Are Cohort Mortality Rates Autocorrelated?

Demography, Vol. 28, No.4, November 1991

Are Cohort Mortality Rates Autocorrelated?*

Andrew FosterEconomics DepartmentUniversity of Pennsylvania3718 Locust WalkPhiladelphia, PA 19104-6297

In this paper the author examines the proposition that heterogeneity in individualfrailty leads to autocorrelation in cohort mortality rates. A simple model is used toconstruct analytic expressions for the covariance of cohort mortality rates at differentages under a number of alternative assumptions about the stochastic processgenerating shocks in mortality. The model then is used to construct a procedure thatuses correlations in cohort mortality rates to estimate the extent of heterogeneity in apopulation without relying on strong assumptions about the distribution of frailty orthe shape of the underlying hazard. The procedure then is used to show that cohortmortality data from France are consistent with a generalized random-effects model inwhich frailty is gamma-distributed.

Although it has long been known that different populations can face substantiallydifferent risks of death in any given period, only recently have researchers begun to explorethe implications of unobserved differentials in death rates for the analysis and interpretationof mortality data. In general these explorations can be divided into two parts: a theoreticalliterature showing that heterogeneity in individual frailty' can have a substantial effect onobserved age patterns of mortality through the process of selection (e.g., Vaupel, Yashin,and Manton 1988) and an empirical literature that uses advanced statistical techniques tomeasure and control for unobserved heterogeneity (Behrman, Sickels, and Taubman 1989;Trussell and Richards 1985).

In view of the strength of results derived from theoretical models, at first it may seemsurprising that existing results from the empirical literature provide little justification for theincreased mathematical and computational burden that these new techniques impose on theresearcher (Trussell and Rodriguez 1990). It has been found, on the one hand, that estimatesof the extent of unobserved heterogeneity are dependent on the assumptions that are madeabout the shape of the underlying hazard and, on the other, that parameter estimates forcovariates of interest are not substantially affected when these new techniques are applied toexisting data. Because it is always possible to find a model with no unobserved factors thatis observationally equivalent to a model incorporating unobserved heterogeneity, there

* Initial stages of this work were carried out at the University of California, Berkeley and were supported bya Population Council Graduate Fellowship. Additional support was provided by a grant from the University ofPennsylvania Research Foundation. I am indebted to Kenneth Wachter for suggesting that a formal analysis of thisquestion might prove interesting and to John Wilmoth for making the French data available. I am grateful toKenneth Wachter, Ronald Lee, John Wilmoth, George Alter, and Samuel Preston for their comments andsuggestions. Any remaining errors are of course my own responsibility.

Copyright © 1991 Population Association of America

619

620 Demography, Vol. 28, No.4, November 1991

would seem little reason to estimate the heterogeneous model unless one had good reason tobelieve that heterogeneity plays an important role in determining observed outcomes.

In some circumstances, however, it seems reasonable to argue that this condition ismet. Consider, for example, the literature examining the role played by prices in generaland famines in particular in determining short term fluctuations in vital rates. After a rise inthe price of grains or other foodstuffs, there is frequently an increase in mortality followedby a fall in mortality to below the usual level (e.g. Galloway 1987). This pattern is easilyunderstood to be a result of selection and therefore to be evidence for heterogeneity inindividual exposure to the risk of death: excess mortality in one period prunes a populationof its least healthy members and thus leads to lower mortality in subsequent years.

In this paper I make use of the fact that the mortality rates of populations fluctuate inresponse to environmental conditions in order to provide new insight into the effects ofheterogeneity in frailty on observed mortality patterns. By noting that heterogeneity imposesa particular kind of structure on the variance-covariance matrix of cohort mortality rates, Iobtain direct measures of the mean and variance of frailty at each age without relying onstrong assumptions about this distribution or about the shape of the underlying hazard. Theapproach then is used to show that cohort mortality data from France are consistent with ageneralized random-effects model in which frailty is gamma-distributed.

Heterogeneity with Mortality Shocks

Much of the existing theoretical literature describing the implications of heterogeneityin individual frailty has incorporated the assumption that the mortality schedules faced byindividuals within a population are proportional to each other (Vaupel, Manton, and Stallard1979). Although this assumption is certainly restrictive, it has proved to be convenient froman analytical perspective.

The model used in this paper incorporates this assumption for any given population,but assumes that the mortality schedules differ across populations. Thus, for example, inexamining a cross-section of cohort mortality rates from different countries it is assumedthat the mortality schedules faced by individuals in a given country are proportional, but thatthe age pattern of mortality is different in each country. Similarly, in examining a time seriesof cohort mortality rates it is assumed that mortality schedules faced by individuals in agiven cohort are proportional, but that the age pattern of mortality in each cohort isdifferent. Because mortality shocks such as epidemics, wars, or famines will affectindividuals in a given cohort at the same age, it seems reasonable to argue that differencesin cohort mortality are primarily a result of differences in mortality experience rather than inthe distribution of frailty.

In formalizing this model I make use of the following four assumptions:

1) Each individual is endowed with a fixed frailty z;2) Each population n is large and faces the same distribution of frailty with a mean of

I, finite variance, and density function fez);3) The mortality rate faced by the individuals of frailty z who are age x in population

n is Z/-Ln(x) where /-Ln(.) is the underlying mortality schedule in population n; and4) The underlying mortality schedule in population n can be written

/-Ln(x) = /-L(x) + un(x)

where /-L(.) is a standard underlying schedule and un(.) is a stochastic process whichhas a mean of 0, is fixed within each interval x to x + I, and is small in the sensethat terms containing un2 are negligible.

Are Cohort Mortality Rates Autocorrelated? 621

In the example suggested above, in which one observes a series of cohort mortalityrates, each cohort may be regarded as a distinct population and ....n(x) refers to the observedmortality schedule of cohort n; thus un(x) represents any shocks in mortality that wereexperienced by cohort n at age x. The standard underlying schedule, ....(x), may be regardedas the expectation of ....n(x). Thus if the underlying mortality schedules ....n(x) were observedfor each cohort, the average ....n(x) would provide a reasonable estimate of ....(x). A positivevalue of un(x) would indicate that the underlying mortality rate at age x in cohort n is higherthan that in other cohorts; it might reflect temporary unfavorable conditions (such asepidemic, war, or famine) or more persistent unfavorable conditions (such as a lack ofaccess to adequate health care or nutrition).

In practice, of course, the underlying mortality schedules are not observed; theobserved mortality rate at age x in population n, iin(x), reflects both the underlyingmortality at age x in that population and the distribution of frailty at that age. As Vaupel etal. (1979) show, one may write

(1)

(2)

x-I

iin(x)=I1(x) +z(x)un(x) - 'Y(x)z(x)Iunra)a=O

where zn(x) is the mean of the distribution of frailty (among those still living at age x) inpopulation n. Because zn(x) will reflect the mortality experience of population n until age x,it will differ across populations. Thus it would seem difficult to uncover useful informationabout the underlying mortality patterns or the distribution of frailty from the observedmortality schedules.

If the un(x) are small, however, the following linear approximation will hold (seeAppendix A):

x-I x-I

=fl(x) + a;(x) - 'Y(x)Iexp( - I'Y(w))unra)a=O w=a

where ii(x) = z(x)....(x) is the mortality schedule that would be observed in a heterogeneouspopulation facing the standard underlying mortality schedule, z(x) is the mean frailty at agex, and 'Y(x) = CV;(x)ii(x) is the product of the coefficient of variation squared of z at x andthe observed mortality in the standard population. In subsequent analysis Equation (2), andany equations derived from it, will be treated as equalities in order to simplify discussion.

Because E(un(x» = 0, Equation (2) has a simple interpretation. The deviation ofobserved mortality in population n at age x (iin(x» from its expected value at age x(E(iin(x» = ii(x» consists of two additive terms. Recall that un(x) represents the deviation ofthe underlying mortality at age x in population n from its standard level. Thus the first term,un(x)=zn(x)un(x), may be regarded as a "shock" in the underlying mortality arising fromenvironmental conditions relevant to that particular population at that particular age. Bycontrast the second term describes how shocks at ages less than x affect observed mortalityat age x through their effect on the distribution of frailty at that age.

An examination of the second term makes clear that past shocks affect current mortalityonly in the presence of heterogeneitr in individual frailty. If there is no variation in frailtyat age x, then 'Y(x) =a;(x) =CVz(x)=0 for all x; thus current observed mortalityiin(x) = ii(x) +un(x) depends only on the current shock. Past shocks only affect currentmortality by changing the composition of the population, something that can only occurwhen the population is heterogeneous.

It is well known that heterogeneity provides a mechanism whereby past mortality


experience affects observed mortality rates. Vaupel et al. (1988) present a number ofsimulations showing how a severe shock, such as a famine, will affect mortality patterns insubsequent periods. Although these simulations are useful for the insight they provide, theyare not particularly helpful in providing a mechanism for interpreting most cohort mortalitydata: in practice one rarely observes a single isolated shock.

Equation (2) provides a possible avenue for approaching actual mortality data. Becausethe deviation of the observed mortality from its average level will depend on past shocks inmortality only in the presence of heterogeneity, it seems reasonable to argue that thecorrelation of observed mortality rates at different ages will be affected by the presence ofheterogeneity. Indeed, this notion underlies a test for heterogeneity suggested by Coale andKisker (1986): they examine the correlation between survivorship from age 55 to 65 and lifeexpectancy at age 85 in a sample of countries with accurate mortality data. Because"heterogeneity should produce a negative relation between these two measures" (p. 392)the observed positive correlation is purported to be evidence against the possibility thatheterogeneity plays an important role in determining age patterns of mortality.

Although Equation (2) suggests that heterogeneity in individual frailty will have aneffect on the correlation of mortality rates at different ages, the sign of this correlation isindeterminate and will depend on the stochastic process generating the shocks. In order toillustrate this point, as well as to gain additional insight into the effect of heterogeneity onmortality correlations, it is helpful to examine a series of three simple models of the processgenerating un(x):

1) Isolated shocks;2) Perfectly correlated shocks; and3) Independent and identically distributed shocks.

I chose these three special cases because they easily yield analytic expressions and becausethey represent extreme examples. More general models of heterogeneity and autocorrelationwill be considered subsequently. Results for each of these models are summarized inTable 1.

Isolated Shocks

Suppose, for a given cohort n, that underlying mortality at age x is elevated withrespect to the standard underlying schedule as a result of famine (un(x1»0) but that in otherage groups the standard underlying force of mortality obtains so that J.1n(x) = J.1(x) for x;=x1.Note that un(x1)enters positively in the equation for fJ.n(x1)but negatively in every other agegroup x>x1 as long as -y(x»O: a cohort that experiences the famine will have lowermortality at subsequent ages than one that does not. As a result, if one were to plot mortalityat age Xl and at some other age y>x for this cohort and for another cohort with standardmortality at all ages, one would obtain a negative relationship.

This point is illustrated with 25 simulated populations in Figure 1. In constructing thissimulation the baseline proportion dying at each of three age groups in the standardpopulation was 0.07 and frailty took on values of 8 and 0.2222; the former occurred in 10%of individuals. Shocks were assumed to be zero except at age 0, where they were generatedaccording to a uniform distribution over the range (- 0.005,0.005). I also assumed that eachpopulation was large so that the observed mortality rate at each age would equal the actualprobability of death. Thus the only source of variation distinguishing these populations isthe size of the mortality shock at age 0.

Deviations of mortality rates from their average level (across populations) for ages 1


Table 1. Summary of Different Processes Generating Shocks in MortalityModel of Shocks Process Equations for Variance- Condition for

Generating Covariance Matrix with y~ x NonpositiveShocks Covariance hetween

Mortality at Ages xand y>x.

Isolated

PerfectlyCorrelated

u/x\)-(O,o:>",(x)=o for x¢x\

u.(O) -(0,0;)II,(X) =",(0)

x<x'<y for unique

x' S.t. X 'y(x ') = 1

Independent(Homoskedastic )

Independent(Heteroskedastic)

GeneralizedAutoregressive

",(x) -i.id. (o,o~

(0) =ry

- - 2-z(x)z(y)y(y)o. x>y

+.a(x)Z(y)y(x)y(Y)O; ifx=y

ptr

GeneralizedRandom Effects

Notation:

VectoIs:

Matrices:

u...v...~ with v. -N(O,E)

[0 y>X] [ 1

r,A.t with I'{%.)'1= ~xL if Y=X , 04[%.)']= -ez(x) if- y(%)z(x) y<x 0

x=y } [ 0 }%=,.-1 and E[x.)')= 2() if X?otherwise u. x x-y

and 2 were then plotted against the mortality rate at age O. As is readily apparent in thefigure, a correlation of - 1.000 exists between the mortality rate at age 1 and each of thesubsequent ages, and a correlation of 1.000 exists between the mortality rates at each of thesubsequent ages. This positive correlation arises because the excess mortality during thefirst year of life for cohorts with an unfavorable shock contributes to lowered mortality insubsequent age groups. The fact that the correlation is perfect to three decimal points showsthat the linear approximation used to derive Equation (2) is reasonable under the conditionssimulated here: if quadratic terms were of a significant magnitude, then the relationshipbetween the plotted mortality rates would be curvilinear and the correlation would not beperfect.

The slopes of the curves in Figure 1 are sensitive to the extent of heterogeneity in the


Age Actual

Predicted

Age 2: Actual

Predicted

0N00

0

0(J) ~

c 00 0

....., 00

>Q)

0 0», 0

....., 00

0 0.....,I....

0:::2:

0

0D0

0I

00N00

o 0.064I

0066 0.068 0.0 70 0072 0.074 0.076

(3)

Mortality at Age 0

Figure 1. Simulated Mortality Patterns and Fitted Lines for Isolated Shocks

population. In order to see this relationship, it is helpful to construct a formal expression forthe covariance between the mortality at different ages under the assumed model. Thus, ifE(Un(X I)2) = <T~ and un(x)=0 V x# XI' then for y~x,

o x<xIz(xl<T;' X=Y=XI

u>xy = COV(l1n(X),l1n(Y)) = [ -z(x)Z(y)-y(Y)<T;' if x=xl<yj.+z(x)z(y)"y(x)-y(y)l)~ xI<x

These terms may be used to derive theoretical expressions for the slopes of the linespresented in Figure 1. For example, the slope obtained when regressing mortality at age Ion mortality at age 0 is

(4)

which will be 0 in the absence of heterogeneity in frailty. Unfortunately, although this andother, similar expressions provide some information about the distribution of frailty, they donot permit identification of z(x) and f-L(x) without further restrictions. The problem is thatbecause the mortality rates are correlated perfectly across age, the variance-covariancematrix for a three-age-group population contains only three independent terms.


Perfectly Correlated Shocks

A somewhat different picture emerges in the presence of perfectly correlatedhomoskedastic shocks. Specifically, consider populations with higher (or lower) thanstandard underlying mortality at all ages and the same distribution of frailty. In this case onemay write the observed mortality

fin(x) =z(x)f.L(x) +z(x)[l- xy(x)]un(O) (5)

and therefore obtain

(7)

u'>xyZ(x)z(y)[1 - xy(x)][l - yy(y)]a~ , (6)

whose sign will be negative whenever the signs of the third and fourth multiplicative termsare different. If f.L(x) is continuous in x, then in order for opposite signs to obtain there mustbe some point x" such that l-x*'Y(x*)=O. Because

fin(x*) = z(x*)f.L(x*) +z(x*) [ 1- x*'Y(x*)]un(O)= fi(x*) ,

the observed mortality schedules for any two populations generated according to this modelmust intersect at x*; thus x* is a crossover point.

Figure 2 is analogous to Figure 1, but refers to data generated under the assumptionthat the mortality shock generated at age 0 also applies at subsequent ages (i.e.

CDo;S,.---,..----,---,..-----,----,-----r-----,------r---...,.----,

o

0076

Age 1: Actual

Predicted

Age 2: Actual

Predicted

0.074

o

00720.0700.068

./

CD~oooL-__l--__L-_---''--_--..J__--'__---'__----'__---'-__---'-__---'

6 0066I

'Tooo

(f) 0co

Mortality at Age 0

Figure 2. Simulated Mortality Patterns and Fitted Lines for Perfectly Corelated Shocks


u(2) =u(l) =u(O». Thus, in the absence of heterogeneity, populations with higherthan-average mortality at age 0 would have higher-than-average mortality at subsequentages as well. It is evident from Figure 2, however, that rather different patterns can emergein the presence of heterogeneity: simulated populations with high mortality at age 0 havehigh mortality at age 1 but low mortality at age 2. A crossover point thus exists betweenages 1 and 2. Note also that in contrast to what was observed in the case of isolated shocks,the observed relationship between the different mortality rates in this model is slightlynonlinear.

Again, insight into the distribution of frailty may be obtained by examining the slopesof these lines. In contrast to the isolated-shock case, however, mortality rates at young andold ages will be correlated positively in the absence of heterogeneity; substitution of "I=0into the expressions for ~lO=WlO/WOO yields a predicted slope of 1. In the presence ofheterogeneity, these slopes will differ from 1 as they do in Figure 2. As with theisolated-shock case, however, it is not possible to identify f.L(x) for x>O by using thetheoretical expressions for the variance-covariance matrix of the mortality rates.

Equation 6 also provides some insight into the test for unobserved heterogeneityproposed by Coale and Kisker (1986). Although a test for heterogeneity based on the sign ofthe correlation in mortality rates at ages 65 and 85 can be justified under the assumption ofperfectly correlated shocks, the test has low power because even in the presence ofsubstantial heterogeneity cohort rates may be correlated positively: 1) no crossover age mayappear in the population (i.e., x'Y(x) exceeds or is less than one for all reasonable values ofx); 2) the selected age groups may be on the same side of the crossover age; and 3) theremay be more than one crossover age. This last possibility arises because the function x'Y(x)is not necessarily monotonic. For example, consider the implications of adding another agegroup to the simulated population. In the simulated population "1(1)< I and 2"1(2» 1, thusyielding a crossover; however, x'Y(x) = xCV;(x),:L(x) will fall back below 1 if the mortalityin that subsequent age is sufficiently low, yielding a second crossover and thus a positivecorrelation between mortality at age 0 and age 3. Finally, the Coale and Kisker test restsheavily on the assumption that the model of perfectly correlated shocks is appropriate to thedata being studied.

Independent and Identically Distributed Shocks

The third case, that of independent and identically distributed shocks, provides thefollowing theoretical covariation for y:2:x:

[- Z(X)Z(YJ'Y (y)(J'~ +xz(x)Z(Y)'Y(xJ'Y(y)(J'~ ] .

wxy= (z(xl+xz(xl'Y(xl)(J'~ lif y>xy=x

An examination of Equation (8) shows that the covariance terms will be negative when

(8)

x'Y(x) - 1=xCV~ (x)fl(x) - 1<0. (9)

Thus if the variance in frailty and in the initial age is low, a negative covariance will result.Under other circumstances a positive correlation will result. The value of x is important indetermining the sign of the observed correlation because mortality rates at older agesincorporate the same history of past shocks: the greater the extent of shared history at thetwo ages, the higher the positive correlation between the rates.


Figure 3 shows plots of mortality deviations at ages I and 2 versus mortality at age 0in a third simulated group of populations. I generated each of these populations using thesame baseline mortality distribution as that used in Figures 1 and 2, but incorporatingindependent and identically distributed shocks at each age. One striking feature of thisfigure in contrast to the previous two is the fact that mortality rates are not correlatedperfectly: this pattern arises because, in contrast to the previous simulations, each agereceives an independent shock. Nonetheless, in the simulated populations, mortality rates atages 1 and 2 are negatively correlated (p= -0.3792 and -0.5209, respectively) with thatat age O.

Estimation

The above discussion makes it clear that even when a great deal is known about theprocess generating shocks in mortality, a test for heterogeneity based on a single correlationwill be uninformative in the absence of controls for shocks in past periods. It is notimmediately clear, however, how these controls should be introduced: it would seem thatone needs estimates of the extent of heterogeneity to determine the shocks, and estimates ofthe shocks to test for heterogeneity.

In order to deal with this problem in a general fashion, it is helpful to introduce someadditional notation. Let ji.n' ji., and Un denote vectors of ji.n(x), ji.(X) , and un(x) for all xrespectively, and let r be a matrix with x,y element:

DO DD

D0

B 0-0- _ Q,-c-;

0~li - - -r:::-- ---0 '6 18 ""--

D DD°D e

0

Age 1: Actual

Predicted

Age 2: Actual

Predicted

0.0760.0740072

6. 0D

0 DDr>; "-<;:» '-'

0.0700.0680.066

(f) 0c 00

0-+--'0

>QJ

0 ~

», 0-+--' 0

00 I

-+--''--02'

n00 D0I

0If)

00

~ 0.064

nooo

Mortality at Age 0

Figure 3. Simulated Mortality Patterns and Fitted Lines for Independent and Identically Distributed Shocks


[° y>x ]f[x,y] = z(x) fory=x

- "'((x)z(x) y<x(10)

which reduces to the identity matrix in the absence of variation in individual frailty (inwhich case "'((x) =°and z(x) = 1 for all x). Thus Equation (2) may be written

The theoretical variance-covariance matrix 0 of the observed mortality rates as a function ofthe variance-covariance matrix I of the vector u of shocks is thus

0= E[(j1n - j1)(j1n- j1)'] = E[fuu'f'] = fIf. (12)

Selection Tests in which Shocks Are Independent and Heteroskedastic

For independent (but not necessarily identically distributed) shocks, I consists of adiagonal matrix; thus, in the absence of unobserved heterogeneity, the variance-covariancematrix of observed mortality rates, 0, also should be diagonal (because in that case I' willbe the identity matrix). If heterogeneity is present, then 0 in general will not be diagonal.Thus if shocks are known to be independent and off-diagonal elements of the actualvariance-covariance matrix of cohort mortality rates (call it M) are significantly differentfrom 0, one may conclude that the population is heterogeneous.

An alternative approach is to obtain estimates of "'((x) by fitting the theoretical (D)variance-covariance matrices (for different values of "'((x» to the observed matrix M. Underthe joint hypothesis that shocks are independently distributed and that there is noheterogeneity, "'((x) should not differ significantly from 0. Note that although in general onecannot determine the sign of the off-diagonal elements of 0, "'((x) will take on positivevalues in the presence of heterogeneity in individual frailty.

To obtain a better understanding of how the matrix M may be used to estimate theextent of unobserved heterogeneity, one may return to the derived expressions for thecovariance of mortality rates at different ages. For example, a closed-form expression forz(1) may be derived by using the expressions for the covariances of mortality of those aged0, 1, and 2:

(13)_ 1320

- 1321(0)

• W02 --,----""---

Wooz(1)=

2W0 1

W --II Woo

W 0 1W 02W 12- Woo

where 1320 is the slope coefficient obtained by regressing mortality at age 2 on mortality atage °and a constant and where 1321(0) is the slope coefficient on mortality at age 1 obtainedby regressing mortality at age 2 on mortality at age 0, mortality at age 1, and a constant.Once the z(x) are estimated, they may be combined with the observed average mortality andother covariance terms in order to obtain estimates of f.L(x) and other parameters of interest.Carrying out the appropriate regressions with the 25 populations simulated to create Figure3 provides a point estimate for z(1) of 0.5294. This figure corresponds closely with theactual figure (based on the numbers used to simulate the data) of 0.5902. When thepopulation consists of more than three age groups or when the shocks are autocorrelated,


closed-form expressions such as that presented in Equation (13) are difficult to obtain;therefore I developed a general numerical procedure, which is discussed in Appendix B.

Initial application of this approach to actual mortality data from France proved to bedifficult. Table 2 presents the variance-covariance matrix (M) of cohort mortality rates forFrench females collected for the period 1899 to 1981.2 It is not surprising that thecovariances are positive because mortality rates at all ages have declined during the relevantperiod. Although this pattern suggests that a model of independent shocks will not fit thedata, it is instructive to proceed under the assumption that the shocks, in fact, areindependent.

Table 3, Column 1, provides maximum-likelihood estimates of CV;(x) = 'Y(x)/Il(x) fordata for French females under the assumption of independent shocks. Estimates of CV;(x)are presented instead of estimates of 'Y(x) so that parameter estimates can be compared withthose which would arise in the case of gamma-distributed frailty. In that special case thecoefficient of variation is constant across age (Vaupel et al. 1979). As discussed inAppendix B, it proved helpful in practice to constrain the coefficients of variation in the firsttwo age groups to be equal and to proceed similarly for the last two age groups. Thusseparate estimates for the first (0-4) and last (55-59) age groups are not presented.

A somewhat surprising result is that the estimated values of CV;(x) are negative andsignificant. Because under the null hypothesis of no heterogenei~ CV;(x) =0, and under thealternative hypothesis of heterogeneity in individual frailty CVz(x»O it is evident that theestimated model must be misspecified in some way. Moreover the goodness of fit (GOF)test is rejected at the I% level. 3 A likely source of this misspecification is the assumptionthat cohort mortality rates are independent; this assumption, as suggested earlier, seemsinconsistent with the readily apparent secular decline in mortality. The problem may becharacterized as follows: The slope of a regression of mortality at some age x on mortalityat age 0, under the assumption of independent shocks, should be wox/woo = - <T;(x)j.L(x). Inthe presence of a secular decline in mortality, however, wox/woo will be positive. Thus, infinding parameters that fit the data, the estimation procedure will try to make either <T;(x) orj.L(x) negative. It would seem to be necessary to work with a model in which shocks areallowed to be correlated positively within any given cohort.

Selection Tests In Which Shocks Are Autocorrelated

In order to accommodate the possibility that shocks are correlated positively I considertwo alternative processes:

1) Generalized first-order autoregressive model (GAR)

2) Generalized random-effects model (GRE)

un(x) = 13(x)wn+un(x).

(14)

(15)

The vn(x) are assumed to be distributed as independent normals with variance <T~(x). Also,wn~N(O,1) is independent of the vn(x). Note that these models differ from the standardautoregressive and random-effects models in that the coefficients «(x) and 13(x) are allowedto differ by age.

Using matrix notation, we may write Equations (11) and (12):

~

Tab

le2.

Var

ianc

e-C

ovar

ianc

eM

atri

xfo

rFr

ench

Fem

ales

(in

Fiv

e-Y

ear

Age

Gro

ups)

Age

s0

-45

-910

-14

15-1

920

-24

25-2

930

-34

35-3

94

0-4

44

5-4

950

-54

55-5

9

0-4

3.94

0.36

0.28

0.46

0.50

0.49

0.31

0.25

0.21

0.13

0.12

0.10

5-9

0.36

0.04

0.03

0.05

0.06

0.05

0.04

0.03

0.02

0.01

0.01

0.01

10-1

40.

280.

030.

030.

040.

050.

040.

030.

030.

020.

010.

010.

0115

-19

0.46

0.05

0.04

0.08

0.09

0.08

0.06

0.05

0.05

0.03

0.02

0.02

20-2

40.

500.

060.

050.

090.

130.

090.

060.

050.

040.

020.

020.

0025

-29

0.49

0.05

0.04

0.08

0.09

0.13

0.07

0.05

0.04

0.02

0.02

0.01

0 I'D30

-34

0.31

0.04

0.03

0.06

0.06

0.07

0.07

0.05

0.04

0.02

0.02

0.02

5135

-39

0.25

0.03

0.03

0.05

0.05

0.05

0.05

0.05

0.04

0.03

0.03

0.03

Q IIQ "'I

40

-44

0.21

0.02

0.02

0.05

0.04

0.04

0.04

0.04

0.04

0.02

0.03

0.03

='Cl4

5-4

90.

130.

010.

010.

030.

020.

020.

020.

030.

020.

020.

010.

01=- ~

50-5

40.

110.

010.

010.

020.

020.

020.

020.

030.

030.

010.

020.

02-e

55-5

90.

100.

010.

010.

020.

000.

010.

020.

030.

030.

010.

020.

02~ N ~ Z ? ~ Z Q -e I'D 51 =- I'D "'I .... I,Q I,Q ....


Table 3. Estimates of CV;(x)

Indep. GAR GRE GRE(Fr. Fern.) (Fr. Fern.) (Fr. Fern.) (Simul:CV == 10)

5-9 -6.73** -0.09 -2.14* 7.04**(0.66) (0.74) (0.94) (1.03)

10-14 -6.61 ** -0.66 0.94 4.95(0.77) (0.69) (1.47) (9.22)

15-19 -6.26** -1.33* 6.39 * 5.83*(0.76) (0.81) (1.23) (4.34)

20-24 - 5.23** -2.12* 3.17* 12.06**(0.72) (1.16) (2.21) (4.26)

25-29 -4.97** -4.63** 4.43* 15.21**(0.72) (1.31) (2.48) (4.89)

30-34 -3.59** -1.55* 9.59** 5.08(0.51) (0.88) (2.04) (4.39)

35-39 -2.70** -0.75* 12.80** 7.05*(0.39) (0.51) (1.95) (4.37)

40-44 -1.83** -0.31 16.47** 13.52**(0.28) (0.27) (1.75) (4.48)

45-49 -0.89** -0.11 9.18** 12.51**(0.12) (0.11) (1.34) (5.31)

50-54 -0.46** 0.06 11.08** 13.73**(0.06) (0.09) (2.01) (4.10)

In Lik. 4416.0 4673.1 4764.2 27010.1Test 'Y(x) == 0 679.1 ** 17.4* 445.7** 50.8**d.f. 10 10 10 10GOF test 649.2** 134.4** 46.1 37.2d.f. 60 49 48 48

Standard errors in parentheses.Significance levels: ** p<O.OI, * P < 0.1.Parameter estimates for «(x), 13(x), and <T~(x) are not shown.

AUn ==Un (16)

un==l3wn +un • (17)

respectively, where vn and 13 are vectors of vn(x) and l3(x) respectively, and A is a matrixwith x.y element:

x== y ]x=y-l

otherwise(18)

Also, let ~ be the variance-covariance matrix of Vn (i.e., ~ is a diagonal matrix with<T~(x) as the X

1h diagonal element). Thus the theoretical variance-covariance of theresiduals

n =E«fln - fl)(fln - fl)/)=rA-l~A-l'f' and r(~+I3I3')r'

for the autoregressive and the random-effects models respectively.


Table 3 provides estimates of CV;(x) for the French female data, using theautoregressive (Column 2) and random effects (Column 3) models. It is evident that theautoregressive model is unsatisfactory in a number of respects. As was the case for themodel that assumed independent shocks, estimates of the coefficient of variation arenegative and the GOF test is rejected at the I% level.

By contrast, the random-effects model seems to fit the data rather well. The GOF testis not rejected, and only one of the estimated CV;(x) is negative. In addition, this modelyields substantial evidence of heterogeneity in individual frailty: the joint hypothesis that'V(x) = 0 is rejected soundly.

An unexpected result is that the estimates of CV;(x) are all of approximately the samemagnitude. This finding is interesting because such a pattern is consistent with thehypothesis that frailty is gamma-distributed. In order to illustrate this point, I simulated datausing a random-effects model with gamma-distributed frailty." The resulting estimates ofCV;(x) are presented in Table 3 Column 4, and are encouraging: with one exception,estimates are within two standard errors of their true value (10). Moreover, there is astriking similarity between the results obtained for the simulated (Column 4) and the actual(Column 3) data.

This pattern is of some interest because although the gamma distribution is often usedin analyses of unobserved frailty, it is typically selected for its tractability rather than fromany belief that the gamma distribution is the most appropriate representation of the actualpattern. Although this result may depend on the assumption that shocks follow arandom-effects model, it is worthy of note that the assumed process is extremely flexibleand fits the data well. The only major assumption about the distribution of frailty that hasbeen made here is that the distribution is the same for all cohorts.

Discussion

In this paper I have constructed a measure of the distribution of frailty without relyingon strong assumptions about this distribution or about the shape of the underlying hazard.These standard assumptions have been replaced with alternative ones: that a relativelysimple stochastic process generates the underlying mortality schedule for each populationand that the deviation of the underlying mortality schedule of any given population from theaverage schedule is small.

By applying the same estimation strategy to three different models of mortality shocks,I have shown that the variance-covariance matrix of cohort mortality rates can be quiteinformative about the nature of the underlying stochastic process as well as about thedistribution of frailty. In particular, I find that the French data are inconsistent with a modelbased on independent or autoregressive shocks. but consistent with a generalizedrandom-effects model with gamma-distributed frailty.

The assumptions incorporated into this model are relatively weak; they are notinnocuous, however. In particular, one implication of using the random-effects model isthat the age pattern of the secular decline in mortality cannot be used to distinguishbetween a heterogeneous and a homogeneous population because the effects of such adecline on the age pattern of mortality will be absorbed into l3(x). As a result, estimates ofthe extent of heterogeneity rest primarily on the impact of short-run fluctuations inmortality. This reliance creates no difficulty if frailty is fixed over time, as the modelassumes; if individual frailty is persistent in the short run but changes over time,however, then estimates of the distribution of frailty based on short-run fluctuations inmortality may tell us little about the role of frailty in explaining long-run changes in theage pattern of mortality.


Although this approach provides a new and (I hope) useful mechanism for analyzingfluctuations in cohort mortality rates, certainly one may use reasons other than thosebased on heterogeneity to explain the tendency of mortality to decline during periodsimmediately following a famine or other shock. For example, an epidemic of a diseasemight lower subsequent mortality by decreasing exposure to that disease for eachindividual (by increasing the size of the pool of nonsusceptible individuals) rather than bypruning the population of its weakest members. When such factors play a role, theapproach developed in this paper would probably over estimate the extent ofheterogeneity.

Finally, although this study has focused on the analysis of aggregate mortality data,there is no reason why a similar approach cannot be applied to individual-level data withcovariates or to failure-time data other than mortality. Indeed, covariates are likely toprovide additional insight into the extent of heterogeneity, particularly when they varyacross time or when they apply to populations subject to substantial environmental oreconomic fluctuations.

Appendix A. Decomposition of Observed Mortality RatesFollowing Vaupel et al. (1979), let ZfL(.) represent the standard age pattern of mortality for an

individual with fixed frailty z; let f(z) with a mean of 1 and finite variance denote the standard densityof frailty at birth. The observed standard mortality at age x, fL(X), is then equal to z(X)fL(X), where z(x)is the mean frailty at age x. Since the probability that an individual with frailty of type z will surviveto age x is

x

e - JZfL(a)da e

lix.z}» 0

x

-z JfL(a)da

o =l(x)"

(20)

the density of frailty at age x is

00

!(x,z)=j(z)l(x,z)/ J!(w)l(x,w)dwo

(21)

and therefore the mean frailty is

(22)J

oo -w J~(a)dawfiw)« 0 dw

oz(x) = _

J~w)e -w ~~(a)dadWo

Similarly, the mean frailty at age x for the nth population in the model is

(23)

00

J w!(w)ln(x,w)dwozJx) = _

00

J!(w)ln(x,w)dwo


00

Jwf(w)l(x,w)e- w S:.!aidao 0 dw

Joo - w S:(aidaf(w)l(x,w)e 0 dw

o

Assume [u (a)I<B so that a first-order Taylor expansion in Un(x) = gun(a)da for in(x) may be written

_ i(x) - Un(x)(3~(x) + i(xl) + o(B 2)

z(x) = --'--'--I-"'---'U-'-n-'(x"':"')i-'-(x-)-+-'-o-'-(B-':2'-)--'----'-

= (i(x) - UJx)(a~ (x) + i(xl»(l + Un(x)i(x)) +o(B 2)

=i(x)-a~ (x)Un(x)+o(B2)

(24)

where a;(x) is the variance of z at x in the standard population and 0(B2) is used to represent functions

A(B) with the property that limB~oB - 2A(13 2) is bounded. The observed mortality rate at age x inpopulation n may then be written

x

= z(x)f-L(x) + z(x)un(x)- a~ (x)f-L(x)Jun(a)da +o(B2)

ox

=j1(x)+i(x)uJx)-'Y(x)i(x)Jun(a)da+o(B2)

o

x x

= j1(X)+un(x)+'Y(x)Jexp( - J'Y(w)dw)un(a)da + o(B2)

o a

(25)

where 'Y(x) = CV;(x)j1(x) is the product of the coefficient of variation squared of z-at x and theobserved mortality in the standard population. The final expression in Equation (25) follows from thefact that

and therefore

di(x)/dt = _ f-L(x)a~ (x)

i(x) i(x)-cv~ (x)j1(x) = -'Y(x) (26)

x

i(x)=exp( - J'Y(a)da)o

(27)

because zeD) = I. Because un(x) is assumed constant within each age interval, an approximation toEquation (25) may be written in terms of summations, yielding Equation (2) in the text.

Are Cohort Mortality Rates Autocorrelated?

Appendix B. Estimation Procedure

In matrix notation

and

where, for the autoregressive model,

AUn=lJn and fl=[A-1lA- 1' [ ,

and, for the random-effects model,

u; = 13wn +lJn and fl = f(l +13l3')['

635

(28)

(29)

(30)

(31)

with E(vnvn')= I a diagonal matrix.Let 8 denote the vector of parameters that is to be estimated (consisting of «(x), 'Y(x), and 8~(x)

in the case of the autoregressive model and 13(x), 'Y(x), and 8~(x) in the case of the random-effectsmodel), Then, since vn is assumed to be normally distributed, the log-likelihood function is of theform

InL«l1n);8,1L) = - ~Iln2'lT+ Inlfl(8)1 + (I1n -11)'fl(8) -1(l1n-11)n

I _~= c - 2(Mnlfl(8)1 + tr(fl(8) 1~(l1n -11)(l1n-11)'»

n

(32)

where c is a constant. Since the maximum-likelihood estimate of 11 is the average mortality vector, theabove likelihood function can be concentrated with respect to to give

NInL(M;8) = c - 2'lnlfl(8)1 + trfl(8) - 1M)

where M is the estimated variance-covariance matrix of the I1n'The associated first-order condition is

(33)

alnL(M;8)

a8-N vec(fl_M)'(fl-l®fl-l/vec(fl)

a8(34)

and the probability limit of the average information matrix is

(35)

I obtained parameter estimates by first concentrating the likelihood function with respect to o(x) and13(x) in the autoregressive and random-effects models respectively. In order to do this I require analyticsolutions for the maximum likelihood estimates n(x) and 13(x) given I' and I.

In the case of the autoregressive model, this presents little difficulty. If H=[-IM[-I', thenu(x) = H[x - I,x]/H[x - I,x - I], These estimates can be used to construct A and therefore fl.

In the case of the random-effects model, concentration is more difficult. First, note that

(36)


where h = 1/(I + j3'I - 113). Let A be the largest eigenvalue of the positive definite matrix

(37)

and rthe associated eigenvector scaled so that r'r = 1. Also let 13 = (A_l)1/2II/2r so that j3'I -113 = A- Iand h = 1/A. Then

olnL(M;8)

013(38)

Thus maximum-likelihood estimate for 13, given I and I', can be obtained by finding the eigenvaluesand eigenvectors of H, choosing the largest eigenvalue (which yields the highest likelihood), andtransforming the corresponding eigenvector.

The results from two likelihood ratio tests are presented in Table 3. The first is a joint test of thehypothesis that 'Y(x) = O. This test was carried out by first maximizing the likelihood with respect to I,with the 'Y(x) constrained, and then maximizing the unconstrained model. The test statistic is twotimes the difference between the likelihoods of the two models, and is distributed as a chi-squaredistribution with degrees of freedom equal to the number of 'Y(x) parameters estimated.

The second test is a measure of the goodness of fit and is obtained by comparing the fitted modelwith an unconstrained model in which the estimated variance-covariance matrix n coincides with theactual variance-covariance M. The log-likelihood of the unconstrained model is thus

N- "2 [ln2'll'+ InlM! + s](39)

where s is the dimension of M. Because s = 12 here, there are s(s + 1)/2-3s +2 = 48 degrees offreedom in the case of the random-effects model and 49 in the case of the autoregressive model.

Notes

1 In the study of mortality unobserved heterogeneity refers to variation in mortality risk acrossindividuals with the same observed characteristics (such as age and sex). In keeping with the existingliterature (see, e.g. Vaupel, Manton and Stallard 1979), frailty refers to the mortality risk of anindividual relative to that of other individuals with the same characteristics; a frail individual is onewith higher-than-average mortality risk and a robust individual is one with lower-than-average risk.

2 Because the mortality experience of some of the cohorts considered is not available, the estimatedcovariances reflect different numbers of observations. Covariances were not computed for ages greaterthan 60 because too few observations were available. Note also that 5-year mortality hazards wereconstructed by summing up annual mortality hazards, and should be interpreted accordingly.

3 This test is a likelihood ratio test that measures the extent to which additional parameters wouldenhance the fit between the actual and estimated n. A formula and justification are provided inAppendix B.

4 In the simulation (x) was extracted from the French data and used to compute f1(x)/,.,.,(x) byinverting the standard formula for gamma-distributed frailty with mean I and variance Ilk:

f1(x) = k,.,.,(x)/(k+ f ,.,.,(w)dw)o

(1)

for k=O.1. (Vaupel et al 1979). The parameters l3(x) and lTv(x) were chosen as follows: l3(x)=0.000 I f1(x) and ITvex) = 0.0 I f1(x). Normal errors were generated in order to obtain mortality schedules for100 populations. The f1n(x) were computed, given k, by applying the above formula to the ""'n(x).

Are Cohort Mortality Rates Autocorrelated?

References

637

Behrman, Jere R., Robin Sickles, and Paul Taubman. 1990. "Age Specific Death Rates withCovariates: Sensitivity to Sample Length and Unobserved Frailty." Demography 27(2): 267-84.

Coale, Ansley 1. and Ellen E. Kisker. 1986. "Mortality Crossovers: Reality or Bad Data?" PopulationStudies 40: 389-401.

Galloway, Patrick R. 1987. "Population, Prices, and Weather in Preindustrial Europe." Unpublisheddoctoral disseration, University of California, Berkeley.

Trussell, James and Toni Richards. 1985. "Correcting for Unobserved Heterogeneity in HazardModels: An Application of the Heckman-Singer Procedure to Demographic Data." pp. 242-276in Sociological Methodology, edited by Nancy Tuma. Jossey-Bass, San Francisco.

Trussell, James and German Rodriguez. 1990. "Heterogeneity in Demographic Research." pp.111-132 in Convergent Issues in Genetics and Demography, edited by Julian Adams, David A.Lam, Albert I. Hermalin, and Peter E. Smouse. New YorkILondon: Oxford University Press.

Vaupel, James W., Kenneth G. Manton, and Eric Stallard. 1979. "The Impact of Heterogeneity inIndividual Frailty on the Dynamics of Mortality." Demography 16(3) 439-54.

Vaupel, James W., Anatoli Yashin, and Kenneth G. Manton. 1988. "Debilitation's Aftermath:Stochastic Process Models of Mortality." Mathematical Population Studies 1(2): 21-48.

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Are Cohort Mortality Rates Autocorrelated?

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