Table Of ContentMathematicalPopulationStudies,12:183–198,2005
Copyright#Taylor&FrancisInc.
ISSN:0889-8480print=1543-5253online
DOI:10.1080/08898480500301751
Comparison of Four Methods for Estimating Complete
Life Tables from Abridged Life Tables Using Mortality
Data Supplied to EUROCARE-3
P. Baili
A. Micheli
Unit ofDescriptive Epidemiology and HealthPlanning, Istituto perlo
Studio e laCura dei Tumori, Milan,Italy
A. Montanari
Department of Statistics, Universityof Bologna,Italy
R. Capocaccia
National Centerof Epidemiology, HealthSurveillance and Promotion
Department, Istituto Superioredi Sanita`,Rome,Italy
Toestimatemortalityduetocancer,itisnecessarytohavemortalitydatabyyear
ofageinthepopulationofcancerpatients.Whensuchdataarenotavailable,esti-
matingone-year(complete)lifetablesfromfive-year(abridged)lifetablesisneces-
sary. Four such methods—Elandt–Johnson, Kostaki, Brass logit, and Akima
splinemethods—arecomparedwithrespectto782empirical completelifetables
pertainingto19Europeanregionsorcountries,from1954to2000.Abridgedlife
tablesarefirstderivedfromtheempiricalones,thenusedtoproduceone-year-life
tablesbyeachofthefourmethods.Thesereconstitutedcompletelifetablesarethen
compared with the empirical complete life tables. Among the four methods, the
Elandt–Johnson demographic method produces the best reconstitutions at
adultages,specificallythoseagesatwhichobservedcancersurvivalneedstobe
corrected.
Keywords: abridgedlifetables;completelifetables;EUROCARE-3;generalmortality
INTRODUCTION
Cancerrelativesurvivalistheratiooftheobservedsurvivalincancer
patients to the survival over the same period in the age- and
AddresscorrespondencetoDr.PaoloBaili,IstitutoNazionaleperloStudioelaCura
deiTumori,ViaVenezian1,20133Milan,Italy.E-mail:[email protected]
183
184 P.Bailietal.
sex-matched general population (Capocaccia et al., 2003; Hakulinen,
1982). General survival, usually available in life tables, must be
known in order to estimate relative survival (Micheli et al., 1999,
2003). EUROCARE-3 is a European Commission project to estimate
relative cancer survival in participating European cancer registries.
All registries participating in EUROCARE-3 (Capocaccia et al.,
2003) therefore provide information on general mortality for their
areas. For the purposes of calculating relative survival, complete life
tables for men and for women, for each calendar year from 1978 to
2000,arerequiredforeachregistryarea.However,themortalitydata
provided vary considerably in detail, and for many registry areas it is
necessary to estimate complete general mortality rates from limited
data using mathematical interpolation and extrapolation procedures
(Micheli et al., 2003).
Alifetableisconsideredcompletewhenitcontainsmortalityinfor-
mation for each year of age between 0 and x (in our case x¼99);
otherwise the life table is abridged. We compare four methods of
estimating complete life tables from abridged ones. We start with
authentic complete life tables for both sexes, for the period 1954 to
2000 inclusive, from 19 European populations. From these complete
life tables we extract abridged life tables in which mortality infor-
mation is grouped into five-year age classes. We then apply each of
the methods to estimate complete life tables from the abridged ones
and compare the result with the original complete data. The methods
usedareElandt–JohnsonandJohnson(Elandt–JohnsonandJohnson,
1980), Kostaki (Kostaki, 2000), Brass logit (Brass, 1971), and Akima
(Akima, 1991).
We define:
q : death probability at age x from complete life tables
x
q :deathprobabilityinageinterval[x,xþn)fromabridgedlifetables
n x
l : total number of survivors at age x
x
Al : total number of survivors at age x from abridged life table
x
Cl : total number of survivors at age x from complete life table
x
d : total number of deaths at age x from complete life tables
x
d :totalnumberofdeathsinageinterval[x,xþn)fromabridgedlife
n x
tables
e : life expectancy at age x.
x
METHODS
TheKostakiandBrassmethodsuseinformationfromotherlifetables
(called standard life tables and defined a priori) in order to estimate
the complete life tables. These standard life tables could be either
(a) life tables of the region or country including the registry area
FromAbridgedtoCompleteLifeTable 185
studied, for each calendar year, or (b) life tables for the registry area
but for a different calendar year.
A. The Elandt–Johnson Method
This method yields a complete life table from an abridged one using
‘‘smoothing’’ formulae and three interpolation schemes depending on
agex(Elandt–JohnsonandJohnson,1980).Thestartingpointisthetotal
number of survivors Al at age x, obtained from the abridged life table,
x
fromwhichthesurvivorsofthecompletelifetableCl areestimated.
x
Forages0–9thecoefficientsinTable1andAl arecombinedlinearly.
x
Clearly Cl ¼Al . If Al ¼Al , there are no deaths in the interval
5 5 x xþ5
[x,xþ5),andinthecompletelifetablethedeathprobabilityq iszero
x
at each age.
TABLE 1 Coefficients Used toCalculate Cl (cid:1)Cl
2 9
Al Al Al Al Al Al
1 5 10 15 20 25
Cl2 0.562030 0.717600 (cid:1)0.478400 0.283886 (cid:1)0.100716 0.015600
Cl3 0.273392 1.047199 (cid:1)0.531911 0.299200 (cid:1)0.103747 0.015867
Cl4 0.096491 1.108800 (cid:1)0.328533 0.172800 (cid:1)0.058358 0.008800
Cl6 (cid:1)0.041667 0.798000 0.354667 (cid:1)0.152000 0.048000 (cid:1)0.007000
Cl7 (cid:1)0.048872 0.561600 0.665600 (cid:1)0.240686 0.072758 (cid:1)0.010400
Cl8 (cid:1)0.037281 0.333200 0.888533 (cid:1)0.244800 0.070147 (cid:1)0.009800
Cl9 (cid:1)0.018379 0.140800 1.001244 (cid:1)0.160914 0.043116 (cid:1)0.005867
Al :totalnumberofsurvivorsatagexfromavailableabridgedlifetable.
x
Cl : total number of survivors at age x from complete life table to estimate
x
Source:Elandt–Johnson,1980.
For ages 10–74 the coefficients in Table 2 and Al are combined
x
linearly.
TABLE 2 Coefficients Used toCalculate Cl (cid:1)Cl
10 74
Al Al Al Al Al Al
5m(cid:1)10 5m(cid:1)5 5m 5mþ5 5mþ10 5mþ15
Cl5mþ1 0.008064 (cid:1)0.07392 0.88704 0.22176 (cid:1)0.04928 0.006336
Cl5mþ2 0.011648 (cid:1)0.09984 0.69888 0.46592 (cid:1)0.08736 0.010752
Cl5mþ3 0.010752 (cid:1)0.08736 0.46592 0.69888 (cid:1)0.09984 0.011648
Cl5mþ4 0.006336 (cid:1)0.04928 0.22176 0.88704 (cid:1)0.07392 0.008064
.m¼2 fo.r .cl11(cid:1)cl14
. . .
where . . .
m¼14 for cl70(cid:1)cl74
Al5mþj:totalnumberofsurvivorsatage5mþjfromavailableabridgedlifetablewith
j¼(cid:1)10;(cid:1)5;0;5;10;15.
Cl5mþI:total number ofsurvivorsatage 5mþi fromcompletelifetabletoestimate
withi¼1;...;4Source:Elandt–Johnson,1980.
186 P.Bailietal.
For ages 75 and above, the Elandt–Johnson method uses the
Gompertz survival distribution:
SðxÞ¼eRað1(cid:1)eaxÞ ¼b1(cid:1)cx where x>0;R>0;a>0;b¼eRa and c¼ea
ð1Þ
with age x and parameters a and R.
The parameters b and c (functions of parameters a and R) are
estimated for each age x using Eq. (2). Starting from the logarithms
of the ratio of the two adjacent values of Al , this system yields the
x
parameters bx and cx for age x. The estimates stop at bbbx and ccbx when
x¼X(cid:1)10 where X is the oldest age in the abridged life table for
which Al is available.
x
8><ccbx ¼(cid:1)yy12(cid:2)(cid:1)1=5 where8>><yy1 ¼¼llooggf10gAAAllxlxþxþ55 ð2Þ
>:bbbx ¼10cc^xxðcc^y5x1(cid:1)1Þ >>:x2¼75;8f010;g.A.l.xþ;1X0 (cid:1)10
After estimating these parameters we calculate the survivals
according to Eq. (1):
1(cid:1)cc^xþi i¼1;...;4 forx¼75;80;...;X(cid:1)15
SSbðxþiÞ¼bbb x where
x
i¼1;...;ð109(cid:1)XÞ forx¼X(cid:1)10 ð3Þ
Starting from the survival SSbðxÞ we estimate the total numbers of sur-
vivors Cl in the complete life tables:
x
SSbðxþiÞ i¼1;...;4 forx¼75;80;...;X(cid:1)15
Cl ¼Al where
xþi x
SSbðxÞ i¼1;...;ð109(cid:1)XÞ forx¼X(cid:1)10 ð4Þ
For example:
ð1(cid:1)cc^89Þ
SSbð89Þ bbb 85
Cl ¼Al ¼Al 85 for X¼90; 95;or 100
89 85SSbð85Þ 85bbbð1(cid:1)cc^8855Þ
85
ð1(cid:1)cc^89Þ
SSbð89Þ bbb 75
Cl ¼Al ¼Al 75 for X¼85
89 75SSbð75Þ 75bbbð1(cid:1)cc^7755Þ
75
Appendix A gives an application of this method in adult ages.
FromAbridgedtoCompleteLifeTable 187
B. The Kostaki Method
Kostaki (Kostaki, 2000, 2001) gives a simple non-parametric method
relating the n-death probabilities q of an abridged life table to the
n x
one-year death probability of a standard complete life table defined
a priori.
Thehypothesisofthismethodisthatineachageinterval½x; xþnÞ,
theforceofmortalityl oftheabridgedlifetableisaconstantmultiple
x
of the force of mortality Sl of the standard life table in the same age
x
interval. We therefore estimate a constant K for each age interval
n x
½x; xþnÞ using:
lnð1(cid:1) q Þ
K ¼ n x ð5Þ
n x
n(cid:1)1
P lnð1(cid:1)Sq Þ
xþi
i¼0
where q istheprobabilityofdeathfromtheabridgedlifetableforthe
n x
interval½x; xþnÞandSq istheprobabilityofdeathfromthestandard
x
complete life table at age x, with x2½x; xþnÞ.
The complete death probabilities are estimated as follows:
q ¼1(cid:1)ð1(cid:1)Sq ÞnKxx^ ð6Þ
x x
using:
K for x2½1;4(cid:2)
4 1
K for x2½5;9(cid:2)
5 5
. . .
. . .
. . .
K for x2½95;99(cid:2)
5 95
C. The Brass Logit Method
Thismethod(Brass,1971)isusedtoestimatelifetablesinpopulations
for which only two data points are known. It assumes the linear
relation (7) between the ‘‘logit’’ functions (8) of the survivors of the
reallifetablel andthestandardlifetable l (definedaprioriforeach
x S x
age x).
logitð1(cid:1)l Þ¼aþb logitð1(cid:1) l Þ ð7Þ
x S x
(cid:3) (cid:4)
1 1(cid:1)l
with logitð1(cid:1)l Þ¼ ln x ð8Þ
x
2 l
x
188 P.Bailietal.
To determine the parameters a and b with the two simultaneous
linear equations, only two values of the independent variable
logitð1(cid:1)l Þ need to be known. We choose l and l . After estimating
x 1 50
theparametersaandb,allsurvivorsl arereconstructedusingEq.(9)
x
derived from Eq. (8):
1
l ¼ ð9Þ
x 1þe2ðaþblogitð1(cid:1)SlxÞÞ
D. The Akima Method
Akima suggests two techniques (Akima, 1970, 1991) for the interp-
olation of univariate single-valued functions. Both techniques are
implemented inthe softwareR2.0.1 under thenamesof‘‘Akima orig-
inal’’ (Akima, 1970) and ‘‘Akima improved’’ (Akima, 1991). We choose
the Akima improved technique as it gives better results for our pur-
poses (data not shown). The Akima spline interpolates the total num-
berofsurvivorsAl atagexwithapiecewisefunctioncomposedofaset
x
of third-degree polynomials (Akima, 1991).
If Al ¼Al , the interpolated distribution of the total number of
xx^ xx^þ5
survivors has the same Cl ¼Al for each age x in the interval
x xx^
½xx; xxþ5Þ.
b b
DATA
Westartwith782authenticcompletelifetablesforbothsexes,forthe
period 1954 to 2000 inclusive, and from 19 European countries or
regions. TheselifetablesarearchivedintheEUROCARE-3database,
but their original sources are mainly the national statistical offices of
the countries participating in EUROCARE.
For two countries, Norway and Sweden, the life tables are grouped
intotwosetsofcalendaryears(1954–1975and1976–1997forNorway,
1958–1977 and 1978–1999 for Sweden) in order to compare the four
methods in time.
For the Kostaki and Brass methods, which require standard life
tables, we use standard life tables of the same population and sex,
and the calendar year (or group of years) immediately succeeding
theyearofinterest,exceptforthelastyearwhereweuseasstandard
the year or group of years immediately prior to that year. For the
ItalianareasweuseasstandardlifetablesthoseforthewholeofItaly
for the same calendar year as that studied.
ThechoiceofstandardlifetableismoreimportantintheBrasslogit
becauseitreliesononlytwodatapointsfromtheoriginalabridgedlife
FromAbridgedtoCompleteLifeTable 189
table,whereastheKostakimethodfeelstheeffectofthischoicelessas
the data for each five year interval of the abridged life table are
coupled with those from the standard life table in the same interval.
In fact the results of the Brass logit are highly sensitive to the stan-
dard life table used, while the results obtained with Kostaki method
are less sensitive to the choice of standard life table.
RESULTS
The death probabilities, for each year of age, estimated by the four
methodsarecomparedwiththoseoftheauthenticcompletelifetables,
using the measure:
(cid:5)(cid:5)Yqqbx(cid:1)qx(cid:5)(cid:5): ð10Þ
whereYqq isthedeathprobabilityatagexestimatedbymethodYand
bx
q is the real death probability at age x.
x
From the comparison in Table 3 of the four methods applied to 782
European life tables in terms of the minimum value of Eq. (10), the
Elandt–Johnson method is the best, in that its estimates are closest
tothetruemortalityfor28.1%ofallages.ItisfollowedbytheKostaki
method (26.8%) and the Akima method (26.7%). Comparing the
Elandt–JohnsondirectlywiththeKostakimethod(columnA1inTable3)
and then directly with the Akima method (column A2 in Table 3),
the former comparison provides mortality estimates closer to reality
for 56% of ages and the latter for 52%. Furthermore, for national
populations the Elandt–Johnson gives better results in comparison
with the Kostaki method. In Norway and Sweden, where we compare
the situations before and after 1975 for Norway and before and after
1978 for Sweden, the Elandt–Johnson method gives the most accurate
results. Another result from Table 3 is that there are no considerable
differences between the methods in estimating survivals in men and
women.
When local Italian areas are considered individually, the Kostaki
method gives the best results, while the Elandt–Johnson and Akima
methods have similar results.
The Elandt–Johnson method is compared with Kostaki method in
Table 4andwithAkimamethodinTable5bythepercentagesofages
for which the former is better than the latter in both tables. Once
again Table 4 shows that for all local Italian populations considered,
mortalities estimated by the Kostaki method are always closer to real
mortalities than those estimated by the Elandt–Johnson method. For
national populations the Kostaki method is better for age classes 1–5
es.en % 484956574852464552525257514851505251505248504954
anLifeTablwhichaGiv Elandt-JohnsonvsAkima(A2) 10381073122512511902079649421026102011411234101295710189821022101193498110871146391429
opefor % 626263615558525846435052464345434245424359605351
782EurofAges) andt-nsonvsaki(A1) 1340134713631322219229107612159108531097112490484289784982588378580513391362420405
toal Elohost
dt JK
eo
ppliofT % 293128253129293122222523232423242024242131292926
ablesAentage Akima(D) 640666601551123113608649444445535500457469461473394469445398707657233208
Tc
eer % 1818971311181725241918262524222624192018171413
LifdP s 564235585831460173806639
CompleteAges(anShown Bras(C)% 243924382519271628529428372535304932472942294030513149314833433351314736343737224024392811339
matingmberofalityis Kostaki(B) 519519538578112114576519599638624639594620614653653612681691511548225258
MethodsofEstin,theTotalNutheTrueMort Elandt-Johnson(A)% 624296072884539887411082712431520255532744222419215962763829415213952042521423214162142622407224222265329676302212822729
Fourulatioateto (cid:3)bN 21782178217821783963962079207919801980217821781980198019801980198019801881188122772277792792
sofPoptim 9 9 8 7 9 7 7 7 7 0
nofResultforEachClosestEs aPeriod 1978199... 1978199... 1978199... 1978199... 1978199... 1978199... 1978199... 1978199... 1978199... 1976–1999 1978200... 1976–1999
TABLE3ComparisoForEachMethodandMethodProvidedthe Population Denmark-MenDenmark-WomenEstonia-MenEstonia-WomenFinland-MenFinland-WomenItaly-MenItaly-WomenFlorence(I)-MenFlorence(I)-WomenGenoa(I)-MenGenoa(I)-WomenMacerata(I)-MenMacerata(I)-WomenRomagna(I)-MenRomagna(I)-WomenTurin(I)-MenTurin(I)-WomenVeneto(I)-MenVeneto(I)-WomenNetherlands-MenNetherlands-WomenEindhoven(Nl)-MenEindhoven(Nl)-Women
190
525053535054585965645352535250535150 52 aki, ma,
st ki
o A
283534446347984548 2 K (
1131091161141071157475252511010711511296101110109 3970 hod( thod
t e
e m
Norway-Men19541975217859827532244352061328127158...Norway-Women217861128534254051962829126758Norway-Men19761997217863829537254141958927132461...Norway-Women217865630482224091963129137263Scotland-Men19781999212856427579273831860228121957...Scotland-Women212859928505244131961129133063Slovakia-Men1981–19981287555432642110783612885967Slovakia-Women1287566442742111393342687468Basque(E)-Men1975–19963962195554141331102831179Basque(E)-Women396207526817621152928973Granada(E)-Men19781998207960929535263641857127126661...Granada(E)-Women207962630508243631758228130463Navarra(E)-Men19781999217866731492234422057726138964...Navarra(E)-Women217864330586273691758027133461Sweden-Men19581977192550126482253661957630111058...Sweden-Women192554828475253671953528113959Sweden-Men19781999217864630507234111961428129559...Sweden-Women217863029566263531662929131760 Totals7681221552282061527141411820504274268156 aCalendaryearsconsideredinthisstudy:y1y2indicatesalllifetablesfromy1toy2inclusiveareinvestigated....y1–y2indicatesthatnotalllifetablesintheperiody1–y2areinvestigated.bTotalnumberofagesforwhichestimatedlifetablesarecomparedwithreallifetable.(A)NumberofindividualagesforwhichElandt-JohnsonmethodisbestintermsofminimumvalueofEq.(10).(B)NumberofindividualagesforwhichKostakimethod(Kostaki,2000)isbestintermsofminimumvalueofEq.(10).(C)NumberofindividualagesforwhichBrassmethod(Brass,1971)isbestintermsofminimumvalueofEq.(10).(D)NumberofindividualagesforwhichAkimamethod(Akima,1991)isbestintermsofminimumvalueofEq.(10).(A1)NumberofindividualagesforwhichtheElandt–Johnsonmethod(Elandt–Johnson,1980)isbetterthanKostakim2000)intermsofminimumvalueofEq.(10).(A2)NumberofindividualagesforwhichtheElandt–Johnsonmethod(Elandt–Johnson,1980)isbetterthanAkima1991)intermsofminimumvalueofEq.(10).
191
192 P.Bailietal.
TABLE 4 Comparison ofElandt–Johnson and KostakiMethods by Age
Classes.TheTableShowsforEachPopulationConsidered,thePercentagesof
Individual Ages, GroupedintoAge Classes, forwhich the Elandt–Johnson
Methodis Betterthanthe Kostaki Method
No.of Ageclass(%)
individual
Population Perioda yearsb 1–5 6–10 11–20 21–50 51–75 76–99
Denmark-Men 1978...1999 22 37 46 54 66 64 65
Denmark-Women 22 49 56 60 62 65 62
Estonia-Men 1978...1999 22 48 44 35 70 71 64
Estonia-Women 22 37 43 40 66 58 74
Finland-Men 1976–1999 5 16 24 28 65 61 61
Finland-Women 5 44 40 52 68 64 53
Italy-Men 1978...1998 21 15 34 31 62 70 40
Italy-Women 21 36 50 47 70 71 42
Florence(I)-Men 1978...1997 20 45 46 54 46 50 39
Florence(I)-Women 20 34 40 47 46 46 37
Genoa(I)-Men 1978...1999 22 39 32 39 51 54 57
Genoa(I)-Women 22 40 41 44 51 53 59
Macerata(I)-Men 1978...1997 20 48 53 52 47 49 36
Macerata(I)-Women 20 46 52 44 45 43 36
Romagna(I)-Men 1978...1997 20 42 36 41 49 48 42
Romagna(I)-Women 20 38 47 46 43 46 39
Turin(I)-Men 1978...1997 20 42 45 35 42 48 36
Turin(I)-Women 20 45 50 45 47 47 39
Veneto(I)-Men 1978...1996 19 28 26 36 44 40 49
Veneto(I)-Women 19 40 41 42 49 43 36
Netherlands-Men 1978...2000 23 10 55 49 65 64 61
Netherlands-Women 23 26 48 64 63 65 58
Eindhoven(Nl)-Men 1976–1999 8 3 43 40 60 67 49
Eindhoven(Nl)-Women 8 13 38 58 60 64 34
Norway-Men 1954...1975 22 15 41 54 62 64 62
Norway-Women 22 29 54 58 63 59 58
Norway-Men 1976...1997 22 40 55 39 69 67 59
Norway-Women 22 50 59 62 65 65 62
Scotland-Men 1978...1999 22 39 50 44 63 59 59
Scotland-Women 22 54 56 62 65 63 61
Slovakia-Men 1981–1998 13 28 40 24 68 74 89
Slovakia-Women 13 46 48 42 69 68 86
Basque(E)-Men 1975–1996 5 56 28 58 91 94 79
Basque(E)-Women 5 24 20 56 87 89 59
Granada(E)-Men 1978...1998 21 52 55 61 63 64 58
Granada(E)-Women 21 60 61 62 65 63 62
Navarra(E)-Men 1978...1999 22 55 54 56 66 63 69
Navarra(E)-Women 22 51 53 58 65 61 62
Sweden-Men 1958...1977 20 20 49 46 63 65 59
Sweden-Women 20 24 53 57 62 61 63
(Continued)
Description:Comparison of Four Methods for Estimating Complete. Life Tables from Abridged
Life Tables Using Mortality. Data Supplied to EUROCARE-3. P. Baili. A. Micheli.
