ACTBL
Produces population and cohort life tables.
Required Arguments
IMTH — Type of life table. (Input)
IMTH = 0 indicates a population (current) table. IMTH = 1 indicates a cohort table.
N — Number of age classes. (Input)
NPOP — Population size. (Input, if IMTH = 0; not used otherwise)
For IMTH = 0, the population size at the beginning of the first age interval. The value is somewhat arbitrary. NPOP = 10000 is reasonable. Not used if IMTH = 1.
AGE — Vector of length N + 1 containing the lowest age in each age interval, and in AGE(N + 1), the endpoint of the last age interval. (Input)
Negative AGE(1) indicates that the age intervals are all of length ∣AGE(1)∣ and that the initial age interval is from 0.0 to ∣AGE(1)∣. In this case, all other elements of AGE need not be specified. AGE(N + 1) need not be specified when IMTH = 1.
A — Vector of length N containing the fraction of those dying within each interval who die before the interval midpoint. (Input)
A common choice for all A(I) is 0.5. This choice may also be specified by setting A(1) to any negative value. In this case, the remaining values of A need not be specified.
IPOP — Vector of length N containing the cohort sizes during each interval. (Input)
If IMTH = 0, then IPOP(I) contains the size of the population at the midpoint of interval I. If IMTH = 1, then IPOP(I) contains the size of the cohort at the beginning of interval I. When IMTH = 0, the population sizes in IPOP may need to be adjusted to correspond to the number of deaths in IDTH . See the “Description” section of the KAPMR routine for more information.
IDTH — Vector of length N containing the number of deaths in each age interval. (Input, if IMTH = 0; not used otherwise)
If IMTH = 1, IDTH is not used and may be dimensioned of length 1.
TABLE — N by 12 matrix containing the life table. (Output)
The rows of TABLE correspond to the age intervals.
|
Col. |
Description |
|
1 |
Lowest age in the age interval. |
|
2 |
Fraction of those dying within the interval who die before the interval midpoint. |
|
3 |
Number surviving to the beginning of the interval. |
|
4 |
Number of deaths in the interval. |
|
5 |
Death rate in the interval. If IMTH = 1, this column is set to NaN (not a number). |
|
6 |
Death rate in the interval. If IMTH = 1, this column is set to NaN (not a number). |
|
7 |
Proportion dying in the interval. |
|
8 |
Standard error of the proportion dying in the interval. |
|
9 |
Standard error of the proportion of survivors at the beginning of the interval. |
|
10 |
Expected lifetime at the beginning of the interval. |
|
11 |
Standard error of the expected life at the beginning of the interval. |
|
12 |
Total number of time units lived by all of the population in the interval. |
Optional Arguments
IPRINT — Printing option. (Input)
If IPRINT = 1, the life table is printed. Otherwise, no printing is done.
Default: IPRINT = 0.
LDTABL — Leading dimension of TABLE exactly as specified in the dimension statement in the calling program. (Input)
Default: LDTABL = size (TABLE,1).
FORTRAN 90 Interface
Generic: CALL ACTBL (IMTH, N, NPOP, AGE, A, IPOP, IDTH, TABLE [, …])
Specific: The specific interface names are S_ACTBL and D_ACTBL.
FORTRAN 77 Interface
Single: CALL ACTBL (IMTH, N, NPOP, AGE, A, IPOP, IDTH, IPRINT TABLE, LDTABL)
Double: The double precision name is DACTBL.
Description
Routine ACTBL computes population (current) or cohort life tables based upon the observed population sizes at the middle (IMTH = 0) or the beginning (IMTH = 1) of some userspecified age intervals. The number of deaths in each of these intervals must also be observed.
The probability of dying prior to the middle of the interval, given that death occurs somewhere in the interval, may also be specified. Often, however, this probability is taken to be 0.5. For a discussion of the probability models underlying the life table here, see the references.
Let ti, for i = 0, 1, …, tn denote the time grid defining the n age intervals, and note that the length of the age intervals may vary. Following Gross and Clark (1975, page 24), let di denote the number of individuals dying in age interval i, where age interval i ends at time ti. If IMTH = 0, the death rate at the middle of the interval is given by ri = di/(Mihi), where Mi is the number of individuals alive at the middle of the interval, and hi = ti ‑ ti‑1, t0 = 0. The number of individuals alive at the beginning of the interval may be estimated by Pi = Mi + (1 ‑ ai)di where ai is the probability that an individual dying in the interval dies prior to the interval midpoint. When IMTH = 1, Pi is input directly while the death rate in the interval, ri, is not needed.
The probability that an individual dies during the age interval from ti−1 to ti is given by qi = di/Pi. It is assumed that all individuals alive at the beginning of the last interval die during the last interval. Thus, qn = 1.0. The asymptotic variance of qi can be estimated by
When IMTH = 0, the number of individuals alive in the middle of the time interval (input in IPOP(I)) must be adjusted to correspond to the number of deaths observed in the interval. Routine ACTBL assumes that the number of deaths observed in interval hi occur over a time period equal to hi. If di is measured over a period ui, where ui ≠ di, then IPOP(I) must be adjusted to correspond to di by multiplication by ui/hi, i.e., the value Mi input into ACTBL as IPOP(I) is computed as
Let Si denote the number of survivors at time ti from a hypothetical (IMTH = 0) or observed (IMTH = 1) population. Then, S0 = NPOP when IMTH = 0, and S0 = IPOP(1) for IMTH = 1, and Si is given by Si = Si−1 ‑ δi−1 where δi = Siqi is the number of individuals who die in the i‑th interval. The proportion of survivors in the interval is given by Vi = Si/S0 while the asymptotic variance of Vi can be estimated as follows.
The expected lifetime at the beginning of the interval is calculated as the total lifetime remaining for all survivors alive at the beginning of the interval divided by the number of survivors at the beginning of the interval. If ei denotes this average expected lifetime, then the variance of ei can be estimated as (see Chiang 1968)
where var(en) = 0.0.
Finally, the total number of time units lived by all survivors in the time interval can be estimated as:
Example
The following example is taken from Chiang (1968). The cohort life table has thirteen equally spaced intervals, so AGE(1) is set to ‑5.0. Similarly, the probabilities of death prior to the middle of the interval are all taken to be 0.5, so A(1) is set to ‑1.0. Since IPRINT = 1, the life table is printed by ACTBL.
USE ACTBL_INT
IMPLICIT NONE
INTEGER IMTH, IPRINT, LDTABL, N, NPOP
PARAMETER (IMTH=1, IPRINT=1, N=13, NPOP=10000, LDTABL=N)
!
INTEGER IDTH(13), IPOP(13)
REAL A(1), AGE(1), TABLE(13,12)
!
DATA AGE/-5.0/, A/-1.0/
DATA IPOP/270, 268, 264, 261, 254, 251, 248, 232, 166, 130, 76, &
34, 13/
!
CALL ACTBL (IMTH, N, NPOP, AGE, A, IPOP, IDTH, TABLE, &
IPRINT=IPRINT)
!
END
Output
Life Table
Age Class Age PDHALF Alive Deaths Death Rate
1 0 0.5 270 2 NaN
2 5 0.5 268 4 NaN
3 10 0.5 264 3 NaN
4 15 0.5 261 7 NaN
5 20 0.5 254 3 NaN
6 25 0.5 251 3 NaN
7 30 0.5 248 16 NaN
8 35 0.5 232 66 NaN
9 40 0.5 166 36 NaN
10 45 0.5 130 54 NaN
11 50 0.5 76 42 NaN
12 55 0.5 34 21 NaN
13 60 0.5 13 13 NaN
Age Class P(D) Std(P(D)) P(S) Std(P(S)) Lifetime
1 0.007 0.00522 1.000 0.00000 43.19
2 0.015 0.00741 0.993 0.00522 38.49
3 0.011 0.00652 0.978 0.00897 34.03
4 0.027 0.01000 0.967 0.01092 29.40
5 0.012 0.00678 0.941 0.01437 25.14
6 0.012 0.00686 0.930 0.01557 20.41
7 0.065 0.01560 0.919 0.01665 15.62
8 0.284 0.02962 0.859 0.02116 11.53
9 0.217 0.03199 0.615 0.02962 10.12
10 0.415 0.04322 0.481 0.03041 7.23
11 0.553 0.05704 0.281 0.02737 5.59
12 0.618 0.08334 0.126 0.02019 4.41
13 1.000 0.00000 0.048 0.01303 2.50
Age Class Std(Life) Time Units
1 0.6993 1345.0
2 0.6707 1330.0
3 0.6230 1312.5
4 0.5940 1287.5
5 0.5403 1262.5
6 0.5237 1247.5
7 0.5149 1200.0
8 0.4982 995.0
9 0.4602 740.0
10 0.4328 515.0
11 0.4361 275.0
12 0.4167 117.5
13 0.0000 32.5
