The Multistate Life Table with Duration-Dependence

Working Paper

THE MULTISTATE IlFE TABLE WITH DURATION-DEPENDENCE

Douglas A. WolJ

May

1987 IP-87-46

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

THE MULTISTATE L13FE TABLE WITH DURATION-DEPENDENCE

Douglas A. W o w

May

1987 WP-87-46

Working Papers are interim reports on work of the International Institute f o r Applied Systems Analysis and have received only limited review. Views o r opinions expressed herein do not necessarily represent those of the Institute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A - 2 3 6 1 Laxenburg, Austria

The classical l i n e a r multi-state model is r e p r e s e n t e d by a n equation due to Kolmogorov, and applied to demography by Andrei Rogers. For many purposes i t gives a r e a l i s t i c r e p r e s e n t a t i o n of phenomena, especially in problems in which t h e population is nearly homogeneous. In t h a t r e s p e c t i t resembles t h e o r d i n a r y life t a b l e , of which i t is a generalization. But like t h e life t a b l e i t a c t s a s though all of t h e individuals of a given c a t e g o r y have t h e identical probability, s o t h e statisti- cally o b s e r v e d a v e r a g e r e p r e s e n t s e a c h and e v e r y individual in i t s category.

No demographer h a s e v e r r e g a r d e d t h i s as quite s a t i s f a c t o r y ; all recognize t h a t individuals within a given c e l l a r e d i f f e r e n t from one a n o t h e r and t h e a v e r a g e of t h e c e l l does not apply to individuals. In a given g r o u p e v e r y couple may have one chance in 3 of divorcing; o r else lj9 of couples may divorce 3 times e a c h . The o v e r a l l probability t h a t a couple will divorce is t h e same in t h e t w o c a s e s , but t h e inference a b o u t what will happen to a random couple in t h e f u t u r e i s v e r y different f o r t h e two. Y e t to t a k e into account t h i s distinction involves difficulties, both of d a t a and of t h e model f o r dealing with t h e d a t a .

James Vaupel a n d Anatoli Yashin of this program have made g r e a t p r o g r e s s in dealing with t h i s question, and t h e i r work will be b r o u g h t t o g e t h e r in a volume now being p r e p a r e d .

The p r e s e n t p a p e r sets out t h e t h e o r y of a p r o c e d u r e for taking account of a p a r t i c u l a r kind of heterogeneity-that associated with t h e length of time in a s t a t e . Insofar a s people a r e l e s s likely t o d i v o r c e t h e longer t h e y have been married, and if divorce rates by duration a r e known, s e p a r a t e transition matrices c a n be set up f o r d i f f e r e n t durations. Douglas Wolf ingeniously shows how t h e s e s e p a r a t e transi- tion matrices c a n b e combined in a single matrix, and t h e analysis c a r r i e d out sim- ply and without f u r t h e r r e f e r e n c e t o duration.

Thus what follows h a s a special significance f o r IIASA's population program, in t h a t i t combines lines of thought t h a t g o back to t h e multi-state model introduced by Rogers, and on which many IIASA p a p e r s w e r e based in t h e period 1975-83, and t h e work on heterogenelky of Vaupel a n d Yashin, t h a t h a s been c e n t r a l to IIASA's program in more r e c e n t y e a r s .

Nathan Keyfitz

L e a d e r , Population Program

Acknowledgement

The a u t h o r has benefited from numerous discussions with Wolfgang Lutz, and from useful comments by Nico Keilman and Andrei Rogers on a n e a r l i e r d r a f t . The U.S. National Institute of Child Health and Human Development made available t h e June 1985 CPS d a t a , and Laszlo Zeold and Margaret K e r r helped p r e p a r e t h e d a t a f o r analysis.

-

iii

-

THE BIULTISTATE LIFE TABLE WITH DURATION-DEPENDENCE

INTRODUCTION

In r e c e n t y e a r s techniques f o r constructing multistate increment-decrement life tables have been extensively developed, and have been fruitfully applied t o such demographic phenomena as migration (Rogers and Willekens, 1986), fertility (Suchindran and Koo, 1980) and marital-status dynamics (Espenshade, 1983; 1986).

The a p p r o a c h allows one t o p r e s e n t a useful t a b u l a r summary of a complex demo- g r a p h i c p r o c e s s , one in which individual units may make a number of transitions among p a i r s of d i s c r e t e s t a t e s o r statuses. One r e s t r i c t i v e f e a t u r e of the ap- p r o a c h i s t h e Markov assumption-namely t h a t a g e s p e c i f i c transition intensities depend only on t h e s t a t u s c u r r e n t l y occupied.

U s e r s of t h e technique have recognized t h a t t h e Markov assumption i s overly r e s t r i c t i v e (see, f o r example, Wijewickrema and Alli, 1984; Espenshade, 1986), y e t t o d a t e t h e r e a p p e a r s t o have been no development of a method of life table con- s t r u c t i o n incorporating such "duration dependencen.'

This p a p e r d e s c r i b e s a new way t o formulate a multistate life t a b l e such t h a t transition intensities v a r y by both a g e and duration of time in status. The solution proposed h e r e i s motivated by a d e s i r e to utilize insofar as possible t h e mathemat- i c s t h a t have been developed f o r the usual multistate life table case-that is, t h e c a s e in which duration-dependence does not a p p e a r . I t t u r n s out to be r a t h e r easy t o i n c o r p o r a t e the generalization, provided t h a t we are willing t o introduce dura- tion in a v e r y specific way: in p a r t i c u l a r , w e use age-specific rates t h a t v a r y by

"duration-category a t l a s t birthday". The term "duration-category" will be ex- plained below. There are undoubtedly o t h e r possible solutions to t h e problem of constructing a duration-dependent life table; t h e v i r t u e s of t h e approach

%here has, however, developed a discrete-time semi-Markov approach based upon renewal equa- tions, which like the present approach can handle duration-dependent transitions. S e e Littman and Mode (1977) and Mode (1980); for some applications s e e Hennessey (1980) and Rajulton (1985).

d e s c r i b e d h e r e i s t h a t i t uses existing mathematical tools, and r e q u i r e s inversion of matrices no l a r g e r than t h o s e encountered in t h e s t a n d a r d multistate case.

The method generalizes t h e l i n e a r model whose e a r l y development i s d u e pri- marily to Rogers (1975), and in p a r t i c u l a r r e l i e s o n what h a s been termed t h e

"linear integration hypothesis" (Hoem a n d Funck Jensen, 1982). The l i n e a r model h a s been c r i t i c i s e d , and shown t o produce nonsensical r e s u l t s i n s o m e cases (see Hoem and Funck Jensen, 1982; Nour and Suchindran, 1983; a n d Keilman and Gill, 1986). Nonetheless t h e l i n e a r model enjoys widespread use and evidently p e r f o r m s s a t i s f a c t o r i l y in most applications, and s o i t seems r e a s o n a b l e to a d o p t i t as t h e s t a r t i n g point f o r a m o r e g e n e r a l model. But the shortcomings of t h e basic model no doubt p e r t a i n to t h e more g e n e r a l one d e s c r i b e d h e r e , as well.

Before proceeding, i t i s worthwhile to consider why one might want to incor- p o r a t e duration-dependence i n t o a life t a b l e in t h e f i r s t place. A simple answer t o t h i s question is t h a t a life t a b l e which i n c o r p o r a t e s t h e duration dimension i s con- s i d e r a b l y more informative than is t h e usual life table. W e c a n , f o r example, cal- c u l a t e t h e s h a r e of a l l person-years lived i n a given s t a t u s t h a t are lived p r i o r t o t h e f i r s t a n n i v e r s a r y , between t h e f i r s t and second a n n i v e r s a r y , and so f o r t h . In applications such as t a b l e s of working life, t h i s additional information h a s particu- l a r significance: w o r k e r s typically gain in firm-specific human c a p i t a l e a r l y in t h e i r t e n u r e , so t h e d e g r e e of concentration of work e x p e r i e n c e at low t e n u r e s c a n b e used as a n index of r e s o u r c e s devoted to training costs. Also, since c u r r e n t a g e plus c u r r e n t d u r a t i o n are sufficient to determine a g e of most r e c e n t transition, a life t a b l e which disaggregates s u r v i v o r s h i p at e a c h a g e by duration as well as s t a t u s c a n b e used to study i n t e r c o h o r t differences in survivorship.

Another, and a more compelling, r e a s o n i s t h a t a duration-dependent life t a b l e may produce d i f f e r e n t r e s u l t s t h a n t h e usual a p p r o a c h , especially with respect to t h e s t a t u s distribution at a given a g e ( t h e L(z) figures). The difference c a n a r i s e when period d a t a are used, f o r a population in which t h e c u r r e n t duration-in- s t a t u s distribution d e p a r t s from t h a t f o r t h e life t a b l e (stable) population. Infer- e n c e s from a duration-dependent life t a b l e may thus p r o v e t o b e more a c c u r a t e , and even small d i f f e r e n c e s c a n p r o v e t o b e important i n p r a c t i c a l applications.

The following section d e s c r i b e s t h e formulation of a d u r a t i o n 4 e p e n d e n t multi- state life table, and provides formulas f o r t h e calculation of transition probabili- ties. Most of t h e discussion i s devoted t o t h e derivation of s u r v i v o r s h i p f i g u r e s f o r a population at a sequence of e x a c t a g e s 0 , l . .

.. .

This i s followed by a discus- sion of s e v e r a l summary indices d e r i v e d from t h e s u r v i v o r s h i p figures. W e then

consider briefly t h e d a t a requirements of t h e proposed model, and conclude with a n illustration: a simple marital s t a t u s life t a b l e based upon r e c e n t U S d a t a .

THE

MODEL

A.eLiminaries. W e use as a s t a r t i n g point a s t a n d a r d formulation of t h e multi- state life t a b l e (MSLT), employing essentially Keyfitz's (1979) notation. Thus, l e t M ( z ) b e t h e matrix of transition intensities between p a i r s of states in t h e set 1,

...,

^{n ,} between a g e s z and z + 1 . The contents of M ( z ) a r e depicted in (1); ele- ments m i j ( z ) c o r r e s p o n d to transition rates into state i f r o m state j between ex- act a g e s z and z + l , while elements m6,(z) are death rates i n state j between t h e s e ages.

The fundamental r e s u l t used in t h e sequel i s t h e following:

where L(z) is a n a r r a y r e p r e s e n t i n g survivorship (numbers, or proportions) in states 1 , .

...

^{n at} e x a c t a g e z . The time unit used in this calculation i s a single period. Other life-table functions of interest-such as life expectancies, and s o on-can b e derived from t h e L(z) a r r a y s . The derivation of (2) i s discussed in s e v e r a l s o u r c e s , including R o g e r s and Willekens (1986); Keyfitz (1986); and Willek- e n s et a l . (1982).

I n c o r p o r a t i n g d u r a t i o n - d e p e n d e n c e . W e now consider a n extension of t h e above formulation to t h e case in which transition rates v a r y by duration of expo- s u r e to r i s k as w e l l as by a g e . W e denote t h e model a "duration-dependent multi- state life table" (DDMSLT). That is, w e suppose t h a t a t e a c h a g e , z , t h e r e is a s e p a r a t e matrix of transition rates of t h e form found in ( I ) , pertaining to p e r s o n s in duration c a t e g o r i e s d

=

0 , l ,

...,

^z

.

A person of a g e z will be in duration c a t e g o r y d if t h e most r e c e n t a n n i v e r s a r y in t h e s t a t e c u r r e n t l y occupied w a s t h e

d - t h a n n i v e r s a r y . Obviously d S z at e a c h a g e

z .

Note t h a t "duration c a t e g o r y " h a s a r a t h e r s p e c i a l relationship to "duration"

in this formulation. A t e x a c t a g e

z ,

someone i n duration c a t e g o r y d has been in t h e i r c u r r e n t s t a t u s at l e a s t d , but less than d + l , time units (years). A t a g e z

+&

⁽⁰

<

^&

<

I ) , t h i s p e r s o n may have passed t h e d + l t h anniversary, depend- ing on t h e e x a c t timing of t h e previous transition. Y e t w e classify individuals only with r e s p e c t to t h e duration c a t e g o r y occupied on a given birthday.

In view of t h e way in which age- and duration-dependent transition rates are defined h e r e , t h e essence of t h e proposed model i s as follows. First, t h e L ( z ) ma- t r i x of survivorship according to s t a t u s occupied, i s modified to accommodate duration-dependent rates. Then, t h e r a t e s , suitably a r r a n g e d , are manipulated us- ing essentially t h e s a m e mathematics as in t h e usual MSLT c a s e , yielding L ( z + l ) . Someone in state i , and duration c a t e g o r y d , at age z , and who s u r v i v e s to a g e z +1 in state i , h a s necessarily advanced t o duration c a t e g o r y d +l. Thus, t h e ele- ments of L ( z +1) are r e l a b e l l e d at a g e z + I , to r e f l e c t t h e advancement or "promo- tion" in duration. This p r o c e s s continues until t h e terminal a g e of t h e life t a b l e h a s been r e a c h e d .

I t should be recognized t h a t i n t h e expanded formulation duration h a s not been i n c o r p o r a t e d into t h e state space. If duration were to b e incorporated into t h e state s p a c e , w e would be r e q u i r e d to contend with quantities described as t h e

"rate of movement f r o m state i , d "-with d indexing duration categories-"to state i',d'". Instead, w e are c o n c e r n e d h e r e with quantities described as t h e "rate of movement f r o m state i to i ' , given t h a t duration at last anniversary w a s d a t ex- act a g e z

".

The distinction i s r a t h e r fine-and t h e v e r b a l description of t h e rates used h e r e i s somewhat cumbersome-but t h e formulation adapted h e r e g r e a t l y fa- c i l i t a t e s computation, as s h a l l b e s e e n .

W e f i r s t develop t h e a p p r o a c h f o r a simple case i n which flows out of all states are governed by duration-dependent transition rates (or, m o r e simply, a l l states are "duration-dependent states"), with o n e set of rates at e a c h duration c a t e g o r y up t o age w (the maximum a t t a i n a b l e birthday). The model r e q u i r e s t h a t at a g e z , w e have a sequence of m a t r i c e s Md(z), Mo(z), Ml(z),

..., M,

( z ) , e a c h of which i s in t h e f o r m of (1). The s u b s c r i p t s A, 0 , 1,

...

refer to duration categories.2 As noted b e f o r e , c a t e g o r y d refers t o those whose last a n n i v e r s a r y in t h e c u r r e n t s t a t u s Zlndividual elements o f Md(Z) now bear three subscripts: mlld(z) i s the rate o f j + i move- ment between exact ages Z and Z 4-1, given that the duration category in s t a t e j a t age z i s d .

w a s t h e d t h a n n i v e r s a r y .

Duration c a t e g o r y A plays a special r o l e in t h e model. This is t h e c a t e g o r y entered if a transition o c c u r s between ages z and z + l . In words, an off-diagonal element of MA(z) i s t h e "rate of j-to-i movement between a g e s z and z +1, given t h a t a k - t o - j move h a s t a k e n place s i n c e e x a c t a g e z

".

Someone who has e x p e r i - enced a transition i n t o s t a t u s i between a g e s z and z + l will be in duration c a t e g o r y 0 at e x a c t a g e z +l. T h e r e f o r e , what we are calling duration c a t e g o r y A might as easily (but not as tidily) b e called c a t e g o r y "-1".

Calculations f o r t h e DDMSLT are g r e a t l y facilitated if t h e rates are a r r a n g e d in the following way. F i r s t , l e t DMd ( z ) denote t h e matrix of Md ( z ) , with i t s off- diagonal elements r e p l a c e d by zeros. Second, let CMd ( z ) be t h e matrix Md ( z ) with i t s main diagonal elements r e p l a c e d by z e r o s . Then Md ( z )

=

CMd ( z )

+

DMd ( z ) . A l l t h e matrices aWd ( z ) and CMd ( z ) , d

=

A,O,l,

...,

z , a r e , of c o u r s e , n-by-n matrices.

The full matrix of age- and duration-dependent r a t e s , analogous to (1) but now denoted M* ( z ) , i s defined as follows:

M* ( z ) i s t h u s a n (nz +2n)-by-(= + 2 n ) matrix of rates. Moreover, i t s diagonal i s t h e duration-dependent c o u n t e r p a r t to t h e corresponding diagonal in t h e usual MSLT case: f o r a given duration category, d , t h e diagonal element of M* ( z ) equals m 6jd ( z )

+

myd (2). The matrices Mo(x), Ml(z),

...,

have t h u s been pulled

i f j

a p a r t , with t h e i r off-diagonal elements appearing in a band across t h e t o p of (3), while t h e i r diagonal elements a p p e a r as t h e diagonal of (3).

In o r d e r to conform to M* ( z ) , t h e L ( z ) a r r a y of survivorship figures must b e a column v e c t o r of t h e form

t h e symbol

" 1 "

indicating grouping by duration c a t e g o r y ; a n element Ltd(x) r e p r e s e n t s the number ( o r proportion) of t h e r a d i x population in state i , duration

c a t e g o r y d

,

a t e x a c t a g e z

.

The f i r s t n elements of 1 ( z ) a r e z e r o s , correspond- ing to duration c a t e g o r y A. I t i s impossible t o occupy c a t e g o r y A a t e x a c t a g e z ; r a t h e r , as noted above, transitions occurring between a g e s z and z + 1 are tan- tamount t o moves into c a t e g o r y A.

Now, l e t 1 * ( z ) b e t h e r e s u l t of t h e following operation, analogous t o t h a t given in (2):

Since

M

* ( z ) i s a matrix with ( n z + 2 n ) rows and columns, t h e computational requirements n e c e s s a r y to calculate I * ( z ) may a p p e a r formidable. However, this t u r n s out not t o be so. To simplify notation, l e t Y

=

(I

+ -

M ' ( z ) ) and

2

Z = ( I - -). The p a t t e r n s of z e r o and nonzero elements in both Y and

Z

a r e 2

t h e same as in M * (2). Then Y c a n be written in partitioned form as

where

R =

MA(z); S

=

[CMo(z) CMl(z)

-

CM=(z)]; and

S i s thus n-by-(hz + n ) , while T i s a diagonal ( h z +n)-by-(hz + n ) matrix. Z can be similarly partitioned, and written as

I t c a n easily be verified t h a t

%ere we consider the simple situation of a single radix state, in which case 1 ( z ) i s a vector.

More generally, L(Z) i s a matrix with a s many columns as there are initial statuses.

Since T is diagonal, i t s inverse i s trivially easy t o calculate. Thus, only f o r R , a n n-by-n matrix, i s a matrix-inversion algorithm r e q u i r e d . This matrix inversion problem i s of the same computational o r d e r as in t h e MSLT with t h e same state- s p a c e , but without duration-dependence.

Using (6) we c a n r e w r i t e (5) as

The data-storage requirements associated with (8) a r e admittedly much g r e a t e r than in the usual MSLT case. However, since T and W a r e both diagonal, they c a n b e s t o r e d as vectors [of length nz + n ] ; t h e diagonal matrix product T ' W can b e obtained directly and similarly s t o r e d as a vector.

The v e c t o r I*(=) contains, in i t s f i r s t n elements, t h e a r r a y of survivors t o a g z x +1 who made transitions between a g e s x and x + l . These individuals neces- s a r i l y a r e in duration c a t e g o r y 0 at a g e x + l . The next n elements contain the ar- r a y of survivorship t o a g e x + 1 of those in duration c a t e g o r y 0 at a g e x ; t h e s e in- dividuals have not made a transition between ages x and x +1, and t h u s have ad- vanced t o duration c a t e g o r y 1 ; and s o on. Thus, I* (x ) is of t h e form

The v e c t o r L*(x) must b e manipulated into the form given by (4). in o r d e r t h a t t h e computation c a n proceed t o t h e next age. In o t h e r words, t h e f i r s t n elements of L(x +1) must be z e r o s , t h e next n must denote s u r v i v o r in duration-category z e r o , and s o on. To d o this, we merely augment L ' ( x ) above with n zeros, and rela- bel t h e augmented v e c t o r 1 ( x + I ) . The manipulation r e q u i r e d c a n be expressed in matrix form as

where A is a n (nz +3n )-by-(= + 2 n ) matrix of t h e form

Combining (5) and ( l o ) , we c a n r e p r e s e n t t h e basic DDMSLT calculation as

IncLuding s t a t e s whose r a t e s a r e n o t d u r a t i o n - d e p e n d e n t . The previous section p r e s e n t e d t h e essential mathematics of t h e DDMSLT, but only f o r t h e case in which a l l transition rates are d u r a t i o n d e p e n d e n t . However, such a simple case is often i n a p p r o p r i a t e in p r a c t i c e . For example, in a marital s t a t u s life t a b l e t h e r e c a n b e no independent a g e and duration-dependence in rates of e n t r y into f i r s t union; similarly with f i r s t employment in a working-life table. In both exam- ples, t h e analyst would typically study a synthetic c o h o r t whose life begins in a purely age-dependent state (e .g

.

"never married" or "never worked").

In c o n t r a s t , if t h e states are geographic regions, and t h e transitions of in- terest are migration p a t t e r n s , i t is possible to b e born i n t o a d u r a t i o n d e p e n d e n t state. H e r e , though, a n o t h e r s p e c i a l point must b e recognized. If i t i s possible to b e b o r n into a duration-dependent s t a t e , then at e x a c t a g e z one should e x p e c t to see individuals i n duration c a t e g o r y O,l,

...

^{, z}

.

Those i n c a t e g o r y d e n t e r e d t h e i r c u r r e n t s t a t u s between t h e i r z -d -1th and z 4 t h birthdays. In g e n e r a l we antici- p a t e t h a t t h e e x a c t durations of those in duration c a t e g o r y d are distributed (ap- proximately evenly) on t h e interval [d , d + l ) . Y e t those in duration c a t e g o r y z have i n f a c t n e v e r moved, and t h e i r e x a c t duration in s t a t u s i s a l s o z , t h e s a m e as t h e i r e x a c t age. This distinction must b e b o r n in mind, especially when p r e p a r i n g t h e input d a t a f o r a DDMSLT model.

Only a slight modification of t h e a p p a r a t u s described above i s n e c e s s a r y , if s o m e states are not duration-dependent. Suppose t h a t nl states are not duration- dependent, while n 2 are duration-dependent. W e simply a r r a n g e t h e L ( z ) a r r a y s , and t h e

M'

( z ) matrices, such t h a t t h e non-duration-dependent states a p p e a r f i r s t . Now, t h e Mb(z) submatrix h a s t h e s t r u c t u r e

where Mfl)(z)-which i s nl-by-n l-represents flows within t h e set of non- duration-dependent s t a t e s , M ^{f 2 )} ( z )-which i s n l-by-n l--represents flows from duration-dependent to non-duration-dependent s t a t e s , and so on. However, since t h e f i r s t nl states d o not depend on duration, t h e r e are no corresponding subma- t r i c e s ~ 6 l l ) ( z ) ,

- .

^, ~ ! ~ ~ ) ( z ) or M ~ ~ ' ) ( z ) ,

. .

, Mi2')(=). Thus, M1(z), t h e full

matrix of rates used in t h e DDMSLT, c h a n g e s to accommodate t h e set of non- duration-dependent rates i n only t h e following ways: across t h e t o p , we find t h e s e q u e n c e Mi") ( x ) , Mf2) ( x ) , M0(12) ( x ) ,

- . .

^, M2(lZ)(x); down t h e l e f t s i d e , we find (again) hff1)(x), t h e n M ~ ~ ' ) ( X ) , a n d t h e n z e r o s . With t h e rates a r r a n g e d t h i s way, t h e computational a p p r o a c h d e s c r i b e d a b o v e s t i l l applies.

m e n - e n d e d d u r a t i o n c a t e g o r i e s . In some a p p l i c a t i o n s d u r a t i o n of e x p o s u r e to r i s k will b e c a t e g o r i z e d i n t h e r e s t r i c t e d set O , l ,

...,

u , w h e r e u i s a n open- e n d e d uppermost c a t e g o r y . I t may b e , f o r example, t h a t t h e o r y s u g g e s t s t h a t t h e h e t e r o g e n e i t y within a c o h o r t s o r t s itself o u t i n t h e f i r s t few y e a r s , a f t e r which susceptibility to r i s k i s c o n s t a n t . I t may also b e t h e case t h a t a v a i l a b l e d a t a are insufficient t o r e v e a l s t a t i s t i c a l l y significant d i f f e r e n c e s in rates of movement by d u r a t i o n f o r l a r g e values of d u r a t i o n .

The e f f e c t of using a n open-ended u p p e r d u r a t i o n c a t e g o r y i s t h a t t h o s e who are in s t a t e i , d u r a t i o n c a t e g o r y u , at a g e z , a n d who d o n o t e x i t to some o t h e r state between a g e s z a n d x +1, r e m a i n in d u r a t i o n c a t e g o r y u at a g e x + l ; t h e r e are n o f u r t h e r "promotions". This modification obviously m a t t e r s only at a g e s z

>

u

.

F o r a g e s x

>

u

,

t h e dimensions of 1 ( x +1) a n d 1 ( 2 ) are t h e same, contain- ing nu +n elements. In a l l o t h e r r e s p e c t s , t h e calculations d e s c r i b e d a b o v e are t h e same, e x c e p t t h a t t h e matrix A, now i s a n ( n u + n ) - b y - ( n u + n ) matrix of t h e f o m

f o r x

>

u . In t h i s form, A, causes t h o s e newly promoted i n t o d u r a t i o n c a t e g o r y u to b e a d d e d t o t h o s e a l r e a d y in c a t e g o r y u .

P a n s i t i o n s w h i c h p r e s e r v e d u r a t i o n . A s a final v a r i a n t form of t h e model.

we c o n s i d e r t h e e x i s t e n c e of moves between states which d o n o t set t h e d u r a t i o n clock back to z e r o ; i.e. t h e y d o not c h a n g e t h e s t a t u s v a r i a b l e f o r which t h e d u r a - tion clock i s running. One example of s u c h a s i t u a t i o n i s a combined m a r i t a l s t a t u s a n d f e r t i l i t y model, in which m a r i t a l s t a t u s t r a n s i t i o n s depend upon m a r i t a l s t a t u s d u r a t i o n ( a s well as a g e ) , while p a r i t y t r a n s i t i o n s e i t h e r depend upon m a r i t a l

s t a t u s duration (and a g e ) or on a g e only. In such a model a b i r t h , unaccompanied by a marital status change, leaves marital duration unchanged. A s a n o t h e r (and formally equivalent) example, a table of working life might distinguish t h e employ- ment s t a t u s e s "never worked", "employed", "unemployed", and "withdrawn from t h e l a b o r force", with rates of movement between employment and unemployment duration-dependent, while simultaneously keeping t r a c k of t h e number of job changes experienced; h e r e , increments to t h e count of job s h i f t s function exactly as p a r i t y in t h e marital s t a t u s / f e r t i l i t y example given above. Still a t h i r d example c a n be c o n s t r u c t e d using t h e s a m e four-state employment model, but distinguishing those with a single c o n c u r r e n t employer from those holding a secondary job.

H e r e , transitions between t h e states "single c o n c u r r e n t employer" and "multiple jobholder" c a n t a k e place without setting t h e duration-of-employment clock back to zero. In e a c h of t h e s e examples, t h e state s p a c e i s in s o m e sense two- dimensional (marital s t a t u s plus p a r i t y ; employment status plus cumulative number of employers, and so on).

Examples such as t h e s e c a n b e f i t i n t o t h e basic framework d e s c r i b e d above, and complicate t h e mathematics only slightly. In p a r t i c u l a r , t h e matrix

M'

( z ) i s now block-diagonal instead of diagonal, t h e size of e a c h block depending on t h e number of "duration-preserving" states t h a t are included in t h e model. Otherwise, t h e matrix [and t h e computational a p p r o a c h embodied in equation ( 8 ) ] i s essential- ly t h e s a m e . A c o n c r e t e illustration i s provided i n t h e appendix.

Sammary Indicators Based on the Model

The sequence L (O), L ( I ) ,

. . .

^, L ( z ) ,

. . .

obtained as d e s c r i b e d above c a n s e r v e as t h e basis f o r calculating a n extensive set of r e l a t e d quantities, including generali- zations of t h e person-years lived and expectancy calculations associated with t h e usual MSLT.

The a r r a y of survivorship f i g u r e s at e a c h a g e produced by equation (10) i s grouped by duration r a t h e r t h a n by state occupied, and hence i s somewhat awk- ward f o r purposes of displaying and i n t e r p r e t i n g the r e s u l t s . A m o r e convenient a r r a n g e m e n t is one i n which survivorship at e a c h a g e i s grouped by c u r r e n t s t a t e , and by duration within s t a t e , yielding t h e following matrix

r:

The LiA(z) e n t r i e s , which are always z e r o and t h e r e f o r e convey no r e a l informa- tion, are nonetheless included in (12) because t h e y facilitate t h e l a t e r calcula- tions.

In (12), a g e r u n s from l e f t t o r i g h t r a t h e r t h a n from t o p to bottom in t h e usual case. Within e a c h column of

r

survivorship i s shown by duration within states.

More generally, t h e r e would b e such a n r m a t r i x f o r e a c h r a d i x considered.

CaLcuLation ofL ('z) and r e l a t e d q u a n t i t i e s . In t h e usual MSLT t h e L ( z ) a r r a y provides t h e person-years lived between a g e s z and z +1 in e a c h state. In t h e DDMSLT, t h i s information i s disaggregated by duration c a t e g o r y , f o r duration- dependent states. For simplicity consider a single column r ( z ) s e l e c t e d from

r.

^{W e}

a l s o make use of t h e corresponding column r ( z ) , consisting of t h e e n d ~ f - y e a r ( p r i o r t o relabelling) survivorship f i g u r e s produced by equation (5), but rear- ranged as i n (12). That i s , t h e f i r s t row of contains

liA

^(0).

..., liA

( z ) , . .

.

^-or,

equivalently, LlO(l)

,...

, Llo(z +I),... -the a r r a y of s u r v i v o r s t o a g e z + 1 who a r e , at a g e z +1, in s t a t u s 1 and duration c a t e g o r y zero. Employing t h e usual l i n e a r s u r -

1

-

vivorship assumption w e c a n calculate

L

' ( z )

=

-[L ( z )

+ r*

(z)]. H e r e , t h e a s t e r i x 2

indicates t h a t

L

( z ) is a provisional quantity. N o w , t h e d t h e n t r y in

L

( z ) gives t h e number of person-years lived between a g e s z and z +1, by someone in s t a t u s 1 , duration c a t e g o r y (2, at e x a c t a g e z . The n + d t h e n t r y gives t h e corresponding f i g u r e f o r someone in s t a t u s 2 , duration c a t e g o r y d ; and s o on.

If someone i s in state i , duration c a t e g o r y d , at a g e z , and survives to a g e z +I and remains in s t a t e i t h e n p a r t of t h a t person-year of e x p e r i e n c e must be al- located t o c a t e g o r y d , and p a r t t o c a t e g o r y d+1. In keeping with t h e linear s u r - vivorship assumption, w e allocate one-half a person-year's experience t o e a c h category. However, rows 1 , n +1, 2 n +l,... , of L * (2)-which equal, respectively,

1 1

- [ I lA(z)

+

1 1 0 ( ~ +I)], -[LzA(z)

+

1 20(z +I)],

...

-consist of e x p e r i e n c e lived ex-

2 2

clusively in duration c a t e g o r y zero. Recall t h a t l i A ( z )

=

0 --one cannot occupy duration c a t e g o r y A on one's birthday-while li o ( z +1) contains t h e new a r r i v a l s in s t a t u s i as of t h e z +lst birthday. Thus, these rows of L * ( z ) must be allocated ex- clusively t o duration c a t e g o r y z e r o .

Let L ( z ) b e t h e column v e c t o r [Llo(z),Lll(z) L i d ( z ) . .

.

L ( z ) ] ' where Lid ( 2 ) r e p r e s e n t s t h e number of person-years lived in s t a t e i , duration c a t e g o r y d , between a g e s z and z +I. The reasoning of t h e preceding p a r a g r a p h s suggests t h a t w e obtain L ( z ) as follows:

and

f o r d

>

0. A compact matrix expression f o r (13) i s

where B is a v e r t i c a l concatenation of n repetitions of t h e w-by-w matrix

A matrix L containing, i n succession, t h e column v e c t o r s L (O),L ( I ) ,

...,

L (w -1) would display t h e distribution of t h e life t a b l e population's e x p e r i e n c e by state and

duration c a t e g o r y , y e a r by y e a r from b i r t h t o t h e terminal a g e considered, from t h e p e r s p e c t i v e of a specified initial s t a t u s . A s in t h e MSLT, w e c a n g o on to calcu- l a t e T y ( z ) , t h e total number of person y e a r s lived in e a c h status/duration c a t e g o r y from e x a c t a g e s y t o z , using

Of p a r t i c u l a r i n t e r e s t i s To(w), t h e t o t a l lifetime person-years of e x p e r i e n c e , or t h e expectation of life at b i r t h , a l s o denoted Eo. Again, all t h o s e quantities impli- citly condition on a single specified initial s t a t u s .

The lifetime e x p e r i e n c e of t h e population in state i i s given by

while t h e p r o p o r t i o n of t h i s e x p e r i e n c e t h a t i s lived in between t h e d t h and d + l t h a n n i v e r s a r y i s

If i i s a state which c a n b e r e e n t e r e d , and from which e x i t o c c u r s f a i r l y rapidly, then a substantial proportion of t h e total expectation of time s p e n t in i may p r e - c e d e t h e f i r s t a n n i v e r s a r y of e n t r y into i

.

Using (14), i t i s possible t o trace out a frequency distribution, by duration c a t e g o r y , of t h e life-table population's lifetime e x p e r i e n c e in a given status.

A d d i t i o n a l s u m m a r y i n d i c a t o r s . The information contained in t h e s u r - vivorship matrix c a n b e used to d e r i v e f u r t h e r summary indicators, unique to t h e DDMSLT. First, w e c a n approximate t h e a v e r a g e duration of c u r r e n t time-in-status ( o r , in renewal-theoretic language, backwards r e c u r r e n c e times) at e a c h age. A t a g e z , those in s t a t u s i , duration c a t e g o r y d , have on a v e r a g e been in state i f o r approximately d

+-

1 y e a r s . Using t h i s assumption, w e can compute a v e r a g e back-

2

wards r e c u r r e n c e times f o r s t a t u s i as

t h e l a s t t e r m reflecting t h e f a c t t h a t those in duration c a t e g o r y z at a g e z must have been i n s t a t u s i continuously since b i r t h .

I t i s a l s o possible to compare t h e survivorship of successive a g e c o h o r t s of e n t r a n t s into a given status, f o r example those e n t e r i n g status i at a g e s a l and a 2 , by simply reading along t h e a p p r o p r i a t e diagonals of

r.

In t h e f i r s t instance, t h e survivorship is given by t h e sequence Li o(al), Li , ( a l + l ) , Li2(al + Z ) ,

- -

; in t h e second instance, i t i s t h e sequence Li 0 ( a 2 ) , Li ,(a2+1), Li 2(a2+Z),

...;

and so on. Fi- nally, t h e median time to e x i t (by any reason) c a n b e approximated, simply by find- ing d' such t h a t Lid .(z +d ') w 1 Li o ( z ) .

D a t a Bequirements

The input d a t a r e q u i r e d f o r a DDMSLT might b e provided by s e v e r a l potential s o u r c e s . Retrospective event-history d a t a , collected in s u r v e y s with adequate sample sizes, c a n often b e used t o estimate t h e n e c e s s a r y rates directly. A migra- tion s u r v e y might, f o r instance, collect t h e dates, origins, and destinations of a l l moves made during t h e previous 12 months. In combination with date-of-birth d a t a , age- and duration-category-specific r a t e s , as defined previously, c a n b e tabulated.

Information on second (and h i g h e r + r d e r ) moves within the p e r i o d can a l s o provide d i r e c t estimates of t h e MA(z ) rates.

A second potential s o u r c e of t h e necessary d a t a i s a population r e g i s t r a t i o n system, such as t h a t found in s o m e European and Scandinavian countries. In such a system t h e population can b e classified according to y e a r of b i r t h (yob) and y e a r of l a s t event h o e ) as of t h e beginning of a c a l e n d a r y e a r , and e v e n t s experienced during t h e y e a r by e a c h yob-yoe combination can be counted. From t h e s e counts, in combination with a suitable assumption about exposure (i.e. t h e distribution of t h e midyear population) t h e n e c e s s a r y rates c a n b e computed directly. Data of this sort, from Finland, has in f a c t been used t o c o n s t r u c t a marital status/parity DDMSLT, some r e s u l t s from which are r e p o r t e d in Lutz and Wolf (1987). Similar d a t a are a l s o d e s c r i b e d and utilized by Keilman and Gill (1986), who a l s o provide a more elegant a p p r o a c h to calculating t h e rates.

A t h i r d potential s o u r c e of d a t a i s a vital-event r e g i s t r a t i o n system such as t h a t found in t h e United S t a t e s . For example, t h e S t a n d a r d Certificate of Divorce, Dissolution of Marriage or Annulment used in t h e U.S. Divorce Registration Area provides f o r the r e g i s t r a t i o n of divorces according t o e a c h f o r m e r spouse's y e a r of b i r t h , and t h e y e a r of m a r r i a g e (although t h e published divorce d a t a are not ta- bulated according t o both time concepts simultaneously).

Although Vital-Event r e c o r d s might s e r v e a s a s o u r c e of o c c u r r e n c e d a t a f o r rates in t h e form r e q u i r e d by t h e DDMSLT, t h e analyst would still f a c e t h e problem of assembling t h e requisite exposure data. This is complicated by t h e p r e s e n c e of a second time dimension--duration, in addition to age-to b e allocated t o t h e ap- p r o p r i a t e units. If, f o r instance, a midyear s u r v e y were t a k e n in which a g e and duration-category of c u r r e n t s t a t u s were a s c e r t a i n e d , i t would b e n e c e s s a r y to al- locate t h e population in a given age/duration-category combination to f o u r dif- f e r e n t yob/yoe combinations. Conversely, i t c a n easily b e shown t h a t a given yob/yoe "cohort" passes through f o u r d i f f e r e n t a g e ( a t l a s t birthday)/duration ( a t l a s t anniversary) combination during a calendar y e a r .

The rates used in t h e DDMSLT c a n be viewed a s a finite-valued approximation t o intensities defined on a continuous s e t of a g e s and durations, and denoted a s m i j ( a , d ) . The approximation embodies a simplifying assumption, namely t h a t t h e rates a r e constant o v e r subregions of a-d s p a c e with unit a r e a . The subregions happen t o b e parallelograms. A more n a t u r a l simplifying assumption, p e r h a p s , is t h a t t h e r a t e s a r e constant o v e r unit subregions defined by age-at-last-birthday, d u r a t i o n a t - l a s t a n n i v e r s a r y i n t e g e r s , t h a t i s unit s q u a r e s . Let us denote rates of t h e latter form as mijad, t h e use of s u b s c r i p t s reflecting t h e i n t e g e r n a t u r e of t h e a g e and duration arguments. Since in some applications rates defined on unit s q u a r e s may be available, o r more easily calculated, i t i s worth considering how to t r a n s l a t e from them t o t h e r a t e s w e have denoted mijd(a)-that is, rates in t h e form r e q u i r e d f o r t h e DDMSLT.

The translation i s r a t h e r s t r a i g h t f o r w a r d . F i r s t , since someone who e n t e r s s t a t e i between a g e s a and a +1 necessarily has a (continuous) duration-in-status less t h a n 1, t h e following equation holds:

NOW, a t e x a c t a g e a , people in duration c a t e g o r y d (in t h e s e n s e used in t h i s pa- p e r ) c a n b e assumed to b e distributed more o r less uniformly on the [d , d +1) i n t e r - val of continuous duration-in-status. In t h e coming y e a r , about half t h e i r e x p o s u r e will t h u s b e lived beyond t h e d +1th anniversary. Thus, f o r d

>

0, w e can write

If a n open-ended duration i n t e r v a l i s used in t h e DDMSLT, (15) holds only f o r d

>

^{u ;} f o r d r u t h e two a l t e r n a t i v e approximations are again equal.

An

Example

S e v e r a l of t h e f e a t u r e s of t h e DDMSLT a r e illustrated h e r e , using as an exam- ple a n a b b r e v i a t e d and g r e a t l y simplified marital s t a t u s model, employing d a t a f o r U.S. females f o r t h e 1980-84 period. Only f o u r statuses-single (S), first-married

(M), divorced @), and widowed (W)-are considered. Divorce and widowhood are t r e a t e d as absorbing s t a t e s . Of t h e t h r e e possible marital-status transitions, only t h a t from married to divorced i s t r e a t e d as duration-dependent.. Only e x p e r i e n c e s from a g e 15 to 5 0 are t r e a t e d .

F o r two of t h e transitions-single to married, and m a r r i e d t o divorced-rates were calculated from r e s p o n s e s given to t h e m a r r i a g e and f e r t i l i t y history ques- tionnaire appended t o t h e June 1985 C u r r e n t Population Survey (CPS). True occurrence-exposure rates were computed, with e x p o s u r e , in integer-valued a g e and duration c a t e g o r i e s , measured in person-months. F o r combinations of a g e and duration i n t e r v a l s with few o c c u r r e n c e s a n d / o r l i t t l e e x p o s u r e , aggregation o v e r a g e and duration was used to impose a modest amount of r e g u l a r i t y on t h e d a t a . The DDMSLT calculations were still performed f o r single-year-of-age/single- duration-category combinations, but with rates t r e a t e d as constant for s o m e g r o u p s of age/duration c a t e g o r i e s ; f o r example, a single r a t e was calculated f o r divorce rates among women aged 30-49, in y e a r s 0-4 of marriage (the m o s t extreme case of grouping used). Finally, divorce rates at a given a g e were t r e a t e d as con- s t a n t at durations 15 y e a r s and o v e r .

Selected duration-dependent divorce rates used in t h e example are plotted in Figure 1. The lines show t h e rates f o r duration c a t e g o r i e s 0 , 5, and 15+. The t h r e e lines plotted make c l e a r t h e substantial variability found in age-specific di- v o r c e r a t e s , according to t h e d u r a t i o n of marriage. The grouping scheme d e s c r i b e d above i s also r e f l e c t e d in Figure 1: t h e duration-zero r a t e s , for exam- ple, behave as a step-function a f t e r a g e 22. I t should b e noted t h a t o u r purpose h e r e is not to i n t e r p r e t or explain such differences--as, f o r example, period ef- f e c t s , or age-of-marriage e f f e c t s , or p u r e duration e f f e c t s ( f o r e f f o r t s in this direction s e e , f o r example, Thornton and Rodgers, 1987)-but merely to recognize and t a k e account of t h e differences, whatever t h e i r origin.

F o r p u r p o s e s of comparison, divorce rates depending on a g e only. calculated from t h e same CPS occurrence-counts and e x p o s u r e d a t a , a r e plotted in Figure 2.

No grouping or smoothing w a s imposed on t h e s e data. The divorce rates (which p e r t a i n to f i r s t m a r r i a g e s only, f o r t h e period 1980-84) s e e m s somewhat low com- p a r e d t o t h e U.S. Vital S t a t i s t i c s d a t a f o r 1982 (which p e r t a i n to all marriages);

Figure 1. Selected duration-dependent divorce rates, by age: U.S. females, 1980-84.

Age

u n d e r r e p o r t i n g of d i v o r c e s h a s been noticed in previous administrations of t h e CPS Marital and Fertility History Supplement (see Espenshade and Wolf, 1985), and may be operating h e r e as well.

The illustrative model i s f u r t h e r specified by t h e use of 1982 death rates f o r a l l females ( t r e a t e d as constant o v e r marital s t a t u s e s and durations) and widow- hood rates-using t h e 1982 d e a t h rate of men two y e a r s o l d e r than t h e married woman's c u r r e n t age-from U.S. Vital S t a t i s t i c s s o u r c e s .

In Table 1 are shown t h e columns 3 2 0 ) , r(25),. ..,r(50) produced by t h e input rates just described. The r a d i x f o r this matrix is 100000 single women aged ex- actly 15 y e a r s old. The s t a t u s "married,dM means "married, in duration c a t e g o r y d". A t a g e 20, m a r r i e d women are found only in duration c a t e g o r i e s 0 ,

...,

4; at e a c h successive 5-year observation period they a r e a r r a n g e d o v e r 5 additional dura- tion categories. I r r e g u l a r i t i e s in t h e duration distribution, at e a r l y durations, be-

Figure 2. Age-dependent d i v o r c e r a t e s : U .S. females, 1980-84.

l i l j l l l l i l l 1 j i l l l ( l i l i / i i l i j i i i i

- - - -

I I

: ^I

-

3

-

-

3

I 5

- -

1

'I

-

-

(-al 3 ^'I I

I W i r

- ^\

^,P,

^p\ -

⁹

'$

*,,{

^I#--,-,

. -

-

^I _'I

- -

r14 ^'I

l u A ^'? _{5.. =...}

-

= '\. 9

""5..

4

rn ³

-

^*

^L-=j-\.. -

-

! f l i J l ! j 1 1 1

p--=..

f i , l l l l ! f ! l ! 1 ! 1 1 1 1 l 1 1 1

9

'-'

15

r5 ^{&J r5c}

35 40 45

L 3

gin to a p p e a r in t h e r(40) column, a consequence of t h e unsmwthed age-at- m a r r i a g e rates used in t h e calculation of t h e table.

The final column of Table 1 gives t h e status-specific expectation of life (times 100000) from a g e 15 to 50; t h u s 12.70 y e a r s (per woman) are expected to b e lived in t h e s t a t u s single, 0.59 y e a r s are expected to b e lived in t h e s t a t u s widowed, and 3.29 y e a r s are e x p e c t e d to b e lived divorced ( r e c a l l t h a t widowhood and divorce are t r e a t e d as absorbing s t a t e s ) . Of t h e t o t a l expectation of life in t h e married state (17.96 y e a r s ) , 0.85 years-4.7 percent-is lived in duration category z e r o , 0.83 years-4.6 percent-is lived in duration category one, and so on, with a declin- ing p e r c e n t a g e lived in e a c h successive duration category. This frequency p a t t e r n r e f l e c t s t h e combined e f f e c t s of t h e p a t t e r n of a g e at f i r s t marriage and the dif- f e r e n t i a l d i v o r c e r i s k s by marital duration. In a t r u e increment-decrement table, with r e - e n t r y i n t o t h e married state a possibility, t h e frequency p a t t e r n of marital-duration e x p e r i e n c e would a l s o r e f l e c t t h e a g e p a t t e r n of higher-order

Table 1. Selected life table functions, duration-dependent marital s t a t u s life table f o r United States.

Total Status l(20i l(25) l(30) 1(35) l(401 l(451 l(5Oj Person-years

Lived Single ;

Ididowed Divorced Harried,O Harried, 1 Harried,2 Harried,3 Harried,4 Harried,5 Harried,&

Harried,7 Harried,8 Harried,9 Married,lO Harried,ll Harried,l2 Harried,l3 Harried,lS Harried,l5 Harried,lb Harried,l7 Harried,l8 Harried,l9 Harried,20 Harried,21 Harried, 2 2 Harried,23 Harried,24 Harried,25 Harried,26 Married,27 Harried,28 Harried,23 Harried,30 Harried,31 Harried,??

Harried,33

marriages.

For purposes of comparison, a n o r d i n a r y MSLT f o r t h e same marital s t a t u s model, differing only in t h e use of purely age-dependent divorce r a t e s (illustrated in Figure 2). A t each age, t h e two t a b l e s a r e identical with r e s p e c t t o t h e s t a t u s never-married. And, f o r t h e f i r s t s e v e r a l ages, the two tables a r e identical, o r nearly s o , in all o t h e r r e s p e c t s as well. T h e r e a f t e r , t h e proportions married and divorced differ, as a consequence of controlling f o r duration of marriage. The

p r o p o r t i o n m a r r i e d i s h i g h e r at all a g e s in t h e DDMSLT t h a n in t h e MSLT (with t h e e x c e p t i o n s of e x a c t a g e s 1 9 a n d 21)-as much as 1.5 p e r c e n t h i g h e r . G r e a t e r rela- t i v e d i f f e r e n c e s are found f o r t h e p r o p o r t i o n d i v o r c e d ; i n t h e DDMSLT t h e p r o p o r - tion d i v o r c e d i s as much as 9.2 p e r c e n t lower ( a t e x a c t a g e 3 2 ) t h a n t h e MSLT.

D i f f e r e n c e s t h i s l a r g e c a n , of c o u r s e , b e c r i t i c a l in some p r o j e c t i o n applications, suggesting t h e i m p o r t a n c e of accounting f o r d u r a t i o n e f f e c t s when t h e r e q u i s i t e d a t a c a n be assembled.

Table 2. S e l e c t e d t r a n s i t i o n p r o b a b i l i t i e s (times 1 0 0 ) from m a r i t a l status DDMSLT;

v a r i o u s initial statuses.

Initial s t a t u s

Age 15 Age 2 5 Age 35

Subsequent s t a t u s

Age 40: S 16.0 36.9

- -

^84.8

- - -

D

.

1 4 . 3 4.9 1 1 . 2 16.7 0.4 4.2 2.9 2.6

Age 45: S 14.0 32.4

- -

74.3

- - -

D 15.5 6 . 1 1 2 . 7 18.0 1.1 6.8 4.5 4.3

Age 50: S 13.4 30.8

- -

70.8

- - -

D 15.8 6.7 1 3 . 0 18.3 1.8 8.1 5.0 4.8

S e l e c t e d t r a n s i t i o n p r o b a b i l i t i e s from t h e DDMSLT are displayed i n Table 2 . With t h e e x c e p t i o n of t h e S + S p r o b a b i l i t i e s , a l l depend i n some way o n t h e p r e s - e n c e of duration-dependent rates in t h e analysis. The d u r a t i o n e f f e c t s are most pronounced f o r t h e M + D t r a n s i t i o n s from a g e 25: newly-married women at a g e 2 5 are much l e s s likely t h a n 2 5 y e a r old women in t h e i r fifth y e a r of m a r r i a g e , to b e d i v o r c e d at a g e s 40, 45. a n d 50. In c o n t r a s t , newly-married women at a g e 35 are more likely t h a n 35-year old women i n t h e i r fifth, or t e n t h , y e a r of m a r r i a g e , to b e d i v o r c e d at a g e s 40, 45, or 50.

Finally, we c a n compute a v e r a g e backwards r e c u r r e n c e times, which i n t h i s example i s merely t h e a v e r a g e d u r a t i o n of m a r r i a g e , f o r t h o s e c u r r e n t l y m a r r i e d , at e v e r y a g e . The a v e r a g e s are plotted in Figure 3 , which r e v e a l s t h a t t h e a v e r - a g e d u r a t i o n of c u r r e n t m a r r i a g e r i s e s slowly at f i r s t , r e a c h i n g approximately 7 y e a r s at a g e 30, a n d r i s i n g essentially Linearly from a g e s 3 0 to 50.

Figure 3. Average d u r a t i o n of m a r r i a g e , f o r t h o s e - c u r r e n t l y m a r r i e d .

S - r y

A method f o r generalizing t h e multistate, increment-decrement life t a b l e , t o include rates which v a r y by d u r a t i o n of e x p o s u r e to r i s k as well as by a g e , h a s b e e n p r o p o s e d . The method builds upon t h e l i n e a r approximation or l i n e a r in- t e g r a t i o n hypothesis developed p r i m a r i l y by R o g e r s a n d h i s colleagues. A computationally-efficient a r r a n g e m e n t of t h e n e c e s s a r y rates h a s b e e n p r e s e n t e d , o n e which r e q u i r e s inversion of m a t r i c e s no l a r g e r t h a n t h o s e o n e would e n c o u n t e r in t h e c o r r e s p o n d i n g multistate life t a b l e without duration-dependence.

The p r o p o s e d method hinges on t h e u s e of rates classified a c c o r d i n g to a g e , ( a t l a s t b i r t h d a y ) a n d duration-category-at-last-birthday, simultaneously.

Duration-category, in t u r n , i s simply a classification of t h e continuous duration- in-status c o n c e p t a c c o r d i n g t o durationat-last-anniversary. The e s s e n c e of t h e a p p r o a c h i s , f i r s t . t h a t given t h e way in which t h e rates are defined t h e y are piecewise c o n s t a n t ( o v e r unit i n t e r v a l s ) , a n d , second, t h a t s u r v i v o r s h i p in a given

status/duration-category from one birthday t o t h e n e x t implies advancement o r promotion t o the next d u r a t i o n c a t e g o r y .

Provided t h a t t h e n e c e s s a r y d a t a c a n be assembled. t h e method outlined h e r e yields a considerably r i c h e r a r r a y of indices of lifetime e x p e r i e n c e than does t h e usual life table. This r i c h e r a r r a y includes a n allocation of status-specific life ex- pectancies according to duration c a t e g o r y , median waiting times in e a c h s t a t u s . mean time-in-status (backwards r e c u r r e n c e times) at e v e r y a g e , and t h e ability t o compare t h e s u r v i v o r s h i p of d i f f e r e n t groups according to t h e i r a g e s of e n t r y into a given status.

The technique w a s i l l u s t r a t e d with a simple 4-state marital-status model, only one transition of which (marriage t o divorce) w a s t r e a t e d as duration-dependent.

Even in this simple example, in which a r e s t r i c t e d a g e r a n g e w a s considered, t h e new method w a s found t o produce r e s u l t s at considerable v a r i a n c e with t h e conven- tional a p p r o a c h . A t some a g e s , t h e p r o p o r t i o n divorced was as much as 9 p e r c e n t lower with t h e more g e n e r a l model. Given t h e widespread use of t h e o r d i n a r y mul- t i s t a t e life table in a wide r a n g e of substantive applications, t h e method proposed h e r e would s e e m t o b e of considerable p r a c t i c a l importance as well.

A p p e n d i x

T h e DDMSLT with D u r a t i o n - P r e s e r v i n g States

Consider t h e c i r c u l a r p a t t e r n of possible flows in an unlabelled set of five s t a t e s , depicted with 2's in Figure A.1. Suppose t h a t 1 -, 2 and 5 ^-, 1 moves are duration-dependent, and t h a t all o t h e r moves depend on t h e duration since moving

into state 2. In t h i s illustration, a 1 ^-, 2 move i s like a change of marital (employ- ment) s t a t u s , while 2 ^-, 3 and 3 -, 4 moves are like p a r i t y progressions (job shifts), as discussed in t h e t e x t .

Figure A . 1 Flows in hypothetical state s p a c e .

Origin state

Destination state

In constructing t h e M a ( z ) matrix, w e must a r r a n g e t h e rates such t h a t 2 ^-, 3 and 3 -, 4 moves p r e s e r v e t h e duration of time since a r r i v a l in state 2. This re- q u i r e s a r r a n g i n g t h e r a t e s as in equation (A.1). To simplify, t h e dependence on z (age) h a s been d r o p p e d from t h e notation, a n d t h e main diagonal e n t r i e s are

5

r e p r e s e n t e d in s h o r t h a n d ; t h a t i s f j d

=

mjbd ( 2 )

+

mijd ( 2 ) . The only way in

i =l

which M * ( z ) in (A.l) d i f f e r s from M * ( z ) shown in equation (3) of t h e t e x t i s t h e way in which t h e duration-specific submatrices Mo(z),

... ,Mz

(2) are "pulled a p a r t " when forming M * ( 2 ) . Rates of duration-preserving moves now a p p e a r in t h e diagonal block of M * ( 2 ) r a t h e r t h a n in t h e band a c r o s s t h e top. Reverting t o t h e notation f o r t h e partitioned M a ( 2 ) matrix, a s used in t h e t e x t , t h e submatrix T i s no longer a simple diagonal matrix, but r a t h e r a block-diagonal matrix. I t s i n v e r s e must b e computed block by block; again, however, no matrix l a r g e r than 5 - b y 3 (the number of s t a t e s ) must b e inverted. Given the slightly more complex form of T

(and the corresponding increase in computational requirements) all the r e s t of the procedures laid out in the main t e x t still apply.

References

Espenshade, T.J. (1983) Marriage, Divorce and Remarriage from Retrospective Data: A Multiregional Approach. Environment a n d P l a n n i n g A 15:1633- 1652.

Espenshade, T.J. (1986) Markov Chain Models of Marital Event Histories. Current k r s p e c t i v e o n Aging a n d t h e LUe Cycle, Volume 2, pages 73-106.

Espenshade, T.J. and D.A. Wolf (1985) SIPP Data on Marriage, Separation, Divorce, and Remarriage: Problems, Opportunities, and Recommendations. Journal of Economic a n d Social Measurement 13(3&4):229-236.

Hennessey, J . (1980) An Age-Dependent, Absorbing Semi-Markov Model of Work His- t o r i e s of t h e Disabled. Mathematical Biosciences 51:283-304.

Hoem, J . and U. Funck Jensen (1982) Multistate Life Table Methodology: A Probabil- i s t Critique. Pages 155-264 in Multidimensional Mathematical Demography, edited by Kenneth Land and Andrei Rogers. New York: Academic P r e s s . Keilman, N. and R. Gill (1986) Dn the E s t i m a t i o n of Multidimensional Demo-

g r a p h i c Models w i t h P o p u l a t i o n R e g i s t r a t i o n Data. Working P a p e r No. 68.

Voorburg : Netherlands Interuniversity Demographic Institute.

Keyfitz, N

.

(1979) Multidimensionality in Population Analysis. Sociological Metho- dology -80, pp. 191-218. San Francisco: Jossey-Bass.

Littman, G. and C. Mode (1977) A Non-Markovian Model of t h e Taichung Medical IUD Experiment. Mathematical Biosciences 34:279-302.

Lutz, W. and D. Wolf (1987) Fertility and Marital Status Changes o v e r t h e Life Cy- cle: A Comparative Study of Finland and Austria. P a p e r p r e p a r e d f o r presen- tation at t h e European Population Conference, J y v k k y l a , Finland, June 11-16.

Mode, C. (1982) Increment-Decrement Life Tables and Semi-Markovian P r o c e s s e s from a Sample Path Perspective. Pages 535-565 in Multidimensional Mathematical Demography, edited by Kenneth Land and Andrei Rogers. N e w York: Academic P r e s s .

Nour, E. and C.M. Suchindran (1984) The Construction of M u l t i s t a t e Life Tables:

Comments on t h e Article by Willekens e t a l . P o p u l a t i o n S t u d i e s 38:325-328.

Rajulton, F. (1985) Heterogeneous Marital Behavior in Belgium, 1970 and 1977: An Application of the Semi-Markov Model t o Period Data. Mathematical Biosci- ences 73:197-225.

Rogers, A. and F. Willekens (Eds.) (1986) Migration a n d Settlement: A Multire- gional Comparative S t u d y . Dordrecht: D. Reidel Publishing Co.

Suchindran, C.M. and H.P. Koo (1980) Divorce, Remarriage and Fertility. P a p e r presented at t h e Annual Meeting of t h e Population Association of America.

Thornton, A. and W.L. Rodgers (1987) The Influence of Individual and Historical Time on Marital Dissolution. Demography 24(1):1-22.