Towards advanced methods for computing life tables

8/20/2019 Towards advanced methods for computing life tables

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Working Papers

04 /2010

The views expressed in this working paper are those of the authors and do not

necessarily reflect the views of the Instituto Nacional de Estadística of Spain

First draft: December 2010

This draft: December 2010

Towards advanced methods for computing life

tables

Sixto Muriel de la Riva

Margarita Cantalapiedra Malaguilla

Federico López Carrión


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Towards advanced methods for computing life tables

Abstract

INE Spain puts compiled data and literature in a series of tests and comparative analyses

in order to select the best-suited methodologies for computing life tables, trying to reach

an optimal use of the available data on deaths, better international comparability of

mortality indicators and an enhanced approach to measuring mortality over small

territorial areas.

Keywords

Mortality tables; risks of dying; regional mortality tables; different methods to build

mortality tables

Authors and Affiliations

Sixto Muriel de la Riva, Margarita Cantalapiedra Malaguilla and Federico López Carrión

Directorate of Population Statistics, National Statistics Institute of Spain

INE.WP: 04/2010 Towards advanced methods for computing life tables


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To w a r ds a d v a nce d m e t ho d s f o r co m p ut i n g l i f e t a b l e s

S i xt o M ur ie l d e l a R iv a, M ar ga ri ta C a nt al ap ie dr a M al ag ui ll a and Federico López

Carrión

D i re c to r at e o f P o pu l a ti o n S t at i s ti c s, Na t io n al S t at i s ti c s I n st i t ut e o f S p ai n

Abstract L if e t ab le s ar e, o f c o ur s e, a l on gs ta nd in g t as k of go ve rn me nt st at is ti cs . N o w, a

m a j o r e f f or t of r es e a r ch a nd m et h o d ol o g i c al i n n o va t i o n s h o u l d o p e n t h e w a y t o

o pt im al u ti li sa ti on of th e a va il ab le d at a o n d ea th s, b ett er in te rn at io na l

c o mp a ra b il i ty o f m o rt al i ty i n di c a to r s, a nd a f i ne r a p pr o ac h t o t h e m e as u re m en t

o f m o r t al i ty o v er s m al l t e rr i to ri a l a r ea s .

I N E S pa i n pu t s c o m p i le d da t a a n d l i t er a t u r e i n a s e r i es o f te s t s a n d c o m p a ra t i ve

a n al y s es in or d er to s e l ec t t h e b e st -s u i te d m e th o do l og i es fo r t h e m e nt i on e d

o bj ec ti ves . The o utc om e o f t hi s tes ti ng s tage was a two -pa rt r ep or t on o ur

m e t h od o l og i c a l r e s e ar c h on l i f e t a b le s .

T h e f i r s t p a r t t r i e s s h o ws t hr e e d i f f er e n t a p p ro a c h es t o c o n s t r uc t i n g a c o m p le t e

a nn ua l l if e t ab le f or t he S pa in r es id en t p op ul at io n, i n a g ro wi ng s op hi st ic at io n

s c al e: t he f ir st o ne a ss um es t h at d ea th s ov er t he r ef er en ce y ea r at e ac h a ge a re

d is t ri bu te d a bs ol ut el y u ni fo rm ly ; t he s ec o nd , a ss u me s u ni fo rm d i s tr ib ut io n o f

d ea th s o ve r t he y ea r a t ea ch a ge w it hi n a g iv en g en er at io n (u ni fo rm it y i n L ex is

t ri an gl es ), f ol lo wi ng t h e p ro to co ls o f t he H um an M or ta li ty D at ab as e 1

; the t hird

a n d g en u in e m e th o d u s es ob s er v at i on a l d at a f r om th e v i ta l s t at is t ic s o n t h e d a te

o f o c c u r re n c e o f e a c h r e c o r de d d e a t h t o ge t t h e b e s t a p p ro a c h t o p o p ul a t i on a t

r i s k s i n s pe c i f ic m o r ta l i t y r a t es .

B e si d es , t h is pa r t s h ow s t he c o ns i d er a bl e d i st a nc e b et w ee n u n if o rm i ty a n d t h e

t e mp o ra l d is t r ib u ti o n ( wi t hi n th e o bs e r ve d y ea r ) of d ea t hs i n lo w a n d a d va n ce

a ge s. I t i s f ol lo we d t ha t t he t hi rd m et ho d p ro vi d es t he m os t a cc u ra te

m e a s ur e m e nt o f th e b i o m et r i c fu n c t io n s o f a l i f e t a b le .

T he s ec on d p ar t o f t he r ep or t f oc us o n l if e t ab le s f or s ub -n at io na l t er ri to ri es

(d evo lv ed r eg ion s a nd p rov in ces , NU TS -2 a nd NU TS -3) . I t anal ys es t hre e

p os si bl e wa ys t o c on st ruc t lif e ta bles a t d is ag gr egat ed te rr ito rial l eve ls

( a bb r ev i at e d f i ve - ye ar l y a g e r a ng e s l i fe t a bl e s, f u ll l i fe t a bl e s wi t h f i ve y e ar l y a g e

r a n ge s o u t pu t a n d s m o o t hi n g s ur v i v or f u nc t i o n of r e gi o n al l i f e t a b l e a c c o r di n g t o

1 Collaborative project sponsored by the University of California at Berkeley, United States and the Max Planck Institute for

Demographic Research, Rostock, Germany. / www.mortality.org


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n a ti o na l d a ta ) . I t a i ms to pr o vi d e a n u n de r st a nd i ng of th e s h or t -t er m ( a nn u al )

i n ci d en c e o f mo r ta l it y i n e ac h t er r it or y w hi l e m i ni m is i n g t h e e ff e ct o f r e po rt i ng

r a n do m n e ss a n d o f p o s s i bl e i n c o ns i s t e nc i e s b e t we e n o b s e rv e d d e at h s a n d t h e est im at es used as ref erence populat ion f igures. Advant ages and disadvant ages

o f e ve r y p r o po s e d m et h o d ar e d e e pl y a n a ly s e d a nd c o m m e nt e d .


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1 Int ro ductio n

L if e t a bl es c o mp u ti n g c o ns t it u te s a t r ad it i on a l w or k in t h e f i el d of d em o gr a ph ica n al ys i s an d p ro j ec t io n s, f o r t he m a jo r it y o f t he o f fi c ia l s t at i st i ca l o f fi ce s . I n t he

p r es e nt p a p er , s e ve r al m e th od o lo g ic a l o p ti on s f o r t h e c a lc u lu s o f l if e t a bl es a r e

s h o wn an d di s c us s e d, b o t h f o r t h e n a t io n a l a n d f o r t h e r e g io n a l l e v el .

Thi s w ork has the fol l ow ing obj ecti ves:

1 . T h e m a xi mu m u s e o f t h e d e mo g ra p hi c i nf o rm a ti on g e ne r at e d b y t h e S p ai n

r es id en t p op ul at io n S ta ti st ic al S ys te m, s o a s t o o pt im is e t he a cc ur ac y o f

m o r t al i t y m e as u r e me n t s f o r a l l t h e p o p ul a t io n s c o n s id e r ed .

2 . T o a ss ur e t he n ec es sa ry m et ho do lo gi ca l c on si st en cy i n t he d ef in it io ns a n d

t h e a s su m pt i on s u n de r ly i ng t h e c a lc u lu s of al l t h e b io m et r ic f un c ti on s of

t h e c o m p le t e l i fe t a b le , a n d a l s o i n t h e d e f in i t io n o f i t s d e r iv e d i n d ic a t o rs .

3 . A s fa r as p os si bl e, t o r ef le ct a n nu al b eh av io ur o f m or ta li ty i nc id en ce i n t he

S p ai n r es i de n t p o pu la t io n , a v oi d in g t h e i n fl u en c e o f r a nd o mn es s i n

observed data.

4 . T o i mp ro ve t he a cc ur ac y w he n m ea su ri ng m or ta li ty i nc id en ce at th e m os t

a d va n ce d a g es , g iv e n th e p r ev a il in g a g in g p r oc e ss o f t h e S p ai n P o pu la t io n

a n d t h e i m p or t a n t r e c en t a d v a nc e s i n l i f e e x p e ct a n cy a t t h o se a g e s.

5 . T o p ro vi de m ea su re s o f m or ta li ty i nc id en ce f or su b n at io na l p op ul at io ns ,

i .e ., r e gi on s a n d p ro v in c es , i n a r e gu la r b a s is .

6 . T o s up po rt i nt er na ti on al c om pa ra bi li ty o f t he m or ta li ty i nd ic at or s a ndf a c il i ta t e t h e ir i n t er p r e ta t i on b y u s e r s.

T h re e d if f er e nt o p ti on s a r e t e st e d f or t h e c o ns t ru ct i on o f c o mp l et e l if e t a bl es f o r

t h e t o ta l , m a le a n d fe m al e S p ai n re s id e nt p o pu l at io n s. A n ot h er t h re e o p ti on s a re

t es te d f or t he r eg ion al l if e t ab le . A ll o f t he m m ak e us e o f t he r eg is te re d d at a o n

d e at h s o c cu rr e d am on g r es i de n ts i n S pa in d u ri ng a o n e- y ea r i nt e rv a l a n d th e

c or re sp on di ng p op ul at io n s to ck s ex po se d t o r is k o f dy in g. Y ea r 2 00 6 w as u se d

fo r t he n atio na l lif e ta ble s an d ye ars 20 05 a nd 20 06 f or t he r eg ion al an d

p r o vi n c ia l s l i fe t a b le s .


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2 . M et ho ds f or c om pu t in g n at io n al l if e t a bl es

I n t hi s f ir s t pa r t o f t h e p a pe r t h re e m et h od o lo g ic a l a pp r oa ch e s ar e p re s en t ed f o rt h e c o mp ut a ti on o f li f e t a bl es a t th e n a ti o na l l e ve l. D if f er e nc e s a m on g t h e th r ee

o pt io ns a re d ue t o th e t hr ee d if fe re nt h yp ot he si s ma de a bo ut t he t im e l iv ed b y

the observed populati on between ages x and 1x + .

I n th e f i r st o pt i o n, u n i f or m di s t ri b u t io n o f d e a th s i ns i d e t h e a g e i n t er v a l )1x,x( + i s

s u pp os e d. F o r t h e s ec o nd op t io n , b as e d o n t h e p r ot o co l s o f t h e Hum an M ort alit y

Dat abase 1

pr oj ec t, u ni fo rm d is tr ib ut io n wi th in t he t wo c or re spo nd in g

g e n er a t i on s o f a n n u al d e a th s i n t h e a g e i n t er v a l )1x,x( + i s as s um e d. I n th e t h ir d

o pt io n, o bs er ve d d at a o n t he d at e o f o cc ur re nc e o f e ac h r eg is te re d d ea th o f

an nu al V it al St atis tic s ar e us ed. Fo r al l t he opt io ns b ir th d at es ar e s up pos ed

u n if o rm ly d is t ri b ut e d a lo n g ea c h y e ar .

I n g en e ra l , t h e f u nc t io n s o f t h e p e ri od l i fe t a bl e a r e d ef i ne d a s f o ll o ws :

The m o rt a li t y r a te )s,x,t(m , f or ye ar t, a ge x an d s ex s , i s a q uo ti en t w he re t he

numerator i s the number of deaths occurri ng in the year t among indi vi dual s at

age x and s ex s , a nd th e d en om in at or is th e t ot al t im e, i n y ea rs , l iv ed by t he

individuals of age x and sex s , during the year, i.e. , the person-years of

individuals of age x and sex s duri ng year t .

N o w, l et a f ic t it i ou s g e ne r at i on w it h a n i nc i de n ce o f m o rt a li t y a cc o rd i ng to th e

m o rt a li ty r a te s t ha t h as b e en d e fi ne d . S o :

The death probability )s,x(q is th e p ro b ab il it y o f a n i nd i vi d ua l o f s e x s of the

f i c ti t io u s g e n e ra t i on t h a t r e ac h e s t h e a g e x l i ves unti l next bi rthday.

The survivors )s,x(l is th e n u m b er o f in d i vi d u al s f r o m t h e f i c ti t io u s g e n er a t i on

that reach the x a ge f or ea c h s ex s .

The deaths )s,x(d i s t h e n u m be r o f d ec e a se s f r om t h e f i c t it i o us g e n e ra t io n a t

each age x a n d f or e a ch s e x s .

Survivors )s,x(l and deaths )s,x(d are obtai ned recursi vel y:

000.100)s,0(l =

)s,x(q)s,x(l)s,x(d = y )s,x(d)s,x(l)s,1x(l −=+ for += 100,99,...,1,0x

The a v er a ge n u mb e r o f y e ar s l i ve d i n t h e l a st ye a r o f l i fe f or t ho se i nd iv id ua lsd y in g i n t h e a g e i nt e rv a l )1x,x( + i s denoted by )s,x(a .

The s t a t i o na r y p o p u la t i o n )s,x(L i s t he t ot al t im e l iv ed fo r a ll t he i nd iv id ua ls of

the fi cti ti ous generation that are x years ol d and sex s .

1 D e pa r t m en t o f D e mo g r ap h y a t U n iv e r s it y o f Ca l i f o rn i a , B e r k el e y (U S A ) a n d M a x P l a nc k

I n s t it u t e f o r D e m og r a p hi c R e s ea r c h .


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The t ot al y ea rs l iv ed )s,x(T i s th e t ot al a mo un t of ye ar s liv ed f or a ll t he

i ndi vi dual s of the fi cti ti ous generati on aged x o r more.

)s,x,t(L)s,1x,t(T)s,x,t(T ++= for += 100,99,...,1,0x

F in al ly , t he l if e e xp ec ta nc y is the mean number of years left to li ve for an

i nd iv i du al of a ge x a nd s ex s .

)s,x,t(l / )s,x,t(T)s,x,t(e = , for += 100,99,...,1,0x

2 . 1 O pt io n 1 . - A ss um p ti on o f u ni f or m d is t ri bu ti o n o f d e at h s i n t he a g e

i n te r va l ( x ,x + 1 )

T he m ai n ob je ct iv e o f op ti on 1 i s to p er fo rm a s en si ti vi ty a na ly si s, i .e ., t o sh ow

t h e e f f ec t o f t h e a l t er n a t iv e o p t io n s i n c o m p ar i s on t o u n i fo r m d i s tr i b ut i o n.

U nd er t hi s as su mp ti on , t he d en om in at or o f )s,x,t(m i s e st im at ed by t he

expressi on )s,x,1t(P21)s,x,t(P21 ++ , a ss um in g u ni fo rm di st ri bu ti on o f t he

b i r th d a ys o f a l l i n d i vi d u a ls i n t h e p o p u la t i on n o t d y i ng o ve r t he y e a r a t a g i v en

ag e.

Hence:

)s,x,1t(P21)s,x,t(P21

)s,x,t(D)s,x,t(m

++=

Where:

t i s t h e r ef er en ce y ea r, p er io d o r c al en da r t im e.

x i s t h e a g e o r c om p l et e d ye a r s , += 100,99,...,1,0x .

s i s s ex , wh ic h ma y b e m ale , f em ale o r b ot h s ex es .

)s,x,t(D i s th e n u m b er o f d e a th s b et w ee n r es i d en t s i n Sp a i n d u r in g t he y e a r t , a t

age x and sex s .

)s,x,t(P i s the stock of resi dent populati on i n Spai n at 1s t

Ja nu ar y o f t he y ea r t ,

with age x a nd s ex s .

I n ad di ti on , f or t hi s op ti on t he f ol lo wi ng k e y e xp re ss io ns f or s om e o f th e

bi ometri c functi ons are adopted:

T h e e s t im a t ed p r ob a bi l i ty o r r i s k o f d y in g )s,x(q , w il l be :

)s,x(m)21(1

)s,x(m)s,x(q

⋅+= 99,...,1,0x = and .1)s,100(q =+

The s t a t i on a r y p o p ul a t i on at age x and sex s , for 99,...,1,0x = , o n t he as su mp ti on

of uni form di stri buti on of deaths over the year, i s esti mated as:


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)s,x(d2 / 1)s,1x(l)s,x(L ++= for 99,...,1,0x = .

F in a ll y , t h e a g gr e ga t e o f ye a rs l iv e d b y i nd i vi d ua ls o f th e f ic t it io u s g e ne r at i onr e a ch i n g t h e a g e o f 1 00 , f r o m t h a t a g e t o t he i r d e a t hs , i s ap p r o xi m a te d by :

)s,100(m

)s,100(l)s,100(L

+

+=+

2 . 2 O pt i on 2 . - A s su m pt i o n o f un i fo r m d i s tr i b ut i o n o f de a t hs b y ag e an d

generation

T h is a lt e rn at iv e f o ll ow s th e p r ot o co ls o f H u ma n M or t al i ty D a ta b as e 1

for the

c on st ru ct io n o f l if e t ab le s. I n t hi s ca se , t he d en om in at or o f )s,x,t(m is estimated

a s s um i n g u n if o r m d i st r ib u t io n o f d e a t h s a m o ng i n d i vi d u al s o f t h e s a m e a g e a n dg e n er a t i on ( L ex i s t r ia n g ul e ) b y t h e e x p re s s io n :

))s,1g,x,t(D)s,g,x,t(D(6 / 1)s,x,1t(P2 / 1)s,x,t(P2 / 1 −−+++

Where:

g i s the generati on of indi vi dual s completi ng x years i n year gxt,e.i,t =−

)s,g,x,t(D i s t h e n u m be r o f i nd i v id u a ls d y i n g i n y ea r t , a t a ge x , f r o m g e n er a t io n

g and sex s .

Hence:

))s,1g,x,t(D)s,g,x,t(D(61)s,x,1t(P21)s,x,t(P21)s,1g,x,t(D)s,g,x,t(D)s,x,t(m

−−⋅++⋅+⋅−+= , f or += 100,99,...,1,0x .

F o r t h is o pt i on th e f o ll o wi ng k ey e x pr e ss i on s f o r s o me o f t h e f un c ti on s a r e

adopted:

T h e e s t im a t ed probability or r i sk o f dy i ng w il l b e o b ta in e d b y:

)s,x,t(m))·s,x(a1(1

)s,x,t(m)s,x(q

−+= for 99,...,1,0x = and .1)s,100(q =+

being a(x,s) approximated by:

)s,1g,x,t(D)s,g,x,t(D

)s,1g,x,t(D·3 / 2)s,g,x,t(D·3 / 1)s,x(a

−+

−+=

F or th e o pe n- en de d g ro up t he a ve ra ge t im e l iv ed in t he r ef er en ce y ea r t i s

esti mated for ages of 100 and above and sex )s,100(a,s + , as:

1

D ep a rt m en t o f D em o gr a ph y a t U n iv e rs i ty o f C a li f or n ia , B e rk e le y ( U SA ) a nd M a x P l an c kI n s t it u t e f o r D e m og r a p hi c R e s ea r c h .


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)s,100(m

1)s,100(a

+=+

2 .3 Opt io n 3 .- Re al d is tri bu ti on o f d at e o f d ea th e ve nts is t ak en i nt o

account

I n t hi s c as e , t h e d e no mi n at o r o f )s,x,t(m c o m e s f r o m t h e e x p r es s i on :

∑+++i

i)s,x,b(t,s)x,1,1/2·P(ts)x,(t,P·2 / 1

Where:

)i,s,x,t(b is de fi ne d a s t h e d i ff e re n ce ( i n y e ar s) b e tw ee n t h e d a te o f d ea t h a nd t h e

d at e o f bi rt hd ay ( in ye ar t ) of each i ndi vi dual i , of sex s , dying in year t atcompleted age x .

H en ce :

∑+++=

i

i)s,x,b(t,s)x,1,1/2·P(ts)x,1/2·P(t,

s)x,D(t,s)x,(t,m , fo r += 100,99,...,1,0x .

F o r t h is o p t i o n th e f o ll o wi n g k e y e x p r es s i o ns f o r s o m e o f t h e b i o me t ri c f u n ct i on s

a r e a d op t ed :

The probability or risk of dying, )s,x(q will be:

)s,x,t(m))·s,x(a1(1

)s,x,t(m)s,x(q

−+=

Bei ng the a ve ra ge n um be r o f y ea rs l iv ed in th e l as t y ea r o f l if e for t hose

i n d iv i d ua l s d yi n g i n t he a g e i n t e rv a l )1x,x( + , )s,x(a , a p pr o xi m at e d b y t h e a v er a ge

o f t he t im e ( ye ar s) l iv ed i n t f o r t he i nd iv id ua l i d y in g i n th e a g e i n te r va l )1x,x( +

d u ri n g t h e o b se r ve d y e a r t , )i,s,x,t(a :

)s,x,t(D

)i,s,x,t(a

)s,x,t(a

)s,x,t(D

1i

∑== , 99,...,1,0x = .

B es i de s , f o r t h e o p en - en d ed g r ou p t h e a v er a ge t im e l iv e d i n t h e r e fe re n ce y e ar t

i s es t i ma t e d f o r a g e s o f 1 0 0 a n d ab o v e a n d se x s , )s,100(a + , as:

)s,100(m

1)s,100(a

+=+


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2 . 4 D i sc us si on

T h e d if f er e nc es a mo ng t he s p ec i fi c m o rt a li t y r a te s c o mp u te d by e a ch o f t h e

a pp ro ac he s t es te d w er e n eg li gi bl e e xc ep t f or t he m os t a dv an ce d a ge s ( ab ov e 8 0

y e a rs o l d ) wh e r e s l i gh t d i f f er e n ce s e m er g e d , b o t h , f o r m a le s a n d f o r f e ma l e s.

m(x)*1000 according to option1, option2 and option3. Males

0,01

0,1

1

10

100

1000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

m (x)option1 m (x)option2 m (x)option3

Differe nces in m(x) according to options 1, 2 and 3. Males

-8

-6

-4

-2

0

2

0 10 20 30 40 50 60 70 80 90 100

o pt io n3 -o pt io n1 o pt io n3 -o pt io n2

Dif ferences in t housandths


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m(x)*1000 according to option1, option2 and option3. Females

0,01

0,1

1

10

100

1000

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96

m (x)option1 m (x)option2 m (x)option3

Differences in m(x) according to options 1, 2 and 3. Females

-3

-2

-1

0

1

2

3

0 10 20 30 40 50 60 70 80 90 100

option3-option1 option3-option2

Dif ferences in t housandths

Menti oned di fferences at advanced ages refl ect that deaths devi ate from uni form

d i s tr i b ut i o n b e t we e n t h e a g e s x and 1x + : i n f a ct , a t a d va nc e d a ge s , d e at h s a re

c o nc e nt r at e d a m on g t he i n di v id u al s o f th e y o un g es t g e ne r at i on o f t he t wo w hi c h

c o mp le t e d e at h s a l t h e a ge i n te r va l )1x,x( + d u r in g t h e y e a r t .

T h e d if f er e nc e s s h ow n a m on g t h e s p ec if i c m o rt a li ty r a te s a re l ik e wi se v is i bl e f o r

t h e p r o b ab i l it i e s o f de a t h.


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q(x)*1000 according to option1, option2 and option3. Males

0,01

0,1

1

10

100

1000

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

q(x)option1 q(x)option2 q(x)option3

Differences in q(x) according to options 1, 2 and 3. Males

-4,5

-4

-3,5

-3

-2,5

-2

-1,5

-1

-0,5

0

0,5

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Differences in

thousandths



13/28

11

q(x)*1000 according to option1, option2 and option3. Females

0,01

0,1

1

10

100

1000

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

q(x)option1 q(x)option2 q(x)option3

Difference s in q(x) according to options 1, 2 and 3. Females

-3,5

-2,5

-1,5

-0,5

0,5

0 5 10 15 20 25 3 0 35 40 45 50 55 60 65 70 75 80 85 90 95 100

opt ion3-opt ion1 opt ion3-opt ion2

N e v er t h el e s s, t h e c a l cu l at i o n o f t he s e p r o b ab i l it i e s o f d ea t h is a l s o i nf l u en c e d b y

d i f fe r e nc e s i n e s t i ma t ed a v e ra g e n u m be r o f y e ar s l iv e d i n t he l a s t ye a r o f li f e f o r

t ho se i nd iv id ua ls d yi ng i n t he a ge i nt er va l )1x,x( + , )s,x(a . In particular , the

d if f er e nc e s a re n ot e wo r th y i n t he a pp r ox im a ti o n by e a c h m et h od o f )s,0(a . N ot

e v en o pt io n 2 p r od u ce s a g oo d ap p ro x im a ti on o f ob s er v ed r ea l it y . T h e m o st

accurate approxi mation i s provi ded by opti on 3.


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12

a(x) according to option1, option2 and option3. Males

0

0,5

1

1,5

2

2,5

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

a(x)

a(x)o pt io n1 a(x)o pt ion2 a(x)o pt io n3

Differences in a(x) according to options 1, 2 and 3. M ales

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Differences in units



15/28

13

a(x) according to option1, option2 and option3. Females

0

0,4

0,8

1,2

1,6

2

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

a(x)

a(x)o pt io n1 a(x)o pt ion2 a(x)op tio n3

Differences in a(x) according to options 1, 2 and 3. Females

-0,3

-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Differences in units


Fin ally , th e dif fer enc es e me rg in g in eac h of t he mo rt ality ta ble f unc ti ons

r e s ul t in g fr o m e a c h m e t ho d h a v e o n l y a n e g li g i bl e e f f ec t on t h e r e s ul t in g va l u es

o f li f e e x p e ct a n c y a t b ir t h , a s s ho w n i n th e f o l lo w in g c h ar t s .


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14

Life expectancy according to options 1, 2, 3. MalesAge

option1 option2 option3 option3-option1 option3-option20 77,589 77,591 77,595 0,005 0,004

5 72,971 72,973 72,978 0,006 0,005

15 63,072 63,074 63,079 0,006 0,005

25 53,407 53,408 53,413 0,006 0,005

35 43,778 43,779 43,784 0,006 0,005

45 34,427 34,428 34,433 0,006 0,005

55 25,668 25,670 25,674 0,007 0,005

65 17,709 17,711 17,715 0,007 0,005

75 10,825 10,828 10,831 0,006 0,003

85 5,830 5,835 5,836 0,006 0,001

95 3,131 3,145 3,146 0,015 0,001

e(x) Differences

Differences in e(x) according to options 1, 2 and 3. Males

-0,010

0,000

0,010

0,020

0,030

0,040

0,050

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Differences in years


Life expectancy according to options 1, 2, 3. FemalesAge

option1 option2 option3 option3-option1 option3-option2

0 84,089 84,091 84,094 0,005 0,003

5 79,418 79,420 79,424 0,006 0,004

15 69,487 69,489 69,492 0,005 0,004

25 59,614 59,615 59,619 0,006 0,004

35 49,773 49,774 49,778 0,006 0,00445 40,125 40,127 40,131 0,006 0,004

55 30,743 30,745 30,748 0,006 0,004

65 21,656 21,658 21,662 0,006 0,004

75 13,280 13,283 13,286 0,006 0,003

85 6,757 6,761 6,762 0,005 0,001

95 3,192 3,201 3,192 0,000 -0,008

e(x) Differences


17/28

15

Differences in e (x) according to options 1, 2 and 3. Females

-0,015

-0,010

-0,005

0,000

0,005

0,010

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Differences in years


O pt io n 3 , s ho ws a s li gh t i mp ro ve me nt i n l if e e xp ec ta nc y a t e ac h a ge , a s a r es ul t

o f th e s li gh t de cr ea se i n mo rt al it y r at es i n th e u pp er a ge s. T he l if e e xp ec ta nc y

g a in s a r e m o re s i gn if i ca n t a t t h os e a g es , w he r e p r es e nt i m pr o ve me nt s o f s u ch

i ndi cator are concentrated.


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16

3 . M et ho ds fo r c om pu t at io n o f l i fe ta bl e s a t l ow er t e rr it or ia l l ev e ls

T he s ec on d pa rt o f th is p ap er a dd re ss es t he q ue st io ns f or a n ac cu ra tem ea su re me nt of mo rt al it y a t l ow er t er ri to ri al l ev el s, t ry in g t o m in im iz e t he

l im it a ti on s or i gi na t ed b y t h e s m al l s i ze o f th e o b se r ve d p o pu la t io n .

T hr ee d if fe re nt o pt io ns a re t es te d t o co ns tr uc t an nu al m or ta li ty t ab le s fo r t he

r e gi on s a n d t he p r ov i nc e s o f S p ai n. F o r t he f ir s t a pp r oa c h, a b br e vi at e d l if e t a bl es

b y f iv e- ye ar a ge i nt er va ls a re c om pu te d. F or t he s ec on d o pt io n in a f ir st s te p,

c o mp le t e l if e t a bl es fo r a ll s u b n a ti on a l p o pu la t io n s c o ns i de r ed a r e c al c ul at e d; i n

a s ec on d s te p, a gg re ga te d r es ul ts by f iv e- ye ar ag e i nt er va ls ar e o bt ai ne d. T he

t h ir d m e th o d c o ns i st s o f a d ju s ti n g t h e s u rv iv o rs s e ri es o f t h e c o mp le t e m or t al it y

t a b le s us i n g t h e Brass logit s m odel 1

t o d er iv e t he r es t of t he l if e t ab le s er ie s.

3 . 1 O pt i on 1 . - A bb r ev i at e d l i fe t a bl e

U n de r t h is o pt i on , t h e d e no m in a to r of th e s p ec i fi c m o rt a li t y r at e , )s.n,x,t(m , i s

c o m pu t e d j u s t a s s um i ng u n if o r m di s t r ib u t io n o f b ir t h d ay s o f a ll i n d iv i d ua l s o f

t he p op ul at io n no t d yi ng o v er t he y ea r at t he a ge x b y m e an s of t he f o ll o wi ng

e x p re s s io n :

∑+++i

i)s,n,x,b(t,s)n,x,1,1/2·P(ts)n,x,(t,P·2 / 1

Where:

)s,n,x,t(P i s t h e s t oc k o f r e s id e n t p o p u la t i on i n th e c o n s id e r ed r eg i o n, a t J a n u ar y

the 1s t of year t , i n t he a ge in te rv al [ )nx,x + and sex s .

)i,s,n,x,t(b i s d e f in e d a s t h e d i ff e r en c e ( i n ye a r s) b e t we e n t he d a t es o f d e at h a n d

b ir th i n t he y ea r t , o f e a ch i nd iv id ua l i , o f s ex s , d ying in ye ar th e t wit h ag e in

t h e i n te r va l [ )nx,x + .

Hence:

∑+++=

i

i)s,n,x,b(t,s)n,x,1,1/2·P(ts)n,x,1/2·P(t,

s)n,x,D(t,s)n,x,(t,m for += w...5,1,0x .

Where:

)s,n,x,t(D D (t , x, n, s ) is t h e n um b er o f d e at hs o f i n di v id u al s w it h a g e i n t he i n te r va l

[ )nx,x + , o f s ex s , a m o ng t h e r e s id e n t p o p u la t i on i n t h e c o n si d e re d re g io n du r i n g

t h e y e ar t .

1 B r as s , W il li a m. M e th o ds f o r e s t im a ti n g f er t il it y a n d m or t al it y f r om l im it e d a nd

d e f ec t i v e d a t a .( 1 9 75 )


19/28

17

T he n, t he f oll owi ng k ey e xp re ss io ns f or so me o f t he b io met ric f un ct io ns ar e

adopted:

The probability or r i s k o f dy i ng, )s,n,x(q is derivated by:

s)n,x,s))·m(t,n,a(x,-n·(11

s)n,x,n·m(t,s)n,(x,q

+= for 99,.,2,1,0x = .

1s),(100q =+

Bei ng he a ve ra ge n um be r o f y ea rs l iv ed in t he l as t ye ar o f l if e for those

i nd iv id ua ls d y in g i n t he a ge i nt er va l )nx,x( + , )s,n,x(a , a pproximat ed by the

a v er a ge o f t he t im e ( y ea r s) l i ve d i n t f o r t h e i n d iv i d ua l i d y in g in t h e a g e i nt e rv a l

)nx,x( + d u r in g t h e o b se r ve d y e ar t , )i,s,n,x(a :

s)n,x,D(t,

i)s,n,x,a(t,

s)n,x,(t,a

s)x,N(t,

1i∑== , 5w,...,5,1,0x −= .

B es i de s , f o r t h e o p en - en d ed g r ou p t h e a v er a ge t im e l iv e d i n t h e r e fe re n ce y e ar t

i s esti mated for ages of w and above and sex s , )s,w(a + , as:

)s,w(m

1)s,w(a

+=+

3 . 2 O p t io n 2 . - A g g r e g a te d r e s u l ts f r o m c o m p l e te l i f e t a b l e s

T he st art in g- po int f or t his a lt er na ti ve is a c omp let e l if e t ab le , c ons tr uc te d, f or

e a ch su b -n a ti on a l p o pu la t io n, a c co r di ng to t he t h ir d o p ti o n c o nc er n in g t o ta l

p o p ul a t io n , a s d e s cr i b ed in t h e f i r st pa r t o f t h e p r e s en t p a p er . F r o m t h e s u r v iv o r s

f u nc t io n o f th is c om p le t e l if e t a bl e, a l if e t a bl e w it h ag g re ga t ed r es u lt s by g r ou ps

o f a g es i s d e f in e d , a s d e s cr i b ed i n t h e f o l lo w i n g po i n ts .

The s u rv i vo r s f u nc t i on, )s,x(l , f or t h e l if e t ab le w it h a gg re ga te d r es ul ts t ak es t h e

values of the same function of the complete life table for ages

w,5w,...,15,10,5,1,0x −= .

T h en , t he deaths seri es, )s,n,x(d , c a n be c a lc u la t ed a s:

)s,nx(l)s,x(l)s,n,x(d +−= for +−= w,5w,...,15,10,5,1,0x and 4,1n = o r 5 ( a s a p p li c ab l e ).

F ro m t he d ea th s an d t he s ur vi vo rs f u nc ti on s, t he p r o b ab i l i ty o f d y i n g w it hi n t he

a g e i n te r va l )nx,x[ + , )s,n,x(q , i s e st i ma t ed b y t he e x pr e ss i on :

s)l(x,

s)n,d(x,s)n,(x,q = for 5w,...,15,10,5,1,0x −= .

1)s,w(q =+


20/28

18

A nd , t he s t a t i on a r y p o pu l a t io n at ages )nx,x[ + and sex s , )s,n,x(L i s devel oped

f r o m t h e s a m e f u n ct i o n o f t h e s t a r ti n g c o m p le t e t a b le , )s,x(L , a s f o l lo w s :

∑+≤≤

=

nxyx

)s,y(L)s,n,x(L , f or +−= w,5w,...,15,10,5,1,0x .

Besides, a ve ra ge n um be r of ye ar s l iv ed in t he l as t ye ar o f l if e b y i nd i vi du a ls o f

sex s d yi ng wi t hi n t h e a ge i nt e rv a l )nx,x[ + d ur in g y ea r t , )s,n,x(a , i s o bt ai ne d

from:

s)n·d(x,

s)L(x,-s)n·l(x,-1s)n,(x,a = for 5w,...,15,10,5,1,0x −= .

)s,w(m

1s,w(a

+=+

E v en t ua l ly , t h e m or t al i ty r at e s seri es, )s,n,x(m , is d educ ed f rom t he f ollowin g

formula:

s)n,L(x,

s)n,d(x,s)n,(x,m = for +−= w,5w,...,15,10,5,1,0x .

3 . 3 O p t io n 3 . - C o m pl e t e l if e ta b l e s a d j u s t ed b y th e Br a s s l o g i t s m e t h od

The Brass m et hod e st ab lis he s a f un ct io na l r el at io n be tw ee n t he s ur vi vo rs

functi on, )h,s,x(l , o f t he c omp let e life t able o f e ach ter rit or ia l un it h and t he

s u r v iv o r f u nc t i on of t he c o m p le t e l i f e t a b l e f o r S p a in , )Spain,s,x(I :

)h,s,x()Spain,s,x(it)·logh,s()h,s()h,s,x(itlog ε+β+α=

Where:

)h,s(α and )h,s(β ar e t he p ar am et er s o f t he r eg re ss io n m od el a nd )h,s,x(ε i s the

r a nd o m er r or t e rm i n t h e m o de l, a n d:

)h,s,x(l

)h,s,x(l000.100·ln(

2

1)h,s,x(itlog

−= )

)Spain,s,x(l

)Spain,s,x(l000.100·ln(

2

1)Spain,s,x(itlog

−= )

T h e p a ra m et e rs o f t h e l in e ar m o de l b e tw ee n l o gi s ti c t r an sf o rm at i on o f t h e

s ur viv or f un ct io ns f or t he r ef er en ce t er ri to ria l u ni t h a re esti mated by Least

S qu ar ed M et ho d . T h en e s ti ma t ed s ur v iv or s fu n ct i on , )h,s,x(l̂ , r e su lt s f r om t he

estimated model.

A ft e r t h at , t he f o ll ow i ng e x pr e ss i on s f or s o me o f t he k e y b i om e tr ic s f un c ti on s o f

t he c om pl et e l if e t ab le a re a do pt ed :


21/28

19

)h,s,1x(l̂)h,s,x(l̂)h,s,x(d̂ +−=

)h,s,x(l̂)h,s,x(d̂)h,s,x(q̂ = , for 99,...,1,0x = .

000.100)h,s,0(l̂ =

)h,s,100(l̂)h,s,100(d̂ +=+

1)h,s,100(q̂ =+

In a dd it io n t o t ha t, t he r ob us tn es s o f i nf an t m or ta lit y i n a ll t er ri to ri al l ev els

consi dered al l ow s to i mpute the p ro b ab i li t y o f dy i ng a t ag e 0 w i th t h e o b s er v e d

value.

B es id es , t he aver ag e n um be r of y ea rs li ved in th e l as t y ear of li fe fo r t hos ei n d iv i d ua l s dy i ng i n t h e a g e i n t er v a l )1x,x[ + , )h,s,x(a , i s ta ke n 1 /2 , e xc ep t fo r th e

a ge 0 w he re t he o bs er ve d v al ue o ve r s tu di ed po pu la ti on du ri ng th e y ea r t i s

t ak en .

F i na l ly , t h e s t a t i on a r y p o p u la t i o n, )h,s,x(L , i s calcul ated as fol low s:

)h,s,x(d̂)h,s,x(â)h,s,1x(l̂)h,s,x(L̂ ⋅++= , 99,...,1,0x = .

A n d f o r t he o p e n -e n d e d a g e g r o u p is a p p ro x i m at e d by :

∑

∑

≤≤

≤≤=+

99x95

99x95

)Spain,s,x(l

)i,s,x(l̂

)i,s,100(L̂

3 . 4 D i sc us si on

T he o pt io n 1 s uf fer s fr om th e l im it at io n o f n ot is ola ti ng th e e ff ec t o f t he a ge

s tr u ct u re w it h in e a ch a g e i nt e rv a l. O n t he c o nt r ar y , o p ti o n 2 e l im i na t es t h e e f fe ct

o f th e a g e s t ru ct u re o f th e po p ul a ti o n, e v en w it h in e v er y a g e in t er v al .

I n a d d it i o n, t h e d e v ia t i o ns i n t h e r i s ks o f d e a th b et w ee n t h e t w o m e t h od s mu s t

t h er e fo r e a r is e f r om t h is li mi t at io n o f o p ti o n 1 w it h r e sp ec t t o o p ti o n 2 , w hi c h

l ea ds t o a l es s a cc ur at e m ea su re me nt o f t he i nc id en ce o f m or ta li ty i n t her ef er en ce p op ul at io n i nd ep en de nt ly o f t he a ge s tr uc t ur e o f t he p op ul at io n. T he

d e v ia t i on s i n c re a s e w i th d i s ta n c e f r o m u n i fo r m it y o f t h e d i s tr i b ut i o n o f d e a th s i n

e a ch a g e i n te r va l. T h e e f fe ct i s i m pe rc e pt ib l e i n y o un g a n d a du l t a ge g r ou p s, b u t

b e c o me s v i s ib l e i n t h e u p p er a ge i n t er v a ls .

T he re f ol lo w s om e e xa mp le s e xh ib it in g t he d if fe re nc es b et we en th e r es ul ts

discussed:


22/28

20

Comunidad de Madrid. Females. 2005.

Differences in q(x). Method 2 minus M ethod 1

-50

-40

-30

-20

-10

0

10

0 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

dif q(x)M2-M1

Thousandths

Age group

Canarias. M ales. 2006.

Differences in q(x). Method 2 minus M ethod 1

-6

-4

-2

0

2

4

6

0 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

dif q(x)M2-M1

Thousandths

Age group

T h es e d e vi at i on s in t he r i sk s of d ea t h o b ta i ne d by e ac h o pt io n in t h e u p pe r ag e

i nt er va ls en ta il t ha t o pt io n 1 q ua nt if ie s l if e e xp ec ta nc ie s a t a ll a ge s t ha t a re

s l ig h tl y a b ov e t h e v a lu e s p ro d uc e d b y o p ti on 2 :

Comunidad de Madr id. Fem ales. 2005.

Differences in e (x). Method 2 minus Me thod 1

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0,01

0 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

dif e(x)M2-M1

Years

Age group


23/28

21

Canarias. Males . 2006.

Differences in e (x). Method 2 minus Me thod 1

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

dif e(x)M2-M1

Years

Age group

H ow ev er , t he r an do mn es s o f d at a o n d ea th s b y a ge r ec or de d i n a g iv en ye ar in

t he s ma ll er r eg io ns a nd p ro vi nc es , a nd t he g re at er r el at iv e e rr or t ha t t he

p o p ul a t io n e st i m at e s us e d fo r s uc h t e r r it o r ia l u n i ts m a y h a v e, a t a dv a n c ed a g es

e sp ec ia ll y, d o o f co ur se d is to rt t he v is ib il it y o f t hi s ef fe ct . T hi s i s t he c as e o f

t e r r it o r i es s u c h as T e r ue l w h er e t h e d ev i a ti o n s b et w e en r i s ks o f d e at h a n d l if e

e x p ec t a nc i e s r e su l t in g f r o m t h e t w o m e th o d s a r e h a r d er t o i n t e r p re t :

Teruel. Males. 2006.

Differences in q(x). Method 2 minus Method 1

-10

-5

0

5

10

15

20

25

0 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

dif q(x)M2-M1

Thousandths

Age group


24/28

22

Teruel. Males. 2006.

Differences in e(x). Method 2 minus Method 1

-0,14

-0,12

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

dif e(x)M2-M1

Units

Age group

B u t bo t h o p t i on s h av e t h e d r a wb a c k o f pr o v id i n g l e s s d e t ai l ed r e s ul t s t h a n a

c omp le te ta ble b y s impl e ag es . H owe ve r, t he y m it ig at e t he r an domn ess of

r es ult s th at a ny c omp le te li fe t ab le f or a s ma ll p op ula ti on d is pl ay s an d, i n

g en er al , t he y a vo id t he p ro bl em s o f a v er y lo w o r n ul l n um be r o f d ea th s

o b se r ve d a t s o m e s i mp le a g es i n s u ch p o pu la t io n s.

O n t he o th er h an d, o pt io n 3 a ll ow s f or co ns tr uc ti ng a c om pl et e l if e t ab le . T he

m e th o d e m er g es f r om a n a dj u st me n t of o b se r ve d d a ta d e si g ne d t o el im i na t e

i nt r in s ic r a n d om n es s a n d t o a v oi d i n co n si st en c ie s i m po r te d b y t h e p o pu la t io n

e st i ma t es u se d , w h il e a c hi ev i ng t h e a im o f m ea s ur i ng th e i nc i de n ce o f mo r ta l it y

f o r t h e r e fe r en c e p e ri o d. I f we c o mp a re t h e r es u lt s of t hi s me th o do l og i ca l o p ti on

t o t he r es ult s o f a c omp le te l if e t ab le c on str uc te d d ir ec tl y, a t a ges u nd er 9 0, n o

s ig ni fi ca nt d if fe re nc es a re p er ce pt ib le i n t he r is ks of d ea th , a s a g en er al r ul e.B es i de s , d e vi at i on s i nc r ea s e w it h a ge a n d d e cr e as e w it h p op u la t io n s iz e . A b ov e

a ge 9 0, t he d if fe re nc es a re l ar ge r, e sp ec ia ll y i n t he s ma ll er p op ula ti on s. B y wa y

o f e x am p le , w e p r es e nt t h e d if f er e nc e s b et w ee n t h e r i sk s o f d e at h r e su lt i ng f r om

o p ti on 3 a n d t h os e p r ov id e d b y a c o mp l et e l if e t a bl e c a lc u la t ed d i re ct l y f o r s m al l

p o p ul a t io n s , s u c h a s t he f e m al e p o p u la t io n o f So r i a i n 20 0 6 :

Soria. Fem ales. 2006.

Differences in q(x). Method 3 m inus Observed

-100

-50

0

50

100

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

dif q(x) M3-ObservedAge

Thousandths


25/28

23

R e s pe c t t o l i fe e x p e ct a n ci e s a t bi r t h r es u l t in g f r o m op t i on 3 a n d f r om a c o m pl e t e

li fe t ab le c on st ru ct ed d ir ec tly , t he d if fe re nc es a r e n eg lig ib le i n m os t c as es . I n

fac t, t hey e xce ed s ix mo nt hs o nly in m os t e xt rem e c ases . F urth er mo re , thed i f fe r e nc e s a r e a l s o s e e n t o d e c r ea s e w i th r e f e r e nc e p o p ul a t io n s i z e .

Soria. Fem ales. 2006.

Differences in e(x). Method 3 minus Observed

-0,40

-0,20

0,00

0,20

0,40

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

dif e(x) M3-Observed

Years

Age

I n a d di t io n t o a l l t h at , t h e m a in p r ob le m o f o p ti o n 3 i s t h e d i st o rt io n i mp o rt e d

i nt o li f e e x pe c ta n ci es a t th e m o st a dv a nc e d a ge s wi t h r e sp ec t to o b se r ve d l if e

e x p ec t a nc i e s u p o n d i r ec t c o n s tr u c ti o n .T h i s i s ob s e r ve d , f o r e x a mp l e , i n t h e m a le

popul ati on of Catal uña i n 2006:

Catalunia. Males. 2006.

Differences in e (x). Method 3 minus Observed

-0,40

-0,20

0,00

0,20

0,40

0,60

0,80

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

dif e(x) M3-Observed

T h e d i st o rt i on c re a te d b y t h e B r as s me t ho d in t he u p pe r ag e i n te r va ls i s a bs e nt

f r o m o p t io n s 1 a n d 2, g i v e n t h e r e s u lt s o bt a i n ed e v en i n th e s m a ll e r po p u la t i on s :


26/28

24

Lugo. Females. 2005.

Differences of the methods versus the observed.

-0,60

-0,40

-0,20

0,00

0,20

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

dif e(x) M3-Observed dif e(x)M2-Obs dif e(x)M1-Obs

Age

Years

Almeria. Males. 2005.

Differences of the methods versus the observed.

-8,00

-6,00

-4,00

-2,00

0,00

2,00

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

dif e(x) M3-Observed dif e(x)M2-Obs dif e(x)M1-Obs

Age

Years

T h is m e th o do l og i ca l a pp r oa c h, o n t h e o t he r h a nd , i s t h e o n ly o n e t h at p r ov id e s

mo rt ality c ur ves and o th er fu nc tions by s imp le ag es , while av oi din g t he

u n de s ir a bl e e f fe ct s o f t h e i nt r in s ic r a nd o mn e ss o f o bs e r ve d a n nu a l d e at h s i n

s ma l l p o pu la t io ns a n d p o t e nt ia l i n co ns is t en ci es b e tw e en p o pu la t io n e s ti m at es

u s ed a s a r e fe r en c e a n d th e r e su lt s o bt a in e d.


27/28

25

4 Conclu sion s

A cc o rd i ng t o t h e d is c us s io n s r e la t ed t o n a ti on a l l if e t a bl e s, t h e t hr e e o p ti o ns a r ed es ig ne d to c a pt ur e t he a nn ua l b eh av io ur . Th ey a ls o k ee p co ns is te nc y in t h e

d e fi ni t io n s an d c al cu la t io n a s um p ti o ns s t at e d. N e ve r th e le s s, o p ti on 3 i s t h e o n e

t ha t ma xi mi ze s th e u se o f th e a va il ab le d at a f or m Sp an is h st at is ti ca l s ys te m,

g e tt i ng m a xi m un a cc u ra c y i n m e as u ri n g t h e i nc i de nc e o f m o rt a li t y a n d t h e l i vi n g

t i me o v er t he w ho le r a ng e o f ag e s, s p ec i al ly i n ad v an c ed a g es .

H o we v er , t h is t h ir d o pt io n f or t h e n at i on a l l if e t a bl es i s n ot s u it a bl e f or r e gi on a l

l if e t a bl es d u e t o t h e r a nd o m fa ct o rs t h at p u ni s h sm a ll s i ze p o pu la t io n s. B es i de s,

f i rs t a nd t h ir d o pt i on p r o po s ed f o r r eg i on al l if e t a bl es a r e r e je c te d s in c e t h ey a r e

n o t c o ns is t en t w it h na t io n al l if e t a bl e s a n d t h ey p r es e nt s om e o t he r m e nt io n ed

h an di ca ps . O n t he o th er h an d, t he s ec on d o pt io n k ee ps a ll t he g oo d p ro pe rt ie s

f o ll o we d b y t h e c h os e n o p ti o n f o r n a ti on a l t a bl e s.


28/28

References

J.R .W ilm oth , K .A nd reev, D .Jda nov , a nd D .A .G le i. M et hods P rot oc ol f or t he H u m an M o rt a li t y D a ta ba s e.

ht t p://www.m ort alit y.org/Public/Docs/M et hodsProt ocol.pdf

J u li o V i nu es a a n d D ol o re s P u ga . T é cn i c as y e j er c i ci os d e D e m og r af í a. ( 2 00 7 )

B ra ss , W il li am . M et ho ds f o r e st im at in g f er ti li ty a nd m o rt al it y f r om l im it ed a nd

d e f ec t i v e d a t a. ( 1 97 5 )

I N E. T a bl as d e m o r ta l id a d. M e to do l og í a. ( 2 00 9 )

ht t p://www.ine.es/daco/daco42/m ort alidad/m et odo_9107.pdf

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Towards advanced methods for computing life tables