1
Department of StatisUcs,t University of North Carolina.. Chapel. lJi,ll...
The Nssaroh of this authOl' and the pl'epaz'a:tion of the manu-
N. C. 2751.4.
. saript tJas supported by NSF Gzaant GP-2362l).
2
N:C•
Depazt1ment of Soc{,oZcgy.. l./nivel'Sity of. N01'th Crnro"Lina/j Chapel. Bin..
•
.
.ATlt-E SERIES APPROAOi
.
.
TO lHE LIFE TABLE
by
~dward J. Wegman 1 and
Cri 5 R. Kukuk2
t1nive:rsity of Nonh CaroUna at Chapel Hili
Depa.t'tment of Statistics
Institute of Statistics Mimeo Series No •. 730
;
December. 1970
-.
.
ATIfIE
SERIES APPROAOi 10 11iE LIFE TABLE
by
Edward J. Wegman 1
1.
and
Cr1s R. Kukuk 2
INTRODUCfIOO
"Statistics ll was historically the name given to the science which collects facts about the state.
One of the most useful and best-known collec-
tion of facts about the state is the life table.
In this paper. we will
present a time series approach to estimating the life table.
There are two general forms of the life table:
e
and the current life table.
the cohort life table
The former follows the fortmes of a cohort or
group of individuals born at the same time throughout their lifetimes mti1
the death of the last-surviving
member.
,
The latter version of the life table
describes the mortality experiences of an entire population" at a fixed point
in time.
In either case.
l
x.
lx
is the number of people surviving at age
x.
The
are usually related by the equations
(1.1)
or
(1.2)
lx •
In these equations,
1
e·-
p x-l,x
Pij
1x-l
..
may be interpreted as the conditional probability
Depa:zrtment of StatistiCB~ University of North Ca1'Ot.ina~ Chape1. HiZZ~ "
N. C. 8'1514. The research of this authol' and the prepareation of the manuscript fJaB supported by NSF Grant GP-23S20.
"2
Dspanment of SocioloB'J~ University of·North Carolina~ ChapeZ HiU,~ N.C.
2
of an .individual surviving to age
•
the function
.ex· is
given that he was alive at age 1.
j
normali%ed so that
population distribution.
Lxtx •
1,
If
it is referred to as the
An extensive discussion of the life table and the
related·biometric functions may be found in Keyfitz, (1968).
As early as 1907, Lotka had enunciated the idea of the stationary popu-
lation distribution and suggested the form
(1.3)
lex) •
p(x)Ae
-(}.-ll) x
..
Here lex)
is meant to be a continuous (density) version of l , p(x) is
x
the probability of an individual surviving to age x, and' A and lJ are
respectively the crude birth and death rates.
been exposed to a constant death rate,
If we assume an individual has
throughout his
"}.I,
x years, then
p(x)
so that (1.3) becomes
lex) •
he
-AX
•
In general, this suggests that lx decreases like a negative exponential.
This is approximately true although such a parametric form is generally not
satisfactory to explain all details appearing in real data.
A description
of Latka's work can be found in Lotka, (1956).
The fact that t
by Grenander, (1956).
of
x
is a monotone decreasing function is used to advantage
He gives a non-parametric maximum likelihood estimate
l.
Grenander's type of estimate may be used to estimate
x
any monotone
probability density and has generated more work along these lines.
Chiang, (1960) demonstrates that
chastic Process.
{lx , x. O,1,2, ... } is a Markov Sto-
We will take advantage of this fact in order to use time
series methods to study {lx,
x· 0,1,2, ••• }.
If
{tx '
x· O,1,2, ••• } is
3
,.,
A
a stochastic process, it is reasonable to expect that (1.2) still holds except that the relation is perturbed by an error term,
.tx •
(1.4)
where
E:
p
'
x-l,x
.tx-l + E x'
x
'
Hence we assume
x • 1,2,3, •••
is a sequence of uncorrelated zero meanrando1l1 variables and where
EX
x is uncorrelated with .t.
x t
for
t
P
x- l
,x'
x' < x.
Let us also, benceforth, write
p
x
A process satisfying equation (1.4) is known as a first-order
autoregressive process.
The relation (1.4) is sometimes referred to as a
simple Markov relation.
We shall devote the next section to outlining some
facts about first order autoregressive processes.
In the third section, we
shall carry through some computations using the 1961 Indian Census data to
illustrate our methods.
2.
AUTOREGRESSIVE AND RELATED PROCESSES
In this section, let us write (1.4) as
(2.1)
t •
1,2, •••
in order to keep the notation more consistent with conventional time series
notation.
0·£
{X :
t
We will give, in this section, the mean and covariance structure
t · l,2, ••• }
We shall denote by
and propose estimates of
EXt
p(t).
the expected value of
easy to see
Using this formula recursively, we obtain
(2.2)
EXt
•
• EXo
t • 1,2, •••
X •
t
From (2.1), it is
4
•
If Xo is given as an initial condition, the conditional expectation of Xt
given Xo is
1,2, ...
t •
Again using (2 ..1), we may study the covariance structure.
'both sides of (2.1) by
X_ '
t s
Multiplying
we have
so that by taking the expectation, we have
Subtracting the product of the appropriate means,
e
Applying (2.3) recursively,
eov(xt,Xt -s )
(2.4)
•
pet) p(t-l) ••• p(t-s+l)
V~(Xt
Thus, the system of covariances, depends on the variances.
-s ).
Applying (2.1)
recux:sively leads to
xt •
Subtracting (2.2) from this yields
(Xt-EX )
t
.
•
rfr
P(j~
b-1 J
• (Xo-EXa)
+ til p(t) ... p(t-j+l)tt_j + tt •
j-l
Squaring, taking expected values and recalling
t
e·
f1
t'
and with
X "
t
t' < t,
we have
E:
t
is uncorre1ated with
E:
t
"
.
5
e
Here,
02
VtVt(E
j
)
j
• V~(&j).
In the special case
i8 a constant) Bay
02 t
•
p2t
.
(2.5)
VM(X ) · 0
O
is a constant, say
p,
and
becomes
V~(XO) + 02
Using the formula for a finite geometric
If Xo is known, then
pet)
til p2j •
j-O
SUlD
and the conditional variance is
Finally, the conditional covariance is
In certain cases, particularly in connection with census data, the errors
appear not to have a constant variance, but rather to be proportional, roughly
speaking, to population size.
That is to say, there appears to be a constant
percentage error.
Thus, it is reasonable to let
uncorrelated with
nttt
zero mean and variance
t '" t'
02 •
and with
Xt"
£t· xt_l·n t
t' < t
where
and where
nt
nt
is
has
In this case, (2.1) becomes
(2.6)
For an eqUation of the form (2.6), (2.2) and (2.4) are still true.
ever, (2.5) is no longer true.
xt •
Solving (2.6) recursively leads to
How-
6
Squaring and taking expected values,
EX,2
•
!X
2
o
t
It'rom the second moments and equation (2.2), the variance of
X
t
may be
calculated,
•
If
pet)
If
X
o
is a constant,
is known,
• X2[(p 2+a 2 )t_p 2t] ..
V~ (XtIx)
OO
e
In both (2.1) and (2.6), it is of considerable interest to estimate
If we parameterize
pet)
as an
pet).
t-th degree polynomial, we may use standard
In (2.1), let us
least squares procedures to determine the coefficients.
assume . £t
has a constant variance and let us also assume we have. n
aerVations
x ' ••• ,x _ •
O
n l
ob-
We then wish to minimize
Taking the partial derivative with respect to
and equating to zero, we
~
have
(2.7)
k • 0, _•• ,t.
Solving (2.7) simultaneously gives an estimate of
&j'
hence, an estimate of
the estimate of
3
pet).
If
pet)
j . 0 .... ,t
and,
is' assumed to be a.constant, then
A
tn-l
tn-l
0 ·p is p. [lot-l XtXt_l]-[l.t_l
X
2
_ ]
t 1
-1
•
7
Let us rewrite (2.6) as
Rere we wish tQ minimize
Taking partial derivatives with respect to
~
and equating to zero. we
have
(2.8)
Again. solving (2.8) simultaneously gives the estimate of
is constant. the estimate of
special case' pet)
tn
-1
Lt_1XtXt-l.
of
p.
An interesting feature of
~
p
8
0
• P
p(t).
is
In the
3. (n_1)-1.
is that it is an unbiased estimate
vtVt(~). (n_1,-1.0'2 where 0'2 is the variance of "to
In addition,
We close this section by noting that we describe the fitting of a polynomial
p(t)·
I;..
j
oa t •
j
dent but not orthonormal.
The set of functions
j
{e }
are lineax:ly indepen-
1'1: is frequently advisable to investigate the
fitting of an orthonormal serles, rather than one which is merely linearly
independent.
3.
ftw..VSIS OF lHE INDI,6H POPULATI<i-4 CoOOBY
PGE
The data to which we shall refer in this section is taken from papers
published in India on the 1961 Indian census.
See Census of India. (1963).
We shall present our technique for arriving at the population distribution.
l..
x
but first we
. would like to mention the procedure followed by the Indian
Census officials.
8
e
The population count of India by the stated age of the respondant is
found in Table 1.
Perhaps the most noticeable facet of this census data 1s
the marked preference shown for the ages divisible by S.
This age heaping
effect is well-known and represents a considerable distortion of the true
situation.
To eliminate this peaking, the Indian census officials smoothed
the.ir data by an eleven term. moving average.
They next formed 5 year totals
(with some special adjustments for the very young and the very old). smoothed
the result with a weighted 3 term moving average and finally interpolated by
the well-known fifth difference osculatory interpolation formulae due to
Kozakiewicz.
We have several objections to these procedures.
First of all, from the
point of view of time series an eleven term moving average is a low pass
fUter, filtering out variations with period smaller than 11 years.
Since
we are primarily interested in filtering the peaks which occur at 5 year intervals, a five term moving average is sufficient •. (Clearly, one wants to
use the filter which least changes the data, but still removes the unwanted
periodicities.)
A second objection is that the ordinary arithmetic moving
average and the procedure of forming 5 years totals are both procedures which
tend to reshape the data so that the altered data will have a tendency to
cluster along a straight line.
In Section 1, we observed, in the case of a
stationary population, the population distribution tends to decrease as a
negative exponential, hence, very much in a non-linear fashion.
(In the case
of a nonstationary population where the total population size increases, the
population distribution will drop off even more quickly than in the stationary
case.)
In view of this situation, the data analyst should avoid as much
88
•
possible procedures which linearize the data.
We believe that the two
It
We have ezpe:l'imented tJJi.th 11- and 6- term moving averago[;" using these to
fiZter a curve knavn to decrease ezponentiatly. The fittered data.. in both
oases.. fsl:t abovs the origino.7. cu:rve. The percentage difference· was respeot1.'l)ety
about 8% and about 1% foXt the 11 tem and the S tenn fitters.
9
TABLE
e.
Population count (1961) of India
by
stated age of respondant
0
13,551,710
26
5,472,520
1
11,673,441
13,427,444
14,129,333
13,242,893
15,227',496
13,966,354
12,268,094
13,655,781
9,47f,091
14,366,679
7,115,477
12,729,548
6,922,327
8,094,839
8,403,095
8,184,236
4,641,026
10,225,422
4,370,035
13,596,192
4,536,574
9,314,331
4,427,801
27
3,986,209
7,616,137
2,376,492
18.049.763
1,992,606
6.205,031
2,116,072
2,432,703
14,523,328
3,.626,705
1,920,928
3,822,719
1,532,454
15,496,709
1,350,982
3,334,978
1,287,810
1.352,650
11,580,622
1,643,541
1,207,143'
2,596,1l0
1,002,262
12,352,529
2
3
4
5
6
7
e
1
8
9
10
11
12
13
14
15
16
11
18
19
20
21
22
23
24
25
5.395.409
17,076,873
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70+
960,486
1,970,895
800,709
996.797 .
6,217.166
1.181,324
664,442
1,198.366
. 551,415
8,723,146
626,729
1,022,5 1.0
392,757
453,118
3,493,498.
358,707
306,016
460,037
223.171
8,601..562
10
procedures mentioned above have distorted the shape of their curve sufficiently
so that their estimate of.t is virtually linear from age 20 to age 70.
x
for example Census of India, (1963, page 4).
See
If the moving average is modified by forming a geometric moving average,
i.e.
y
t
•
instead of the arithmetic moving average
two advantages result.
First, if
tial curve. then
t · 2.3•••• } will be the same exponential curve. and
{Yt:
{X t = t · 0,1,2 •••• } truely is an exponen-
second. it is well-known that geometric averages are less affected by extreme
values than arithmetic averages. hence data filtered geometrically is smoother
than data filtered through a corresponding arithmetic filter.
We have used
both the geometric and arithmetic filter and the results may be found in
Figures 1 and 2 respectively.,
:Both have a similar shape. but the curve of
the arithmetically filtered data tends to lie above the curve of the 8eometrically filtered data.
Before we'lllove on to the fitting of an autoregressive process. we should
point out that the procedures followed by the Indian census officials will
tend to keep the total population count constant throughout the whole graduation process whereas the geometric filter most assuredly will not.
However,
we believe it is the shape of the curve that is of interest and not the total
population count.
We should now like to estim.ate t
based on the filtered data. (We
x
shall illustrate'the results for the geometrically filtered data only, but
we shall give. in Table 2. results for the arithmetically filtered data also.)
11
Recall that our autoregreesivemodel is
II
.(.x
•
x-t-x- l +'""'x·
pfl
To give us an idea of the form of
See Figure 3.
p
• ••• •
p.
x
we can plot
It is evident from Figure 3, that
stant and that the errors
nomials to
x-12
x
£
are fairly small.
x
confirms that
p
x
(12
and of the fom (2. 6) •
x
verses
is probably
t x- 1.
a con-
Fitting higher degree poly- .
is effectively a constant.~~
We fitted models both of the form (2.1) with
iance
p
tx
£
x
having constant var-
An analysis of the residuals indicates
that the model of (2.6) is more appropriate.
residuals for the fitting of this model.
Figure 4 is the plot of the
If one assumes that the errors are
. normally distributedt , then one may compute the marginal likelihood product
as Ii function of
P
p.
The ratio of this likelihood product as a function of
to the likelihood product for the least squares (maximum likelihood) es-
timate for model (2.6) is given in Figure 5.
this ratio lends a good deal
·of
Pi~.
ot
The sharpness of tne plot of
confidence to this model and to the estimate
The results of both models using data filtered both ways is sum-
marized in Table 2.
Recall from (2.2). we have
~,.
That p should be a constant in a hum:zn popuZ-ation is indeed !'emarakable.
In deveZcpe~ countrieB~ one e~ects the probability Of Burrviving to the ne:z:t
yeal" to be constantly decreasing. Perhaps" this effect is dUB to relatively
harsh conditio'18 in India" fJhere onZy the very hardy can survive to an otd
age. Being 1U1:t'dy.. the pl'ObobiZity of dying in the next year i6 not signifioantZy diffsttent from that of the generaZly less haJ.?dy younger generations.
:;=- We oonstructed a histogram of the residuals and it appeal'S as though the
errors are probably not no:rm:z'tly distFibuted/l rather .tending more tcMazads a
'tighte:r taited distribution. NonetheZess,.. the ZikeZihood Ntio wiZt. be at
Zsas t appro:::lmate 1,y OOl'1'6ot. In ·the alS e th at fJe fi tted (:1. 6) to the data fi'Ltered by the arithmetic fitter Jl the nottma't assumption is much more. betievabZe.
12
TABLE
2
Mode1 (2. 1) .
e
Geometric
Filter
Arithmetic
Filter
x
Model (2.6)
has variance (12
Estimated P •
x
Estimated 02- •
.9626476654
19420364.46
Estimated P • .9571073391
x
Es timated variance of Ox •
"
.7596421223
Ratio of the Sums of Squares .•
78.18
.
I
Ratio of the Swns of Squares •
1.35.44
Estimated P •
x
Estimated (12- •
.9702683796
21267288.58
Ratio of the Sums of Squares •
156.32
Estimated P • .9673863102
x
Estimated variance of Ox •
.7538528904
Ratio of the Sums of Squares •
81~69
13
•
'Thus.
Etx«px.
If we normalize so that
lation distribution.
"Et
Lx x • 1, we have the expected popu-
Using our estimate
~ from
(2.6) and the geometrically
filtered data, we may plot the estimated expected population distribution.
This is done in Figure 6 along with the original data similarly normalized.
A direct statistical comparison of our methods with commonly used procedures is difficult to make since many commonly used procedures tend to arise
from considerations other than statistical ones.
We believe, however, that
the statistical evidence gives a good deal of confidence in our procedures,
particularly because of the relatively close agreement found in Table 2 in
spite of the variations in the procedures.
This statistical evidence together
with the relative simplicity of our procedure and the previously mentioned objections to other procedures, certainly indicates that the time series approach merits consideration.
REFERENCES
Census of India. 1963.
1961 Census-Age Tables.
Paper No.2 of 1963.
Chiang)' C. L. 1960a. !fA stochastic study of the life table and its applications: I. Probability distributions of the biometric functions."
Bionetrf,08~ 16" 618-635.
___. 1960b. "A stochastic study of the life table and its applications: II.
Sample variance of the observed expectation of life and other biometric
functions." Human Biotogy" 32, 221-238.
.
_--: 1961. itA stochastic study of the life table and its applications: III.
The follow-up study with the consideration of competing risks."
Biomstri.cs~ 1?~ 57-78.
liOn the theory of mortality measurement, Part II."
Skan.. Aktumtieti.dskr." 39.. 125-153.
Grenander, Ulf. 1956.
Keyfitz, N. 1968. Introduction to the Mathematics of Poputation.
Addison-Wesley. Reading, Mass.
Lotka, A. J. 1956 .E"temsnta of Mathematioal, Bio1,ogy.
Dover. New York.
14
•
APPENDIX
Legends for Figures
Figure 1.
Scale on
Figure 2.
Scale
Data i.n Table 1 filtered by a 5 term geometric filter.
vertical axis is xl05 •
Data in Table 1 filtered by a 5 term arithmetic ·filter.
on vertical axis is xI0 5 •
Figure 3.
Scatter diagram
term geometric filter.
Figure 4.
tit
using data filtered by a five
Scale on both axes is xl0 5 •
VB.
lx-l)
Residuals usi.ng model of (2.6) fitted to data filtered by geometric
filter.
Figure 5.
<!x
The arithmetic filter yields siDiilar results.
Ratio of marginal likelihood products using model of (2.6) fitted
to data filtered by geometric f~lter. The arithmetic filter again
yields similar results.
Figure 6.
Original data and expected curve from the model of (2.6) using
data filtered
geome~rically.
population is 1.
Both are normalized so that the total
Scale on the vertical axis is xlO-2 •
FIGtR! 1
•0#
'0
•
~
•
-a
•
'0'
U>
•to
o-f-----..,...----,..----....,.----..,-----"f-----..
12.0
36.0
i8.D
60.0
72.0
• O.
2~.O
FlCUE
-·
2
. .t
(;)
a
(i1
•
.,
WVIVV~
·e""n
-t
e
-..J
·
0
'0
0
.,.J
·
( fI
(;
· o.
I
L2.0
S6.0
21f.O
..
r
1
if}.O
.,
72.0
FIGURE
c
3
..
-
..
:r
- .-
_~..
•
..
-
•
~.
•
•
..
:3'
10
e
•
•
•• •
0)
.-
m
••
..
.
&I)
•
- .'
::r
N
•
•
••
•
•
•
•
.
Q
•
.
'1R
•
..-
..
I/)
..• -..
.
•
•
•
• •
•
•
"
- . '.-.-.. .
r-
.
e
••.. .-
a.
23.3
ZiS:&
.. 65.5
53.2
117.
1110.
•
FIGURE
4
<0001
0-
.....
FIGURE 5
1.0 I
000
1
.0
1
o
0
0
o
1
•
I
o
•
0.91
0
o
I
I
,
o
0,
I
0
o
0.81
1
1
0
•
J
0
•
•
•
0.'/
I
o
•
I
1
I
•
•
o
0.61
I
I
I
•
o
o
1
0.5J
1
'0
o
•
I
I
I
•
o.ltl
o
0'
o
•
1
1
o
I
I
o
o
•o
•
0.31
I
I
I
I
o
•
o
0.21
•o
I
I
o
•
o
o
1
1
o
o
0.11
00
,
o
00
J
o
00
I
I
0.01 •• 0
0.90
o
o
00
00
.000
000
••• 0
I
0.92
O~9lt"
I
I
o
0.96
0.98
1.00
FIGmE
•
6
....en
o
.-.
....J
n>
--
.
C7'
.
'0
(J\
.~
----'f"------t.o,o-r-----t'-----.,.....
o.
12 •S
23 . 0
3'l. S
~6 • 0
13
o
-t"
51 .S
.-;;~
69. 0