• Diarrhea:
• Total number of deaths due to diarrhea = 4375
• 50% of total deaths due to diarrhea = 4375 * .50 = 2187.5
• Cumulative number of deaths due to diarrhea up to age 1 = 1609
• Number of diarrhea deaths in the age interval 1-4 = 646
• Since 1609 + 646 = 2255, which is > 2187.5, we know that the median age will fall in the 1-4 age interval (and close to the maximum age for that interval). Now we need to use interpolation to get the specific age...
• Length of age interval 1-4 = 4
• Therefore, median age at death for diarrhea deaths:
• 1 + [((2187.5 - 1609)/646) * 4] = 4.58

• Cancer:
• Total number of deaths due to cancer = 15,178
• 50% of total deaths due to cancer = 15,178 * .50 = 7589
• Cumulative number of deaths due to cancer up to age 65 = 6012 (from Exercise 12)
• Number of cancer deaths in the age interval 65-70 = 2298
• Since 6012 + 2298 = 8310, which is > 7589, we know that the median age will fall in the 65-70 age interval. Now we need to use interpolation to get the specific age...
• Length of age interval 65-69 = 5
• Therefore, median age at death for cancer deaths:
• 65 + [((7589 - 6012)/2298) * 5] = 68.43

• CVD:
• Total number of deaths due to CVD = 28,352
• 50% of total deaths due to CVD = 28,352 * .50 = 14176
• Cumulative number of deaths due to CVD up to age 75 = 12,594
• Number of CVD deaths in the age interval 75-79 = 4352
• Since 12,594 + 4352 = 16,946, which is > 14,176, we know that the median age will fall in the 75-80 age interval. Now we need to use interpolation to get the specific age...
• Length of age interval 75-89 = 5
• Therefore, median age at death for CVD deaths:
• 75 + [((14,176 - 12,594)/4352) * 5] = 76.82

4. Use these summary measures to describe the age pattern of each cause of death.

• Most diarrhea deaths occur before the age of 5. The median age is a mere 4.58 years old, and 37% of all diarrhea deaths occur in the first year.
• Cancer deaths are pretty rare until about age 40 when they start rising steeply. This is easy to see in the graph of cumulative proportion of deaths by cause where less than 10% of all cancer deaths have occurred through age 45.
• CVD deaths follow a pattern similar to cancer, except the notable increases start about 10 years later. As with cancer, this pattern is easy to see in the graph of cumulative proportion of deaths by cause. The difference between CVD and cancer also shows up clearly in median age at death: cancer is 68.43 and CVD is 76.82.