Soc 215
Demographic Methods
Problem Set 5: Life Tables (From Herb Smith, U of Penn)
1. You are presented with retrospective survey data on the age at first birth for twenty-five women who are age 50 at the time of the interview: Their ages (last birthday) at first birth are 29, 22, 43, 31, 26, 20, 28, 25, 23, 30, 26, 37, 21, 25, 28, 32, 23 (twins), 27, 34, 25, 24, 21, 17, no birth, no birth.
A) Construct a life table where the state of interest is childlessness. That is, being childless is the state akin to being alive in the classic life table, and having a child causes a woman to exit this state (just as mortality is the source of exits in the classic life table). Do not worry about actual mortality: This is a life table pertaining (by definition) to a cohort of survivors to age 50. There is only one source of decrement (exit)–having a child.
Start the life table at age 15 and terminate it at exact age 50. The columns of your life table should be lx, ndx, nqx, nLx, Tx, and ex. Assume that births take place on average at the mid-year of an age interval, i.e., that a woman who gives birth at age 35 is on average 35.5 years old.
Assume further that no births can take place after exact age 50; this has certain implications for “closing out” the life table. In the classic life table, all persons surviving to the beginning of the last (open-ended) age interval are assumed to die eventually. Here they are assumed to “live” (i.e., remain childless) the rest of their lives. To work around this, set T50=0. This implies no “life” (childlessness) after age 49, which would appear to contradict the first three sentences of this paragraph. Instead, consider all calculations of expectation of “life” (years spent childless) to pertain only to the interval 15-49 years of age. I.e., interpret e15 as expected number of years childless through age 49 for a woman at exact age 15.
B) On the basis of this life table, answer the following questions.
a) What is the probability that a childless woman at exact age 30 would still be childless at age 40?
b) On average, what is the expected number of further years of childlessness (through age 49) for a woman who is childless at age 25?
c) Who can expect more years of childlessness prior to age 50–a woman who is childless at age 20, or a woman who is childless at age 35?
d) What fraction of years between (exact) ages 15 and 50 did this cohort spend childless?
2. The following table provides the mid-year population by age for females in the United States in 1985, total deaths by age for U.S. females in 1985, and deaths due to neoplasms (tumors, i.e., cancer) by age for U.S. females in 1985. It also provides a good estimate of average years lived among those women dying in a given age interval.
|
Table 1. Data for U.S. Females, 1985 |
||||
|
Age at last birthday |
Estimated mid-year population (thousands) |
Total number of deaths |
Deaths due to neoplasms |
Average years lived for those dying in the interval |
|
0 |
1831 |
17079 |
97 |
0.086 |
|
1-4 |
6968 |
3099 |
411 |
1.500 |
|
5-9 |
8214 |
1739 |
284 |
2.500 |
|
10-14 |
8339 |
1711 |
268 |
2.757 |
|
15-19 |
9106 |
4239 |
353 |
2.644 |
|
20-24 |
10483 |
5538 |
546 |
2.552 |
|
25-29 |
10869 |
6519 |
965 |
2.588 |
|
30-34 |
10172 |
7985 |
1879 |
2.632 |
|
35-39 |
8967 |
9882 |
3139 |
2.678 |
|
40-44 |
7167 |
12448 |
4849 |
2.706 |
|
45-49 |
5968 |
17080 |
7502 |
2.702 |
|
50-54 |
5661 |
26251 |
11767 |
2.683 |
|
55-59 |
5959 |
42986 |
18756 |
2.671 |
|
60-64 |
5877 |
65825 |
26584 |
2.650 |
|
65-69 |
5176 |
86517 |
29922 |
2.642 |
|
70-74 |
4354 |
113189 |
32387 |
2.631 |
|
75-79 |
3359 |
137554 |
29676 |
2.614 |
|
80-84 |
2177 |
151535 |
23896 |
2.596 |
|
85+ |
1934 |
277506 |
25149 |
6.969 |
A) Construct a female multiple decrement life table showing lx, ndx, nqx, lxi, ndxi, and nqxi, where the i-superscript denotes columns specific to neoplasms as a cause of death. For a radix, use l0=100000. You can assume that nmx=nMx; i.e., you can take the death rates for the life table from the observed period-specific death rates.
B) On the basis of this life table,
a) What fraction of female newborns will die from neoplasms under the U.S. age-cause-specific death rates of 1985?
b) What fraction of those who survive to age 50 will die from neoplasms?
C) Suppose that the death rate due to neoplasms was reduced by half among women ages 50 to 54; i.e., that there were now a new 5m50i equal to half of the old 5m50i (equivalently, that 5D50i, the number of observed deaths in 1985 due to neoplasms, were reduced by half).
a) What would happen to 5m50-i, the death rate in this age interval due to causes other than neoplasms?
b) What would happen to 5m50, the death rate in this interval due to all causes?
c) What would happen to 5q50i, the probability of dying due to a neoplasm during this age interval?
d) What would happen to 5q50-i, the probability of dying due to something other than a neoplasm during this age interval?
e) What would happen to 5q50, the probability of dying due to any cause during this age interval?
D) Calculate the *lx-i column in the associated single decrement life table. This life table presents an estimate of what the survivorship column would look like if neoplasms were eliminated as a cause of death. It is also called the “cause-deleted” life table. Set *l0-i=100000.
Use the assumption that within each n-year wide age interval, the force of mortality function causes of death other than neoplasms--:(x)-i--is proportional to the force of mortality function for all causes combined (:(x)). This assumption gives rise to the following relation:
where R–the power to which the overall probability of surviving the age interval is raised–is equal to the proportion of deaths due to causes other than neoplasms:
.
E) How would elimination of deaths due to neoplasms affect the probability that a newborn would live to age 85 (in a cohort experiencing the mortality of your life table)?