Produces population and cohort life tables.
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 document 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. |
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).
Generic: CALL ACTBL (IMTH, N, NPOP, AGE, A, IPOP, IDTH, TABLE [,…])
Specific: The specific interface names are S_ACTBL and D_ACTBL.
Single: CALL ACTBL (IMTH, N, NPOP, AGE, A, IPOP, IDTH, IPRINT TABLE, LDTABL)
Double: The double precision name is DACTBL.
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 − δ
i−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:
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
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
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