Estimating national disability risk

Theoretical Population Biology 65 (2004) 389–400

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Estimating national disability risk$

Nan Lia,b,*aMorrison Institute for Population and Resource Studies, Stanford University, Stanford, CA 94305-5020, USA

bDepartment of Sociology, University of Victoria, P.O. Box 3050, Victoria, BC, Canada V8W 3P5

Received 18 June 2002

Abstract

In this article, I provide a method to rebuild the active and disabled life expectancy (ALE and DLE) on the basis of ‘current’ death

and disability risks, and to measure disability risk. This method uses national-level data, and is based on two main assumptions. The

first is the Gompertz assumption that death rate rises with age exponentially, and the second is the Cox assumption that death rates

of active status are proportional to those of disabled status across age. Applying this method to the US data, I find that the disability

risk has increased between 1970 and 1990 for both men and women aged 40 and older. Situations in which above assumptions could

be removed are also discussed.

r 2004 Elsevier Inc. All rights reserved.

1. Introduction

Disability data have been collected in censuses, forexample in the US and Canada (Weeks, 1999, p. 52).These data provide the proportions of disabled popula-tion in each age group for both sexes. Combining withmortality data collected from vital statistics, disabilityproportions are used to address the active life expec-tancy (ALE), an indication of living how long and howwell, that is, free from disability.Sullivan (1971) provided the first calculation of the

ALE, which is now called the prevalence method. In thismethod, first the death rates are used to construct anordinary life table; disability proportions are then usedto divide the person-years lived in each age group intodisabled and active status. The total-person-years livedfrom each age are then calculated for both disabled andactive status. Dividing the active total-person-years bythe number of survivors at the corresponding age, the

h was funded by the Morrison Institute for Population

tudies at Stanford University, and by the NIA-funded

the Economics and Demography of Aging at the

California at Berkeley. I thank particularly Marcus

ad Tuljapurkar and Ronald Lee for their generous

thank Tom Burch, Jean-Marie Robine, Robert Schoen,

for their relevant comments. I am solely responsible for

ing author. Max Planck Institute for Demographic

rad-Zuse-Strasse 1, D-18057 Rostock, Germany.

ess: [email protected].

e front matter r 2004 Elsevier Inc. All rights reserved.

b.2003.03.001

result could be interpreted as the number of years livedin active status (ALE), for an average person in thesynthetic cohort, if current death rates and disabilityproportions prevailed through the cohort’s lifetime. Thedifference between the life expectancy (LE) and ALE isnaturally called the disabled life expectancy (DLE).The prevalence method has generated meaningful

studies based on national-level data. For example,Crimmins et al. (1989) found that in the US in 1970smost of increase in LE was DLE, implying the longerlife of Americans was more in disabled years. Theyfound later (Crimmins et al., 1997) that this trend hadbeen reversed in 1980s, and Cambois et al. (2001)reported a similar change in France. These findingscould provide a quantitative basis to study theconjecture (e.g. Fries, 1980) of whether or not peoplewould be healthier when they live longer.In the prevalence method, however, there are two

obvious problems yet to be solved. Life table mayinclude event other than death, but such event shouldhappen in a certain time interval in order for itsmeasures to be compared between different times orregions. The age-specific disability proportions, how-ever, does not measure the risk of disability in a certaintime interval; they are rather records of mixed events,including disability and mortality, accumulated througha long period of history. Comparing the prevalence ALEbetween different times is hence questionable. A declinein the number of individuals who become disabled in acertain time interval, for example, may not reduce the

ARTICLE IN PRESSNan Li / Theoretical Population Biology 65 (2004) 389–400390

disability proportions which were formed earlier, andhence may not lower the prevalence ALE. Using theprevalence ALE to compare disability risks betweendifferent regions is equally problematic; since it may notbe clear to what time interval the comparison refers. Thefirst problem, therefore, is how to make the ALE refer tocurrent disability risk. Further, reducing mortality ofactive people faster than that of disabled will lead tolower disability proportions and hence higher ALE. Inother words, higher ALE does not necessarily reflectlower disability risk. Thus, the second problem is how tomeasure disability risk.The double-decrement (e.g., Katz et al., 1983) and the

multistate models (e.g., Land et al., 1994) could providethe basis to solve the two problems. In fact, themultistate model can describe the dynamics of transi-tions between more health status than only active anddisabled (e.g., Crimmins et al., 1994; Manton and Land,2000). The power of these models, however, is builtupon panel data that record each individual’s healthstatus at two different times. These data are not yetavailable at the national level. Because the sizes of thesampled data used in these models are too small tocalculate age-specific death or transition rates, assump-tions about how these rates change with age have to beintroduced. These assumptions typically include that thedeath and transition rates increase with age linearly orexponentially. For almost 200 years, demographers havebeen using the Gompertz law (see Preston et al., 2001, p.192) that death rate rises with age exponentially fromsome starting age around 50 years. Recent modificationsof this assumption are only for the oldest ages, usually90 years and older (e.g., Horiuchi and Coale, 1990). Theassumptions that the transition rates (between activeand disabled status) increase with age linearly orexponentially, however, remain to be examined.Using prevalence data, the relationship between

mortality and disability can still be disentangled, as inthe method I will propose in this paper. This methoddoes not assume how the transition rates change withage, but requires assumptions that can be examined andhave been widely used. The first assumption is the waythat death rate changes with age, which I choose theGompertz law. The second assumption is about how theratio of death rates of active to disabled status changesover age. I choose the Cox proportional hazard model(Cox, 1972), in which the ratio of death rates of active todisabled status is constant over age, as the secondassumption. The Cox assumption has been used tomodel death rates of different health status (Lee, 1992)and has been working well in small sample data (Lee,1997). This method needs also other two supplementaryassumptions that are related with the data whose agegroup covers n years. The third assumption is about howthe number of population in a certain age group changesover a certain time interval. I assume it changes linearly

over time for 2n years. The fourth assumption is thatdeath rates and disability proportions remain constantfor 2n years. The third and fourth assumptions havebeen used elsewhere when 2n covers fewer than 5 years,and occasionally for longer interval.In this method, the first and second assumptions are

used to estimate the death rates of active and disabledstatus. The estimation is based on minimizing the errorsof using the modeled death rates to describe the deathrates of the active and disabled combined population.After estimating death rates of active or disabledpopulation, however, transition rates cannot yet beestimated without knowing how the number of active ordisabled population changes with age. Using the thirdand fourth assumptions to describe such changes,transition rates are estimated. Assuming the rates ofdeath and transition remain constant for the lifetime ofa synthetic cohort, the first problem is solved and theALE is calculated to describe the current risks ofmortality and disability. After estimating the transitionrates, which describe the age-specific disability risks, thismethod provides a summary measure of disability risk inthe way similar to the use of life expectancy tosummarize the risk of death.For demonstrating the method, examples are pro-

vided using the US data. In these examples, the age-status-specific death rates, the age-specific rates of nettransition from active to disabled status, the ALE andDLE based on current risks of mortality and disability,and the summary measure of disability risk areestimated. These estimates, at national level, have notbeen provided previously by either the prevalencemethod or the multistate model.

2. The method

2.1. The mortality model

Let the death rate and the disabled proportion ofpopulation aged x to x þ n years, obtained from vitalstatistics and census data, be mðxÞ and dðxÞ; respec-tively. For this age group, let the death rates of activeand disabled population be maðxÞ and mdðxÞ; respec-tively, and be described by the following model:

maðxÞ ¼ exp½c1 þ c0ðx � sÞ�; ð1Þ

mdðxÞ ¼ exp½c2 þ c0ðx � sÞ�: ð2Þ

In Eqs. (1) and (2), coefficient c0 makes death rateincrease with age exponentially from a starting age s;according to the Gompertz assumption. That the activeand disabled death rates share the same c0 is based onthe Cox assumption, which implies that the ratio ofdeath rates of active to disabled status is constant over

ARTICLE IN PRESSNan Li / Theoretical Population Biology 65 (2004) 389–400 391

age. Since this ratio cannot be 1 everywhere, constantsc1 and c2 are therefore introduced.The three parameters, c0; c1 and c2; can be estimated

minimizing the imbalance of the following equation forall ages older than s:

mðxÞ ¼ ½1� dðxÞ�maðxÞ þ dðxÞmdðxÞ: ð3Þ

This is because that multiplying the total population ineach age group on both sides of Eq. (3), the right-handside represents the sum of deaths in active and disabledstatus, which should equal the deaths of total popula-tion in the left-hand side. The results of estimating aregiven by (A.1) and (A.6) in the Appendix.

2.2. The net transition rate

The number of the net transition, from active todisabled status, can be defined as the number oftransitions from active to disabled status minus that offrom disabled to active status. The net transition fromdisabled to active statues could be defined similarly ifactive-to-disabled net transition were negative.For the age group of x to x þ n years and the net

transition from active to disabled status, the nettransition rate in a calendar year, denoted by tadðxÞ;can be defined in the same way as the death rate: theratio of the number of net transitions in a calendar yearto the average number of active population in this year.Using the third and fourth assumptions, the net

transition rates are estimated by (A.15) and (A.16) in theappendix as

tadðxÞ ¼1

n1� ½1� dðx þ nÞ�½2� nmðxÞ�

½1� dðxÞ�½2þ nmðx þ nÞ�

� �� maðxÞ;

ð4Þ

and

tadðwþÞ ¼ 1� 0:5½dðw � nÞ þ dðwþÞ�1� dðwþÞ mðwþÞ � maðwþÞ;

ð5Þ

where w is the oldest age.There is an intuitive explanation for Eq. (4) when the

death rate and disability proportion are supposed to beconstant over two successive age groups. In thissituation, the transition rate in (4) is the differencebetween the death rates of total and active population.When the death rate of active population is smaller thanthat of the total population, (4) points to active-to-disabled net transition. On the other hand, higher deathrate of disabled status implies that dðxÞ should declineover x; if there were no transition; and the constantdðxÞ; therefore, concludes that there are net transitionsfrom active to disabled status.

2.3. Stationary population and life expectancy

To estimate the active and disabled life expectancies,assuming that death rates remain constant for asynthetic cohort’s lifetime is necessary. After estimatingthe transition rates, they can be assumed constant forthis synthetic cohort’s lifetime to reflect the disabilityrisk in an n-year interval around current time. Thus, thefirst problem, as noted in introduction, is solved.Because the mortality model is built on the Gompertz

assumption that applies for ages older than s; thesynthetic cohort starts its life from age s: When s isaround 50, the mortality is low and hence the disabilityproportion at age s can be taken as that of age group s;dðsÞ: Thus, the number of active population at thestarting age is written as

laðsÞ ¼ 1� dðsÞ: ð6ÞFrom this number at the starting age and with the

usual assumption that the population distributes linearlywithin an age group, the numbers of active survivors atany age x and the person-years in any age group of x tox þ n years, namely laðxÞ and LaðxÞ; respectively, can bederived from using the decline rates, maðxÞ þ tadðxÞ; asdeath rates in an ordinary life table:

laðxÞ ¼2� n½maðxÞ þ tadðxÞ�2þ n½maðxÞ þ tadðxÞ�

laðx � nÞ;

Laðx � nÞ ¼ n

2½laðx � nÞ þ laðxÞ�;

LaðwþÞ ¼ laðwÞmaðwþÞ þ tadðwþÞ: ð7Þ

Let lðxÞ and LðxÞ be the numbers of total survivorsaged x and the person-years lived in ages of x to x þ n

years, and ldðxÞ and LdðxÞ be the numbers of disabledsurvivors and the person-years, respectively. Using themðxÞ in an ordinary life table, lðxÞ and LðxÞ areproduced. Variables ldðxÞ and LdðxÞ are then obtained asldðxÞ ¼ lðxÞ � laðxÞ;LdðxÞ ¼ LðxÞ � LaðxÞ: ð8ÞThe ALE and DLE at age x; namely eaðxÞ and edðxÞ arethen estimated as

eaðxÞ ¼Pw

y¼x LaðyÞlðxÞ ;

edðxÞ ¼Pw

y¼x LdðxÞlðxÞ : ð9Þ

Notice that in the prevalence method the activeperson-years is LðxÞð1� dðxÞÞ; but the LaðxÞ in (7) isdifferent. The LaðxÞ in (7) describes the age-specificactive population in a stationary population. In thisstationary population, the age-specific disabled propor-tions are LdðxÞ=LðxÞ and in general are not dðxÞ; whichare the current values of the age-specific disabledproportions. This stationary population would be


reached by applying the estimated current death andtransition rates constantly to any initial population,including the current one whose disabled proportions aredðxÞ: In the prevalence method, the age-specific disabledproportions in the stationary population are fixed as ofthe current state: dðxÞ: Such a stationary population isreached maintaining the current death rates (of activeand disabled status combined) constant, with unknowntransition rates and death rates of active and disabledstatus.Basing the ALE on the current risks of mortality and

disability is of course a progress, it cannot, however,distinguish changes in mortality and disability. Forexample, let the death rates of active and disabledcombined population and the net transition rates befixed. When the death rates of active population decline,the death rates of disabled people increase, because mðxÞare fixed. This decline of maðxÞ would result in largerlaðxÞ and hence higher ALE, according to (7) and (9).Therefore, in above example, the ALE is raised byreducing mortality of active population and raisingdeath rates of disabled people. In general, ALE can beincreased by reducing mortality of active populationfaster than that of disabled, without requiring anychange in disability risk.The ALE does not necessarily measure changes in the

disability risk for a certain age and older. On the otherhand, although the net transition rates tadðxÞ measureage-specific disability risks, as vectors they are notcomparable between different times or regions. How tocomparably measure the disability risk for a certain ageand older, therefore, is still a problem.

2.4. The equivalent disability fraction

There are three components in the observed disabledtotal-person-years,

TdðxÞ ¼Xw

y¼x

LdðyÞ: ð10Þ

The first one is the total-person-years that disabledindividuals aged x would live if there were no transitionat older ages, namely T�

d ðxÞ: The life expectancy fordisabled status at age x; denoted by e�dðxÞ; can beobtained using mdðxÞ in an ordinary life table. The firstcomponent is then expressed as

T�d ðxÞ ¼ ldðxÞe�dðxÞ: ð11Þ

The second component is the total-person-years due toactive-to-disabled transition over age x; represented asTadðxÞ: And the third is the total-person-years due todisabled-to-active transition over age x; denoted byTdaðxÞ: For a certain age x; TdðxÞ would equal T�

d ðxÞ ifthere were on transitions over age x; TadðxÞ raises andTdaðxÞ reduces TdðxÞ from T�

d ðxÞ: The TdðxÞ is therefore

decomposed as

TdðxÞ ¼ T�d ðxÞ þ TadðxÞ � TdaðxÞ: ð12Þ

The second component TadðxÞ measures the risk ofentering disability, but excludes the chance of returningactive, which is described by TdaðxÞ: The model in thispaper does not distinguish how long a disabled personhas been disabled. Individuals returning to active statusover age x could therefore be regarded as part of thoseentered disabled status over age x: Thus, among theTadðxÞ person-years of active-to-disabled transition,TdaðxÞ returned back to active status. Therefore, theequivalent total-person-years due to net transition fromactive to disabled status is ½TadðxÞ � TdaðxÞ�; which canbe measured by the right-hand side of the followingequation according to (12),

TadðxÞ � TdaðxÞ ¼ TdðxÞ � T�d ðxÞ: ð13Þ

The right-hand side of (13), ½TdðxÞ � T�d ðxÞ�; depends

also on the mortality of disabled status. Such aninfluence of mortality can be eliminated in the waysimilar to using life expectancy to measure death risk.For individuals aged x; they will die at older but

different ages. The basic idea of life expectancy is to askif the total-person-years over age x maintained invariantand all individuals died at the same age, what would thisage be? The answer is x þ eðxÞ years.Now there are active individuals aged x; some of them

will become disabled before death, and the total-person-years of living in disabled status due to net transition is½TdðxÞ � T�

d ðxÞ�: In order to obtain a summary measureof disability risk for ages x and older, I ask if ½TdðxÞ �T�

d ðxÞ� remained invariant and all active individuals whobecome disabled did so at the same age x; what wouldthe number of these individuals be? This number mustbe a fraction of the active population aged x; laðxÞ:Define this fraction as the equivalent disability fraction atage x; EDFðxÞ; this number is EDFðxÞlaðxÞ: Since theseactive individuals become disabled at age x; they will livee�dðxÞyears in disabled status. Thus, the total-person-years of living in disabled status due to this nettransition is EDFðxÞlaðxÞe�dðxÞ: Because that the total-person-years of living in disabled should hold constant,

EDFðxÞlaðxÞe�dðxÞ ¼ TdðxÞ � T�d ðxÞ: ð14Þ

From (14), the equivalent disability fraction at age x isobtained as

EDFðxÞ ¼ TdðxÞ � T�d ðxÞ

laðxÞe�dðxÞð15Þ

EDFðxÞ is a summary measure of disability risks overage x: It indicates the fraction of active individuals whowould have to become disabled at age x in order to yieldthe given ½TdðxÞ � T�

d ðxÞ�: In other words, the numberof active persons who become disabled at different agesolder than x is not EDFðxÞlaðxÞ but larger. On the other


hand, the years of living in disabled status for each ofsuch person is not e�dðxÞ; but fewer and different. If twopersons living in disabled status for 1 year wereequivalent to one person living in disabled status for 2years, the different ages at which active individualsbecome disabled can be simplified as one age, x years;and the different years in which they lived in disabledstatus can be summarized as a unique number of years,e�dðxÞ: This simplification requires a certain number ofactive individuals, EDFðxÞlaðxÞ; to become disabled,and produces a summary measure of disability risk overage x; EDFðxÞ: Such a summary measure allowscomparing disability risk between different populationsand over different times.The numerator of EDFðxÞ is a measure of combined

risks of disability (entering and leaving disabled status)and mortality (staying in disabled status); its denomi-nator eliminates the effect of mortality. Given e�dðxÞ; alarger ½TdðxÞ � T�

d ðxÞ� implies more transitions fromactive to disabled status and thus higher disability risk.Fixing ½TdðxÞ � T�

d ðxÞ�; that a larger e�dðxÞ does notresult in a bigger total-person-years from the active-to-disabled net transition implies lower disability risk.

2.5. Application and discussion

In this section, I apply above method to the US datain years 1970, 1980 and 1990. The age-sex-specific deathrates, mðxÞ; are obtained from the World Health

0 10 20 30 400

20

40

60

80

100

ag

mal

e di

sabi

lity

prop

ortio

n 1970

1980

1990

10 20 30 400

0

20

40

60

80

100

ag

fem

ale

disa

bilit

y pr

opor

tion 1970

1980

1990

Fig. 1. Disability propo

Organization. The age-sex-specific disability propor-tions are adopted from the article of Crimmins et al.(1997). For these data, a disabled person is defined asnot being able to perform the normal activities of lifeincluding going to school for children, working, keepinghouse or other things that people do. In the article ofCrimmins et al. (1997), there were two types of age-sex-specific disability proportions, one was for institutiona-lized and another was for noninstitutionalized disabledpopulation. For each age group and sex, I sum the twotypes of values together as the age-sex-specific disabilityproportions, dðxÞ: These data are in 5-year age group,thus assumptions three and four require 10 years ofconstant death rates, disability proportions and linearchanges in the number of population in each age group.These disability proportions are shown in Fig. 1. It canbe seen that these proportions are similar between thetwo sexes at ages younger than 65; at older ages female’sdisability proportion rises faster than that of male. Inthe 20 years of period, there was no notable decline inthe disability proportions.I choose the starting age s as 40 years to apply above

mortality model. Using (4a) and (6a), coefficients c0; c1and c2 are estimated. Applying the estimated c0; c1 andc2 in (1) and (2), death rates for active and disabledstatus, maðxÞ and mdðxÞ respectively, are calculated. Themodel values of mðxÞ are then given as the right-handside of (3). The maximum relative errors, calculated by(A.7), are shown in Table 1. Fig. 2 shows the fittings for

50 60 70 80e

50 60 70 80e

rtions of the US.


the situations with the highest maximum relativemodeling error, which are 1970 for male and 1980 forfemale as can be seen in Table 1. Overall, the Gompertzand Cox assumptions work well for the US data as theydid elsewhere. In general, choosing smaller value for sleads to bigger errors and vice versa.Because the Cox assumption is used to model

mortality, the ratio of death rates of disabled to activepopulation is constant over age. A somewhat surprisingresult is that the estimated death rates for disabled werelower than that of active population for males in 1970and 1980, as shown in Table 2.How can lower mortality for disabled status, c14c2;

be estimated in the model? According to (A.6), thecondition for c14c2 is

meanðdðxÞÞXx¼s

mðxÞexp½�c0ðx � sÞ�

4Xx¼s

mðxÞexp½�c0ðx � sÞ�dðxÞ:

Notice that in general dðxÞ increases with age, so thatabove inequality holds when mðxÞexp½�c0ðx � sÞ� has

40 45 50 55 600

0.05

0.1

0.15

0.2

ag

mal

e de

ath

rate

, 19

70

Observed valuesModeled values

40 45 50 55 600

0.05

0.1

0.15

0.2

ag

fem

ale

deat

h ra

te, 1

980 Observed values

Modeled values

Fig. 2. Observed and modeled

Table 1

The maximum relative modeling errors

1970 1980 1990

Male 0.1034 0.0716 0.0388

Female 0.0847 0.0989 0.0843

larger values at younger ages and smaller values at olderages. This requires mðxÞ be larger at younger ages andsmaller at older ages than the estimated Gompertzcurve, in which the increase rate is c0: How could therebe such mðxÞ? Notice that the Gompertz curve has onlyage as explanation variable, and is estimated for all ages,including the younger ages at which disabled people arerare and the older ages with more disabled people. Theanswer to how could there be such mðxÞ; therefore,would naturally be that disabled people have lowermortality.In reality, could the mortality of disabled be lower

than that of active population? The answer depends onmany factors, including how the disability is defined andhow the disabled are take cared of. For the definition ofdisability used here, if disabled husbands were lookedafter by their wives, who were younger and less likely tobe disabled, better than active husbands who weredeemed to take care of others, lower mortality fordisabled status would be possible. Besides, althoughthere is no previous evidence at national level yet, lowermortality for disabled status has been estimated in small

65 70 75 80 85

e

+

65 70 75 80 85

e

+

death rates of the US.

Table 2

The ratios of modeled death rates of disabled to active population

1970 1980 1990

Male 0.78 0.84 1.10

Female 1.29 1.18 1.03


sample studies, for example, the model I of Land et al.(1994).The age-specific disability risks are measured by net

transition rates from active to disabled status. How didthe disability risk change with age? Fig. 3 shows that itrises with age in general. Negative net transition rateswere found for males older than 80 years in 1970, whichwould imply net transition from disabled to active statusif there were no estimating error. If there were nodisabled-to-active net transition at younger ages, itwould be unlikely for such transition to occur at olderages. Therefore, the two negative transition rates inFig. 3 could result from modeling errors as shown inboth Table 1 and Fig. 2.How does the disability risk differ between sexes?

Fig. 4 displays the ratios of net transition rates of maleto female, which indicate that males are more likely tobe disabled than females at ages younger than 65; afterthis age the disability risk is higher for females. For thisobservation, that men may work harder in working agesand widows may suffer higher stress in older ages couldbe an explanation.How does the disability risk compare to the death

risk? Fig. 5 shows the T/D ratios, defined as the ratio ofnet transition rates to death rates of active status. It canbe seen that for males the disability risk is higher at agesyounger than 65; at older ages the death risk is higher.For females, however, the disability risk is higher thanthe death risk at all ages. Further, nothing suggests thatthe disability risk declined faster than that of death risk,

40 45 50 55 60-0.02

0

0.02

0.04

0.06

0.08

a

mal

e tr

ansi

tion

rate

197019801990

40 45 50 55 600

0.05

0.1

0.15

0.2

0.25

ag

fem

ale

tran

sitio

n ra

te

197019801990

Fig. 3. The transition

which can be measured as the decline in the T=D ratios.As long as mortality declines, to reduce the T=D ratioswould be an attractive target, if living healthier wereprior to living longer.To assess living healthier or longer, the values of ALE

and DLE, from using the prevalence method and (9), areshown in Table 3. The two methods lead to the sameconclusion: in 1970s, the longer life Americans gainedwas more in disabled years, but in 1980s the increasedlife was more in active years.The closeness between the numerical results of the two

methods, however, is not guaranteed. Suppose, only forthe sake of illustrating, that extraordinary effort hadbeen made to reduce disability risk and no one becamedisabled within year 1980, and the dðxÞ in 1980 wereresulted from disability risk before 1980. In this case, ifthe method of this paper estimated net transition ratescorrectly, it would conclude that edð40Þ ¼ 0 in 1980.Using the prevalence method, however, the values ofDLE in 1980 remain the same as in Table 3. On theother hand, if what happened in 1980 were notextraordinary reduce in net transition rates but fasterdecline in mortality of active than disabled people, (9)would also report higher ALE.The problem that ALE and DLE do not separate

risks of mortality and disability can be solved by theequivalent disability fraction, EDFðxÞ in (15), whichmeasures disability risk. The values of EDFðxÞ for age40 are shown in Table 4 and for older ages are displayedin Fig. 6.

65 70 75 80 85ge

+

65 70 75 80 85e

+

rates of the US.

ARTICLE IN PRESS

40 45 50 55 60 65 70 75 80 85-0.5

0

0.5

1

1.5

2

2.5

3

age

ratio

of

tran

sitio

n ra

tes

of m

ale

to f

emal

e

+

197019801990

Fig. 4. The ratios of transition rates of male to female, the US.

40 45 50 55 60 65 70 75 80 85-1

0

1

2

3

4

age

mal

e T

/D r

atio

+

+

197019801990

40 45 50 55 60 65 70 75 80 850

1

2

3

4

5

age

fem

ale

T/D

rat

io

197019801990

Fig. 5. The T=D ratios of the US.

Nan Li / Theoretical Population Biology 65 (2004) 389–400396

In 1970s when the longer life that Americans gainedwas more in disabled years, as expected, EDFð40Þincreased for both sexes as shown in Table 5. In 1980swhen the increased life was more in active years,

however, EDFð40Þ for men did not drop but increased.Thus, more increase in active than disabled life does notnecessarily imply a decline in disability risk. In fact, thevalues of e�dð40Þfor male were estimated as 34.8 and 34.7

ARTICLE IN PRESS

Table 3

Active and disabled life expectancies for an average person aged 40

Male Female

1970 Increase in

10 years

1980 Increase in

10 years

1990 1970 Increase in

10 years

1980 Increase in

10 years

1990

LE 31.6 2.1 33.7 1.6 35.3 37.9 2.2 40.1 1.4 41.5

Prevalence ALE 22.3 0.2 22.5 1.2 23.7 26.5 �0.4 26.1 0.9 27.0

Prevalence DLE 9.3 1.9 11.2 0.4 11.6 11.4 2.6 14.0 0.5 14.5

eað40Þ 21.6 0.1 21.7 1.1 22.8 25.4 �0.5 24.9 0.9 25.8

edð40Þ 10.0 2.0 12.0 0.5 12.5 12.5 2.7 15.2 0.5 15.7

Table 4

Equivalent disability fraction at age 40

Male Female

1970 Increase in

10 years

1980 Increase in

10 years

1990 1970 Increase in

10 years

1980 Increase in

10 years

1990

EDFð40Þ 0.208 0.042 0.250 0.012 0.262 0.262 0.030 0.292 �0.012 0.280

40 45 50 55 60 65 70 75 80 850.05

0.1

0.15

0.2

0.25

0.3

age(x)

mal

e E

DF

(x)

+

197019801990

40 45 50 55 60 65 70 75 80 850.25

0.3

0.35

0.4

0.45

0.5

age(x)

fem

ale

ED

F(x

)

+

197019801990

Fig. 6. The equivalent disability fractions ðEDFðxÞÞ of the US.

Nan Li / Theoretical Population Biology 65 (2004) 389–400 397

for 1980 and 1990, respectively, so that the moreincrease in ALE than DLE for male in 1980s may bepartly from the increasing mortality of disabled status.Therefore, targeting on maximizing the ALE (Office ofDisease Prevention and Health Promotion 1991; Melt-zer, 1997) needs cautious rethinking. The ALE can beraised by reducing mortality of active individuals fasterthan that of disabled people, without paying attention toreducing disability risk.

Indeed, reducing disability risk is an issue worthy ofattention. Although it is hard to tell the changes indisability risks when they are measured by the age-specificnet transition rates in Fig. 2, the EDFðxÞ reveal the trendsclearly in Fig. 6. Contrary to the general trend of mortalitydecline over time, the estimates of EDFðxÞ have increasedat all ages in the US between 1970 and 1990, as can beseen in Fig. 6. The increase of disability risk may not beincomprehensible. Declines in mortality would expose


active people to the force that derives disability for longertimes, and hence lead to a potential increase in disabilityrisk. If no corresponding efforts were made to counteractthis influence, disability risk would increase.Contrary also to the mortality difference between

sexes, and to the common sense that men’s behaviorshould cause disability more likely, female’s EDFðxÞ hasbeen estimated higher than that of male at all ages andtimes in the US, as can be seen in Fig. 6. This result isnot conflict to that in Fig. 4 where men’s net transitionrates are higher than that of women in working ages,since EDFðxÞ measures disability risks at ages x andover. Because disability and mortality of active statuesare competing risks, the explanation of why womensuffer higher disability risk may lie in the answer to whywomen experience lower mortality risk.The main purpose of this paper is to develop a

conceptual framework, not to provide a standardprocedure, to estimate national disability risk. Theassumptions made in this paper are optional. Forexample, the third assumption could be that the numberof population in a certain age group changes over a 2n-year interval exponentially, instead of linearly. When thedata of single-year age group were available, the Coxassumption could be removed. That is, death rates couldbe assumed increasing with age exponentially but atdifferent rates for active and disabled status (the c0 in (1)and (2) could be different), because there would be muchmore age groups to be used to estimate one moreparameter. Besides, the estimated transition rates wouldrefer to a period of only 2 years. Further, if the active ordisabled status of each death in the vital statistics couldbe identified by matching with the census data, theGompertz assumption would no longer be necessary, thedeath rates of active and disabled status could becalculated directly. In this situation, the estimatingerrors may come only from the two supplementaryassumptions: the number of population in each agegroup changes linearly over time for 2 years; death ratesand disability proportions remain constant for 2 years.Furthermore, if in census not only whether disabled butalso since when could be asked, the supplementaryassumptions would not be in need, the transition ratescan simply be calculated.

Appendix

A.1. Estimating parameters of mortality model

The coefficients c0; c1 and c2; in (1) and (2), areestimated minimizing the sum of error square:

minXw

x¼s

fmðxÞ � ½1� dðxÞ�exp½c1 þ c0ðx � sÞ�

� dðxÞexp½c2 þ c0ðx � sÞ�g2: ðA:1Þ

Expression (A.1) is nonlinear about the coefficients; itcan be simplified into two steps. Notice that at age s;

mðsÞ ¼ ½1� dðsÞ�expðc1Þ þ dðsÞexpðc2Þ: ðA:2ÞNotice also that ½ð1� dðxÞexpðc1Þ þ dðxÞexpðc2Þ� ¼expðc1Þ � dðxÞ½expðc1Þ � expðc2Þ� does not change sig-nificantly with age, when jexpðc1Þ � expðc2Þj; the differ-ence between the death rates of active and disabledpopulation aged s; is small. The first step is thereforeusing mðsÞ to substitute ½ð1� dðxÞÞexpðc1Þ þdðxÞexpðc2Þ� in (A.1) and the estimation of c0 issimplified as

minXw

x¼s

flogðmðxÞ=mðsÞÞ � c0ðx � sÞg2: ðA:3Þ

Expression (A.3) yields

c0 ¼Pw

x¼sðx � sÞlogðmðxÞ=mðsÞÞPwx¼sðx � sÞ2

: ðA:4Þ

When the value of c0 is obtained from (A.4), c1 and c2can then be estimated simply. To avoid larger relativeerrors at younger ages, whose death rates are smaller,the second step of estimating c1 and c2 can be written as

minXw

x¼s

fmðxÞexp½�c0ðx � sÞ�

� ð1� dðxÞÞexpðc1Þ � dðxÞexpðc2Þg2; ðA:5Þ

expðc1Þexpðc2Þ

� �¼

Pwx¼s

ð1� dðxÞÞ2Pwx¼s

ð1� dðxÞÞdðxÞ

Pwx¼s

ð1� dðxÞÞdðxÞPwx¼s

dðxÞ2

0BB@

1CCA

�1

�

Pwx¼s

ð1� dðxÞÞmðxÞexp½�c0ðx � sÞ�

Pwx¼s

dðxÞmðxÞexp½�c0ðx � sÞ�

0BB@

1CCA:

ðA:6ÞHow well the two steps work can be evaluated using

the maximum relative error,

MRE

¼ max 1� ð1� dðxÞÞexp½c1 þ c0ðx � sÞ� þ dðxÞexp½c2 þ c0ðx � sÞ�mðxÞ

:

ðA:7ÞWhen the maximum relative error is acceptable, thevalues of maðxÞ and mdðxÞ are estimated using (1) and(2), in which c0; c1 and c2 are given in (A.4) and (A.6).

A.2. Estimating net transition rates

To estimate the net transition rate, I trace a syntheticcohort aged x at current time, but for only 2n years tomake the estimate refer to the shortest time interval. Inorder to do so, I assume that the death rate anddisability proportion maintain constant for 2n years,


and that the number of population change linearly for2n years.Let the number of active population aged x be paðxÞ;

the active person-years they lived in the first n years bePaðxÞ; and at the end of the first n years the number ofactive population be paðx þ nÞ: The death and nettransition rates are defined as,

maðxÞ þ tadðxÞ ¼ paðxÞ � paðx þ nÞPaðxÞ

¼ paðxÞ � paðx þ nÞn

2½paðxÞ þ paðx þ nÞ�

; ðA:8Þ

which leads to

paðxÞ ¼2þ n½maðxÞ þ tadðxÞ�

2nPaðxÞ: ðA:9Þ

According to the assumption that active populationchanges linearly in 2n years,

paðx þ nÞ ¼ PaðxÞ þ Paðx þ nÞ2n

; ðA:10Þ

where Paðx þ nÞ is the active person-years in the secondn-year time interval. Let the number of total populationaged x be pðxÞ and the person-years that they lived inthe first and second n years be PðxÞ and Pðx þ nÞ;respectively. Using the assumption that disabilityproportion maintain constant for 2n years, (A.10) iswritten as

paðx þ nÞ ¼ ð1� dðxÞÞPðxÞ þ ð1� dðx þ nÞÞPðx þ nÞ2n

:

ðA:11Þ

Using the assumption that the death rates maintainconstant for 2n years,

Pðx þ nÞ ¼ 2� nmðxÞ2þ nmðx þ nÞ PðxÞ: ðA:12Þ

From (A.11) and (A.12),

paðx þ nÞ ¼ 1

2n

� 1þ ð1� dðx þ nÞÞð2� nmðxÞÞð1� dðxÞÞð2þ nmðx þ nÞÞ

� �PaðxÞ:

ðA:13Þ

BecausePaðxÞ ¼n

2½paðxÞ þ paðx þ nÞ�; thus from (A.9)

and (A.13),

PaðxÞ ¼n

2

2þ n½maðxÞ þ tadðxÞ�2n

PaðxÞ�

þ 1

2n1þ ð1� dðx þ nÞÞð2� nmðxÞÞ

ð1� dðxÞÞð2þ nmðx þ nÞÞ

� �PaðxÞ

�:

ðA:14Þ

Therefore, tadðxÞis solved from (A.14) as

tadðxÞ ¼1

n1� ð1� dðx þ nÞÞð2� nmðxÞÞ

ð1� dðxÞÞð2þ nmðx þ nÞÞ

� �� maðxÞ:

ðA:15Þ

Since tadðxÞ is used in (A.9) to describe the number ofsurvivors in the first n years, the estimated tadðxÞ standsfor the average net transition rate for an n-year timeinterval around the current time.For the age group of ðw � nÞ to w years, the 2n-year

assumptions do not strictly apply because the successiveage group is open. Taking the mðwþÞ and dðwþÞ as thecorresponding values for the closed age group of w toðw þ nÞ years approximately, (A.15) still produces theestimate of tadðw � nÞ:For the last open age group, age w and older, (A.15)

does not work even approximately, because there is nota successive age group. For this age group, stationarypopulation should be introduced to substitute the 2n-year assumptions. In this stationary population, theproportion of active population aged w can beapproximately chosen as the average of its two neighborage groups, 0:5ð1� dðw � nÞ þ 1� dðwþÞÞ: Let thenumbers of population aged w and in the last age groupbe pðwÞ and PðwþÞ; the numbers of active populationaged w and in the last age group are 0:5ð1� dðw � nÞ þ1� dðwþÞÞpðwÞ and ð1� dðwþÞÞPðwþÞ; respectively.The death and transition rates can then be written as

tadðwþÞ þ maðwþÞ

¼ 0:5ð1� dðw � nÞ þ 1� dðwþÞÞpðwÞð1� dðwþÞÞPðwþÞ : ðA:16Þ

Since mðwþÞ ¼ pðwÞ=PðwþÞ; the transition rate of thelast open age group is estimated from (A.16) as

tadðwþÞ ¼ 1� 0:5ðdðw � nÞ þ dðwþÞÞð1� dðwþÞÞ

� mðwþÞ � maðwþÞ: ðA:17Þ

Strictly speaking, the estimate of tadðwþÞ in (A.17)represents the average value of transition rate for therest years of individuals aged w: The average of theserest years, however, is eðwþÞ ¼ 1=mðwþÞ; which ingeneral is less than 10 years if w ¼ 85:

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Estimating national disability risk

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