APPROXHiJATE RELATIONSHIPS Ar 10NG SOt~E
ESTIMATORS OF MORTALITY PROBABILITIES
i
Norman L. Jolmson
Department of Statistias
University of North CaroZina
ChapeZ Hitt~ North CaroZina 27514
Institute of Statistics Mimeo Series No. 1098
December, 1976
This research was supported by the Arnrt Research Office under Grant
No. DAH G29-74-C-0030.
•
APPROXIN'"ATE RELATIONSHIPS .tVDNG SOME
ESTIMATORS OF
~':ORTALI1Y
PPOBABILITIES
by
N. L. Johnson*
University of North Carolina at Chapel Hill
1.
Introduction
Kuzma [1967] and Drolette [1975] have compared by means of quite extensive
numerical calculations, several estimators of
q~
the probability of death in
an interval (given alive at the beginning of the interval).
It is the purpose
of this note to give some approximate algebraic relations among some of these
estimators.
Thse confirm, in general terms, the conclusions reached from the
numerical investigations.
They also provide quite good approximations (to
four decimal places when q < 0.15, apparently) to two rather complicated (in
estimators~
appearance)
in terms of estimators of
s~pler
form.
This also
gives an L'11proved appreciation of the nature of the more complicated estimators.
.
'
2.
Notation
We use the following notation (as in Elandt-Johnson [1976]).
N
= number
d
= number of deaths in the interval
c
= number lldue to be withdrawnl l in the interval
d'
= number
of deaths among these c individuals
w
= number
actually \vithdrat'l11L
alive at beginning of interval
It will be assumed that there are no l;lost to follow
during the interval so that w+d t
·e
*This
= c;
up~ 11
or new entrants
d Y :s d :s N - w.
research was supported by the Army Research
DAB G29-74-C-0030.
Office~
under Contract
2
•
The estimators to be discussed are:
q(2)
where Nt
= {Nt _
1
= N - ttl
+
(see E1veback [1958])
1
¥,
&(3)
= dCN _ }"r)-1
&(4)
=1
(the lI actuarial'! estimator),
(2)
- }(2N-C)-2[{d t2 +4(2N-C) (2N-w-2d)}1/2_ dt ] 2
(Chiang [1961]) ,
q(S)
(1)
= dCN _ k) -.1
2
(see Elveback [1958]).
(3)
(4)
The estimators q(3) and q(5) are much simpler in form than q(2) and q(4) .
3.
The Approximations
Using the method outlined in the Appendix, the following approximations
are obtained.
q(2) ~ }[(1 + ~(3))-1 + {1 + (~_
R)q(3)}-1]q(3)
(5)
where
(6)
(7)
Results of same numerical calculations ilvlstrating the accuracy of these
approximations, using data from Table 2 of Chiang [1961] are presented in
Table 1.
For larger q, (5) may be improved by addition of ~(3){q(3)(1 + ~(3))-1}3,
but values of
.e
q (>.20,
say) for which this is of any importance are so large
that validity of assumptions made in derivation of formulae become of overriding importance (and usually uncertain validity).
3
,
•
Table 1.
,
Approximations to various estimators of q
A(2)
Interval
0-1
1-2
2-3
3-4
4-5
5-6
q
.2434
.1822
.1033
.0860
.0643
.0584
.0438
.0433
.0337
.0466
.0439
.0513
6-7
7-8
8-9
9-10
10-11
11-12
A(4)
Approx.
(5)
.2429
q
.1821
.1033
.0860
.0643
.0584
.0438
.0433
.0337
.0466
.0439
.0513
.1814-
.2425
.1030
.0858
.0641
.0582
.0438
.0432
.0337
.0465
.0438
.0511
Approx.
(7)
.2426
.1815
.1030
.0858
.0641
.0582
.0438
.0432
.0337
.04655*
.0438
.0511
A(3)
A(5)
q
q
.2417
.1810
.1028
.0856
.0641
.0582
.0437
.0431
.0337
.0464.0437
.0508
.2436
.1819
.10.32
.0859
.0642
.0583
.0438
.0433
.0337
.0467
.0440
.0513
* - exactly
Except for the approximation to
q(2)
in interval 0-1. there is nevermore
than 0.0001 difference between approximation and calculated value (to 4
decimal places).
In most cases the values agree to 4 decimal places.
It is interesting to note that for these data
a good approximator for
4.
q(5)
happens to be quite
q(2) .
Discussion
From (7) it is clear that the approximate value for q(4) is always less
..
.A
than q(S).
This is consistent with numerical results obtained by Drolette
[1975] • She also finds that
q(2)
is greater than
q(3).
To see that this is
in accordance with our fomulae, 'tve note that the right hand side of (5) is
,
'\
4
greater than q(3) if and only if R > (1
+
~~(3))(1
definition of R in (6), it follows that R ~ (1
+
+
q(3))-1.
From the
~(3))-1 and this is
less than (1 + ~(3)), so it is possible for the approximate value of q(2)
to be less than q(3), though the least possible value 0vhen w = 0) is
&(3)[1 _ (q(3))3{8
+
4q(3)
2(qC 3 ))Z
+
+
(q(3))3}-1]
> q(3){1 _ i(q(3))3} .
While the labor of calculating q(2) and q(4) is not great, the approximations put forward here provide some insight into the nature of these estimators, as well as quite simple fonnulae for approximate calculation.
In view of the small differences usually found among the numerical values
given by the different fonnulae, and the usually dubious nature of assumptions
involved in setting up the likelihood functions underlying q(2) and q(4) ,
there are good grounds for using simpler formulae (like q(3) or q(S)) except
in very special cases.
APPENDIX
For It
I<
1, we have the expansion
(1-2t)1/2
}t2 _ ~3
=I
_
=1
- ~t - ~(t
.
':' 1 -
t _
1
zt - 1zt (1
+
t2
- t)
_ ~4 _ ~c5 _
+
-1
t3
+
t4
+
t S) -
.
So
T.1is approximation is used in
q(2)
to yield (5).
= (N'/N)[l - {I - 2(N/Ny 2)d}1/2
it4 _ ~tS -
5
,
\
Similarly, we write (ranembering d' = c-w)
~(4)
=1 _
1 2 [4(2N-c)2(l _ 2d-d')
4 (2N-c)"
2N-c
2d-d'
- 4d'(2N-c){l - 2N-c
+
+
d,2
4(2N-c) 2
2d,2
}l/2
J
and expand the square root, to yield (7).
REFERF,NCES
Chiang, C.L. [1961J. A stocllastic study of the life table and its applications:
III The follow-up studies with the considerations of competing risk,
. Biometrios~ ~, 57-78.
Drolette, M.E. [1975J.
135-144.
The effect of
incom~lete follow-u~, Biometric8~ ~
Elandt-Johnson, R.C. [1976J. Various estimators of conditional probabilities
of death in follow-up studies: Summary of results, J. Chron. Dis.~ 29,
to appear.
E1veback, 1. [1958].
Actuarial estimation of survivorship in chronic disease,
J. Amer. Statist. Assoc., 53, 420-440.
Kuzma, J. W. [1967J.
51-64.
A comparison of two life table models.
Biometrics, 23,