The Method of Expected Number of Deaths, 1786-1886-1986, Correspondent Paper
Author(s): Niels Keiding
Source: International Statistical Review / Revue Internationale de Statistique, Vol. 55, No. 1
(Apr., 1987), pp. 1-20
Published by: International Statistical Institute (ISI)
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StatisticalReview(1987),55, 1, pp. 1-20. Printedin GreatBritain
International
International
StatisticalInstitute
?
Correspondentpaper
The
Method
Deaths,
of
Expected
Number
of
1786-1886-1986
Niels Keiding
StatisticalResearchUnit, Universityof Copenhagen,Blegdamsvej3, DK-2200
CopenhagenN, Denmark
Summary
The method of expected number of deaths is an integralpart of standardizationof vital rates,
which is one of the oldest statisticaltechniques.The expected numberof deaths was calculatedin
18th centuryactuarialmathematics(Dale, 1777; Tetens, 1786) but the method seems to have been
forgotten, and was reinvented in connection with 19th century studies of geographical and
occupational variations of mortality (Neison, 1844; Farr, 1859; Westergaard, 1882; Rubin &
Westergaard,1886). It is noted that standardizationof rates is intimatelyconnectedto the study of
relative mortality,and a short descriptionof very recent developmentsin the methodologyof that
area is included.
Key words:Cox regression;Dale, W.; Epidemiology;
Farr;Generalizedlinearmodels;Neison;
Occupational
mortality;Poissonassumptionin epidemiology;Price, R.; Proportional
hazards;
Relativemortality;
Yule.
SMR;
Standardization;
Tetens;Westergaard;
1 Introduction
Current texts on standardizationof vital rates often emphasize the rather deep
historical roots of that subject; see, for example, Benjamin (1968, p. 97), Lilienfeld
(1978), Logan (1982, p. 9) or Inskip, Beral & Fraser (1983). Often Farr (1859), or
occasionally Neison (1844), see Lilienfeld (1978), is credited with the first use of the
concept and, as we shall see below, these authorswere admirablyclear in explainingthe
backgroundfor and methodology of standardization.However, analogous calculations
of expected numbers of deaths were used in the 18th century literature on actuarial
mathematics.We shall in the present paper describe the calculationsby Dale (1777) and
Tetens (1786). Dale may very well be the earliest to explain and use the method (which
he did in full detail). Tetens demonstratedthe calculation(includingcorrectionsdue to
loss to follow-up) and also suggested what we would today call parametricstatistical
models for excess (or relative) mortality.
Standardization of rates was later primarily used in official statistical publications,
where considerations on random errors due to sampling were (and often still are) stated
verbally at the most. It seems to be generally accepted that Yule (1934) was the first to
derive the standard error of standardized rates. However, Westergaard (1882) gave a
rather careful discussion of sampling errors of mortality (and morbidity) statistics. The
occupational mortality study by Rubin & Westergaard (1886), to be presented here, is an
early application of standardization as an analytical-statistical methodology.
Only rather recently have the concept of standardization and the method of expected
numbers of death been put into full theoretical-statistical perspective. We review and
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2
N.
KEIDING
slightly extend the discussionsby Berry (1983) of calculatingexpected mortality,and by
Breslow (1975), Breslow & Day (1975, 1985) and Hoem (1987) of interpretingthe
standardizedmortalityratio as a maximumlikelihood estimatorin a proportionalhazards
model.
Generalizationsto regression models with continuous time and continuous covariates
were developed and surveyed by Breslow et al. (1983), Andersen et al. (1985) and
Breslow & Langholz (1986). We shall conclude this presentationby a brief referenceto
their work.
2 Brief review of standardization
In the simplest setting, standardizationis concerned with k age groups, where a
standardpopulationwith s, individualsin the first group,... , Sk in the kth, in short, age
distribution
s,, .. , sk, and age (-group)-specificdeath rates A,--... , Ak, is comparedto a
studypopulationwith age distributiona1, ... , ak and death rates cx, ..., tak. This means
that the numberof deaths in the standard(study) populationis E siA (E aicri).
The expectednumberof deaths in the study populationif the standarddeath rates are
applied to it is E aAi,, and a comparisonof this with the actual numberof deaths in the
study population is said to be 'corrected'for the dependence of mortalityon age. This
comparisonis often made in terms of the standardizedmortalityratio
actual no. deaths in study population
aoai,
E aiA, expected no. deaths in study population'
usually this is multiplied by 100. An importantproperty of the SMRis that it may be
calculatedon the basis of the total numberof deaths only, without knowledgeof the age
distributionof the dead.
Alternatively one might compare the actual number of deaths in the standard
population with that expected if the study population death rates are applied to the
standardpopulation. This yields the comparativemortalityfigure, or standardizedrate
SMR
ratiosi
S
-
CMF = SRR =
( x 100)
and because this may be interpreted directly as a ratio of weighted means of the
age-specificdeath rates to be compared, (using the (relative) standardage distributionas
weights), the method is called direct standardization.The weights are the same for all
study populationsthat one might want to consider.
The SMR
may be interpretedsimilarly,althoughsomewhatless directly,namelyby using
the weights
aiA Esii
That these are more complicated seems to be the basis for terming the corresponding
method of standardization'indirect'.
3 London in the 1770s: Careless societies for the benefit of old age, RichardPrice
and William Dale
Around 1770 many societies were formed in London with the purpose of providing
annuitiesfor the membersand/or their widows. Almost all of these were however based
on quite insufficientplans, and Richard Price (who is now perhaps better known as the
publisher of Bayes's essay and as the 'ethical father of the American revolution')
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Method of ExpectedNumber of Deaths, 1786-1886-1986
3
published his treatise Observationson ReversionaryPayments (Price, 1771) with the
explicit purpose of demonstratingthe inadequacyof most of these plans. In Price's own
words in a letter to George Walker, August 3, 1771 (quoted from Price (1983, p. 102)):
TheTreatiseI havelatelypublishedhasgoneoff betterthanI expected.TheLondonSocietiesare
in generalalarmed.I wishI mayprovethe meansof eitherbreakingthem,or engagingthemto
reform.Theyhavein theirpresentstatea verypernicioustendency.
The next year another book (Dale, 1772) appearedwith the same purpose but using
more elementary mathematicsand containing a detailed review of each of 11 societies.
(This book was, formallyanonymously,publishedby 'a fellow of one of the societies'.)
WilliamDale had however little luck in reformingthe LaudableSociety of Annuitants
(which he belonged to) so, after having resigned, he published in 1777 (again formally
anonymously)a 'Supplement'(Dale, 1777).containingfurtherdetailed demonstrationsof
the inadequacy of the plan of the society. Dale provided a very detailed and polemic
accountof his debate with the society, but we shall here concentrateon his calculationof
expected mortality.In his own words (p. 6):
The real Mortalityfor seven years past in the LAUDABLE
has been comparedwith the
SOCIETY,
areheretoannexed,(at p. 60) whichwill
expectedMortalityby Dr. Halley'sTable;the particulars
shew that butfew more than HALF
SOmanyhave hithertodied in the Society, as BRESLAW
Mortality
supposedwoulddie.
Dale goes on to point out that
It maybe well presumedthatthe healthiest,suchas havean opinionof the strengthof theirown
constitution,
so as to expectto liveto enjoythe annuity,wouldengagein
the Laudable Society of Annuitants, and that the opposite may be presumed for (the
husbands)joining the LaudableSociety for benefit of Widows.
The calculationis explained in 'Section XIII, Of Mortality, real and expected, in the
Society'. The calculations are set out in a large fold-out table, where the annual
increments,decrementsand one-year agings are given for one-year age classes for all the
years involved; see Table 1 and Plate 1.
A curiouspoint is that Dale, being only concernedwith demonstratingthat the expected
mortality is far higher than the observed, biases his calculations deliberately in the
following two ways: he disregardsmortalityin the year of entranceinto the society, and
he consistently deletes decimals (rather than rounding) in the calculationsof expected
number of deaths. Dale summarizeshis calculationsfor the first seven years in a small
table, here given as Table 2, and corresponding in modern terminology to SMR=
93/163 x 100= 57.
Until furtherevidence may surface,it seems fair to claim that Dale was the firstto carry
through a detailed calculationof the expected number of deaths. In fact, his table and
explanationcould still be used as a thoroughintroductionto standardization.
Dale (1772) reproducesan applicationthat he submittedon September24, 1771 for the
vacant position of Secretaryto the ProvidentSociety (he did not get the job, his analysis
of the society being considered unintelligible containing, as it did, decimals, etc.). He
here introduced himself as 'a person in his 46th year of age, who lately lived with a
nobleman as house steward'. Elsewhere, Dale (1772, preface) emphasizes that he is
without 'the advantage of a liberal education'. The books allow no other interpretation
than being entirely Dale's own enterprise, done out of interest in correcting the
insufficient plans of the 'Societies'.
Price's treatise was reprinted several times. In the fourth edition Price (1783) included a
brief comparison of the rates of mortality (in ten-year age groups) of the members of the
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4
N. KEIDING
Table 1
The calculationof the expectednumberof deaths.Excerptfrom the tablefacingp. 60 of Dale (1777), entitled:
in the LAUDABLE
SOCIETY
'The STATEof MORTALITY
OfANNUITANTS
for Benefit of AGE, Bottom of BARTHOLOMEW-
1766to CHRISTMAS
1775'
TABLE,
LANE,comparedwithDr. HALLEY'S
annuallyfrom CHRISTMAS
Age
20...
21...
22...
23...
24 ...
25 ...
26...
27...
28 ...
29 ...
30...
31 ...
32 ...
33...
34 ...
35 ...
36 ...
37 ...
38 ...
39 ...
40 ...
41 ...
42 ...
43 ...
44 ...
45 ...
46...
47 ...
48 ...
49...
50 ...
1768
A+B=C
1769
E+F=G
0+0=0
2+2= 4
3 + 1= 4
4+ 2= 6
5+ 2= 7
5+4= 9
6+3= 9
2+ 2= 4
1+ 5 = 6
3+ 5= 8
0+1=1
0+5= 5
0+2= 2
4+6= 10
4 + 13 = 17
1 ex.
5 + 10 = 15
1 ex., 1 d. 7 + 8 = 15
7 + 8 = 15
1 ex.
8 + 10 = 18
4 + 14 = 18
6 + 19 = 25
57
141
10 + 5 = 15
10 + 3 = 13
5 + 2= 7
10 + 7 = 17
5 + 5 = 10
19 + 3 = 22
17 + 7 = 24
15 + 10 = 25
10 + 15 = 25
16 + 17 = 33
2 d.
1 d.
1 d.
8 + 11 = 19
13 + 18 = 31
13 + 11= 24
7 + 22 = 29
17 + 13 = 30
9 + 16 = 25
22 + 18 = 40
24 + 11 = 35
24 + 20 = 44
25 + 21 = 46
191
323
13 + 17 = 30
21 + 11 = 32
18 + 14 = 32
13 + 9 = 22
4 + 14 = 18
14 + 15 = 29
10 + 12 = 22
4 + 8 = 12
6 + 7 = 13
5 + 9 = 14
33 + 12 = 45
30 + 13 = 43
32 + 10 = 42
32 + 10 = 42
22 + 7 = 29
17 + 5 = 22
29 + 8 = 37
22 + 16 = 38
12 + 15 = 27
13 + 14 = 27
224
51 ...
52...
53...
54 ...
55...
56...
57...
58...
59...
60...
D
1 d.
H
1 d.
1 ex., 1 d.
1 d.
1 d.
1 d.
1 d.
1 d.
1 d.
... Multipliers
1769
1770
...0-01003
0-01013
0-01194
0-01036
0.01047
0.01234
0.01250
0.01265
0.01282
0.01484
.. .0.01506
-
0-0477
0*0414
0.0628
0-0863
0.1125
0-1138
0*0512
0.0890
0-1204
0-0505
0*0238
0-1036
0-1779
0-1851
0-1875
0-1897
0.2307
0-2671
0-3765
0-7251
1.7924
0-2293
0-2018
0-1103
0-3065
0-1836
0.4116
0.4574
0-4857
0.4955
0.6672
0-2905
0-4814
0-3784
0-5228
0.5508
0-4677
0.7624
0-6800
0-8720
0-9301
3.5489
5-9361
0-6192
0-7494
0-7673
0-5405
0.4532
0-7493
0-5834
0-3268
0-4005
0.4450
0-9288
1-0070
1-0071
1-0319
0.7302
0.5684
0-9812
0-9261
0-8318
0-8583
5-6346
8-8708
0.2296
0-1695
0.1404
0.0331
0*0684
0-1062
0*0367
0-0381
0.1188
0*0826
0-8528
0-5085
0-4563
0.4634
0.2736
0-2832
0-1835
0-0381
0-0396
0.4543
1-0234
3-5533
0.01529
0.01553
0.01577
0-01803
0.01836
0.01871
0.01906
0.01943
0*01982
... 0.02022
0.02064
0.02342
0.02398
0.02457
0.02518
0.02584
0*02652
0.02724
0.03081
. .. 0-03179
352
3+ 4= 7
1+ 4 = 5
1+3 = 4
0 + 1= 1
1+ 1= 2
1+ 2 = 3
0 + 1= 1
0 + 1= 1
1 + 2= 3
1+ 1= 2
14 + 12 = 26
7 + 8 = 15
5 + 8 = 13
4 + 10 = 14
1+ 7 = 8
2+6= 8
3 + 2= 5
1+ 0 = 1
1+ 0 = 1
3+8=11
29
102
1 d.
0.03280
0.0339
0.0351
0.0331
0.0342
0.0354
0*0367
0.0381
0.0396
.. . 0.0413
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Method of ExpectedNumberof Deaths, 1786-1886-1986
5
Table 1 (Continued)
Age
61...
62...
63...
64...
65...
66...
67...
68...
69...
70 ...
1768
A+B=C
D
1769
E+F=G
2+ 3= 5
1+ 2 = 3
0+0=0
0+0= 0
2+0=2
1 + 0 =1
0 + 1= 1
0+0=0
0+0=0
0+2= 2
0 + 1= 1
4
Admitted 240
Deaths
Exclusons
Total deaths & Excl.
No, Deaths expected
H
... Multipliers
1 ex.
0*0431
0*04504
0-04717
0-04950
0*05208
0*05494
0*05814
0*06172
0*06578
... 007746
11
1769
0-0431
-
0.0990
0*0520
-
1770
0-2155
0-1351
-
0-1041
0.0549
-
0.1942
0.5096
11-1261
20-6622
433
6
3
9
11-1261
9
2
11
20-6622
Columnheads given by Dale:
A, Remainingof the past year- 1767
B, Admittedin the Courseof the Year- 1768
C, Total living at Christmas- 1768
D, Numberof Deaths and Exclusions,Anno - 1769
E, Remainingof the past Year- 1768
F, Admittedin the Courseof the Year- 1769
G, Total living at Christmas- 1769
H, Numberof Deaths and Exclusions,Anno - 1770
Note to Table 1 (not given by Dale). The column 'Multipliers'contains the annual death rates accordingto
Halley, and the two columns to the right denoted '1769' and '1770' are the productsof the numberliving at
Christmasthe year before and the multiplier.The sum of these productsis the expected numberof deaths in
that year, whichis also quoted at the bottom of the columnto the left.
Equitable Society to those in London and those at Breslau (Halley's). Only the
proportions for each ten-year age group were quoted, and no proper calculation of
expected numberwas performed.
Price quoted a very similar comparisonearlier published by Morgan (1779), Price's
nephew and actuary to the Equitable Society. Thus these two contributionswere both
later than Dale's and far less detailed.
It should finally be recorded that Price and Dale made frequent (and mutuallyvery
respectful)cross-referencesto each other.
4 Tetens, 1786
J.N. Tetens (1738-1807) from Schleswig-Holstein (then belonging to the Danish
monarchy)graduatedin Rostock, and was professor of philosophy and mathematicsat
Kiel Universitywhen he publishedhis importanttreatise on actuarialmathematicsin two
volumes in 1785 and 1786. Tetens was elected a memberof the Royal Danish Academy of
Science and Letters in 1787 and later assumed Danish governmental positions in
Copenhagen. Like Price, Tetens is nowadays better known as philosopher than as
mathematician,see, for example, Tetens (1971).
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1
+
-"94
5
1=I
0+
3+
=
1CL7+O=7
0
1=21 0+
0=01 5+
aO=
73 t -O=
+O=
O=
S+
=0A 0+
7I 0+
4 1+
0
S+
0+
1+
1+
*=
I1I Iex0+
O=a
O=
0
O=
2
O=
1
1721+
1+
+0+I I+0=f 0+
O=
I A5
0 1+
01 0+
Q=
X+
11
O=
0
O=
-077
2+
O=
I+O=
1
0=4.1
= 0+
36 1+O
5.4
-61
2 2-4149J+
P
674
24041
2 +-
ago
1*+5=
*+=1=
I+i=
*4+I' i
I=3 1
=24
I= 4
9 I= d. S+
*+ J=
*4'
5
47
J +
'+4a=5
A= 7
5=
AI
q' S++ 4
J+
_
I
'ex.
,@,=5
ex
1+
7
44
S= I*
J++ -7E=1N
+
+
35+ 4231
+I=,
+
a+ 1=24
a
u- 9+ +11
Is 41+ 8+ 49 I *6
+ 1= 5
4
65+ +1c 64 4+ a41 + 5=7
0= 27
S7
381+O=
7+ =15
+ 2=b
,;,At
+0=310
Id.
6
8.5+$i=s
39+4=33
6
6+ o
=0
g + =12
. -7+ =
1 115, 1*
---ifIA+ I4d+)+5
=7
=8=
*0I+
1=+ 0+04=01 7+3O= 7323
+
7+=425++ 4=6
7
6+ .jo=
s=i4
I.
1
6+ 2= 1
e.I+
o6+ 9=19
3+ X=49
t=20
Iex
1+
3+3O=4j
7+7=2
+ O4=
+1 . =a
1 +6=1
2+4=5
I
d
+ 0 =2a
=5
7
3154
a=sO
Id.4 so+
2ex. .i
=?
=
,
d 45+6x=2
+jq
,=~
q 37.913+
+ 1 *=63
48+
* O=48
=5
7-4__ =44
0 13
4+L= 4
1
+
1-3
IIA-.+ 16=38
+
A
5.1
4
4
54
5 T1.
w
A
2
:
00
1+
IA. 4
84
L
ONS
1.* 1+
0=43
d
.0114
i
5=
17
4+
I+
0=4
4= S of,
+ =53
1=4
( a
+-
-559
+ ,-0 =45 411 AS7*
1S.- us
+ 3=4
.0
+=4
aof6+
53+ 1=5
I+7*
d.
0
I
+ 1=4
1A4
31+ cs6-5-7
= 58.-2
43
.01)98
46
so+D=Y41
*"
. 1525
13 . 5877
!162.176
j)U34
.16
4.7.4
3+05=0
4=
6
,'S5
1 A.
413O=4
7+
3 1
5
2
+012
C O=
34+
14
3~1_.____
9
1 1.12
.0 .381469
6
-+=4
o1zj
3-0414+-0414
-01036
O= ;7 24 -01.47
32j
ad.7+
53
52
+ o=3
6
47 1
1X34
If
5
7
1AIM
01
.a01
+
id
.37
091
+3493--0975o*
+3
I.
.0127
+0=7
z7696
s
5.
170=ISOd1$7
-,+37
1
)6+ 8=44
17+ 0=17
-1
+
4+6=10 _45
+11=7+55
5+10=15
+31
7
8+0+ 7$=:I
3+0*oj
Plate 1. The tablefacingp. 60
of Dale (1777), reproduced
from the copy at the libraryof
the Universityof Chicago.
.0-z
I
8
O S
='d +7O==17
0+
9
0+ O=
1=100=
3+ O= 13
o6
6
=
3+
.5=1
17
+ O= 1
+ O=081
1Z
3+1=
0=
7?+
4+2=
1
1
no
tow
' .021
+O=Ito
0) +Q=
0 1242+O=3
I+O=
.1
1+ 0= 3
r
+ 4=
114
4.
0+O
.+ O=a1
1 4
1
0=
=a
C4-T17
7667t
-
2
3X++ O=1
I
I
+ 7=
+4=4
4ex
X
+'I+
0+ 0=
I
O
*a
+
I+4
A
S+
-c+
0+
4 O=0
+=3 o
+ =2 v+
++ O= ;
4+~a8
iii g4
30
1
+
o+
+
;e.
+
Id-1+1=41:.
+O= a
+2+
No
++1 0++Ot
Ij I
Is
I
c=
177L.
I .I+O=IId.
++=
0+1=a_+
=
0+
0
+=O 0
+++=
44
0++
+7
r1771.
+ c
d
+.=t
ADE
Ci
APm
annuall f 713
737.-
Q0~ = +
31Its=+. ft0+
2=
=0 ;7,0o+
OS~
?
ANo
ANNUIT
in the LAKUDABLEOCIETYof
HALLEY's TABLE,
Dr.
TV-nSTATEWMURTILITY
6 1769. '
rewith r
1768.
-~
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1
~
-oa ,-m
61,81
7 -4
Method of ExpectedNumberof Deaths, 1786-1886-1986
7
Table 2
Summaryof the observedand expectednumbersof deathsin the
LaudableSocietyof Annuitantsduringthe seven years 1767-1773.
Excerptfrom the table facing p. 60 of Dale (1777), also reproducedby Tetens(1786, p. 33)
No. of deathsin seven years
Expected
Real
Deaths under21 years of age
Deaths from 21 to 30 inclusive
Deaths from 31 to 40 inclusive
Deaths from 41 to 50 inclusive
Deaths from 51 to 60 inclusive
Deaths upwardsof 60 years of age
Totals
2.3084
10-9904
39-0923
62-2487
40-7085
7.5337
162-8820
4
11
23
33
17
5
93
The treatise was motivatedby Tetens's assignmentto assistwith actuarialmathematical
advice in the reconstructionof the widow fund of the duchy of Calenberg (for which
Hannover was the capital). The second volume (Tetens, 1786) contains a series of
methodological 'Versuche iiber einige bey Versorgungs-Anstaltenerhebliche Punkte'
(Notes of importanceto charitableinstitutions).The first of these 'Von der Sterblichkeit
in ausgesuchtenGesellschaften'(On mortalityin selected societies) contains three parts:
first, empiricalbackground,primarilybased on Tetens's own analysisof experiencefrom
the Calenberg widow fund; secondly, general (theoretical) study of deviations of the
mortalityof selected societies from that in ordinarylife; and thirdlythe influenceof these
deviationson such actuarialobjects as annuitiesand widow'spensions.
4.1 Tetens'sempiricalevidence
The Calenberg widow fund was created in 1767 and for this purpose followed until
1783.
The husbandshad to certifytheir good health at entrance(even though Tetens assumes
such certificationto be often ratherunreliable)but no such requirementswere made for
the wives. Still, Tetens believes that the initial health of the wives was generallygood,
because one must expect husbandswith severely ill wives not to want to join the fund.
It is interesting that Tetens emphasizes the transientnature of the selectivity of the
population:
Ausgesuchtwar iibrigensdiese Gesellschaftnur in Hinsichtder dermaligenGesundheitder
nichtin HinsichtderkirperlichenConstitution,....
Mitglieder
beymEintrittin die Gesellschaft,
(Selectedwas, moreover,this societyonly as regardsthe currenthealthstatusof the membersat
entranceintothe society,not as regardsthe corporalconstitution.)
We shall see below how this viewpoint influencedhis modellingof excess mortality.
The basic analysis is to consider half-yearlycohorts of newly accepted wives in the
fund, and for each individualwoman to calculate her death probabilityfor the period
from entrance until 1783 according to the, at the time, well-establishedlife table of
Siissmilch.By additionover five-yearage-at-entrancegroups, expected numbersof deaths
are obtained, and these are comparedwith the observed numbers;see Table 3. Tetens
devotes careful consideration to the 106 (out of 5221) with incomplete follow-up.
Although Tetens documents a complete understandingof the problems about these:
'... haben sie diese Zeit durchlebet, ohne ihren Beytrag zur Todtenliste zu geben...;
(they have lived during this period without contributingto the list of deaths), his data
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N. KEIDING
8
Table 3
Exampleof Tetens'sdocumentationof the expectednumberof
deathsin the Calenbergwidowfund. From Tetens(1786, p. 7)
Durchschn.
Alter.
Anfangszahl.
Wirkliche
Todten.
Erwartete
Todten.t
18'
23
271
32f
37f
421
47
52
57
621
67f
70'
39'
Noch von
14
61
185
226
243
216
127
104
49
20
7
1
1253
17
4
15
27
38
52
47
36
36
24
12
7
0
298
3*
2,289
10,953
41,222
57,797
73,354
77,978
54,351
54,013
30,586
14,872
5,870
0,872
424,157
6,55
301
1270
430,707
* Hier sind bloss die
ganzen Zahlen genommen. In den
folgendensind auch ein paarDecimalziffernhinzugesetzt.
t Commashave been used to denote decimalpoints.
The table shows the observed and expected mortalityfor
the initial 1270 women joining the Calenbergwidow fund in
June 1767. The columns contain: the average age in each
five-year age group, the initial number in that group, the
observed number of deaths and the expected number of
deaths until 1783. The line 'Noch von' containsthe women
who were 'lost to follow-up', in present-dayterminology.
Translationof footnote: 'Here are only quoted integers. In
the followinga couple of decimalsare sometimesincluded'.
There are a total of 30 such tables, reportingthe mortality
experience for each half-yearlycohort. Tetens emphasizes
that he reports the results in this 'abridged'form to save
space; the calculationsseem to have been performedon a
more detailedbasis.
permits only a rather crude method of correction:the age at entrance is assumed to be
distributedas the empirical distributionof completely followed women from the same
cohort, and the 'observed'numberof deaths is approximatedby assumingcertainsurvival
throughhalf of the period from entranceto 1783, and death probabilityin the latter half
of the period equal to those of the same cohort which had complete follow-up.
The main result is that the 743 observed deaths should be comparedto 1048 expected,
and that the ratio 743/1048= 0.7 is ratherconstantover all cohorts considered.There is,
however, a pattern accordingto age at entrance, and because no partitionaccordingto
current age has been made, this point complicates the interpretationfrom a modern
viewpoint.
Tetens performs similar calculationson the husbands (where the problem of loss to
follow-up is more serious), and briefly quotes and extends the above mentioned
calculationsby Dale, Morganand Price and based upon Halley's life table.
4.2 Tetens'stheoreticalconsiderations
In his more general remarks,Tetens emphasizesthat his object of study is the survival
of 'ausgesuchte Gesellschaften' (select societies) whose members are at least initially
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Method of ExpectedNumber of Deaths, 1786-1886-1986
9
healthy, or perhaps initially healthy as well as more permanentlystrong. Or conversely,
unhealthyand possibly also frail.
As an axiom it is assumed that, if the study population mortality is lower than the
standardmortalityat some age, it stays so with increasingage, but it approachesthe latter
and ultimatelybecomes equal to it. In understandingTetens's desire to have convergent
mortalities, due attention should be paid to his above mentioned emphasis on the
transientnature of the selectivity. He seems however to be unawareof the consequences
of heterogeneityin populationswhere the frail die first, wherebythe populationbecomes
on the average strongeras time goes by (Vaupel, Manton & Stallard,1979).
The above axiom also has to be seen in the context of Tetens's definitionof mortalityas
the death probabilityduringone time unit (e.g. a year); in fact, this has to equal one at
the highest possible age for the study as well as the standardpopulation. Tetens uses an
infinitesimal version (what we would call the hazard) in his later mathematical
calculations,but does not seem to notice that no requirementof ultimate convergenceis
needed for mortalitiesdefined as hazards.
Several general propertiesof the survivalcurves of the study and standardpopulations
are now derived, such as the impossibility of intersection, and also that, once the
differencein survivalprobabilityhas startedto decrease, it will continue to do so.
Of particularinterest is what we would call parametricmodels for the excess (or
relative) mortality. These are discussed within the hazard rate framework, and I shall
here use modern notation. Let S(x)[So(x)] denote the conditional survival probability
given survivalto some fixed age xo of entrance into the study [standard]population, so
that S(xo) = So(xo)= 1 and S(oo)= So(oo)=0. Let A(x)[Alo(x)]
be the correspondinghazard
= -D log S(x).
rates,A(x)
The first 'Hypothese', we would say 'model', assumes
= 1+ PSo(x),
AO(x)
and is thus a model for relative mortality.As Tetens proves using series expansions (a
proof via differentialcalculusis elementary)it is equivalentto
S(x)= So(x)e(so(x)-1).
It is seen that, contrary to our present-dayparadigmaticproportionalhazards model,
1 as x --->oo;this model for the hazardratio is perhapsthe simplestwith this
A(x)/Ao(x)-property.
The second model assumes
S(x)= (1 + M)So(x)-So(x)2,
or equivalently
S(x) - So(x)= M(So(x)- So(x);
thus this model is defined in terms of excess survivalprobability.The motivationfor this
model seems to be that it is in some way the simplestsatisfyingthat S(x) = So(x)for x = xo
and x = ooand S(x)
S(xo) for xo <x < oo
Tetens proves (again using series expansions instead of a modern elementary
differentialcalculusapproach)that the model is equivalentto
A(x)
Ao(x)
1 + 2aSo(x)
1 + aSo(x) '
- p
1+
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10
N. KEIDING
and thus again satisfieshis requirementof converginghazards,A(x)/AO(x)
--1 as x-- oo.
As propertiesof this model Tetens notes first that if > 0 (the case of a healthy study
population)then y < 1. One way of seeing this is to notice that
Y = 1 - A~(xo)/Ao(xo)
(4.1)
(this simple representationis not quoted by Tetens). Further,the differenceS(x) - So(x)
is maximal for So(x)= -, that is at the median survival time ('nach Ablauf des
wahrscheinlichenAlters').
Tetens explains in some detail that in order for this model to be useful, experience
should show that ~ is constantover age at entrance(our xo) as well as observationperiod.
If this is not the case, however, Tetens recommendsthe use of an averagevalue of Y
Und dieseZahlwiirdewie ich glaube,gleicheundwohlgrossereZuverlissigkeit
haben,als irgend
eineunsererbestenMortalitits-Tafeln.
thananyof our
(Andthisnumber,I think,wouldhavea similarandpossiblyevenhigherreliability
bestlife tables).
For the Calenberg widow fund data, Tetens then goes on to estimate CYfor those
five-yearage groups in each of the separatehalf-yearcohorts which are based on at least
30 (or in some cases 20) initial individuals.The conclusionis that a value of Y = 0.4 could
be used for the wives. Further computationsare added for the husbands and for the
cross-sectionaldata of Dale, Morganand Price.
This early attempt at deriving and implementinga simple model for excess mortality
deserves mentioning, even though our representation(4.1) shows that can only be
/Y violates the
is constant, which on the other hand
independent of x0 if
A(x)/•A(x)
converginghazardproperty,so that no model can accommodatea common ~ for varying
age at entrance.
5 Some English contributionsin the 19th century
No report touching the history of standardizationshould bypass the important
contributions by 19th century English statisticians. We shall here mention early
contributionsby F.G.P. Neison and WilliamFarr.
5.1 Neison'ssanatorycomparisonof districts(1844)
In a paper read before the StatisticalSociety of London in December, 1843, Chadwick
(1844) pointed out several shortcomingsof (then) currentmethodsof comparingmortality
between different classes of the community and between the populations of different
districts and countries. As a general rule, Chadwickadvocated that the average age at
death as well as the average age of the living be quoted and used for these comparative
purposes. Four weeks later Neison (1844) read a paper to the Society criticizing
Chadwick'sin terms that could enter present-daytextbooks directly:
cannotbe takenas a test of the valueof
Thatthe averageage of thosewhodie in one community
life whencomparedwith that in anotherdistrictis evidentfromthe fact thatno two districtsor
places are under the same distributionof populationas to ages ....
Neison calculatedfor many districtsnot only the averageat death, but
also what would have been the average at death if placed under the same population as the
metropolis.
(Direct standardization!)
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Method of Expected Number of Deaths, 1786-1886-1986
11
As example,
Thistablecontainssomeinterestingresults,andI beg to cite two cases.The averagedeathin the
metropolisis 29-06years,but in the townof Sheffieldit is only23-19years;however,if Sheffield
wereplacedunderthe populationof the metropolisthe averageage at deathwouldbe raisedto
closeto the metropolis.
28-14years,approaching
Neison also remarkedthat
Anothermethodof viewingthisquestionwouldbe to applythe samerateof mortalityto different
populations.
(Indirectstandardization!)
5.2 Farr'sclassicalexplanationof the expectednumberof deaths
W. Farr (1807-83) worked for many years as 'compilerof abstracts'in the office of the
Registrar-Generalof England. Among his many contributionsto mortalitystatisticsis the
routine implementationof Neison's suggestion of using expected number of deaths to
'comparelocal rates of mortalitywith the standardrate'. For the latter, Farr(1859) chose
the average age-specificannual death rates for 1849-53 in the 'healthy districts',which
were operationallydefined as those with average gross mortalityrates of at most foo.
Farr'slucid explanationof the method is classical,using,
Example:Thenumberof boysunder5 yearsof agewas147,390;the annualrateof mortality
in the
thesetwo fractionstogether,147,390x 0-04348= 6367
healthydistrictwas0-04348;andmultiplying
deathswhichwouldhavehappenedat Londonhadthe mortalitybeenat the samerateas it wasin
the healthydistricts.
As conclusion
... on an average,57,582personsdiedin Londonannuallyduringthe fiveyears1849-53,whereas
the deathsshouldnot, at ratesof mortalitythen prevailingin certaindistrictsof England,have
exceeded36,179;consequently
deathstookplaceeveryyearin London.It willbe
21,403unnatural
the officeof the Boardsof Worksto reducethisdreadfulsacrificeof life to the lowestpoint,and
thusto deservewellof theircountry.
5.3 Wasthe methodof expectednumberof deathsreinventedin 19thcenturyEngland?
I have found no referenceto the use of the method of expected numberof deathsin the
18th century actuarial context from the later English authors, who were concerned
primarilywith geographicalor occupationalvariationsof mortality. In fact, I have not
met Dale's name outside of the above mentioned referencesby Price and Tetens, and as
regardsTetens's contributionto the method of expected numberof deaths, I have seen no
other reference than that by Westergaard(1932), who gave a brief reference to the
practicalcalculation,but did not mention the parametricstatisticalmodels. It would be of
considerableinterest in this context to explore Neison's backgroundfurther.
Tetens also contributedother innovationsin actuarialmathematics,and one of these,
the 'commutation method' was independently (so it has to be assumed) invented by
George Barnettin Englandand publishedby FrancisBaily. Hendriks(1851) reintroduced
this contributionof Tetens to the English actuarialcommunity, and in this connection
enquired about any evidence in Price's papers about Price having known about Tetens's
work. Such evidence would certainlybe ratherinterestingin our context, too. (Price died
in 1791, five years after the publicationof Tetens's second volume).
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12
N. KEIDING
6 Westergaard
Harald Westergaard(1853-1936) had degrees from the University of Copenhagen in
both economics and mathematics.Before becoming professorin economics and statistics
at this university,Westergaardhad studied in Englandand other Europeancountriesand
he presented his considerableknowledge of mortalityand morbidityin a prize paper to
the university in 1880. This was quickly published in German, Die Lehre von der
Mortalitiitund Morbilitiit(Westergaard, 1882), and reissued in a completely revised
version in 1901. From the point of view of the method of expected numberof deaths the
firstedition is however the more interesting.
After having defined the basic concepts of mortalitystatisticsin continuoustime, using
differentialcalculusto specify intensities,Westergaardintroducesthe method of expected
numberof deaths in the followingway (pp. 29-30):
Von wesentlichgleicherArt, aber von gr6sserempractischenInteresse,ist der Fall, wo die
demAlternachgegebenist, die Todesfiilleabernicht.(...)
Bev61lkerung
als Massstabnehmen und
Auch hier kann man nun eine bestimmteSterblichkeitstafel
die man untersucht,
berechnen,wie viele Todesfiillenach dieser Tafel in der Bev61lkerung,
eintreffenwiirden.Diese Berechnung
durch
eine
einfache
der Intensititder
erfolgt
Multiplication
mit der Volkszahl.Erhilt man mehrGestorbenenachder Berechungals nachder
Sterblichkeit
so ziehtmanden Schluss,dassdie Sterblichkeit
kleinsei; umgekehrt
Erfahrung,
verhiltnissmissig
mehrTodesfille
dagegen,dassdie Sterblichkeit
grosssei, sobalddie Erfahrung
verhiltnissmissig
Es kommtmehrfachvor, dassdieseMethodedie einzigeist, welche
aufweist,als die Berechnung.
ohne
ein correctesund vollkommenesResultatgiebt. Der summarische
Sterblichkeitsquotient,
Riicksichtauf das Alter berechnet,ist nicht zuverlissig;denn das Alter ist ja eine der
hervortretendsten
Ursachen,die desshalbeliminirtwerdenmuss.Indemmanberechnet,wie viele
nach
erwartetwerdenk6nnen,findetmanauch
Todesfille
irgendeinerNormfiir die Sterblichkeit
einen summarischen
aber in diesem ist auf den Einflussdes Alters
Sterblichkeitsquotienten;
genommen,-dasAlterist zu einerzufilligenUrsachereducirt.
Riicksicht
[Footnote:Wir werdenim folgendendiese Methodeals: 'die Methodeder erwartungsmissig
Gestorbenen'
bezeichnen].
Es ist klar,dassmansichbei der Anwendung
diesesPrincipsnichtebenan das Alterzu halten
braucht.Mankannes iiberallbenutzen,wo maneinerUrsachenachgeht,undwo die Statistikder
undderBev61lkerung
verschiedene
hat.
Todesfaille
Eintheilungen
Die Methodehat indesseneine weit gr6ssereAusdehnung.Wenn man z.B. zwei bekannte
Sterblichkeitstafeln
vergleicht,so diirfte es schwierigsein, eine genaue Uebersichtiiber die
zu erhalten.EinschnellesHiilfsmittel
aber,einesolcheUebersichtzu
beiderseitigen
Abweichungen
indemmanberechnet,wiesich
erreichen,ist danndie Methodedererwartungsmissig
Gestorbenen,
die Sterblichkeit
nachdenbeidenTafelnstellenwird.Die ZusammensetirgendeinerBev61lkerung
zung dieser Bevilkerungmiisstedann von der Naturder Aufgabeabhingen.Eine einfache
in jederTafelzu addiren,denn diese
Anwendungist jene, alle Intensititender Sterblichkeiten
in einerBev61lkerung
zu berechnen,
Aufgabeist keineandere,alsdie, die AnzahlderVerstorbenen
in welcherin jederAltersclasse
gleichvieleMenschenwiren.
Dass dieses VerfahrenauchanderenRichtungenhin grosseVorteilehat, wirdspiter gezeigt
werden.
Translation:
Of essentiallythe same character,but of greaterpracticalinterest is the case where the population
is given accordingto age, but the deaths are not. (...)
Even here it is possible to take a certain life table as yardstickand compute how many deaths
would have appearedaccordingto this table in the populationunderinvestigation.This calculation
amountsto a simple multiplicationof the force of mortailityby the populationsizes. If one gets
more deaths accordingto calculationthan accordingto experience,then one will concludethat the
mortalityis relativelysmall, and converselyif experienceyields more deaths. There are manycases
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Methodof ExpectedNumberof Deaths, 1786-1886-1986
13
where this method is the only way to obtain a correctand complete result. The summarymortality
ratio calculatedwithout regard to age is unreliable;for age is one of the more importantcauses
which therefore has to be eliminated.By calculatinghow many deaths may be expected according
to some mortality standard one also obtains a summarymortality ratio; however, in this the
influenceof age has been taken into consideration-age has been reducedto a randomcause.
[Footnote: We shall in the followingdenote this method 'the method of expected deaths'].
It is obviousthat it is not necessaryto restrictoneself to age when applyingthis principle,anytime
one searchesfor a cause, and mortalityand populationstatisticshave differentclassifications,it can
be applied.
The method has however a much wider application.If one for instancecomparestwo known life
tables it may be difficultto obtain a clear impressionabout the deviations.A quick tool to provide
such an impressionis then the method of expected deaths, where one calculateshow the mortality
of some population would be accordingto the two tables. The composition of this population
should depend on the nature of the problem. A simple applicationof this is to add all forces of
mortalityin each table, since this just amountsto calculatingthe numberof deaths in a population
with an equal numberof personsin each age class.
That this method also has great advantagesin other directionswill be shown later.
We note in passing that the idea of just adding all forces of mortality was reintroduced
by Yule (1934) and later by Day (1976), who applied it to world-wide cancer incidence
comparisons.
6.1 Confoundercontrolandprecision
In his introduction of the method of expected deaths, Westergaard emphasizes what
would today be termed confounder control: 'das Alter ist zu einer zufailligen Ursache
reducirt', age is 'reduced to a random cause' when the incidence or mortality between two
populations is to be compared. If there are more than one 'stoirende Ursache' (we would
say: confounder), Westergaard recommends further subdivisions, since as he sees the
method of expected number of events, it can only eliminate one confounder.
Although confounder control is discussed more thoroughly here than in most earlier
literature, Westergaard's approach to sampling variablity and its interaction with the
method of expected number of events seems to be his particularly innovative contribution. The concept of distribution of observed events is introduced, and the 'mittlere
Fehler' (standard error) is defined as the distance between mode and point of inflection of
the normal density (here called 'die Exponentialformel'). The observed number of deaths
d has mean error V/(Spq), where S is population size and p = 1 - q the death probability.
Westergaard notes that often p is rather small, so that q is almost 1, and the useful
approximation V(Sp) = Vd is obtained: we would call this a Poisson approximation.
Westergaard goes on to explain that one should in practice always investigate the
applicability of this standard error (through derived fractiles of the distribution) and, if
the variability is greater than the 'theoretical', there will be heterogeneities which should
be removed by further subdivision.
6.2
Rubin & Westergaard'sstudy of occupational mortality, 1886
At the request of the director of the Danish National Bank, who wanted a better
statistical basis for current political discussions on social issues, Westergaard (with Marcus
Rubin) carried out an investigation of the mortality, as specified by occupational classes,
of the rural population of the diocese of Fyen, about 170,000 persons (Rubin &
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14
N.
KEIDING
Westergaard, 1886). Unfortunately this report is available in Danish only, save for a brief
German summary without methodological remarks (Westergaard, 1887).
Unusual care has been taken in this investigation to match 'numerator' and 'denominator' populations. The deaths were based upon medical and church records for the
period 1876-83 around the census 1880, from which the classification of the living was
taken. Further, the occupational classifications of both living and dead were scrutinized
and reported by local assistants, primarily schoolteachers.
Of particular interest in our context is the consistent analytical-statistical viewpoint in
the methodological introduction and throughout the discussion of the results. Thus (pp.
56-57, my translation):
Since, however, the data are usuallytoo few, it is necessaryto quote the mortalityratioswith some
reservation,as far as possible combinedwith limitsfor the errorsby whichthe ratiosare vitiated. In
the presentinvestigationthis has been done using the standarderror....
The method by which it is possible in this way to eliminatethe influenceof age, the 'method of
expected deaths' thus consists in calculating the mortality which would have appeared if the
conditionswithin each age group were in close accordancewith the mortalityconditionsaccording
to some known life table, for instance that for the whole populationor the agrarianclass. If then
the total numberof deaths, relativeto the standarderror,shows a considerabledeviationfrom what
would be normal for the relevant group, when the mortalitycorrespondedto the used life table,
then there is reason to believe that there exist causes which particularlyinfluencethe health status
in the particularoccupations,even if this due to the limited size of the data cannot be concluded
with certaintyfor the single age classes of the occupation.
This emphasis on variability reduction does not seem to have been common for the
period. The basic results (we here only quote those for males) are given in Table 4, which
is followed up by detailed occupation-specific discussions, including tables of age-specific
death rates. The standard error plays an integrated part in these discussions.
Table 4
Observedand expectednumberof deaths.Diocese of Fyen, 1876-83, males above 5 years of
age. Translated
from Rubin & Westergaard
(1886, p. 62)
Working*
Obs.
Exp.
Officials
Teachers
Other minorofficials
Tradesmen, shopkeepers
Craftsmen
Seamen, fisherman
Capitalists
69
44
103
54
50
116
gentleman farmers
Servants
Paupers
Others
Total
*
Includingsome dependents
5
11
17
112
119
104
114
1197
96
125
1297
129
96
169
125
-
112
119
1183
129
-
169
-
-
36
-
39
Total
Obs.
Exp.
4
18
19
1093
Landedproprietorsand
Farmers
Smallholderswith land
Smallholderswithoutland
Not working
Obs.
Exp.
-
-
1674
1283
1070
1708
1370
937
737
801
115
670
692
93
519
15
720
17
655
-
495
-
6150
6479
2549
2222
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73
62
122
39
59
61
133
36
2411
2084
1185
2378
2062
1030
519
655
15
720
495
17
8699
8701
Method of Expected Number of Deaths, 1786-1886-1986
15
use of the methodof expectednumber
6.3 Two 20th centuryreferencesto Westergaard's
deaths
of
In what was then called QuarterlyPublicationsof the AmericanStatisticalAssociation,
Westergaard(1916) published a summaryof his statistical ideas (with discussion). The
paper includeda descriptionand applicationof the method of expected numberof deaths,
as well as a discussionof its relation to standardization,which was at the time primarily
used in England.
Another 20th century reference to Westergaardwas made by Woodbury (1922-23)
who, unfortunately, was somewhat inexact on the history of standardization.In a
footnote on p. 369, Woodburymisinterpreteda footnote of Westergaard(1882, p. 287) to
the effect that Ratcliffewas first to use the method. In fact, what Westergaardwrote (in
German!)was that he had appliedthe method on Ratcliffe'sdata. Ratcliffedid not use it.
On the other hand Woodbury'sclaim that Westergaard'developed' the method seems
also rather unlikely. Westergaard (1882) made frequent reference to current English
literature and even if he never gave specific credit concerningthe method of expected
number of deaths, an obvious asssumptionis that he picked it up during his visits to
England.
7 English contributionsin the early 20th century
The considerable English experience concerning standardizationmade some impact
upon the methodological literature. The Royal Statistical Society discussion paper by
Yule (1934) is still often quoted, primarilyfor the followingtwo points: First, assumethat
SMR'Sare computedfor two populations,using the same standarddeath rates. Then Yule
carries no interpretationas an SMRof one
emphasizedthat the ratioof these two SMR'S
to
with
the
other.
respect
Secondly, Yule derived a standarderror estimate of
population
the SMR,which is in fact equivalent to Westergaard's.Under the assumptionthat deaths
are rare, the denominatoris taken as constantand the varianceof the sum of independent
binomial random variables in the numeratoris estimated by the number of deaths: the
Poisson approximation,we would say. We returnto this more specificallybelow.
Yule added a reference in proof to the paper by Pearson & Tocher (1916) which was
concerned with comparing the death-rates of two age-stratifiedsamples. That paper
certainly deserves more recognition than it seems to have in current survival analysis
literature. Thus, one application of standardizationsuggested there is to derive that
standard age-distribution(direct standardization!)which maximizes the difference in
standardizeddeath rates between the two populations. The paper is however a little
beside our general theme and we shall not commentfurtherupon it here.
8 A modem approachto the expected number of deaths
Following Breslow (1975) and in particularBerry (1983), consider an individualwith
death rate (hazard)A(t); that is, survivorshipfunction,
S(t)= exp(- A( ds).
In the conditionaldistributionof the age of death T given that T > u, the probabilitythat
T ~ t (t>u) is
p = [F(t) - F(u)l/[1 - F(u)l = 1 - exp - A(s) ds ,
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16
N.
KEIDING
and furthermore,as can be seen in an elementaryway by partialintegration,E(A) =p,
where the exposure to death, or 'expected numberof deaths', in (u, t], is given by
A = fAt(s)
ds.
Note that this is a randomvariable. For n independentindividualswith death rate Ai(t),
for i = 1,...
, n, we let Di = I{u, < T-< ti} be the indicator that individual i dies in
(ui, ti];then D = , D, is the observednumberof deaths. Obviously,in the conditionaldistri-
bution given that i was alive at ui, for i = 1,... , n, we have E(D) = pi, Var (D) =
E pi(l -pP), and it follows from the above result that, with A = Ai, E(A) = F pi =
E(D) so that 'the expected numberof deaths' has the same expectationas the observed
numberof deaths. As pointed out by Keiding & Vaeth (1986), this is however only true
when the expectationis calculatedunderthe distributionspecifiedby the (standard)death
rates (A•)enteringin A,. In a proportionalhazardframework,Keiding& Vaeth discussed
the bias resultingfrom the widespreaduse of A to estimate a study populationdeath rate
differentfrom that of the standardpopulation.
It is furthermoreseen (Berry, 1983) that
E[(D - A)2] = Var (D - A) = > pi = E(D),
which yields one exact interpretationof the Poisson hypothesis; this is in fact a very
special case of a rather general connection between counting processes and Poisson
processes;see, for example, Aalen (1978) and Andersen et al. (1982).
However, to returnto the classicalstatement (Yule, 1934) that the denominatorof the
SMR may be assumed approximatelyconstant when deaths are rare, it is convenient to
specify a proportional hazards assumption. In the present framework n independent
individuals are considered over individual periods of time (ui, ti], considered in the
conditionaldistributiongiven that i was alive at ui, for i = 1,... , n, but with the same
death intensity OA(t),where A(t) is assumedknown.
It is then readilyseen that the likelihood functionis
so that
L(O)
=
D log L(O) = DIO - A,
CODe-OA
D2 log L(O) = -D/2,
implying that the maximumlikelihood estimator of the relative mortality 0 is given by
=
and that a large samplevarianceestimateof 0 is D/A2. It is interesting
0 DIA, the SMR,
this
estimate
is identical to Yule's (and hence Westergaard's)suggestion,
that
variance
it
is
derived
without the 'Poisson assumption' implying a constant
even though
SMR
and
a
rare
denominatorof
events approximationof the distributionof the numerator.
ElaboratingBerry's (1983) approach,one may note that (for one individual)
Var (A) = 2(q log q + p) -p2 = p3/3 + O(p4),
Cov (D, A) = q log q +p -p2 = -p2/2 +p3/6 + O(p4);
showing first that the variability of A may in fact be ignored for small p, and secondly that
there is a slight but negative correlation between numerator and denominator of the SMR,
deriving from the fact that a higher D means shorter time lived, that is, a lower A.
Working in a similar framework to Berry (1983), Hill, Laplanche & Rezvani (1985)
suggested using the expected number of deaths for the construction of an 'expected
survival curve'.
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Method of ExpectedNumberof Deaths, 1786-1886-1986
17
9 Statisticaltheory of the standardizedmortalityratio
A more direct comparisonwith the classicalmethod of standardization,comparedwith
? 2, is obtained by assuming the standard death intensity A(t) piecewise constant,
assumingthe values , .. , .kover k age groups.
A1,now specifiesthe study group mortalityas •xiin age group i, and
The statisticalmodel
the likelihood approach in this framework has been discussed by Kilpatrick (1962),
Breslow & Day (1975) and in a very detailed survey by Hoem (1987). The multiplicative
model dates back at least to Kermack,McKendrick& McKinlay(1934); see also Liddell
(1960).
Alternative variance approximations,judged by comparison to the so-called 'exact'
results based on the Poisson distribution,were given by Chiang (1961), OPCS (1978, p.
14f.), based upon Bailar & Ederer (1964), and surveyed, with furthercontributions,by
Breslow & Day (1985). Other recent contributionsshowingthat the 'Poissonhypothesis'
is alive and well are by Vandenbroucke(1982), Liddell (1984) and Gardner (1984, e.g.
p. 47).
In derivingthe SMRas maximumlikelihood estimatorwe assumed the standarddeath
rates A•,
, Ik to be known. A generalizationto unknownk, is obvious. For individualj
.?.? i, let Dijbe the indicatorthatj dies while in i; let P,, be the total time lived
and age group
by j in i; and let DO and PO respectively denote the correspondingquantities in the
standardpopulation. On the proportionalhazardsassumptionthe likelihood function is
now
L(O; •,..
.,
k)'
=
If
(OAi) ije -?19'PiP(1)iA
QJe-`iP
HH
i
j
exp [- (OBP+ P9+
= GD(171ADi+)
leading to the likelihood equations
D
0=
D. + D9
Ai=(ioP;
+
P9
(9.1)
,
=
1,...,
k).
(9.2)
As has been noted by other authors, it is instructive to record the intuitive
interpretationof the first iterationsteps in the solution of these equations. Let first 0 = 1
in (9.2), then the first estimate A! of Ai,is the ordinaryoccurrence/exposurerate in age
group i, for i = 1,... , k. Substitutingthis into (9.1) yields as first estimate 01 of 0 the
as standard.
SMR, using the joint empiricalrates kA)
It is obvious that this statistical analysis is a simple example of generalized linear
models (McCullagh& Nelder, 1983). Papers on mortalityrate analysisalong these lines
are by Gail (1978), Breslow et al. (1983) and Frome (1983). In particular,this approach
makes it possible to analyse the dependence of relative mortality on other covariates,
both in the case of known and of unknownstandardmortality.
10 Regression models for relative mortalityin continuoustime, 1986
The method of expected number of deaths is closely connected to the estimation of
relative mortality. We shall close the paper by briefly mentioning some recent
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18
N.
KEIDING
developments in regression models for relative mortality. A particularlyflexible framework consists in assumingthat the relative mortality0(t) is an arbitraryfunction of age,
so that the study population death intensity may be assumed to be of the form 0(t)Z(t),
where A(t) is the known standard mortality. With the further generality of allowing
log-linear dependence on (possibly age-dependent) covariates z(t) = (z'(t),...,
zP(t)),
Andersen et al. (1985) noted that this model may be viewed as a particularcase of the
regressionmodel for survivaldata proposed by Cox (1972). In fact, let z,(t) denote the
covariatevector for individuali at age t, then the death intensityfor individuali at age t is
A(t)O(t) exp [P'zi(t)] = 6(t) exp [P'zi(t) + log A(t)],
with A(t) known. This is the standardCox model except that the regression coefficient
is known and equals 1. Note that
correspondingto the time-dependentcovariatelog
A,(t)
death intensitiesA•,(t);for example,
there is no difficultyin assumingindividualstandard
sex-, calendartime- or region- dependent.
Andersen et al. (1985) illustratedthe methodology by an applicationto the relative
mortality(comparedto Danish nationalmortalitystatistics)of 2193 Danish diabeticsover
a 50-year period (Green et al., 1985), utilizing the kernel estimation methods of
Ramlau-Hansen(1983) for obtainingplots of estimated 0(t). Breslow & Langholz(1986)
gave a detailed furtherdiscussion,includingconsiderationof possibilitiesof reducingthe
computationaleffort in large cohorts. Their study was illustratedby applicationto the
relativelung cancer mortality(again comparedto relevantnationalmortalitystatistics)of
cohorts of 8014 Montanasmelter workersand 679 Welsh nickel refineryworkers.
Acknowledgements
I am very grateful to Anders Hald for several discussions and suggestions concerning the historical
developments,and I also thank ErnstLykke Jensen, BernardJeune, Ian Sutherlandand an anonymousreferee
for useful references.Severallibrariesin Copenhagen(the Royal Library,DanmarksStatistik'sLibrary,and the
libraryat the Laboratoryof ActuarialMathematics,Universityof Copenhagen)efficientlyhandledmy inquiries.
The paperwas completedduringa vist to the Departmentof Biostatistics,Universityof Washington,Seattle,
whose hospitalityI gratefullyacknowledgealong with useful discussionswith Norman E. Breslow. Special
thanks are due to the Librariesat the Universityof Chicago and Washingtonfor providingthe possibilityof
studyingDale's books (incidentally,the following English librariesdo not have the Supplement(1977): The
BritishLibrary;BodleianLibrary,Oxford;CambridgeUniversityLibrary;Royal StatisticalSocietyLibrary;The
Instituteof ActuariesLibrary).Finally, I acknowledgeCatherineHill's help with the Frenchtranslationof the
summary.
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Yule, G.U. (1934). On some pointsrelatingto vital statistics,more especiallystatisticsof occupationalmortality
(with discussion).J.R. Statist.Soc. 97, 1-84.
R6sume
La m6thodedu nombrede d6ces attendusest partieint6grantede la standardisationdes taux vitaux, qui est
l'une des plus anciennestechniquesstatistiques.Le nombrede d6chsattendus6tait calcul6au XVIIIe sidcleen
math6matiquesd'assurance(Dale, 1777;Tetens, 1786). Donc, la m6thodesemble oubli6e pour 8tre retrouv6e
au XIXe si•cle en 6tudes des variationsgdographiqueset professionellesde la mortalit6(Neison, 1844:Farr,
1859;Westergaard,1882;Rubin& Westergaard,1886). On note que la standardisationdes taux est 6troitement
li6e l'6tude de la mortalit6relative. Les d6veloppementsr6centsde la m6thodologiedans ce domaine sont
bribvementdecrits.
[ReceivedDecember1985, revisedFebruary1986]
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