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 _nd_x values in the table.

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

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

_nd_xd = _nd_x * _nC_xd   (total deaths * cause-specific death ratio)

 Exercise 8

This exercise will use data on 1960 Costa Rican females that you have downloaded and worked with in previous exercises.
You will need to use your spreadsheet software.

1. From the life table for 1960 Costa Rican females constructed in Exercise 7 obtain _nq_x, l_x, and _nd_xcolumns. (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 (_nd_x) 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 _nq_x, may be interpeted as the probability that a person of age x will die in that age interval. This quantity is computed as:

nqx = ndx lx

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

Because the probability _nQ_xd 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:

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 (_nd_xd) by the number alive at the beginning of the age interval (l_x).

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 l_x 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:

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

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 l_x 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 table å h = x  ndh

Similarly, let's introduce the following notation:

lxd = End of table å h = x  ndhd

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

From Table 4.3.1, l_65Diarrhea = 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.

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

Use your spreadsheet software for this exercise and continue to build on your previous work.

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.

Once you've completed your calculations, check the answer key below.

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 R_d? We will denote this probability as P_d. This probability is calculated as:

Exercise 11

Use your spreadsheet software for this exercise.

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.

After you have finished, compare your answers with the answer key below.

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.

Exercise 12

Use your spreadsheet software to do this exercise.

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.

You can compare your work to the answer key below.

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 _nd_xd. Recall the formulas used to calculate _nd_xd  as:

_nd_xd = _nd_x * _nC_xd   (total deaths * cause-specific death ratio)

    or

_nd_xd = l_x * _nQ_xd    (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:

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:

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

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.

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.

Exercise 13

Use your spreadsheet software for this exercise.

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.

Once you've finished your work, check the answer key below.

Answers to Exercises

Exercise 8

1. From the life table for 1960 Costa Rican females constructed in Exercise 7 obtain _nq_x, l_x, and _nd_xcolumns. (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 (_nd_x) 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.

Exercise 9

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

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.

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

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.

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.)

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.