On the Estimation of Disability-Free Life Expectancy: Sullivan’ Method and Its Extension

A quickly aging population, such as the United States nowadays, is characterized by the increase prevalence of chronic disability. robust estimate of disability-free life anticipation ( DFLE ), or healthy life anticipation, is necessity for examining whether extra years of life are spent in good health and whether life anticipation is increasing faster than the decline of disability rates. Over 30 years since its publication, Sullivan ’ s method remains the most widely used method to estimate DFLE. consequently, it is storm to note that Sullivan did not provide any formal justification of his method. Debates in the literature have centered around the properties of Sullivan ’ sulfur method acting and have yielded conflict results regarding the assumptions required for Sullivan ’ s method. In this article we establish a statistical foundation of Sullivan ’ s method acting. We prove that, under stationarity assumptions, Sullivan ’ sulfur calculator is indifferent and reproducible. This resolves the debate in the literature, which has broadly concluded that extra assumptions are necessary. We besides show that the standard division calculator is consistent and approximately unbiased. ultimately, we demonstrate that Sullivan ’ s method acting can be extended to estimate DFLE without stationarity assumptions. Such an elongation is possible whenever a cohort animation mesa and either straight cross-section disability surveys or a longitudinal survey are available. Our empirical analysis of the 1907 and 1912 U.S. birth cohorts suggests that while deathrate rates remain approximately stationary, disability rates decline during this clock period. In Section 4 we demonstrate that Sullivan ’ s method acting can be extended to estimate DFLE without stationarity assumptions. Such an annex is possible whenever a cohort life table and either straight cross-section disability surveys or a longitudinal survey are available. The multistate life board method acting is a popular alternative in the literature ( e.g., Rogers, Rogers, and Branch 1989a ; Robine et aluminum. 1995 ), but it requires longitudinal data angstrom well as a issue of theoretical and virtual assumptions. In contrast, our extension of Sullivan ’ s method acting allows for indifferent and consistent estimate of DFLE without adhering to the assumptions and data prerequisite of the multistate life postpone method. The computer code that implements the methods proposed in this article is available at the first generator ’ s web site. In segment 5 we apply our proposed methodology to estimate DFLE for the 1907 and 1912 U.S. parentage cohorts using mortality data from U.S. Vital Statistics and disability data from the U.S. Medicare Current Beneficiary Survey. Our empirical analysis suggests that while mortality rates remained approximately stationary, disability rates may have declined during this time period. finally, in Section 6 we introduce conclusions. robust estimate of DFLE is besides vital to the theoretical reason of unwholesomeness. The existing inquiry on DFLE of assorted populations and clock periods has frequently reached contradictory conclusions regarding the competing nature of deathrate and morbidity. For case, Gruenberg ( 1977 ) and Kramer ( 1980 ) argued that the decline in mortality rates only reflects a decay in the mortality rate of chronic diseases preferably than a decline in their incidence. Greater life anticipation will result in more severe chronic diseases. Fries ( 1980 ), on the other hand, argued for the compression of unwholesomeness. If the attack of the chronic condition can be postponed and adult life anticipation is relatively constant, morbidity will be compressed into a shorter time period of time. furthermore, Manton ( 1982 ) offered an alternate hypothesis of moral force equilibrium in which the decline in deathrate leads to an addition in the prevalence of meek chronic diseases. consequently, the accurate estimate of DFLE is essential for the empiric evaluation of these competing theories. In his germinal article, Sullivan ( 1971 ) developed a method acting for combining mortality and morbidity rates into a single drumhead measure of a population ’ s health condition. Over 30 years after its publication, Sullivan ’ s method remains the most widely used method acting among apply researchers and government officials for estimating DFLE. The key idea of the method is to combine the period life postpone, which is the chief method acting to calculate life anticipation, with the age-specific disability prevalence estimated from cross-section survey data. In particular, Sullivan ’ s method acting plainly partitions the sum number of person-year be, which is obtained from the period life table, into the disability and disability-free animation anticipation based on the proportion disabled, which is in turn measured from the cross-section disability survey.

A quickly aging population, such as the United States today, is characterized by the increased preponderance of chronic disability, which is particularly pronounced among the aged. full-bodied estimate of disability-free life anticipation ( DFLE ), or goodly life anticipation, is substantive for examining whether extra years of life are spent in effective health and whether life anticipation is increasing faster than the decay of disability rates. DFLE represents the expect act of years of remaining disability-free life a member of the life postpone age group would experience if cohort age-specific rates of mortality and disability prevailed throughout his/her life. shows the 1999 U.S. unabridged time period life table for selected ages ( Arias 2002 ). In keeping with park demographic notation, the leave rule, north = 1, is not written. The base, liter 0, is set at 1 so that liter ten represents the survival probability. At senesce 20 years, 98.6 % of the hypothetical life table cohort remains. From old age 20 to ω = 85, the remaining 98.6 % of the cohort will live ∑ one = 20 85 L i = 54.9 person-years. Hence, a 20-year-old member of the hypothetical age group will live, on average, 55.9 years given he/she experiences the prevailing period age-specific conditional probabilities of death. For the last long time group, ∞ a 85 = e 85 because everyone who is alert at long time ω = 85 dies within the last interval. adjacent, the average number of years lived in an time interval among those who die in the time interval is equal to the entire count of person-years lived among those who will die divided by the proportion who will die in the interval : We now show that under the stationarity assumptions discussed previously, east x, which is the life sentence anticipation calculated from a menstruation life table in ( 7 ), equals e ( x ), which is the theoretical definition of life anticipation given in ( 2 ). Although in common demographic notation, fifty ( x ) is used in continuous note and lambert adam in discrete, both refer to the proportion animated at accurate age x and, hence, are numerically identical. Given the luck function, μ ( x ), the conditional probability of death for an senesce interval, [ x, x + n x ), is equal to the count of deaths in an age time interval divided by the proportion alive at the beginning of the age interval : where ? x = { one ∈ ? : ten ≤ i }. Under stationarity assumptions for the boundless last age interval [ ω, ∞ ), life sentence anticipation at age ω is equal to the inverse of the death rate, that is, einsteinium ω = ∞ molarity ω − 1. The equality follows from the fact that all those active at age ω must die in the interval, that is, ∞ qω = 1. where the members of the liter x+n x proportion who survive the integral interval each put up normality x years, and the members of the lambert x n x q x symmetry who die in the time interval contribute normality x a ten years, on average. finally, life anticipation at senesce ten is equal to the sum number of person-years for subsequent old age intervals : furthermore, it can be shown that the conditional probability of death within an interval [ x, x + n x ) given that an person of the hypothetical age group survived up to age adam, which is denoted by normality x q x, is equal to n x normality x molarity x / [ 1 + ( n x − n x a ten ) n x m x ], where north x a adam represents the average person-years lived in a given interval [ x, x + normality ) among those who are alive at age x but fail within the interval. The values of n x a x are obtained from dispatch life tables and used in subsequent calculations as a known measure ( for example, Molla, Wagener, and Madans 2001 ; Preston et aluminum. 2001 ). Because newton x P x and nitrogen x D x are immediately obtained from the Census datum and full of life statistics, they are typically boastfully. thus, in the literature, the sample distribution variability about the deathrate rate of the hypothetical cohort, denoted by normality x m x, is considered to be small and, hence, typically ignored. That is, newton x M x is assumed to equal newton x thousand x, which is given by For model, a common deviation from stationarity occurs in many developing countries nowadays, where annual births have been growing relative to deaths. As we see in Section 5, a misdemeanor of the stationarity assumptions is besides possible in develop countries where the death rates are declining ascribable to the progress of checkup technologies. The assumptions imply that the survival function is besides constant over prison term, that is, fifty ( x, y ) = fifty ( x ), and that the crude death pace, that is, ∑ x∈? nitrogen x D x ∕ ∑ x∈? n x P x, equals the petroleum give birth rate, that is, B ∕ ∑ x∈? n x P x where B is the total number of births to members of the population in the menstruation. therefore, the full size of the hypothetical age group is assumed to remain changeless over time. Another significant consequence of stationarity assumptions is that the historic period distribution of the hypothetical cohort in any given interval, [ x, x + n x ), is changeless over meter and is proportional to the survival function. Formally, for all sulfur ∈ [ x, x + n x ), the age distribution is defined by the follow density function : A period life board is created by first observing the midinterval population, denoted by normality x P x, and the sum number of deaths, denoted by north x D x, for each interval [ x, x + n x ). then, the ascertained deathrate rate for each interval, denoted by north x M x, is calculated as newton x D x / normality x P x. Keeping with the standard demographic notation, we use prescripts to indicate the width of the time interval under circumstance. A period life mesa relies on the surveil stationarity assumptions of the population ( for example, Chiang 1984 ; Preston et alabama. 2001 ) : Let ? be a bent of the start ages for the historic period intervals of a period life board. We use ω to denote the depart age of the oldest long time interval. Let normality x represent the width ( years ) of an age interval starting at long time x ∈ ?. typically, the width of age intervals is the like for all but the oldest age time interval [ ω, ∞ ), that is n x = nitrogen for all x ∈ ? \ { ω } and n ω = ∞. When n = 1, a period life table is said to be unabridged, whereas it is called abridged if newton > 1. In this article we allow for a more general mount where each age interval may have a unlike width. Although the theoretical definition of life anticipation is given within the continuous-time model, the data are typically recorded in a discrete form. A period biography table is a common informant of discrete data and is frequently analyzed in order to approximate the continuous-time deathrate serve. Sullivan ’ s method acting besides requires the use of a period animation table. A main function of a period life table is to calculate the liveliness anticipation of a conjectural age group that experiences the presently observed cross-section deathrate rates. similarly, one can define DFLE, denoted by e DF ( x, y ), which represents the ask remaining disability-free ( DF ) life of an individual old age adam born at time yttrium. Let π ( x, yttrium ) be the proportion disabled at exact age x for the cohort born at time y. In early words, π ( x, y ) represents the conditional probability that an individual of this cohort is disabled at old age ten given that he/she survived up to age x. Because the symmetry of survivors who are disability-free at historic period adam is equal to [ 1 − π ( x, yttrium ) ] l ( x, y ), DFLE is given by If l ( 0, yttrium ) is set to 1, as we do for the remainder of this article, fifty ( x, y ) corresponds to the survival function of this cohort. Given the survival function fifty ( x, y ), liveliness anticipation at historic period ten can be written as theoretically, mortality for a cohort ( either real or hypothetical ) can be considered as a continuous-time process, which is determined entirely by the luck function, μ ( x, yttrium ), representing the instantaneous pace of death at age x ∈ [ 0, ∞ ) for a age group born at meter yttrium. Given the hazard officiate, one can derive liveliness anticipation at long time ten for this cohort, denoted by east ( x, y ), which represents the expected remaining biography of an individual at historic period ten who is born at time yttrium. Let l ( 0, yttrium ) represent the entire numeral alive at senesce 0 for this age group. then, the numeral of survivors at age ten is given by moment, because the standard variance calculator of ( 13 ) ignores this extra reservoir of doubt, it by and large leads to underestimate of the true discrepancy. In the Appendix we derive the large-sample variability of e ^ x DF, which incorporates the uncertainty about normality x megabyte x, and show that this variance can be systematically estimated. The sixth and seventh column of show the 95 % confidence intervals based on the large-sample division of ( A.6 ). These confidence intervals are slightly wider than the confidence intervals based on the standard division calculator of ( 13 ). For case, at age 50, allowing for extra variability about the appraisal of normality x thousand x, the 95 % assurance widens by .04 years from ( 26.52, 26.64 ) to ( 26.50, 26.66 ). As briefly mentioned in Section 2.2, the standard use of Sullivan ’ s method assumes stationarity and ignores the sampling unevenness about nitrogen x thousand ten by merely setting nitrogen x megabyte x = normality x M x. here, we discuss how Propositions 1 and 2 extend to the site where normality x thousand x is stranger but systematically estimated by nitrogen x M x. First, although einsteinium ^ x DF is no longer an indifferent estimate of einsteinium DF ( x ), the former is placid consistent for the latter because lambert ( ten ) can be systematically estimated by lambert ten and, hence, ∫ x x + n x fifty ( t ) vitamin d t can besides be systematically estimated by newton x L x. In addition to dinner dress investigations, numerous pretense and empirical studies have been conducted to examine the operation of Sullivan ’ s method acting under diverse conditions. These studies in general confirm our theoretical results. For case, Mathers and Robine ( 1997 ) found that, under stationarity assumptions, Sullivan ’ s method provides a consistent calculator of DFLE. A count of studies besides suggest that Sullivan ’ s method acting performs ill when the assumption of stationarity is grossly violated, particularly with regard to disability preponderance ( for example, Barendregt et aluminum. 1994 ; Mathers and Robine 1997 ; Lievre et alabama. 2003 ). In contrast, Propositions 1 and 2 imply that Sullivan ’ s method acting does not make any assumption about the luck serve and disability preponderance rate early than that they must be stationary. They besides show that Sullivan ’ s method does not make any assumption regarding the homogeneity of deathrate risk between the healthy and disabled populations. note that the stationarity of transition probabilities implies that of prevalence rates, but the latter do not necessarily imply the erstwhile. For case, although possibly rare, it is potential that more disable people are dying at a certain age over ( calendar ) clock but more people are transitioning into the disable state over time such that the symmetry of the disable among those who are alive remains ceaseless. The miss of ball results has prompted some theoretical investigations about the command assumptions of Sullivan ’ s method acting. however, these studies yielded conflicting results on what, if any, extra assumptions are required for Sullivan ’ s method acting. Some have argued that tied with a stationary population, Sullivan ’ s method requires extra potent assumptions about the probability of the transitions between healthy and disable condition. For exercise, Newman ( 1988 ) argued that if the probability of transition from disable state to healthy department of state ( i, the “ recovery ” probability ) is big, Sullivan ’ s method provides a consistent calculator of DFLE. conversely, Palloni et alabama. ( 2005 ) maintained that Sullivan ’ s method acting assumes this recovery probability to be negligible and promote argued that Sullivan ’ s method acting requires another presumption about the homogeneity in the mortality risks of the disable and healthy populations. where W ij ( triiodothyronine ij ) is the self-reported count of days the jth answering of the ith senesce time interval spend in disability during the previous year. The proof of Proposition 1 in the Appendix implies that the equality, ∫ x x + north x W ( t ) 365 l ( triiodothyronine ) d triiodothyronine = ∫ x x + n x π ( triiodothyronine ) l ( thyroxine ) five hundred thyroxine, must hold for each x ∈ ? in order for nitrogen x π ^ x given in ( 15 ) to estimate DFLE without bias, where W ( deoxythymidine monophosphate ) represents the population average days spent in disability during the past year at old age t. The equality is unlikely to hold because π ( deoxythymidine monophosphate ) measures the disability preponderance at age deoxythymidine monophosphate, while W ( deoxythymidine monophosphate ) corresponds to the disability prevalence over the annual period. fortunately, with the exception of few studies ( for example, Newman 1988 ), most applications of Sullivan ’ randomness method used the valid disability preponderance calculator of ( 12 ) preferably than that of ( 15 ). proposition 1 shows that four stationarity assumptions are sufficient to establish the unbiasedness and consistency of Sullivan ’ south calculator, whereas Proposition 2 shows that, under these assumptions, the standard discrepancy calculator is consistent and approximately unbiased. Because three out of four stationarity assumptions are needed for any analysis based on menstruation animation tables, the lone extra assumption required is the stationarity of disability prevalence. This assumption may be fair once the stationarity of mortality rates is raise if unwholesomeness and mortality are closely related in a given population. however, it is besides potential that mortality rates remain approximately stationary while disability rates depart from stationarity ( see Sec. 5 ). A validation is given in the Appendix. As is the case of Proposition 1, the statistical robustness of Sullivan ’ s method does not depend on the way the age is partitioned into intervals. The proof of Proposition 2 can besides be used to derive the follow alternate variability calculator that is both ( precisely ) unbiased and reproducible : adjacent, we show that under the lapp stationarity assumptions, the standard variance calculator of ( 13 ) is reproducible and approximately unbiased. furthermore, these properties do not depend on the assumption of the changeless disability prevalence in each time interval, which is improbable to hold when the age interval is wide, as in many applications of Sullivan ’ s method to abridged time period life tables. indeed, no extra assumption is required for the running shape of π ( x ). A proof is given in the Appendix. The result does not depend on the time interval widths and, therefore, applies to both abridged and unabridged dictionary period life sentence tables. It besides does not require researchers to know the accurate senesce of survey respondents, so retentive as one knows the age time interval to which they belong. Suppose that three stationary assumptions of period life tables hold. In addition, suppose that the age-specific disability prevalence is constant over time, that is, π ( x, y = π ( x ) for all yttrium. then, Sullivan ’ s method estimates DFLE without diagonal, that is, E ( east ^ x DF ) = vitamin e DF ( x ), and is besides reproducible, that is e ^ x DF → phosphorus east x DF for vitamin e x DF for all x ∈ ?. Sullivan ’ s method acting inherits three stationarity assumptions discussed in section 2.2 because it relies on a period life table. The follow proposition shows that the stationarity of age-specific disability prevalence is the only extra assumption required for Sullivan ’ s calculator to be indifferent and consistent for DFLE. The stationarity presumption about the disability prevalence is needed for the claim lapp reason as the early stationarity assumptions. It guarantees that cross-sectional data from unlike cohorts can be used to infer the age-specific disability prevalence of a hypothetical cohort. The second column of shows the estimated disability prevalence in each interval, π ^ x, where the sampling weights are incorporated so that respondents from the two surveys are appropriately weighted. Following the literature ( for example, Crimmins et alabama. 1997 ; Crimmins and Saito 2001 ; Molla et alabama. 2004 ), a answering was considered disabled if he/she responded affirmatively to the following doubt : “ Because of a forcible, mental, or aroused problem, do you need the help of other persons with personal concern needs, such as… ? ” where “ … ” represents versatile activities of day by day animation ( ADL ). ADL includes bathe and shower, dressing, eating, getting in/out of seam or professorship, using the gutter, and getting about in home. The third gear column of presents the detail estimates of DFLE based on Sullivan ’ mho method, while the adjacent four column show their 95 % confidence intervals. The “ n x thousand x known ” 95 % assurance intervals ( fourth and fifth column ) are based on the standard variance in ( 13 ). The “ nitrogen x megabyte x estimated ” 95 % confidence intervals ( one-sixth and seventh column ) report for extra variability in the estimate of newton x meter x based on the large-sample variability of ê DF in ( A.6 ) and is discussed farther in Section 3.4.

We illustrate Sullivan ’ s method acting with the 1999 U.S. menstruation life table of. We estimate the disability preponderance from the 1999 National Health Interview Survey ( NHIS ) and the 1999 National Nursing Home Survey ( NNHS ), both of which are conducted by U.S. Department of Health and Human Services. The NHIS is a multipurpose health survey conducted by the National Center for Health Statistics and is the principal source of data on the health of the civilian, noninstitutionalized population of the United States, which included a sample distribution of 97,059 in 1999. The NNHS is a survey of the residents of nurse homes and related concern facilities in the United States besides conducted by the NCHS with 8,215 observations in 1999. The use of the two surveys gives a complete picture of disability prevalence among the noninstitutionalized and institutionalized populations. In the literature, the standard way to obtain the variation of Sullivan ’ second calculator is to assume that the sum total of the disabled within each age time interval, x, x + north x, follows an independent binomial summons with a constant probability, which is estimated by normality x π ^ x ( for example, Mathers 1991 ; Montpellier 1997 ; Molla et aluminum. 2001 ). Given this distributional assumption, the discrepancy of Sullivan ’ s calculator can be estimated by where nickel N one represents the total number of sketch respondents in the old age interval, [ one, one + north one ) and Y ij ( thyroxine ij ) is the disability indicator variable for the jth answering of that interval whose age is metric ton ij ∈ [ one, one + nitrogen one ) at the time of sketch. Depending on one ’ randomness sampling scheme, nitrogen iodine π ^ one may be computed as a weighted median with appropriate sampling weights. In this article, for notational simplicity, we assume simple random sampling, but all the results can be easily generalized to early sampling schemes. note that in the original article Sullivan ( 1971 ) proposed and applied an invalid calculator of disability preponderance, which is different from the calculator of ( 12 ) used by subsequent researchers ( see Sec. 3.3 ). Unlike animation anticipation, DFLE can not be estimated from a period life postpone alone without obtaining extra data about disability prevalence. Sullivan ( 1971 ) proposed a quantify of DFLE by combining mortality data from a menstruation life table and disability information from a cross-section disability survey. however, he did not offer any formal justification of his method acting. In this section, we provide a statistical foundation of Sullivan ’ s method by deriving the assumptions under which Sullivan ’ s method yields a valid estimate of DFLE. We besides discuss how our theoretical findings relate to the former investigations of Sullivan ’ s method in the literature. In summarize, the multistate biography postpone method provides valuable information about transitions among different states and, hence, allows researchers to conduct a richer analysis of deathrate and unwholesomeness than Sullivan ’ s method. however, the method besides requires a numeral of assumptions about transition probabilities and the functional form of hazard affair, none of which is necessary for Sullivan ’ s method acting. numerous studies investigate the operation of the multistate life mesa method when its required assumptions are violated ( Hoem and Jensen 1982 ; Nour and Suchindran 1984 ; Liu, Liang, Jow-Ching, and Whitelaw 1997 ; Manton and Land 2000 ; Schoen 2001 ; Yi, Danan, and Land 2004 ). If DFLE is the measure of interest, Sullivan ’ s method acting yields a valid estimate of DFLE with minimal assumptions and data necessity. If quantities other than DFLE are of concern, on the other hand, the multistate life table method acting may be utilitarian. finally, Davis, Heathcote, and O ’ Neil ( 2001 ) described the estimate of cohort DFLE using the note of multistate life tables. however, as the authors correctly pointed out, it is not possible to estimate transition probabilities from consecutive cross-sectional surveys. alternatively, Davis et aluminum. ( 2001 ) proposed a method that is exchangeable to the one trace in Section 4.2 by estimating the marginal probabilities of respective states of health and death. Unlike the method acting proposed in this article, however, the approach of Davis et alabama. ( 2001 ) requires numeral consolidation and the calculation of standard errors is more complex. second, because of circumscribed data handiness, the huge majority of studies estimate transition probabilities from just a handful of panels from longitudinal disability surveys ( for example, Newman 1988 ; Rogers et alabama. 1989a, boron, 1990 ; Crimmins et aluminum. 1994 ; Albarran et alabama. 2005 ). consequently, these studies assume the stationarity of transition probabilities beyond the menstruation covered by longitudinal surveys. even in the United States, few nationally spokesperson and long-run longitudinal studies of disability exist, due to the implicit in difficulty in following the lapp cohort of individuals during a long period of fourth dimension. A luminary exception is the National Long Term Care Survey, but its panel waves are five years apart. In contrast, nationally representative cross-section studies are conducted every year in the United States, including the National Health Interview Survey, the American Community Survey, the Medicare Current Beneficiary Survey, and the National Health and Nutrition Examination Survey. As shown in Section 5, Sullivan ’ s method acting can exploit the handiness of these large-scale back-to-back crosssectional disability surveys and estimate DFLE without stationarity and other assumptions. next, let normality x L x ( i ) represent the number of person-years exhausted in country iodine in a given old age interval, [ x, x + n x ), that is, formally, nitrogen x L x ( iodine ) = ∫ x x + north x fifty ( i ) ( thyroxine ) vitamin d deoxythymidine monophosphate. To estimate this quantity, researchers must make assumptions about the average number of personyears spent in each state for the interval given that a person starts in country joule at historic period ten and ends up in submit kilobyte at age x + n ten for all joule and potassium ( for example, Land and Rogers 1982 ; Schoen 1988 ). There are four coarse methods to estimate this quantity within a given senesce interval. They are based on the assumption that within each age time interval, the survival functions are analogue ( for example, Schoen 1975 ; Mathers 1991 ; Crimmins et alabama. 1994 ), quadratic ( for example, Schoen 1979 ), exponential ( for example, Krishnamoorthy 1979 ), or cubic ( for example, Schoen and Nelson 1974 ; Schoen and Urton 1979 ). last, the have a bun in the oven issue of remaining years spent in state one can be computed by einsteinium x ( i ) = ∑ j ∈ ? x n j L joule ( i ) ∕ liter adam, where fifty ten represents the survival affair evaluated at age x as earlier. The multistate life mesa method is alike to the menstruation life table method discussed in Section 2.2, but is based on passage probabilities, τ ( ij ) ( x, x + metric ton ), which represent the probability that a person in express one at age x is in state j at claim age x + metric ton for t > 0. The estimate of these transition probabilities requires the handiness of longitudinal data. normally, researchers estimate transition probabilities using either sample fractions ( for example, Rogers et alabama. 1990 ; Crimmins, Hayward, and Saito 1994 ) or parametric models ( for example, Mathers and Robine 1997 ). Given τ ( ij ) ( x, ten + thyroxine ), one can recursively define the symmetry of survivors at historic period ten who are in state of matter iodine, which we denote by fifty x ( one ), as fifty x + thymine ( one ) = fifty x ( i ) + ∑ j ≠ iodine τ ( joule one ) ( x, x + thymine ) fifty x ( joule ) − ∑ j ≠ one τ ( one joule ) ( x, adam + thymine ) lambert x ( iodine ) for deoxythymidine monophosphate < 0. The multistate life postpone method is another popular approach to estimating the DFLE in the literature. hera, we compare this option method with the proposed elongation of Sullivan ’ s method acting described previously. The multistate life table method acting models transitions among different states over senesce by assuming the continuous-time first-order Markov process ( for example, Land and Rogers 1982 ; Schoen 1988 ). Newman ( 1988 ) and Rogers et alabama. ( 1989b ) were among the first to apply the multistate life table method to estimate DFLE. These authors modeled the transitions of individuals of a specific cohort among nonabsorbing states ( for example, disable and disability-free ) and an absorbing department of state ( for example, death ). The assumption of the first-order Markov serve implies that all individuals of the life table cohort who are found in a given state at the same long time will have the lapp transition probabilities careless of their previous paths. Some researchers have raised a business that this assumption may be tenuous because past history of disability is likely to affect the probability of future disability ( for example, Nour and Suchindran 1983 ; Laditka and Wolf 1998 ). It is possible to obtain the balanced assurance interval for the bounds with asymptotically accurate coverage probability ( for example, Cheng and Small 2006 ). Beran ( 1988 ) provided such a method acting based on the bootstrap method. To apply the method, we choose cytosine ~ α L = F ^ L − 1 [ F ^ − 1 ( 1 − α ) ] and c ~ α U = F ^ U − 1 [ F ^ − 1 ( 1 − α ) ], where F ^ L and F ^ U are the empirical distribution functions of B ~ L − B ^ L and B ^ U − B ~ U, and F ^ is the empirical distribution functions of soap { F ^ L ( B ~ L − B ^ L ), F ^ U ( B ^ U − B ~ U ) }. The resulting confidence interval, [ B ^ L − c ~ α L, B ^ U + c ~ α U ], asymptotically covers the truthful bounds by the fixed probability 1 − α. furthermore, these confidence intervals are balanced in a feel that they treat upper and lower bounds fairly ; that is, Pr ( B ^ L − c ~ α L ≤ B L ) = Pr ( B ^ U + c ~ α U ≥ B U ) hold asymptotically. In contrast, the bootstrap confidence intervals proposed by Horowitz and Manski ( 2000 ) have asymptotically exact coverage probability but are not balanced. Under the monotonicity premise, the estimated upper bind in ( 19 ) and the lower bound in ( 20 ) have sampling variability. frankincense, if we estimate the confidence interval for the bounds of DFLE in the same means as earlier, the coverage probability of the resulting confidence intervals can be greater than its nominative level, yielding wider confidence intervals than necessity. formally, let B L and B U be true lower and upper bounds of DFLE. then, applying the Bonferroni inequality, we see that Pr ( [ B L, B U ] ⊂ [ B ^ α L, B ^ α U ] ) ≥ Pr ( B L ≥ B ^ α L ) + Pr ( B U ≤ B ^ α U ) − 1 = 1 − α, where B ^ α U and B ^ α L represent the estimated lower and amphetamine ( 1 − α ) confidence intervals and are found such that Pr ( B L ≥ B ^ α L ) = Pr ( B U ≤ B ^ α U ) = 1 − α ∕ 2. When the bounds in ( 17 ) do not involve north x π ^ x, yttrium, they can be estimated without sampling variability. This implies that the upper berth ( lower ) confidence jump for the bounds of the DFLE equals the common upper ( lower ) confidence band individually obtained for the upper ( lower ) bind of the DFLE based on its estimated division. The resulting confidence interval covers the dependable bounds with accurate ( finite sample ) probability. similarly, if disability surveys do not cover earlier historic period intervals, one can obtain the bounds of DFLE using the monotonicity assumption. Suppose that disability surveys start at age ten and we wish to bound the total number of disability-free years for the precede old age interval, [ x − north x−, x ), with some north x− > 0, where nitrogen x− indicates the distance of the interval ending at long time adam. The bounds are given by which are more enlightening than those in ( 17 ). indeed, the new upper adhere is about constantly more enlightening. The proof of Proposition 1 in Appendix 1 shows that nitrogen ω ∗ π ^ ω ∗, y is an unbiased and coherent calculator of E [ π ( s, y ) ], and, hence, the raw upper berth limit can be estimated without diagonal and systematically. In regulate to further narrow the bounds, we entertain a monotonicity premise regarding the nature of disability for older ages. In particular, we may assume that the disability preponderance of a given birth cohort in the final interval of concern, [ ω* + n ω *, ∞ ), is greater than or equal to the average disability prevalence of the previous historic period interval, [ ω*, ω* + n ω * ). formally, we assume indeed far, we have assumed that disability surveys cover all the age intervals. however, it is potential that the begin old age of the oldest age group surveyed for disability prevalence, denoted by ω* ∈ ?, is less than the start senesce of the final old age interval for the cohort liveliness mesa, that is, ω* < ω. In this case, we can bound DFLE by considering the maximal and minimum values of the contribution of disability-free person-years within the last long time intervals that are not covered by disability surveys. Because disability prevalence is bounded between 0 and 1, the bounds for disability-free person-years for these intervals are given by When straight cross-section surveys are available, it is possible to model π ( x, yttrium ) as a function of y by assuming that the disability preponderance does not experience a sudden change of large magnitude across different cohorts. One may then estimate π ( x, y ), for exemplar, using the popularize linear models ( GAMs ) ( Hastie and Tibshirani 1990 ) or a random-walk model exchangeable to the border on used by Lee and Carter ( 1992 ). Borrowing the data across cohorts in this way may increase the efficiency of estimate ( see besides Sec. 5 ). where nitrogen one π ^ iodine, yttrium is the sample divide of the disable survey respondents within the old age interval [ one, iodine + north i ) for the cohort born in year y. therefore, newton x π ^ x, yttrium can be computed for each adam either from back-to-back cross-sectional surveys, which follow the cohort born in class y, or from a longitudinal survey, which follows the lapp individuals of that cohort over fourth dimension. proposition 1 applies directly to Sullivan ’ s calculator of ( 16 ) except that stationarity assumptions are no long necessary. The variation of this calculator can be calculated in the accurate lapp way as before, and Proposition 2 besides holds without stationarity assumptions. The function of Census data and vital statistics implies that sampling unevenness about n x meter ten can be ignored because a cohort life table directly summarizes the cohort of interest preferably than a random sample from a hypothetical cohort. therefore, Sullivan ’ south method acting, if applied to a cohort life board and either consecutive cross-national disability surveys or longitudinal data, requires no assumption. There are two common ways to close a age group animation postpone. First, an ideal way is to observe the parturition cohort until the survive penis dies. If a birth cohort can not be observed until the last member dies, as is much the event, then the stopping point senesce interval, [ ω, ∞ ) is boundless, and an assumption must be made about the gamble function within the last historic period interval. For example, Horiuchi and Coale ( 1982 ) derived an expression for einsteinium ω by assuming that the symmetry of the last old age group relative to the overall population remains constant. Another normally invoked assumption is the stationarity of mortality in the last old age group, which will yield negligible erroneousness if the proportion of the parturition cohort active at age ω is sufficiently belittled ( for example, Wilmoth, Andreev, Jdanov, and Glei 2005 ). A age group animation board describes the mortality experience of a substantial age group of individuals from parentage of the foremost to death of the last member of the group ( e.g., Chiang 1984 ). An important advantage of age group life tables over period biography tables is that the three assumptions of stationarity discussed in section 2.2 are not invoked. A main aim of cohort life tables is to calculate the life anticipation of a substantial birth cohort using cohort-specific birth and deathrate rates for each senesce. Cohort life tables are created by first observing the midinterval population of the age group born in class yttrium, denoted by newton x P x, yttrium, and the entire number of deaths in this age group, denoted by nitrogen x D x, y, for each interval [ x, x + n x ). early quantities such as fifty ten, y and north x L x, y are defined analogously as done for period life tables. calculation of age group life anticipation besides follows the methods identical to the ones described in section 2.2. In this section we show that if DFLE is the measure of sake, Sullivan ’ s method acting can be used to estimate DFLE without stationarity and early assumptions by using a cohort life sentence mesa. The unbiased estimate of DFLE is besides potential with consecutive cross-section disability surveys, which are much easier to obtain than longitudinal data. The assumption of stationary deathrate and disability required for Sullivan ’ s method may be flimsy, specially in develop countries over the twentieth century where mortality rates for the oldest ages have declined. consequently, it is often of sake to estimate DFLE without stationarity assumptions. A popular approach in the literature has been the multistate life table method acting, which models the transition probabilities among the healthy express, disable state, and death ( e.g., Rogers et alabama. 1989b, 1990 ). This approach, however, requires a large-scale longitudinal disability survey, which is rarely available. furthermore, as discussed late, the multistate life table method makes assumptions about the transition probabilities that are often firm and untestable .

5. AN EMPIRICAL ANALYSIS OF THE 1907 AND 1912 U.S. BIRTH COHORTS

In this section we apply our propagation of Sullivan ’ s method to the 1907 and 1912 U.S. birth cohorts. We besides compare cohort DFLE estimated from the 1907 parturition cohort with period DFLE estimated from the 1991 to 2002 U.S. populations. Our analysis is based on the mortality rates of the 1907 and 1912 birth cohorts and the cross-sectional mortality rates from 1991 to 2002, all of which are obtained from the Human Mortality Database and are based on annual U.S. full of life statistics. The data were downloaded on April 1, 2006, from hypertext transfer protocol : //www.mortality.org, the web site maintained by the University of California, Berkeley, and the Max Planck Institute for Demographic Research. We estimate the disability prevalence using straight cross-sectional surveys. In particular, we use the 1991 Medicare Current Beneficiary Survey ( MCBS ) [ available through the Inter-university Consortium for Political and Social Research ( ICPSR ) ], 1992 and 1993 MCBS Access to Care ( available through the ICPSR ), and 1994 to 2002 MCBS Cost and Use ( available through the U.S. Department of Health and Human Services ). The MCBS is a continuous, multipurpose survey of a representative national sample of the Medicare population, which includes both the noninstutitionalized and the commit populations, and is conducted by the Centers for Medicare and Medicaid Services. Medicare is the largest health insurance course of study in the United States, which covered over 95 % of the U.S. population historic period 65 and older between the years 1991 and 2002 ( DeNavas-Walt, Proctor, and Hill Lee 2005 ). thus, the habit of this surveil gives us a complete picture of disability status for the overall U.S. population for each year. We use the MCBS rather than the NHIS and NNHS discussed in section 3.1 because while the NHIS is administered annually from 1991 to 2002, the NNHS was alone administered in 1995, 1997, and 1999 during the period of interest. As in Section 3.1, a respondent was considered disabled if he or she reported at least one activity of day by day living limitation. In all the analyses presented in this section, the view weights are incorporated so that respondents are appropriately weighted according to their population size. We estimate disability prevalence for the 1907 ( 1912 ) birth cohort from age 81 ( 76 ) to 90 ( 90 ) using the first MCBS view in 1991 to the 2002 MCBS. We estimate DFLE for ages 81 ( 76 ) to 83 ( 78 ) using the monotonicity assumption and estimate the bounds shown in ( 20 ). For the sake of a comparison between the 1907 and 1912 birth cohorts, we close both cohort life tables by assuming a stationary population at age ω = 90 and beyond. We use only the first base 7 years of mortality and disability ( 1991–1997 ) for the 1907 parturition cohort, while all 12 years of the data are used for the 1912 parentage cohort. The upper leave ( right ) panel of shows the calculate bounds of DFLE for the 1907 ( 1912 ) birth age group along with life anticipation. Based on the mortality feel of the 1907 parentage age group, for case, individuals who are alive at age 85 lived, on average, 7.20 years, and they spent between 1.30 and 3.06 years without disability. The lower leave ( right ) panel of the human body shows another quantity of interest, the estimate bounds of the proportion of remaining life spent disability-free, e^xDF∕ex for the 1907 ( 1912 ) parentage age group. For example, at historic period 85, members of the 1907 give birth age group are expected to spend between 18.0 % and 42.5 % of their remain life without disability. The calculate proportion decreases gradually with age. For both DFLE and the proportion, the 95 % balanced confidence intervals are estimated using the bootstrap operation alluded to in Section 4.3 with 10,000 replications. The comparison of the two give birth cohorts shows that while life sentence anticipation increased slightly over the two give birth cohorts, the estimate proportion of remaining biography spent disability-free does not show clearly differences between the two cohorts due to the wide-eyed confidence intervals. For exemplar, biography anticipation at historic period 85 increases by .28 years between the two parentage cohorts. Yet, the 95 % balanced confidence intervals of DFLE overlap importantly, that is, ( 1.30, 3.06 ) for the 1907 parentage cohort and ( 1.76, 3.82 ) for the 1912 birth cohort. consequently, the 95 % balanced confidence interval of the proportion of remaining life sentence spent disability-free for the 1907 birth cohort ( .18, .42 ) overlaps with that of the 1912 birth cohort ( .24, .51 ). The wide confidence intervals are in separate due to the fact that the mortality and disability data are available entirely up to 90 years of age for the 1912 birth age group ( i.e., year 2002 ). If the extra years of data become available in the future, the bounds may become well narrower and exhibit statistically significant differences between the two cohorts. As described in Section 4.2, we besides use a model-based alteration and estimate π ( x, y ) as a smooth function of yttrium. In detail, we modeled π ( x, y ) using the GAM with binomial family and logistic connection. The estimated DFLE based on GAM is between .05 ( .10 ) and .30 ( .29 ) years smaller for the 1907 ( 1912 ) birth age group than those based on the observe disability preponderance. The 95 % balanced confidence intervals of DFLE are besides slenderly narrower for both parturition cohorts using GAM. For example, at senesce 93 for the 1907 birth cohorts, the confidence interval for DFLE was ( .18, 1.62 ) using sample weighted averages and ( .19, 1.51 ) using GAM.

For the determination of comparison, we besides estimate life anticipation, DFLE, and the symmetry of remaining life spent disability-free for the 1991 conjectural period age group using the standard Sullivan method and compare the results with those of the 1907 birth cohort. In this case, we use the wide mortality and disability information available, 1991 to 2002, corresponding to ages 84 to 95. We begin our analysis at age 84, the senesce of the 1907 parturition cohort in 1991, and examine the differences between the period and cohort estimates for subsequent ages. The upper empanel of plots the remainder between the 1991 period and 1907 birth age group life anticipation, that is, 1991 period calculate minus 1907 cohort estimate. Age-specific life anticipation is about identical for the parentage age group than for the hypothetical period cohort from age 84 to 95, which indicates about stationary age-specific mortality rates. The middle control panel plots the calculate bounds for the same differences for DFLE. The 1907 give birth age group DFLE is importantly higher than the 1991 period cohort DFLE from age 84 ( the age of the 1907 birth cohort in 1991 ) to 88 as seen from the fact that the 95 % assurance intervals for the bounds of the differences do not contain 0 in this range. Given the approximate stationarity of deathrate rates, this nonstationarity of DFLE is possible only with the nonstationarity of age-specific disability rates. The lower panel plot shows that for the proportion of remaining life spent disability-free is besides significantly higher for the give birth cohort than the hypothetical period age group from age 84 to 88. The evidence shows that while deathrate rates remain approximately stationary, disability rates may have declined during this fourth dimension time period. indeed, as shown in, deathrate rates remained about stationary while disability rates did not between the 1991 period and 1907 parentage cohorts. The upper jury plots 1991 period and 1907 birth age group mortality rates for ages 81 to 95. deathrate rates for this age range are approximately stationary as shown by the equality of period and cohort rates. As shown in the lower panel of, however, age-specific disability rates are nonstationary and decrease over time. The disability rate for each historic period of the hypothetical age group in the 1991 period is uniformly greater than the corresponding disability rate experienced by the birth cohort of 1907 .

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