W O R K I N G P A P E R
Debilitation's Afternatk Stochastic Procees Models of Mortality
James W. kupel Anatoli I. Yrrshin Kenneth C. hianion
November
19 8 6 WP-86-74
I n t e r n a t i o n a l l n s t ~ t u t e for Applied Systems Analysis
NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHORS
Debilitation's Attermath: Stochastic Process Models of Mortality
James W. V a u p e l Anatoli I. Y i h i n
Kenneth G. Munton
November 1986 WP-86-74
Working P a p e r s are interim r e p o r t s o n work of t h e International I n s t i t u t e f o r Applied Systems Analysis a n d h a v e r e c e i v e d only limited review. Views or opinions e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e o r of its National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria
Foreword
One of t h e important directions of r e s e a r c h in IIASA's Population Program i s focused on understanding t h e e f f e c t s of heterogeneity in human mortality. Various models of constant "frailty" developed in t h e a u t h o r s ' e a r l i e r p a p e r s c a p t u r e w e l l t h e selection e f f e c t s . They, however, c a n not explain t h e debilitation phenomena which are often o b s e r v e d in t h e analysis of s t a t i s t i c a l mortality data.
The p a p e r i s devoted t o t h e analysis of s t o c h a s t i c p r o c e s s models of mortality which c a n explain both selection and debilitation p r o c e s s e s in t h e evolution of c o h o r t mortality. The r e l a t i v e importance of e a c h p r o c e s s i s analyzed. The exam- ples of various regimes of mortality evolution are demonstrated.
Anatoli Yashin Deputy L e a d e r Population Program
Contents
Page
Introduction
Models of Mortality Gompertz and Makeham Exhaustion and Aging
Fixed Frailty: P u r e Selection Model When Everyone's Frailty Changes P u r e Stochastic Indisposition Putting I t All Together
A Decomposition of t h e F o r c e of Mortality Varieties of Disastrous Experience
Disentangling Debilitation and Selection Discussion
References
Debilitation's
Attermath:Stochastic Process Models of Mortality
J a m e s W. VaupeL*, A n a t o L i I. Y a s h i n * * , K e n n e t h G. Manton***
INTRODUCTION
Wars, famines, epidemics, a n d depressions debilitate as w e l l as decimate and t h e lingering morbidity consequences of a calamity may e l e v a t e mortality levels f o r y e a r s a f t e r w a r d s (Kermack et al. 1934a and b; Livi-Bacci 1962; Forsdahl1977;
P r e s t o n and van d e Walle 1978; Okuba 1982; Horiuchi 1983; M a r m o t et al. 1984;
Waaler 1984; Lawrence et al. 1985; Fogel 1986; Caselli et al. 1985 and 1986; H e a r s t et al. 1986). Healing and r e c u p e r a t i o n , f o s t e r e d p e r h a p s by social and public health programs, may restore t h e debilitated to normal health. Furthermore, death may p r u n e t h e population of t h e m o s t debilitated; t h i s i s t h e well-known pro- cess of selection in a heterogeneous population modeled by Beard (1963), Vaupel et al. (1979), and o t h e r s reviewed in Vaupel and Yashin (1985). A s a r e s u l t , death rates among t h e r e c o v e r e d or s e l e c t e d s u r v i v o r s may decline to normal or even below-normal levels.
The dynamic i n t e r a c t i o n of debilitation, r e c u p e r a t i o n , and selection i s compli- c a t e d by aging. Disasters may have a s t r o n g e r debilitating e f f e c t at some a g e s t h a n o t h e r s ; w e will r e f e r t o t h i s phenomenon as vulnerability. The evidence in t h e various a r t i c l e s cited above suggests t h a t t h e childhood and adolescent y e a r s are p a r t i c u l a r l y vulnerable ones. Death rates tend to r i s e exponentially with a g e , so at o l d e r a g e s t h e r e may not b e time f o r full r e c u p e r a t i o n b e f o r e d e a t h s t r i k e s . Selection accelerates with a g e because t h e r a t e of selection i s p r o p o r t i o n a l to t h e level of mortality.
*James W. Vaupel, Humphrey Institute of Public Affeirs, University of Minnesota, 301 1 9 t h Avenue South, Minneapolis, MN 55455, U S A
**Anatoll I. Yashin, Populetion Program, IIASA, A-2361 Lexenburg, Austria
**.Kenneth C . Manton, Center for Demographic Studles, Duke Unlverslty, Box 4732 Duke Statlon, Durham, N.C. 27706, USA
In this p a p e r we make use of a stochastic differential equation model, pro- posed by Woodbury and Manton (1977) and developed by Yashin et al. (1985). t o disentangle and clarify t h e evolving interplay among debilitation, recuperation, selection, vulnerability, and aging. We motivate t h e model by beginning with Gompertz's differential equation model and then adding complications s t e p by s t e p . The completed model leads t o a formula t h a t decomposes t h e mortality rate a t any a g e into two additive components which we call t h e baseline mortality r a t e and ex- c e s s mortality rate. The r e l a t i v e change with a g e of t h e e x c e s s mortality rate can, in t u r n , be decomposed into f o u r additive components which we c a l l t h e f o r c e s of vulnerability, debilitation, recuperation, and selection. To gain some insights about when one of these f o r c e s predominates and about t h e interactions among t h e f o r c e s , w e p r e s e n t t h e r e s u l t s of some computer simulations.
MODELS OF MORTALITY
Two d i s p a r a t e s e t s of mortality models have been developed, f o r different purposes and reasons. The f i r s t s e t of models, which might be called descriptive o r graduation models, w e r e developed t o d e s c r i b e empirical mortality p a t t e r n s without attention t o underlying physiological o r environmental processes. A s dis- cussed by Keyfitz (1982), such models a r e useful:
-
'To smooth t h e data",-
'To make t h e r e s u l t more precise",-
'To c o n s t r u c t life tables",-
'To aid inferences from incomplete data",-
'To facilitate comparisons of mortality", and-
'To aid forecasting".
The multi-parameter c u r v e s of Thiele (1872) and Heligman and Pollard (1980), t h e graduation methods of Reed and Merrell (1939) and Greville (1943) and more r e - c e n t spline approaches, t h e model life tables of Coale and Demeny (1966), Leder- mann (1969), and Petrioli and Berti (1979), and t h e relational transformations of Brass (1975), Zaba (1979), and Ewbank e t al. (1983) a l l f i t into this category.
Mortality models of t h e second kind s t a r t with some biologically plausible pro- c e s s t h a t i s hypothesized t o determine t h e a g e t r a j e c t o r y of mortality. Then t h e mortality c u r v e i s calculated from t h e process, e i t h e r by derivation of a formula o r by numerical approximation. I t i s sometimes forgotten t h a t Gompertz (1825)
pioneered t h i s a p p r o a c h . As discussed in t h e next section, Gompertz s t a r t e d with a differential equation t h a t described t h e p r o c e s s of "indisposition" o v e r a g e and then derived h i s familiar mortality c u r v e from t h i s d i f f e r e n t i a l equation. The sub- sequent mortality m o d e l s of Makeham (1867), Armitage and Doll (1954)' S t r e h l e r and Mildvan (1960), S a c h e r a n d Truco (1962), Beard (1963). Woodbury a n d Manton (1977), Vaupel et al. (1979)' Economos (1982), a n d Moolgavkar (1986), a l l w e r e based on biologically justified p r o c e s s e s on t h e individual level (e.g., involving loss of vitality or accumulation of environmental insults) or o n t h e population level (e.g., selection resulting from heterogeneity among individuals in t h e i r frailty).
In t h i s set of p r o c e s s models t h e model proposed by Woodbury and Manton (1977) i s useful f o r o u r purposes. I t includes t h e k e y elements of debilitation, r e c u p e r a t i o n , a n d selection. In addition, i t can b e specified t o include vulnerabil- ity (i.e., d i f f e r e n t mortality e f f e c t s of d i s a s t e r at d i f f e r e n t a g e s ) and t h e lasting impact of t e m p o r a r y e x t e r n a l conditions, like w a r s and famines, on physiological indisposition. In t h e following sections w e e l a b o r a t e t h e Gompertz model of human mortality to d e r i v e a univariate form of t h e g e n e r a l Woodbury-Manton model.
GOMPERTZ A N D MAKEHAM
Following Gompertz (1825), suppose t h a t "the a v e r a g e exhaustions of a man's power to avoid d e a t h were s u c h t h a t at t h e end of equal infinitely small i n t e r v a l s of time, h e lost equal portions of his remaining power to oppose destruction", so
where p ( x ) i s t h e f o r c e of mortality at a g e z a n d a is some scaling p a r a m e t e r . Given a n initial value
t h e solution follows t h a t
A n a t u r a l generalization of this a p p r o a c h is to l e t
The p a r a m e t e r a. r e p r e s e n t s t h e constant change in t h e f o r c e of mortality,
whereas a i i s t h e proportional change. The solution i s
If a. < 0 , but
and letting
this is equivalent t o Makeham's model:
These familiar models of Gompertz and Makeham a r e often adequate f o r the analysis of mortality data. However, t o s e p a r a t e t h e a g e p r o c e s s of deterioration from t h e a g e p r o c e s s of mortality a more complicated model i s needed.
EXHAUSTION AND AGING
To distinguish Gompertz's exhaustion p r o c e s s from t h e changes in mortality due t o aging, l e t Y ( z ) r e p r e s e n t what Gompertz r e f e r r e d t o as "exhaustion", "in- disposition", and "inability to withstand destruction". If t h e f o r c e of mortality p ( z ) is inversely proportional t o Y(z), as Gompertz assumes, then t h e r e i s little point in distinguishing indisposition from t h e f o r c e of mortality. A more complicat- ed relationship might, however, make sense.
Suppose, f o r instance, t h a t t h e r e i s some optimal s t a t e where t h e f o r c e of mortality i s minimal; t h e f o r c e of mortality increases as a n individual's condition deviates from t h i s optimum, e i t h e r in a positive o r negative direction; t h e f o r c e of mortality i s hardly affected if t h e deviation is small but a l a r g e deviation r e s u l t s in a disproportionately l a r g e i n c r e a s e In t h e f o r c e of mortality. Under t h e s e biologi- cally plausible suppositions, i t may be reasonable t o l e t
where &(z) might be i n t e r p r e t e d as t h e baseline f o r c e of mortality under optimal
conditions, where t h e indisposition Y ( z ) measures t h e deviation in conditions f r o m t h e optimal, and where t h e vulnerability A ( z ) determines t h e level of e x c e s s m o r - tality resulting from t h i s indisposition.
Two s p e c i a l cases a r e of i n t e r e s t . The value of A ( z ) might be constant. This simplifying assumption implies t h a t t h e g r e a t e s t r e l a t i v e i n c r e a s e in mortality lev- els produced by a given level of indisposition o c c u r s at t h e a g e s where t h e abso- lute level of baseline mortality i s lowest, a n implication t h a t may b e plausible given t h e evidence on t h e disproportionate impact of d i s a s t e r s on c h i l d r e n and adoles- cents.
Alternatively, A ( z ) might b e equal to p 0 ( z ) . Then
In t h i s c a s e , y 2 ( z ) measures e x c e s s r i s k i n t h e usual p r o p o r t i o n a l h a z a r d s formu- lation.
If indisposition changes with a g e such t h a t
t h e n Y ( z ) is given by a formula analogous to (4) and
Depending on t h e signs of t h e p a r a m e t e r s , t h i s t r a j e c t o r y f o r t h e f o r c e of mortality c a n t a k e on a v a r i e t y of s h a p e s , e v e n when & ( z ) and A ( z ) are constant and equal to & and A . An interesting case involves negative Yo and a l with posi- tive a o , k , and A . Given
(a),
t h e positive value of a . might b e i n t e r p r e t e d as r e p r e s e n t i n g debilitation, whereas t h e negative value of al might be i n t e r p r e t e d as r e p r e s e n t i n g r e c u p e r a t i o n or homeostasis. These p a r a m e t e r values produce a t r a j e c t o r y t h a t i s reminiscent of human mortality c u r v e s , with declining mortality in infancy, rapidly rising mortality in middle age, and s o m e leveling off at advanced ages.This model, however, implicitly assumes t h a t t h e r e i s no heterogeneity in f r a i l t y among individuals and hence no selection. To c a p t u r e t h e e f f e c t s of selec- tion. s o m e additional f e a t u r e s have to b e added to t h e model.
FMED FRAILTY:
PUBE
SELECTION MODELThe familiar h e t e r o g e n e i t y model with c o n s t a n t , p r o p o r t i o n a l h a z a r d s ,
where p ( z . 2 ) i s t h e f o r c e of mortality of individuals a g e z with f r a i l t y z a n d ~ ( z ) i s t h e f o r c e of mortality f o r s t a n d a r d individuals with f r a i l t y 1 , i s r e a d i l y extended to:
where f r a i l t y z could b e identified with t h e s q u a r e of indisposition Y a n d where p,,(z) i s some baseline f o r c e of mortality. Then p ( z ) , t h e o b s e r v e d f o r c e of mor- tality among surviving individuals, which i s given by
where X d e n o t e s a g e at d e a t h , c a n b e e x p r e s s e d as
where z ( z ) i s t h e a v e r a g e f r a i l t y of surviving individuals, defined by
As shown by Vaupel a n d Yashin (1984a), t h e p r o c e s s of s e l e c t i o n resulting from t h e h i g h e r d e a t h rates of f r a i l e r individuals implies t h a t
Thus,
p(z)
increasingly d e v i a t e s downward from It(z) with a g e .This model of fixed f r a i l t y i n c o r p o r a t e s both aging a n d s e l e c t i o n but i t fails to explicitly c a p t u r e debilitation a n d r e c u p e r a t i o n . Since fixed f r a i l t y i s assumed i n many empirical s t u d i e s (including Manton et al. 1 9 8 1 a n d 1986, Heckman a n d S i n g e r 1984, a n d T r u s s e l a n d R i c h a r d s 1985). it would seem useful to investigate what ef- f e c t debilitation a n d r e c u p e r a t i o n might h a v e and when t h e s e e f f e c t s could b e ig- nored. F u r t h e r m o r e , i n t e r e s t i n c a t a s t r o p h e s r e q u i r e s a t t e n t i o n to debilitation and r e c o v e r y . Thus i t i s a p p r o p r i a t e to combine models s u c h t h a t f r a i l t y both c h a n g e s o v e r a g e (and time) a n d v a r i e s a c r o s s individuals.
WHEN
EXEKYONE'S FRAILTYCHANGES
Consider, then, t h e model
but now assume t h a t Yo i s a random variable t h a t d i f f e r s from individual t o indivi- dual. Suppose t h a t t h e change o v e r a g e in indisposition i s described by
noting t h a t now t h e p a r a m e t e r s a and a may v a r y with a g e (and time). To develop a p p r o p r i a t e methods of analysis f o r this model, i t i s useful t o s t e p back and consid- e r t h e a r b i t r a r y , p e r h a p s random, p r o c e s s Y(x) and not just t h e p a r t i c u l a r pro- c e s s in (17). Then
where, as b e f o r e ,
where X denotes a g e of death 'and where m ( x ) and y ( x ) a r e t h e conditional mean and variance of Y(x) among surviving individuals; w e use t h e notation y ( x ) r a t h e r than a 2 ( x ) t o emphasize t h a t y ( x ) i s not a usual unconditional variance but a con- ditional variance. To d e r i v e (18), note t h a t
The f i r s t term in t h i s expression is, by definition, y ( x ) , t h e second t e r m is just m2(x), and t h e t h i r d term h a s a value of z e r o .
The problem now becomes a problem of determining m ( x ) and y(x). For the p r o c e s s in Y(x) described by (17) i t follows, as a special case of t h e r e s u l t s in Yashin et al. (1985), t h a t if Yo i s normally distributed with mean m o and variance yo, then
and
The evaluation of t h i s p a i r of d i f f e r e n t i a l equations c a n b e a p p r o x i m a t e d by com- p u t e r numerical methods. Interestingly, t h e conditional d i s t r i b u t i o n of Y(x) among t h e surviving at a g e
z
i s normal (with mean m ( x ) a n d v a r i a n c e y ( z ) ) .The model developed above i s b a s e d o n t h e assumption t h a t individuals' initial indispositions c h a n g e deterministically o v e r time. This may b e a n a p p r o p r i a t e as- sumption in s t u d i e s focusing on evolving e x t e r n a l conditions t h a t a f f e c t all t h e in- dividuals in a c o h o r t more o r less t h e s a m e way. However, t h e model f a i l s t o cap- t u r e t h e impact of t u r b u l e n t d i s t u r b a n c e s t h a t a f f e c t d i f f e r e n t individuals dif- f e r e n t l y .
PURE STOCHASTIC INDISPOSITION
In many situations i t may b e r e a s o n a b l e t o allow t h e indisposition of o n e indivi- d u a l to c h a n g e with a g e reLative t o t h e indisposition of a n o t h e r individual. A s indi- viduals g e t s i c k , g e t well, s t o p smoking, start drinking, etc., t h e i r r e l a t i v e indispo- sitions may change, and famines, wars, epidemics, and d e p r e s s i o n s may h a r m some individuals more t h a n o t h e r s .
A s a simple case of changing r e l a t i v e indispositions, c o n s i d e r t h e p r o c e s s
where W i s a Wiener ( o r Brownian motion) p r o c e s s a n d where b i s a p a r a m e t e r t h a t may c h a n g e o v e r a g e (and time). The Wiener p r o c e s s i s a continuous time, continu- o u s p a t h s t o c h a s t i c p r o c e s s with independent, normally d i s t r i b u t e d increments s u c h t h a t
and
Thus, (21) implies t h a t if a n individual h a s some indisposition I'(x ,) a t a g e XI, t h e n t h e individual's indisposition at a g e x2 will b e normally d i s t r i b u t e d with a mean of
1 2
Y(xl) and a v a r i a n c e of
/
b 2 ( x ) d z .1 1
Given t h i s formulation, i t c a n b e shown, a s a s p e c i a l c a s e of t h e r e s u l t s in Yashin et al. (1985), t h a t if, as b e f o r e , a n individual's c h a n c e of d e a t h at a g e x and indisposition Y i s given by
t h e n t h e conditional distribution of vulnerability Y ( z ) among s u r v i v o r s at a g e z is Normal with mean m ( z ) and variance y ( z ) d e s c r i b e d by
and
where, as b e f o r e ,
p ( z ) =
& ( z )+
X ( z ) ( m 2 ( z )+
7 ( z ) )PUTTING
I T
ALL TOGETHERA model t h a t includes t h e various elements discussed so f a r of changing m o r - tality and vulnerability with a g e , heterogeneity among individuals in t h e i r innate indisposition, and both deterministically and stochastically changing individual in- disposition would b e
and
with Y ( 0 ) normally d i s t r i b u t e d with mean mo and v a r i a n c e yo and W(0) equal to zero. Note t h a t t w o p a r a m e t e r s a l ( z ) and a ; ( z ) are used in t h e formulation. The idea is t h a t both t h e s e p a r a m e t e r s are non-negative and t h a t a i ( z ) (along with a o ( z ) ) r e p r e s e n t s t h e effects of debilitation whereas a ; ( z ) r e p r e s e n t s t h e homeostatic healing and r e c u p e r a t i o n . This i s a simple expedient, b u t effective, at l e a s t f o r exposition and f o r gaining insights into t h e e f f e c t s of debilitation vs. re- cuperation.
I t follows from Vaupel et al. (1979) a n d Yashin (1986) t h a t even in t h e c a s e of changing individual indisposition, t h e o b s e r v e d population t r a j e c t o r y of t h e f o r c e of mortality is given by
and hence by
where z ( x ) i s t h e a v e r a g e f r a i l t y (i.e.. s q u a r e d indisposition) at a g e z among t h o s e surviving. A s noted e a r l i e r , t h e r e s u l t
holds f o r any p r o c e s s Y(x) s u c h t h a t
When Y(x) i s d e s c r i b e d b y (29) a n d when Y(0) I s normally d i s t r i b u t e d with mean m o and v a r i a n c e yo, t h e n , as shown b y Yashin e t a l . (1985),
and
d ( 2 )
The p r e v i o u s equations f o r dm(x) a n d
7
given in (19) a n d (20) a n d (25) a n d-
d x dz
(26) c a n b e s e e n to b e s p e c i a l c a s e s of (33) a n d (34).
Equations (33) a n d (34) c a n b e solved i n various s p e c i a l c a s e s , b u t in g e n e r a l t h e values of m ( z ) and y ( x ) h a v e t o b e calculated using numerical approximation methods. This is r e a d i l y done with t h e h e l p of a p e r s o n a l computer using d i f f e r e n t equations t o c a l c u l a t e t h e values of m ( x
+
A) a n d y(x+
A), f o r some sufficiently small increment A, given t h e values of m ( z ) a n d y ( z ) .ADECOMPOSITIONOFTHEFORCE OFPORTALXTY
The model l e a d s t o a two-stage decomposition of t h e f o r c e of mortality G(x).
F i r s t , t h e baseLine mortality rate & ( x ) c a n b e s e p a r a t e d from t h e e x c e s s mortali- t y r a t e given by X(x)?(x). Second, formulas (32). (33), and (34) imply t h a t t h e r e - lative rate of c h a n g e i n t h i s e x c e s s mortality rate c a n b e decomposed i n t o f o u r components:
where
and
In this decomposition p,(z) c a p t u r e s t h e impact of change o v e r a g e ( o r time) in vulnerability, p d ( z ) c a p t u r e s t h e impact of debilitation, p,(z) c a p t u r e s t h e impact of r e c u p e r a t i o n , and p, ( z ) c a p t u r e s t h e impact of selection. N o t e t h a t if a o , a l , and b equal z e r o , t h e r e i s no debilitation and p d ( z ) equals z e r o . On t h e o t h e r hand, if X(z) is z e r o or if t h e population i s homogeneous (i.e., y ( z ) equals z e r o ) t h e n t h e r e i s no selection and p, ( z ) is z e r o . W e will r e f e r t o t h e p's as t h e f o r c e s of vulnerability, debilitation, r e c u p e r a t i o n , and selection.
Because both pd ( z ) and p, ( z ) depend on m ( z ) and 7 ( z ) , t h e t w o p r o c e s s e s in- teract. T h e r e c a n b e selection with n o debilitation-if pol a l l and b are z e r o and
ro
is positive. This i s t h e familiar case of a heterogeneous population with fixed f r a i l t y . T h e r e c a n a l s o b e debilitation with no selection-if a. or a are positive and both yo and b are zero. This i s t h e case when t h e populaiion is homogeneous in f r a i l t y at all ages. But if f r a i l t y i s changing in a heterogeneous population, t h e n debilitation at any a g e will a f f e c t selection, i.e., p, ( z ) , a t l a t e r a g e s and selection at any a g e will a f f e c t debilitation, i.e., pd ( z ) , at l a t e r a g e s .Given t h e formulation of t h e model, t h e f o r c e of r e c u p e r a t i o n a f f e c t s m ( z ) and y ( z ) , as d e s c r i b e d in (33) and (34), and t h u s a f f e c t s t h e f o r c e of debilitation and selection. On t h e o t h e r hand, s i n c e t h e f o r c e of r e c u p e r a t i o n , as given by (38), depends only o n a; ( z ) , t h i s f o r c e i s not d i r e c t l y a f f e c t e d by t h e f o r c e of de- bilitation or selection. A t some level not explicitly included in t h e model t h e r e could, however, b e s o m e linkage. For instance, a d i s a s t e r t h a t causes debilitation might invoke social aid t h a t i n c r e a s e s t h e value of a; ( z ) and hence f o s t e r s r e c u - peration.
VARIETIES
OF
DISASTROUS EXPERIENCETo gain some insights about t h e model. we wrote a simulation program t h a t r u n s on a n
IBM
PC. Table 1 summarizes t h e p a r a m e t e r s of 1 0 mortality regimes t h a t govern t h e life c h a n c e s of a hypothetical c o h o r t as i t a g e s o v e r time. In e v e r y regime, & ( z ) , t h e baseline f o r c e of mortality, i s assumed to b e t h e s a m e . The various parzlmeters are given in t h e t a b l e a n d in t h e n o t e s to t h e table. The regimes w e r e s e l e c t e d to i l l u s t r a t e ideas r a t h e r than t o r e p l i c a t e empirical obser- vations.Table 1. Alternative mortality regimes.
Parame t e r s
during d i s a s t e r Regime
i i i iii i v v v i vii viii ix
X
No* Ln all instanoes m o i s one, & ( z ) i s given by . 0 0 0 1 e . ~
+
.Ole*, and p ( z,Y)
i s given by p 0 ( z )+
X(Z)Y'(Z ) / SO) where X(z ) equals S X e 0 ( z )+
.002 and wherez0
i s ascaling factor equal to m
+
70. This scaling insures that p(0) i s the s a m e in all the r eI
gimes. In regime viii, a l i s z e r o before a g e 20 and .05 afterwards. In all regimes, a. and a are zero and Xo i s one, e x c e p t during a disaster. Disasters last from a g e 10 through a g e 19.
Figure 1 p r e s e n t s e i g h t p a i r s of mortality t r a j e c t o r i e s , labelled (i) to (viii), t h a t c o r r e s p o n d
to
t h e f i r s t e i g h t mortality regimes listed in Table 1. In e a c h c a s e , t h e solid c u r v e gives t h e t r a j e c t o r y when t h e r e i s no d i s a s t e r and t h e do&d c u r v e gives t h e t r a j e c t o r y when , t h e r e i s a d i s a s t e r . A s noted in Table 1 , a disas- ter starts at a g e 1 0 and lasts through a g e 19.Figure l ( i ) i l l u s t r a t e s t h e consequences of a d i s a s t e r in a mortality regime where t h e r e i s n o heterogeneity, no debilitation, and no r e c u p e r a t i o n . The disas- ter comes and goes, with s e v e r e immediate e f f e c t s b u t no aftermath.
Force of Mortality (Log Scale)
Force of
Mortality Figure 1 (ii)
(Log Scale)
1
Figure 1 (iv)Figure 1. Varieties of disastrous experience. (See t e x t f o r explanation.)
-
I I I I
.001
-
I I I I
Force of Mortality (Log Scale)
Force of Mortality (Log Scale)
r
Figure 1 lvlr
Figure 1 (vii)r
Figure 1 (vi)r
Figure 1 (viii)If t h e r e i s heterogeneity in frailty in t h e population at b i r t h , as is t h e case in Figure l ( i i ) , t h e n both with and without a d i s a s t e r , t h e e f f e c t s of selection r e d u c e t h e level of observed mortality. A d i s a s t e r r a i s e s mortality levels and hence ac- c e l e r a t e s t h e d e a t h of f r a i l e r individuals. A s a r e s u l t , t h e i n c r e a s e in mortality rates during t h e d i s a s t e r is somewhat moderated. In addition, t h e level of mortali- ty a f t e r t h e d i s a s t e r i s lowered. Because selection now o p e r a t e s m o r e rapidly in t h e advantaged c o h o r t , t h e t w o mortality t r a j e c t o r i e s gradually converge.
If t h e d i s a s t e r r a i s e s everyone's indisposition by t h e s a m e amount and if t h e population i s homogeneous, t h e case in Figure l ( i i i ) , then t h e r e s u l t of t h e d i s a s t e r i s a permanent i n c r e a s e in t h e level of mortality. If, however, t h e population i s heterogeneous, as i n Figure l ( i v ) , t h e i n c r e a s e d f o r c e of selection after t h e disas- ter r e s u l t s in s o m e convergence in t h e mortality t r a j e c t o r i e s .
Figure l ( v ) a l s o d e s c r i b e s mortality t r a j e c t o r i e s f o r a heterogeneous c o h o r t , but now t h e d i s a s t e r does not r a i s e e a c h individual's indisposition by t h e s a m e ab- solute amount b u t by t h e s a m e proportion. The d i s a s t e r , in increasing t h e vari- a n c e in indisposition among individuals as well as t h e level of indisposition, sub- stantially a c c e l e r a t e s t h e selection p r o c e s s . The mortality t r a j e c t o r i e s , as a consequence, show marked convergence.
The c o h o r t s whose mortality i s d e s c r i b e d in Figure l ( v i ) are both initially homogeneous. The c o h o r t a f f e c t e d by t h e d i s a s t e r becomes heterogeneous as a r e s u l t of t h e d i s a s t e r : t h e d i s a s t e r c a n b e thought of a time of turbulence produc- ing random changes in indisposition. The selection caused by t h e acquired h e t e r o - geneity, coupled with t h e e x i s t e n c e of f o r t u n a t e individuals whose indisposition i s r e d u c e d during t h e d i s a s t e r , so markedly a f f e c t s t h e subsequent mortality t r a j e c - t o r y of t h e s t r i c k e n c o h o r t t h a t a f t e r a g e 70 or so t h i s c o h o r t h a s a m o r e favor- a b l e mortality e x p e r i e n c e .
A s illustrated by t h e s e s i x figures, d i s a s t e r s c a n be c a p t u r e d e i t h e r by changes in t h e vulnerability function h or by changes in t h e p a r a m e t e r s of t h e in- disposition p r o c e s s , a o , a l , or b . The lingering mortality consequences of a disas- ter produced in any of t h e s e ways c a n b e moderated by homeostasis or r e c u p e r a - tion, as r e p r e s e n t e d by t h e p a r a m e t e r a ; . Figures l ( v i i ) and (viii) i l l u s t r a t e this in t h e case of t h e kind of d i s a s t e r p o r t r a y e d in Figure l ( v i ) , a time of turbulence t h a t r e s u l t s in a c q u i r e d heterogeneity.
If homeostasis o p e r a t e s from b i r t h on, t h e n , as shown in Figure l ( v i i ) , t h e mortality c u r v e f o r t h e c o h o r t not s u f f e r i n g t h e d i s a s t e r i s substantially lower t h a n t h e c u r v e s shown in p r e v i o u s figures. The e f f e c t of t h e homeostasis parame- t e r i s to gradually r e d u c e e v e r y o n e ' s indisposition from i t s initial level of o n e to- ward t h e optimal level of z e r o . This might r e f l e c t h e a l t h p r o g r e s s made o v e r t h e c o u r s e of t h e c o h o r t ' s life. A d i s a s t e r t h a t creates s u b s t a n t i a l h e t e r o g e n e i t y in- creases t h e l e v e l of mortality, b u t as a r e s u l t of homeostasis ( o r r e c u p e r a t i o n ) t h e r e i s r a p i d c o n v e r g e n c e of t h e new mortality t r a j e c t o r y toward t h e t r a j e c t o r y of t h e f o r t u n a t e c o h o r t .
In c r e a t i n g Figure l(viii) i t w a s assumed t h a t r e c u p e r a t i o n follows a d i s a s t e r , being produced both by n a t u r a l physiological r e c o v e r y a n d by v a r i o u s s o c i a l interventions. Hence, t h e c o h o r t not a f f l i c t e d by t h e t u r b u l e n t times of t h e disas- ter does n o t benefit from r e c u p e r a t i o n . I t s t r a j e c t o r y i s t h e same as t h e t r a j e c - t o r y f o r t h e advantaged c o h o r t in Figure l ( v i ) . The a f f l i c t e d c o h o r t b e n e f i t s s o substantially from t h e f o r c e of r e c u p e r a t i o n t h a t i t s mortality t r a j e c t o r y falls below t h e o t h e r c o h o r t ' s t r a j e c t o r y less t h a n twenty y e a r s a f t e r t h e d i s a s t e r . The e f f e c t s of s e l e c t i o n , r e c u p e r a t i o n , a n d random lowering of indisposition f o r some f o r t u n a t e individuals during t h e d i s a s t e r combine to yield a v e r y f a v o r a b l e mortal- i t y t r a j e c t o r y from a g e 40 on.
DISENTANGLING DEBILITATION
AND
SELECTIONA s noted e a r l i e r , d e m o g r a p h e r s f o r many y e a r s h a v e b e e n i n t e r e s t e d in t h e e f f e c t s of debilitation vs. selection. Confusion h e r e i s e a s y given t h e i n t r i c a t e in- t e r a c t i o n of debilitation and selection. To gain some insights i n t o t h e n a t u r e of t h i s i n t e r a c t i o n , i t i s useful to c a r e f u l l y d i s s e c t t h e immediate a n d lingering ef- f e c t s of a d i s a s t e r . Figures 2(i) t h r o u g h (vii) p r o v i d e a n illustration f o r mortality regime ix in Table 1. In t h i s mortality regime, t h e r e is n o homeostasis a n d t h e vul- nerability p a r a m e t e r
Xo
i s c o n s t a n t a n d equal to one. These simplifications facili- tate comprehension of t h e e f f e c t s of debilitation and s e l e c t i o n r e s u l t i n g from a combination of a b s o l u t e , p r o p o r t i o n a l , a n d random c h a n g e s i n indisposition in a h e t e r o g e n e o u s population.Figure 2(i) displays t h e mortality t r a j e c t o r i e s with a n d without t h e d i s a s t e r . The e f f e c t s of t h e debilitation c a u s e d during t h e d i s a s t e r a n d t h e s e l e c t i o n follow- ing t h e d i s a s t e r are s u b s t a n t i a l . Note t h a t t h e mortality t r a j e c t o r i e s c o n v e r g e , b u t t h a t t h e r e i s no c r o s s o v e r . Differential s e l e c t i o n n e c e s s a r i l y p r o d u c e s some
Force of 1.
Mortality (Log Scale 1
Figure 2(i): Trajectories of j7
-
I I I I
Force of -25 Selection
Figure 2(iii): Trajectories of p S
Force of
Debilitation 2 5
.20
.15
.10
.O 5
0
-
Figure2(ii): Trajectoriesof pdII II I1 I
I
- I \
I '\
I \
I \ I \ - I I
I I
I I
I I
I
II
II ;
- 1
II I
I
I I I I IL--
--
Net Force .25 of Debilitation Minus Selection
Figure 2(iv): Trajectories of pd -
I I
1 \1
I \\
I
\
I II 1
I I I II 1
I I
LC---
I I I I
Figure 2. Aspects of disaster.
Figures Z(i)-(vii) display the trajectories of
2,
pd , p o , pd -ps , m , y, and m 2+7, respectively.Mean 2 lndiswsition
0
Mean Frailty 5
4
Figure 2(v): Trajectories of m Variance
in 6
Indisposition
Figure 2(vii): Trajectories of mL + 7
1
convergence, but i t does not have to r e s u l t in a c r o s s o v e r (Vaupel and Yashin 1985).
Figure 2(ii) plots t h e f o r c e of debilitation, as defined by formula (3'7) o v e r age. For both c o h o r t s , t h e r e is some debilitation b e f o r e t h e d i s a s t e r , resulting from t h e random changes in indisposition produced by t h e positive value of t h e p a r a m e t e r b . The s t r o n g e s t period of debilitation i s confined t o t h e decade of t h e d i s a s t e r , but t h e r e i s some debilitation t h e r e a f t e r produced by random changes in indisposition. Since t h e value of t h e p a r a m e t e r b is constant a f t e r t h e d i s a s t e r , formula (3'7) implies t h a t t h e i n c r e a s e in t h e f o r c e of debilitation i s a t t r i b u t a b l e t o t h e declining value of mean f r a i l t y (i.e., of Similarly, t h e low value of t h e f o r c e of debilitation f o r t h e afflicted c o h o r t , especially in t h e y e a r s immediately following t h e d i s a s t e r , is a t t r i b u t a b l e t o t h e high value of mean f r a i l t y during t h i s period.
A s shown in Figure 2(iii), f o r both c o h o r t s t h e r e i s some selection b e f o r e t h e d i s a s t e r , largely in infancy when t h e mortality r a t e is high: t h i s selection r e s u l t s from t h e initial heterogeneity of t h e population, as implied by t h e positive value of t h e p a r a m e t e r 70. Selection a c c e l e r a t e s during t h e d i s a s t e r as population hetero- geneity i n c r e a s e s , and selection continues t o o p e r a t e a f t e r t h e d i s a s t e r , with in- creasing f o r c e as t h e level of mortality i n c r e a s e s with a g e . The high mortality rate s u f f e r e d by t h e afflicted c o h o r t gradually r e d u c e s t h e d i f f e r e n c e in mean f r a i l t y between t h e two c o h o r t s : this differential selection p r o d u c e s t h e conver- gence with a g e in t h e f o r c e s of selection f o r t h e two c o h o r t s .
Figure 2(iv) displays t h e difference between t h e f o r c e of debilitation and t h e f o r c e of selection, f o r t h e two cohorts. When this d i f f e r e n c e i s positive, i t c a n b e said t h a t debilitation predominates; when i t is negative, selection predominates.
For t h e c o h o r t t h a t does not s u f f e r t h e d i s a s t e r , selection predominates at a l l ages, although t h e f o r c e s of selection and debilitation are in rough balance (and are both small) from childhood through a g e 40 o r so. For t h e c o h o r t t h a t s u f f e r s t h e d i s a s t e r , debilitation predominates only during t h e d e c a d e of t h e d i s a s t e r , and, as a r e s u l t of t h i s debilitation, t h e f o r c e of selection is substantial at all a g e s a f t e r t h e d i s a s t e r . A s t h i s example makes t r a n s p a r e n t , selection should b e thought of not as a n a l t e r n a t i v e t o debilitation, but as a consequence of any debilitation t h a t i n c r e a s e s population heterogeneity.
Figures 2(v), (vi), a n d (vii) show t h e change with a g e in m , t h e mean level of indisposition, 7, t h e v a r i a n c e in indisposition, and
z,
t h e mean level of f r a i l t y (i.e., indisposition squared). Mean indisposition falls somewhat during t h e f i r s t decadeas a r e s u l t of selection in t h e initially heterogeneous population. For t h e afflicted c o h o r t , t h e mean almost doubles during t h e d i s a s t e r but, as a r e s u l t of r a p i d selec- tion, t h e mean falls below one again around a g e 40 and falls below t h e mean f o r t h e advantaged c o h o r t around a g e 60. Variance in indisposition shows a similar pat- tern, reaching a peak of 8 f o r t h e afflicted c o h o r t , although without a c r o s s o v e r . As a r e s u l t , t h e t r a j e c t o r y of mean frailty, which equals mZ+71 shows a n analogous p a t t e r n f o r t h e advantage c o h o r t of s t e a d y decline and f o r t h e afflicted c o h o r t of a s h a r p rise to a peak, t h i s time close to 5, and t h e n a somewhat less r a p i d fall as t h e f r a i l victims of t h e d i s a s t e r die.
Figures 3(i) through (vii) provide a second set of illustrations of t h e interac- tions of various f a c t o r s in t h e mortality model. As discussed above, t h e impor- t a n c e of what might be called a cohort's memory of p a s t d i s a s t e r s , as r e f l e c t e d in c u r r e n t and f u t u r e mortality r a t e s , i s reduced by selection and by homeostasis o r recuperation. Stochastic change in individuals' indisposition a l s o leads to forget- fulness, because t h e more turbulent t h e s e changes a r e , t h e less c o r r e l a t i o n t h e r e will b e between a n individual's indisposition at two different ages. To e x p l o r e this phenomenon, mortality regime
z
in Table 1 w a s used to d e r i v e t h e diagrams in Fig- u r e 3. In t h i s regime, b , t h e p a r a m e t e r of stochasticity, i s set at a value of .5 at all ages. The d i s a s t e r i s modeled by setting ao, t h e d r i f t p a r a m e t e r , equal to . I ; this kind of d i s a s t e r w a s previously analyzed in Figures l ( i i i ) and (iv).Note in Figure 3(i) t h a t in c o n t r a s t to t h e t r a j e c t o r i e s in Figure l ( i i i ) and (iv) t h e mortality t r a j e c t o r y of t h e afflicted c o h o r t i s only modestly higher, and only f o r a relatively s h o r t period, than t h e mortality t r a j e c t o r y f o r t h e advantaged cohort. Also note t h a t t h e t w o mortality t r a j e c t o r i e s are qualitatively similar t o previous mortality t r a j e c t o r i e s : t h e substantial turbulence in this mortality re- gime i s not a p p a r e n t in t h e t r a j e c t o r i e s of population force of mortality.
Figures 3(ii) and (iii) display t h e t r a j e c t o r i e s of the f o r c e s of debilitation and selection, respectively. Debilitation i n c r e a s e s as a r e s u l t of t h e d i s a s t e r and then falls below t h a t f o r t h e advantaged c o h o r t , f o r t h e s a m e r e a s o n s discussed e a r l i e r . What i s new h e r e is t h e v e r y r a p i d i n c r e a s e in t h e f o r c e of debilitation with a g e , although t h e underlying cause of this, t h e decline in t h e value of mean f r a i l t y , i s t h e same as discussed b e f o r e . The f o r c e of selection a l s o r i s e s v e r y rapidly with a g e , and f o r both t h e f o r c e of debilitation and selection t h e afflicted a n d advan- taged c o h o r t ' s t r a j e c t o r i e s quickly converge.
Forw of Mortality (Log Scale)
Selection
1.
r
Figure 3(iI: Tmjectories of ii Force of DebilitationNet Forw
of Minus Selection
.25r
Figure 3lvil: Trajectories of pd-
psFigure 3. Aspects of mortality in a turbulent regime.
Figures 3(i)-(vii) display the trajectories of
p,
p d , p , , pd -p,, m , 7 , and nr '+?, respectively.Maan
Indisposition 2
0
Frailty 5
4
Figure 3(v): Trajectories of m Variance in 6
Indisposition
Figure 3lvii): TrejectoriaofmZ + y
Figure 3(d l : Trojectoria of y
A s shown in Figure 3(iv), t h e f o r c e of selection manages, a f t e r a g e 20 o r 30, t o k e e p ahead of t h e f o r c e of debilitation, s o t h a t t h e rapid i n c r e a s e in both f o r c e s r e s u l t s in a balance t h a t somewhat f a v o r s selection. The r a p i d i n c r e a s e in t h e f o r c e of selection c a n be i n t e r p r e t e d as resulting from t h e r a p i d i n c r e a s e in t h e f o r c e of debilitation, but t h e f o r c e of selection does not lag behind t h e f o r c e of debilitation, but s t a y s a h e a d of it.
The s t r e n g t h of t h e f o r c e of selection d r i v e s t h e mean value of indisposition t o z e r o , as shown in Figure 3(v). The variance in indisposition r i s e s t o a peak but then falls off as t h e f o r c e of selection exceeds t h e f o r c e of debilitation, as shown in Figure 3(vi). T h e r e i s only a single c u r v e in this figure because t h e d i s a s t e r does not a f f e c t t h e variance in indisposition but only t h e level of indisposition.
The combination of t h e s e two t r a j e c t o r i e s , in the form of m2+7, produces t h e tra- jectories f o r mean f r a i l t y displayed in t h e final figure, Figure 3(vii). In this t u r - bulent mortality regime, memory of t h e d i s a s t e r is no longer a p p a r e n t a f t e r age 50.
EXTENSIONS AND OTHER APPLICATIONS
The model of mortality p r e s e n t e d and illustrated above can be extended in various w a y s and applied t o o t h e r kinds of population problems. This section adumbrates a few possibilities.
Our c o n c e r n h e r e h a s been conceptual advance and insight, r a t h e r than sta- tistical estimation and inference. Elsewhere, however, we discuss how Woodbury- Manton stochastic models c a n be used in empirical studies (see, e.g.. Woodbury e t al. 1979, Manton and S t a l l a r d 1984, and Yashin e t al. 1985). In many of t h e s e appli- cations i t i s a p p r o p r i a t e t o distinguish s e v e r a l different, interacting s t o c h a s t i c p r o c e s s e s r e l a t e d t o various physiological, behavioral o r environmental f a c t o r s t h a t may b e continuously observed, partially observed, o r unobserved. For-
I
tunately, t h e univariate-process model described h e r e i s readily generalized t o a multivariate-process model with various kinds of d a t a (Woodbury and Manton 1977, Yashin et al. 1985).
Our focus h a s been on d i s a s t e r s , but t h e model could a l s o b e used t o study o t h e r a s p e c t s of mortality, including t h e typical s h a p e of mortality t r a j e c t o r i e s . A remarkable f e a t u r e of most human mortality c u r v e s i s t h e bump in mortality r a t e s , usually c e n t e r e d around a g e 20 o r 25. In developed countries today t h i s bump i s g r e a t e r f o r males than f o r females and c a n be largely explained by violent d e a t h s
resulting from accidents, homicide, and suicide. But a bump also a p p e a r s in mor- tality t r a j e c t o r i e s of c o h o r t s born more than a century ago and in mortality tra- jectories f o r less developed countries (see, e.g., P r e s t o n e t al. 1972). The model presented h e r e can be specified t o produce such as bump. By choosing less extreme p a r a m e t e r values, t h e v e r y prominent bumps in some of the figures dis- cussed above can be reduced t o realistic size. I t seems plausible t h a t at least some of t h e e x c e s s mortality bump i s caused by a kind of debilitation t h a t o c c u r s during t h e adolescent and e a r l y adult y e a r s as a r e s u l t of individuals being confronted with environments t h a t they are not fully p r e p a r e d t o deal with. F o r various phy- siological, behavioral, and environmental reasons, as an individual r e a c h e s matu- r i t y a gap may develop between e x t e r n a l demands and i n t e r n a l capabilities and inclinations, a g a p t h a t f o r most individuals i s reduced with age, partially as a r e s u l t of learning and t h e acquisition of wisdom and caution. I t may p r o v e informa- tive t o apply t h e kind of stochastic model presented h e r e t o analyze t h e bump in mortality in various countries and times.
In addition, the model could be used
to
analyze t h e effects of lifetime depriva- tion and t h e e f f e c t s of p r o g r e s s o v e r time in reducing mortality levels. F o r a wide variety of different specifications of t h e model, t h e mortality t r a j e c t o r y of a disadvantaged c o h o r t will converge toward t h e t r a j e c t o r y of a n advantaged cohort. Similarly, equal r a t e s of p r o g r e s s at all ages in reducing the underlying f o r c e of mortality on t h e individual level will r e s u l t in declining rates of p r o g r e s s with a g e in reducing t h e observed, population f o r c e of mortality. Thus, t h e r e will a p p e a r t o be convergence between t h e mortality t r a j e c t o r y of a c o h o r t not bene- fitting from mortality p r o g r e s s and t h e t r a j e c t o r y of a c o h o r t t h a t does benefit.1
Essentially what is needed t o produce these p a t t e r n s of convergence is a higher level of mortality, at l e a s t before some age, f o r t h e disadvantaged c o h o r t and some heterogeneity in frailty, e i t h e r innate o r acquired with age.
This brings us
to
t h e question of how useful fixed frailty models a r e . Much of the theoretical work on heterogeneous population ( a s reviewed by Vaupel and Yashin 1985) as w e l l as nearly all the empirical work (including Manton e t al. 1981 and 1986, Heckman and Singer 1984, and Trussel and Richards 1985), has r e s o r t e d t o t h e simplifying assumption t h a t a n individual's f r a i l t y ( o r r e l a t i v e r i s k ) does not v a r y , at least o v e r t h e period being studied. That is, f r a i l t y i s not necessarily assumed t o be fixed from b i r t h , but frailty i s assumed t o b e constant a f t e r , say, age 6 5 if t h e analysis focuses on mortality r a t e s at older ages. This may be realis- tic. Furthermore, as o u r model suggests (and as discussed by Vaupel and Yashin1985), t h e assumption of fixed f r a i l t y may b e a r e a s o n a b l e e x p e d i e n t if e v e r y o n e ' s f r a i l t y c h a n g e s b y roughly t h e same a b s o l u t e o r p r o p o r t i o n a l amount: much of t h i s effect c a n b e c a p t u r e d in t h e baseline f o r c e of mortality & ( X ) . Finally, if t h e a c t i o n of homeostasis i s t o quickly r e s t o r e individuals t o t h e i r b a s e level of f r a i l t y , deviations from t h i s b a s e may not b e significant.
The k e y question t h e n i s w h e t h e r s u b s t a n t i a l s t o c h a s t i c c h a n g e s in f r a i l t y in- fluence mortality t r a j e c t o r i e s in a way t h a t c a n n o t b e c a p t u r e d by models t h a t as- sume fixed f r a i l t y . Our model, a n d t h e v a r i o u s a n a l y t i c a l e x p l o r a t i o n s w e h a v e en- gaged in using i t , i n d i c a t e t h a t debilitation a s s o c i a t e d with s t o c h a s t i c c h a n g e s i n f r a i l t y p r o d u c e s s u b s e q u e n t s e l e c t i o n t h a t h e l p s c o u n t e r - b a l a n c e t h e e f f e c t s of t h e debilitation. F u r t h e r m o r e , t h e main q u a l i t a t i v e r e s u l t s in t h e t h e o r y of h e t e r o - geneity, c o n c e r n i n g mortality c o n v e r g e n c e a n d t h e deviation of population t r a j e c - tories from underlying individual t r a j e c t o r i e s , remain valid in most cases e v e n given s u b s t a n t i a l s t o c h a s t i c debilitation. Nonetheless, s t o c h a s t i c c h a n g e s in f r a i l - t y c a n , as i l l u s t r a t e d b y t h e f i g u r e s p r e s e n t e d a b o v e , p r o d u c e mortality t r a j e c - t o r i e s q u i t e d i f f e r e n t from t h e t r a j e c t o r i e s p r o d u c e d when f r a i l t y i s fixed. Thus, mortality a n a l y s t s , e s p e c i a l l y in t h e i r e m p i r i c a l r e s e a r c h , may find i t useful to em- ploy a s t o c h a s t i c - p r o c e s s model when t h e y h a v e r e a s o n t o believe t h a t individual f r a i l t y may b e changing t u r b u l e n t l y .
DISCUSSION
Does a c o h o r t r e m e m b e r mortality p a s t ? A r e c u r r e n t a n d f u t u r e mortality rates e x p e r i e n c e d by a c o h o r t influenced by p r e v i o u s mortality rates o r , more d i r e c t l y , b y p r e v i o u s rates of morbidity a n d d e p r i v a t i o n ? I s t h e r e , in s h o r t , a c o h o r t e f f e c t d i s t i n c t f r o m a g e a n d p e r i o d e f f e c t s , a n d if s o , what i s t h e n a t u r e of t h i s e f f e c t ? These questions h a v e long b e e n important in demographic thought a n d remain of c e n t r a l c o n c e r n ( R y d e r 1965, H o b c r a f t , Msnken, a n d P r e s t o n 1982). The model p r e s e n t e d h e r e i s useful in conceptualizing a n d comprehending t h e complex n a t u r e of possible c o h o r t e f f e c t s a n d age-period-cohort i n t e r a c t i o n s . The under- lying a g e - p a t t e r n of mortality i s c a p t u r e d by t h e baseline mortality function, po.
The vulnerability, d r i f t , homeostasis, a n d s t o c h a s t i c i t y p a r a m e t e r s c a p t u r e e f f e c t s t h a t occur at s p e c i f i c a g e s or times. Because o u r focus w a s o n a single c o h o r t , w e did not distinguish between a g e a n d time in o u r analysis, b u t i t i s r e a d i l y possible to e x p l i c i t l y make t h e s e p a r a m e t e r s functions of a g e a n d of time.
Our model and i l l u s t r a t i v e r e s u l t s s u g g e s t t h a t i t may n o t b e p r o d u c t i v e t o conceptualize or model a g e , p e r i o d , a n d c o h o r t mortality e f f e c t s as t h r e e indepen- d e n t f a c t o r s and t h a t i t i s p a r t i c u l a r l y questionable t o assume t h a t t h e s e e f f e c t s are not only independent b u t also c o n s t a n t f o r e a c h a g e , p e r i o d , o r c o h o r t . Time- specific incidents of high mortality o r morbidity p r o b a b l y a f f e c t c o h o r t s of dif- f e r e n t a g e s d i f f e r e n t l y a n d t h e s e incidents p r o b a b l y h a v e lingering consequences t h a t gradually d e c a y as t h e r e s u l t of r e c u p e r a t i o n , selection, a n d s t o c h a s t i c c h a n g e s in individual f r a i l t y . F u r t h e r m o r e , debilitation a n d s e l e c t i o n should not b e thought of as independent f a c t o r s . Debilitation t h a t i n c r e a s e s population h e t e r o g e n e i t y will r e s u l t i n subsequent selection; selection, b y a l t e r i n g t h e d i s t r i - bution of population h e t e r o g e n e i t y , will influence t h e impact of debilitating events.
C o r r e c t l y conceptualizing t h e s e f a c t o r s is important f o r demographic t h e o r y a n d f o r understanding h i s t o r i c a l p a t t e r n s of mortality. In addition, a p p r o p r i a t e models of how c o h o r t s remember p a s t mortality and morbidity c a n c o n t r i b u t e t o public-health decisionmaking. If, f o r instance, high l e v e l s of morbidity in child- hood c a n b e linked to high levels of mortality at o l d e r a g e s , t h e n e f f o r t s to r e d u c e morbidity (and a s s o c i a t e d mortality) i n childhood will have t h e double benefit of a n immediate e f f e c t a n d a delayed e f f e c t . Understanding t h e magnitude of lingering mortality e f f e c t s c a n t h u s h e l p in determining t h e b e n e f i t s of a l t e r n a t i v e public- h e a l t h i n t e r v e n t i o n s a n d in t a r g e t i n g t h e s e interventions t o a c h i e v e maximum benefits.
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