Estimating age specific mortality: a new model life table system with flexible standard mortality schedules

Transcription

1 Estmatn ae specfc mortalty: a new model lfe table system wth fleble standard mortalty schedules Author afflaton: Hadon Wan (Ph.D.) Unversty of Washnton, Seattle, Unted States hadon@uw.edu Alan D. Lopez (MS,Ph.D.) Unversty of Queensland, Australa a.lopez@sph.u.edu.au Chrstopher JL Murray (MD, DPhl) Unversty of Washnton, Seattle, Unted States cjlm@uw.edu 1 P ae

2 To estmate the ae pattern of mortalty for populatons where vtal restraton nformaton s not avalable, a common procedure s to estmate the mortalty rates at one or several aes and to etend the estmate to all aes throuh the use of a model lfe table. A model lfe table s a set of ae-specfc death rates that toether are beleved to represent a partcular level of mortalty. Sets of model lfe tables are collectons of the models arrayed by level of mortalty. The three most wdely used model lfe table systems are the Coale-Demeny model lfe table the Unted Natons model lfe tables for less developed countre and the Modfed Lfe Table system as used by the World Health Oranzaton. However, most of the current models have shortcomns that lmt ther applcablty. The prncpal potental weakness s naccurate or unrepresentatve data n the observed lfe tables that form the emprcal bass of the models. Ths weakness s partcularly serous when dealn wth countres wth a hh prevalence of HIV/AIDS, snce ths dsease has manfested tself only recently and prmarly n countres that lack adeuate systems of measurement. As a result, most current model lfe table systems often yeld mplausble results n hh mortalty stuatons. For eample, when applyn the Modfed Lot Lfe Table system to a set of chld and adult mortalty estmates for 187 countres between 1970 and 2010, we fnd that 0.11 % of the ae specfc mortalty estmates are below zero. In addton, althouh most current model lfe table systems are mult-dmensonal and use multple entry parameters such as 5 0 and e 10 (lfe epectancy at ae 10), few can drectly accommodate broad ndcators of adult mortalty such as 45 15, the probablty of dyn between aes 15 and 60. As both 5 0 and become more readly avalable for most countre a model lfe table system that utlzes both ndces drectly s an urent need. We propose that an deal model lfe table system should have the follown desrable attrbutes. Frst, a model lfe table system should be parsmonous and reure only a few entry parameters to enerate a full lfe table. Second, t should adeuately capture the rane of ae patterns of mortalty observed n real populatons and yeld hh predctve valdty, not just measured by summary ndces such as lfe epectancy at brth, but more mportantly by ae specfc mortalty rates. Thrd, t should provde satsfactory estmates of ae specfc mortalty for countres wth hh levels of mortalty, especally those plaued by the HIV/AIDS epdemc. Fnally, a model lfe table should enerate ae specfc mortalty wth a plausble tme trend, and the partal dervatve of entry parameters such as 5 0 and should be postve wth respect to ae specfc mortalty. These are the prncples that our new model s bult upon. Emprcal lfe table database The performance of a model lfe table system depends partly on the emprcal lfe table database t s based upon. Our new model lfe table system s bult upon a database wth 7,294 emprcal lfe tables. Two major data sources provde about 95.4% of these lfe tables. The Human Mortalty Database (HMD)( has over 8,400 lfe tables from 37 countres or areas datn back to ,163 lfe tables from ths database dated between 1950 and 2008 are ncluded n our database. We ecluded lfe tables of poor ualty as assessed by the Human Mortalty Database. The other major source of emprcal lfe tables s the collecton of vtal restraton data adjusted usn death dstrbuton methods revsed by Murray and colleaues (2010). In total, we nclude 2,799 lfe tables from varous vtal 2 P ae

3 restraton sources adjusted by the death dstrbuton methods. We also nclude 14 lfe tables from Matlab, Banladesh, and 19 lfe tables from US countes rouped by HIV seroprevalence. Provncal level data from South Afrca, , are also ncluded n our database. These 115 lfe tables have HIV seroprevalence ranes from 0.5% to 18.4% based on estmates provded by Actuaral Socety of South Afrca. Our emprcal lfe table database covers 84 countres and areas over the perod of 1950 to Almost half (46.4%) of our lfe tables come from developed countres of Western Europe, North Amerca, Australa, and Asa. A snfcant advantae of our database s the ncluson of more lfe tables from developn countres than any other model lfe table system. We have ncluded 532 lfe tables from developn countres n Asa, amon whch 42 lfe tables are from the Inda Sample Restraton System (earler lfe tables from SRS only o up to ae 70+; we apply a model, to be descrbed n the method secton, to etrapolate ae specfc mortalty rates up to ae roup 80-84). Another 1,021 lfe tables from Latn Amercan countres are also ncluded. Perhaps most mportantly, we have ncluded 237 lfe tables from Sub-Saharan Afrca. These lfe tables come from Botswana (2 lfe tables), Maurtus (90 lfe tables), South Afrca (143 lfe tables whch nclude 115 provnce level lfe tables), and Zmbabwe (2 lfe tables). Amon the lfe tables ncluded n our database, 1,153(15.8%) are from populatons wth over 0.1% HIV seroprevalence amon adults wth aes 15 to 59. Method Our model s descrbed by the follown euaton: lot ˆ n = lot n ˆ 1, + β 1, + ˆ γ (lot (lot lot lot ) + ˆ β ) + ˆ γ 2, 2, (lot (lot lot lot ) ) (1) In the above euaton, 5 0 s the probablty of dyn from brth to ae 5; s the probablty of dyn from ae 15 to 60; represents se; represents ae (0,1,5,10,,80); n s the lenth of the ae roup to +n; and n ndcates probablty of dyn n the ae roup to +n. Ths euaton proposes that the lot transformed ae specfc probablty of dyn n a taret lfe table (p) can be represented as a functon of the correspondn lot transformed ae specfc probablty of dyn n a standard lfe table (s) and the dfferences n probablty of dyn from ae 0 to 5 n lot scale and the dfference n probablty of dyn from ae 15 to 60 n lot scale amon two pars of lfe tables: c & and p & c. Lfe table c s the estmated counter-factual lfe table when HIV seroprevalence s zero. When there s no HIV/AIDS epdemc n a populaton, lfe tables p and c are dentcal. The model s based on an emprcal observaton where the dfferences n ae specfc probabltes of dyn n lot scale between two lfe tables are hhly correlated wth dfferences n 5 0 or n lot scale. Coeffcents 1, ˆβ and 2, ˆβ are estmated usn euaton 2 wth zero-hiv lfe tables n our database. lot lot 1, 2, = β (lot lot ) + β (lot lot ) + ξ n n (2) Proper selecton of the standard lfe table s crucal. Instead of usn a snle standard lfe table for each se, we use country-tme specfc and reon (.e. lobal burden of dsease reon) specfc standard for 3 P ae

4 each lfe table not affected by HIV/AIDS n our database (other areated standard lfe tables by dfferent eoraphcal or epdemolocal clustern crtera are also possble). Country-tme specfc standard lfe tables are used whenever an emprcal lfe table from the same country wthn a 15-year tme frame s avalable n our database. 69,265 unue pars of lfe tables are matched usn our lfe table database wth tme las between 1 and 15 years. Reon specfc standard lfe tables are enerated by collapsn all zero-hiv lfe tables n our database from the same lobal burden of dsease reon by se. We then par up all zero-hiv lfe tables n our database wth the enerated reon specfc lfe tables. Ths ves us an addtonal 5,976 pars of lfe tables. The values of 5 0 and serve as ponts of entry (or entry parameters) for ths new model lfe table system. To enerate a standard lfe table (model ae pattern of mortalty) from our emprcal lfe table database, we calculate the Mahalanobs dstance between the taret lfe table and all zero-hiv emprcal lfe tables of the same se n our database based on 5 0 and (n lot scale). The Mahalanobs dstance between two sets of 5 0 and are defned as: D M ( Q ) = ( Q O) T S 1 ( Q O) where O s a multvarate vector representn entry parameters 5 0 and n lot scale. Q = (lot( 50 ),lot( 4515 )) s a multvarate vector that corresponds to an emprcal lfe table n our lfe table database. We choose Mahalanobs dstance over Eucldean dstance due to the fact that and are hhly correlated n lot space (the correlaton coeffcents are 0.58 and 0.87 for males and female respectvely), and Mahalanobs dstance takes the covarance matr of 5 0 and n lot scale nto consderaton when calculatn the dstance between any par of lfe tables. We smply averae the l seres (probablty of survvn from ae 0 to ae ) from the 30 most smlar lfe tables as evaluated by the Mahalanobs dstance to enerate a standard lfe table. Prorty s ven to lfe tables from the same country or lobal burden of dsease reon. The counterfactual lfe table, or the fnal lfe table, when HIV/AIDS s non-estent n the populaton, s estmated usn a standard lfe table, whch s enerated usn the aforementoned. Currently, perhaps the most challenn ssue for all estn model lfe table systems s provdn plausble ae specfc mortalty estmates ven an HIV/AIDS epdemc. Both wdely used model lfe table system the Coale-Demeny model lfe tables and the Modfed Lot Lfe Table system, are larely based on emprcal lfe tables from the pre-hiv era. In addton, they do not provde an nterated soluton to the problem of ncorporatn HIV/AIDS. The 1,153 emprcal lfe tables n our database wth at least 0.1% HIV seroprevalence provde a unue opportunty for us to drectly estmate the mpact of HIV/AIDS on ae specfc mortalty. Here, the mpact of HIV/AIDS on ae specfc mortalty, as measured by coeffcents ˆ1, γ and 2, ˆ γ, s estmated usn Euaton 3, where we model the dfference n ae specfc mortalty n lot scale between an observed lfe table, affected by HIV/AIDS and a counter-factual lfe table, usn the dfference n 5 0 and (also n lot scale) between the same par of lfe tables. We enerate an HIV-free base lfe table, usn counterfactual 5 0 and values and estmated parameters from Euaton 2. 4 P ae

5 lot n lot n = γ 1, (lot lot ) + γ 2, (lot lot ) + δ (3) = = e ˆ θ3 HIVserot 3 e ˆ η3 HIVserot 3 (4) (5) The counterfactual chld mortalty and adult mortalty ndce 5 0 and respectvely, are estmated usn a set of herarchcal med effects models as descrbed n euatons 6 and 7. ln = θ + θ ln( GDP ) + θ Maternal. Edu. 0 1 GDP ln45 15 = η0 + η1 ln( ) + η2.. + η3 (7) j, t 3 + ξ j + υt + ε In Euatons 6 and 7, j refers to country, refers to se, and t refers to tme between 1970 and Maternal. Edu s the averae maternal educaton level for country j at tme t; Ave. Edu s the averae educaton level between ae 15 and 60 for se of country j at tme t; product measured n nternatonal dollars for country j at tme t; and GDP s the ross domestc HIVsero s the HIV seroprevalence for country j at tme t. We compled a database for 191 countres wth estmates of 5 0 (Rajaratnam et al. 2010a), (Rajaratnam et al. 2010b), GDP n nternatonal dollars (Insttute for Health Metrcs and Evaluaton 2010), maternal educaton (Gakdou et al. 2010), ae specfc level of educaton (Gakdou et al. 2010), and HIV seroprevalence (the Jont Unted Natons Proramme on HIV/AIDS (UNAIDS) 2008) for years between 1970 and A 3-year la s appled to the HIV seroprevalence varable to better capture the effect of HIV/AIDS. Wth the estmated coeffcent ˆ θ 3 and ˆ3 η, the HIV-counterfactual c, and c, could easly be calculated usn euatons 5 and 6 To llustrate the effect of HIV/AIDS on the ae pattern of mortalty, we use the lfe table for South Afrcan males n 2004 as an eample. Usn our new model lfe table system descrbed n Euaton 1, we frst enerate a base lfe table (c) usn the counterfactual entry parameters where HIV/AIDS prevalence s set to zero. We then estmate ae specfc mortalty ven dfferent levels of HIV/AIDS epdemc. The output from the Modfed Lot Lfe Table system for the same lfe table s also ncluded. Our estmated ae specfc mortalty closely resembles both level and ae pattern of the observed mortalty. The ae dfferentals of HIV/AIDS mpact are precsely captured. Impact of HIV/AIDS on Ae Pattern of Mortalty South Afrca / Male / Ave Edu + θ HIVsero 3 HIVsero j, t 3 + ξ + υ + ε j t (6) probablty of dryn n lo scale Ae HIV: 0% HIV: 5% HIV: 10% HIV: 15% HIV: 20% HIV: 25% current model observed Modfed Lot 5 P ae