2010 1 13 1 3 2 3 3 3 4 6 4.1.................................... 6 4.2................................ 7 4.3.............................. 7 4.4............................ 8 4.5.................................. 8 5 8 5.1............................ 9 5.2..................... 13 5.3....................................... 13 5.4...................................... 15 5.5...................................... 16 6 16 7 18 8 28 1
2009 12 21 2009 12 23 1.1 R 2009 12 24 1.2 Greville 2009 12 24 1.21 Greville 2009 12 25 1.3 demogr Epi web URL 2009 12 29 1.4 typo R 2010 1 13 1.5 2
1 2 2 3 24 20 24 21 3
2% 12 30% CT (Ai) A3 *1 WHO 10 1 5 2 30 I I II 1 3 1 15 14 25 5 1 *2 *1 http://www.mhlw.go.jp/toukei/manual/index.html web *2 I ICD-10 WHO 4
19 National Death Index 11.8 3 23 10.2 0 60 19 11 30 [http://www.soc.nii.ac.jp/jes/news/pdf/20071206seifu.pdf] National Death Index 2 National Health Statistics. National Death Index. [http://www.cdc.gov/nchs/ndi.htm] 2003 1 National Death Index National Death Index pp.10-11 20 2008 8 28 http://www.scj.go.jp/ja/info/kohyo/pdf/kohyo-20-t62-6.pdf II 5
4 2009 12 22 2006 4.1, 1998 Charnov, 1992;, 1992 10 6
4.2 K K Charnov (1992) K 50 60 100 4.3 (Hawkes et al., 1998; Alvarez, 2000) Charnov (1992) αm Charnov 7
(1) α (2) (3) αm (b) αb 17.9 14.3 6.0 8.3 0.46 0.28 0.063 0.52 13.9 9.3 3.0 6.3 0.45 0.21 0.126 0.79 17.9 13.0 4.8 8.2 0.46 0.27 0.087 0.70 32.9 17.3 2.8 14.5 0.44 0.21 0.142 2.05 (1) 1/M = 0.4 ω 0.1 M ω (Charnov, 1992) (2) (3) 4.4 Westendorp (1998) 17 19 60 70 80 DNA 4.5 2 Wilmoth J 2009 12 22 Why mortality falls over time? Death human population Recognition Reaction Reduction Death 3R Wilmoth 3R-theory of mortality decline 5 8
1 *3 5.1 CDR Crude Death Rate 7 1 1000 10 1 ADR 10 1 10 1 1 Age-specific Death Rate Chamberlain, 2006 Age-Specific Mortality Rate ASMR ADR DSMR 10 1 Directly Standardized Moratality Rate ADR 1000 *3 9
SMR ADR ADR CDR CDR ADR DSMR Standardized Mortality Ratio ISMR Indirectly standardized mortality rate CDR CDR ADR CDR SMR 1994 Smith 1992 Life expectancy Average life span ADR 10 1.0 5.3 Health expectancy Healthy life expectancy HALE DALE PMI Health Adjusted Life Expectancy QALY Quality Adjusted Life Years) Disability Adjusted Life Expectancy DALY (Disability Adjusted Life Years) 1 1 Proportional Mortality Index Proportional Mortality Indicator 50 10
100000 ICD International Statistical Classification of Diseases and Related Health Problems; 10 ICD-10 *4 1981 34 18 12 6 Staetsky, 2009) Wilmoth 2009 12 22 PMR (Proportional Mortality Ratio) YLL Yeas of Life Lost Graham http://www.aist-riss.jp/software/riskcat/ RiskCaT-LLE IMR Infant Mortality Rate 1000 1 1979 1984 1 1000 *4 1995 ICD-9 ICD-10 11
*5 1000 1000 NICU 1995 22 22 28 22 22 1000 3 5 Toddler Mortality 2005 2005 10 25.4 1999 33 10 2006 10 4.9 1985 15.8 1999 6.1 PIH 2005 4 20 12 *5 1966 30 1980 12
5.2 5.3 *6 0 q x 10 *7 x x T x x l x x x + 1 [x, x+1) q x l x (1 q x /2) x x + 1 L x x x m x x d x x q x q x = m x /(1 + m x /2) *8 1 1 2 3 6 1 7 5 (abridged life table) Greville 5q x = 5m [ x ]] 1/5 + 5 m x [1/2 + 5/12 5m x ln( 5m x+5 ) ln( 5 m x ) 5 (Ng and Gentleman, 1995) 5 *6 http://www.mhlw.go.jp/toukei/saikin/hw/life/life08/index.html 2009 7 16 20 *7 *8 x N x x x + 1 x N x + d x /2 x + 1 d x q x q x = d x N x + d x /2 = d x /N x = m x /(1 + m x /2) N x /N x + d x /2/N x 13
ln( 5 m x+5 ) ln( 5 m x ) 5 5 m x *9 l x l x Gompertz *10 Siler R nls() optim() Siler h(t) = a 1 exp( b 1 t) + a 2 + a 3 exp(b 3 t) l x µ x = 1 dl x l x dx x µ x *9 Greville Ng and Gentleman (1995) Greville TNE (1943) Short methods of constructing abridged life tables. Rec. Am. Inst. Actuar., 32: 29-43. (1963) nq(x) = nm(x) [ ] 1 n + 1 n m(x) 2 + n 12 { nm(x) ln c} c n m(x) Gompertz nm(x) = Bc x US ln c 0.080 0.104 0.09 (2006) nq x = nm [ x 1 n + 1 n m x 2 + n 12 { nm x log e ( n mx+n ) n 1 } nm x Ng and Gentleman Namboodiri and Suchindran (1987) nq x = nm x (1/n) + n M x [(1/2) + (n/12)( n M x k)] k 0.09 Keyfitz Applied Mathematical Demography n µ(x + t)dt = n n m x + n3 12 n m 2 x (log n m x ) 0 Greville n 3 1 *10 http://www.toukei.metro.tokyo.jp/seimei/2005/sm-gaiyou.htm Gompertz ] 14
5.4 frailty Kaplan-Meier R survival *11 [parish record] *11 http://phi.med.gunma-u.ac.jp/swtips/survival.html 15
5.5 Age-Period-Cohort APC R Epi apc.fit() apc.plot() plot.lexis() Carstensen, 2007) 6 Graunt (1662) DeMoivre (1725) l(x) x l(x) = l(0) (1 x/a) l(x) x a Graunt Gompertz-Makeham 3 Thiele 5 Siler 8 Mode-Busby Helligman and Pollard Mode-Jacobson Gage and Mode, 1993 3 Denny (Denny, 1997) * 12 Siler h(t) = a 1 exp( b 1 t) + a 2 + a 3 exp(b 3 t) Gage (1991) Gage 1 *12 http://phi.med.gunma-u.ac.jp/demography/denny.html 3 a, b, c 1 l(x) = x (1 + a( + b x e 105 x 1 + c(1 e 2x ) 105 x ))3 16
2 3 a 1 Coale and Demeny *13 2 Brass (1968) *14 (1) (2) (3) (Gavrilov and Gavrilova, 1991 (1) s 0, s 1, s 2,..., s n,... 0, 1, 2,..., n,... λ 0 µ 0 (2) λ µ Le Bras, 1976 µ 0 (3) µ(x) = µ 0 + µλ 0 (1 exp( (λ + µ)x))/(µ + λ exp( (λ + µ)x)) λ << µ x Gompertz-Makeham frailty (Mori and Nakazawa, 2003) *13 1966 1983 4 [ ] 25 *14 l x l 0 (x) l x l s 2 a, b 1 2 ln( 1 l 0(x) ) = a + b l 0 (x) 2 ln( 1 l s(x) ) l s (x) 17
11 0.4% 1891 30% S (Mori and Nakazawa, 2003) 1920 1995 λ 0 λ 0 7 Excel R Excel R R demogr * 15 Epi * 16 *15 http://www.jstatsoft.org/v22/i10/paper *16 http://staff.pubhealth.ku.dk/~bxc/epi/ 18
demogr Coale and Demeny Epi Age-Period-Cohort mortality.r(1) # mortalityj.r # rev. 1.0 19 December 2009 (C) Minato Nakazawa <minato-nakazawa@umin.net> # sample data definition and standardization # References: # http://www.stat.go.jp/data/nenkan/02.htm # 20 2 # Mortality data of Japanese in Japan. # 2006 S60modelpopJ 60 # S60modelpopJ <- c(8180,8338,8497,8655,8814,8972,9130,9289,9400,8651,7616, 6581,5546,4511,3476,2441,1406,784)*1000 AC <- c(paste("[",0:16*5,"-",0:16*5+4,"]",sep=""),"[85-]") # same as follows: # AC <- c("[0-4]", "[5-9]", "[10-14]", "[15-19]", "[20-24]", # "[25-29]", "[30-34]", "[35-39]", "[40-44]", "[45-49]", # "[50-54]", "[55-59]", "[60-64]", "[65-69]", "[70-74]", # "[75-79]", "[80-84]", "[85-]") names(s60modelpopj) <- AC S60M <- c(6042, 1155, 1011, 3179, 3397, 3167, 4237, 7110, 10234, 15063, 24347, 30747, 30884, 38240, 55100, 65593, 59125, 48786) names(s60m) <- AC H02M <- c(4532, 844, 760, 3204, 3466, 2916, 3264, 5449, 9769, 14218, 20161, 32925, 42742, 42664, 51737, 69320, 67916, 67451) names(h02m) <- AC H07M <- c(3929, 752, 716, 2413, 3640, 3203, 3297, 4413, 8236, 15616, 21905, 30491, 47188, 59828, 60927, 68504, 77924, 87750) names(h07m) <- AC H12M <- c(2933, 438, 493, 1721, 2875, 3271, 3749, 4621, 6840, 13141, 24103, 31848, 42214, 60962, 76413, 73947, 73533, 102177) names(h12m) <- AC H17M <- c(2291, 409, 361, 1220, 2303, 2887, 3915, 4915, 6806, 10577, 19546, 34233, 43403, 55261, 80198, 99338, 89502, 127261) names(h17m) <- AC 19
mortality.r(2) <nminato@med.gunma-u.ac.jp> S60F <- c(4792, 636, 638, 1033, 1272, 1558, 2496, 4017, 5650, 7644, 11504, 14828, 19961, 26490, 40891, 55657, 64448, 80930) names(s60f) <- AC H02F <- c(3451, 533, 482, 1149, 1329, 1361, 1774, 3102, 5542, 7510, 10097, 14616, 19986, 27267, 38076, 58203, 71633, 110407) names(h02f) <- AC H07F <- c(3111, 483, 468, 949, 1447, 1393, 1832, 2426, 4578, 8520, 11041, 14241, 21122, 29261, 41516, 56924, 79939, 141519) names(h07f) <- AC H12F <- c(2336, 300, 251, 676, 1160, 1546, 1847, 2425, 3639, 6595, 11740, 14144, 18466, 28096, 40115, 57053, 73527, 171735) names(h12f) <- AC H17F <- c(1811, 246, 229, 582, 1067, 1283, 2037, 2554, 3432, 5177, 9418, 15346, 18855, 25568, 40627, 60024, 84683, 225778) names(h17f) <- AC S60P <-c(7459, 8532, 10042, 8980, 8201, 7823, 9054, 10738, 9135, 8237, 7933, 7000, 5406, 4193, 3563, 2493, 1433, 785)*1000 names(s60p) <- AC H02P <- c(6493, 7467, 8527, 10007, 8800, 8071, 7788, 9004, 10658, 9018, 8088, 7725, 6745, 5104, 3818, 3018, 1833, 1122)*1000 names(h02p) <- AC H07P <- c(5995, 6541, 7478, 8558, 9895, 8788, 8126, 7822, 9006, 10618, 8922, 7953, 7475, 6396, 4695, 3289, 2301, 1580)*1000 names(h07p) <- AC H12P <- c(5904, 6022, 6547, 7488, 8421, 9790, 8777, 8115, 7800, 8916, 10442, 8734, 7736, 7106, 5901, 4151, 2615, 2233)*1000 names(h12p) <- AC H17P <- c(5578, 5928, 6015, 6568, 7351, 8280, 9755, 8736, 8081, 7726, 8796, 10255, 8545, 7433, 6637, 5263, 3412, 2927)*1000 names(h17p) <- AC ASMR ADR R 20
mortality.r(3) S60T <- S60M+S60F H02T <- H02M+H02F H07T <- H07M+H07F H12T <- H12M+H12F H17T <- H17M+H17F <nminato@med.gunma-u.ac.jp> S60ASMR <- S60T/S60P; S60CDR <- sum(s60t)/sum(s60p) H02ASMR <- H02T/H02P; H02CDR <- sum(h02t)/sum(h02p) H07ASMR <- H07T/H07P; H07CDR <- sum(h07t)/sum(h07p) H12ASMR <- H12T/H12P; H12CDR <- sum(h12t)/sum(h12p) H17ASMR <- H17T/H17P; H17CDR <- sum(h17t)/sum(h17p) CDRs <- c(s60cdr, H02CDR, H07CDR, H12CDR, H17CDR) mortality.r(4) DSMR <- function(asmr) { if (length(asmr)!=18) { print("age class is inadequate."); NA } else { sum(asmr*s60modelpopj)/sum(s60modelpopj) } } DSMRs <- c(dsmr(s60asmr),dsmr(h02asmr),dsmr(h07asmr),dsmr(h12asmr),dsmr(h17asmr)) pdf mortality.r(5) pdf("mortalityj.pdf",width=8,height=8) plot(1:5,cdrs,type="l",col="black",xlab="year",axes=f,ylab="mortality", ylim=c(0,0.01)) axis(1,1:5,c("s60","h02","h07","h12","h17")) axis(2,seq(0,0.01,by=0.002)) lines(dsmrs,col="red",lty=2) legend("topright",col=c("black","red"),lty=c(1,2),legend=c("cdrs","dsmrs")) dev.off() 21
(mode==1) Greville (mode==2) 22
lifetable.r(1) <nminato@med.gunma-u.ac.jp> # lifetable.r # rev. 1.0 23 December 2009 (C) Minato Nakazawa <minato-nakazawa@umin.net> # function definition to make a life table. # included data is lifetable <- function(mx,class=5,mode=1) { nc <- length(mx) if (length(mx)!=nc) exit qx <- numeric(nc) if (mode==1) { qx <- mx/(1+mx/2) } else { for (i in 1:(nc-1)) { qx[i] <- mx[i] / (1/class+mx[i]*(1/2+class/12*(mx[i]-(log(mx[i+1])-log(mx[i]))/class))) / class } qx[nc] <- mx[nc] / (1/class+mx[nc]*(1/2+class/12*mx[nc])) / class } dx <- numeric(nc) lx <- numeric(nc) Lx <- numeric(nc) lx[1] <- 100000 for (i in 1:(nc-1)) { dx[i] <- lx[i]*qx[i]*class lx[i+1] <- lx[i]-dx[i] Lx[i] <- (lx[i]+lx[i+1])/2*class } Tx <- cumsum(lx[nc:1])[nc:1] ex <- Tx/lx data.frame(mx,qx,lx,lx,tx,ex) } 23
lifetable.r(2) <nminato@med.gunma-u.ac.jp> qxjmh20 <- c(0.00266, 0.00038, 0.00027, 0.00019, 0.00014, 0.00012, 0.00012, 0.00010, 0.00009, 0.00008, 0.00008, 0.00009, 0.00010, 0.00012, 0.00016, 0.00020, 0.00027, 0.00034, 0.00040, 0.00047, 0.00052, 0.00056, 0.00059, 0.00061, 0.00061, 0.00062, 0.00063, 0.00064, 0.00067, 0.00069, 0.00071, 0.00072, 0.00075, 0.00078, 0.00083, 0.00089, 0.00096, 0.00104, 0.00113, 0.00123, 0.00134, 0.00146, 0.00158, 0.00170, 0.00184, 0.00201, 0.00223, 0.00250, 0.00278, 0.00306, 0.00333, 0.00362, 0.00395, 0.00434, 0.00478, 0.00527, 0.00578, 0.00636, 0.00700, 0.00765, 0.00833, 0.00907, 0.00989, 0.01078, 0.01172, 0.01264, 0.01351, 0.01453, 0.01582, 0.01741, 0.01925, 0.02130, 0.02359, 0.02620, 0.02935, 0.03294, 0.03696, 0.04135, 0.04608, 0.05137, 0.05722, 0.06371, 0.07072, 0.07819, 0.08653, 0.09543, 0.10549, 0.11693, 0.13037, 0.14353, 0.15716, 0.17126, 0.18586, 0.20094, 0.21650, 0.23256, 0.24910, 0.26612, 0.28362, 0.30158, 0.32000, 0.33886, 0.35814, 0.37782, 0.39789, 1.00000) names(qxjmh20) <- c(sprintf("%d",0:104),"105-") qxjfh20 <- c(0.00247, 0.00032, 0.00023, 0.00016, 0.00011, 0.00009, 0.00009, 0.00008, 0.00008, 0.00007, 0.00007, 0.00006, 0.00006, 0.00007, 0.00009, 0.00012, 0.00015, 0.00018, 0.00021, 0.00024, 0.00026, 0.00027, 0.00028, 0.00029, 0.00030, 0.00031, 0.00031, 0.00031, 0.00033, 0.00035, 0.00037, 0.00040, 0.00042, 0.00043, 0.00045, 0.00048, 0.00053, 0.00059, 0.00064, 0.00069, 0.00073, 0.00079, 0.00085, 0.00092, 0.00100, 0.00109, 0.00119, 0.00131, 0.00143, 0.00156, 0.00168, 0.00181, 0.00194, 0.00209, 0.00225, 0.00242, 0.00259, 0.00280, 0.00306, 0.00333, 0.00358, 0.00381, 0.00403, 0.00428, 0.00460, 0.00498, 0.00542, 0.00595, 0.00659, 0.00734, 0.00821, 0.00922, 0.01036, 0.01161, 0.01300, 0.01456, 0.01641, 0.01858, 0.02108, 0.02397, 0.02724, 0.03103, 0.03558, 0.04088, 0.04712, 0.05405, 0.06168, 0.07024, 0.07997, 0.09082, 0.10278, 0.11551, 0.12859, 0.14236, 0.15698, 0.17234, 0.18848, 0.20540, 0.22312, 0.24165, 0.26099, 0.28115, 0.30211, 0.32387, 0.34641, 1.00000) names(qxjfh20) <- c(sprintf("%d",0:104),"105-") 20 q x q x 24
lifetable.r(3) clifetable <- function(qx) { nc <- length(qx) lx <- numeric(nc) dx <- numeric(nc) Lx <- numeric(nc) lx[1] <- 100000 for (i in 1:(nc-1)) { dx[i] <- lx[i]*qx[i] lx[i+1] <- lx[i]-dx[i] Lx[i] <- (lx[i]+lx[i+1])/2 } Tx <- cumsum(lx[nc:1])[nc:1] ex <- Tx/lx data.frame(qx,lx,dx,lx,tx,ex) } <nminato@med.gunma-u.ac.jp> clifetable(qxjmh20) 60 Greville lifetable.r(4) lifetable(s60asmr,class=5,mode=2) 25
<nminato@med.gunma-u.ac.jp> mx qx lx Lx Tx ex [0-4] 0.0014524735 0.0014455260 100000.00 498193.1 7602349.2 76.023492 [5-9] 0.0002099156 0.0002098010 99277.24 496125.8 7104156.1 71.558761 [10-14] 0.0001642103 0.0001641547 99173.09 495662.0 6608030.2 66.631280 [15-19] 0.0004690423 0.0004685105 99091.70 494878.2 6112368.3 61.683960 [20-24] 0.0005693208 0.0005685192 98859.57 493595.3 5617490.1 56.822927 [25-29] 0.0006039882 0.0006031087 98578.55 492149.6 5123894.8 51.977786 [30-34] 0.0007436492 0.0007423446 98281.28 490494.4 4631745.2 47.127440 [35-39] 0.0010362265 0.0010337771 97916.49 488317.2 4141250.8 42.293701 [40-44] 0.0017388068 0.0017318457 97410.37 484943.1 3652933.6 37.500459 [45-49] 0.0027567075 0.0027393406 96566.87 479527.7 3167990.5 32.806183 [50-54] 0.0045192235 0.0044715872 95244.22 470897.5 2688462.8 28.227043 [55-59] 0.0065107143 0.0064121767 93114.76 458110.4 2217565.3 23.815401 [60-64] 0.0094052904 0.0092050297 90129.42 440276.5 1759454.9 19.521427 [65-69] 0.0154376342 0.0149082601 85981.20 413883.1 1319178.3 15.342637 [70-74] 0.0269410609 0.0253626487 79572.05 372633.3 905295.2 11.377050 [75-79] 0.0486361813 0.0436233035 69481.26 309518.8 532661.9 7.666268 [80-84] 0.0862337753 0.0714027853 54326.25 223143.2 223143.2 4.107465 [85-] 0.1652433121 0.1124108423 34931.02 0.0 0.0 0.000000 R optim() Siler Gompertz-Makeham q(x) optim() Denny l(x) R 26
lifetable.r(5) <nminato@med.gunma-u.ac.jp> # H20 Siler Gompertz-Makeham Denny clx <- function(qx) { nc <- length(qx); lx <- numeric(nc); dx <- numeric(nc) lx[1] <- 1 for (i in 1:(nc-1)) { dx[i] <- lx[i]*qx[i]; lx[i+1] <- lx[i]-dx[i] } lx[nc] <- 0 lx } Siler <- function(a1,b1,a2,a3,b3,t) { rval <- a1*exp(-b1*t)+a2+a3*exp(b3*t); ifelse(rval<0,0,rval) } fsilerm <- function(x) { sum((siler(x[1],x[2],x[3],x[4],x[5],ages[1:105])-qxjmh20[1:105])^2) } (rs <- optim(rep(0,5),fsilerm)) GompertzMakeham <- function(a,b,c,t) { rval <- A + B*C^t; ifelse(rval<0,0,rval) } fgompertzmakehamm <- function(x) { sum((gompertzmakeham(x[1],x[2],x[3],ages[1:105])-qxjmh20[1:105])^2) } (rg <- optim(rep(0.1,3),fgompertzmakehamm)) Denny <- function(a,b,c,t) { 1/(1+a*(t/(105-t))^3)+b*sqrt(exp(t/(105-t))-1)+c*(1-exp(-2*t)) } fdennym <- function(x) { sum((denny(x[1],x[2],x[3],ages[1:105])-clx(qxjmh20[1:105]))^2) } (rd <- optim(rep(0,3),fdennym)) layout(t(1:2)) plot(ages,qxjmh20, main="models fitted for Japanese males\n qx in 2008",ylab="qx") lines(ages,siler(rs$par[1],rs$par[2],rs$par[3],rs$par[4],rs$par[5],ages), col="blue",lty=1,lwd=2) lines(ages,gompertzmakeham(rg$par[1],rg$par[2],rg$par[3],ages), col="red",lty=2,lwd=2) legend("topleft",lty=1:2,lwd=2,col=c("blue","red"), legend=c("siler","gompertz-makeham")) plot(ages,clx(qxjmh20), main="models fitted for Japanese males\n qx or converted lx in 2008",ylab="lx") lines(ages,clx(siler(rs$par[1],rs$par[2],rs$par[3],rs$par[4],rs$par[5],ages)), col="blue",lty=1,lwd=2) lines(ages,clx(gompertzmakeham(rg$par[1],rg$par[2],rg$par[3],ages)), col="red",lty=2,lwd=2) lines(ages,denny(rd$par[1],rd$par[2],rd$par[3],ages),col="black",lty=3,lwd=2) legend("bottomleft",lty=1:3,lwd=2,col=c("blue","red","black"), legend=c("siler","gompertz-makeham","denny")) 27
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(1992). [ ] (2006). Keyfitz N, Caswell H (2005) Applied Mathematical Demography. Science+Business Media, Inc., New York. 3rd Ed., Springer (1994). (1980) 1966., 20: 7-16. Mori Y, Nakazawa M (2003) A new simple etiological model of human death. Journal of Population Studies (Jinko-Gaku-Kenkyu), 33: 27-39. Ng E, Gentleman JF (1995) The impact of estimation method and population adjustment on Canadian life table estimates. Health Reports, 7(3): 15-22. Smith DP (1992) Formal Demography. Plenum Press. Staetsky L (2009) Diverging trends in female old-age mortality: A reappraisal. Demographic Research, 21: 30. [http://www.demographic-research.org/volumes/vol21/30/21-30.pdf] (1998). (1968). (2006) Excel. Westendorp RGJ, Kirkwood TBL (1998) Human longevity at the cost of reproductive success. Nature, 396: 743-746. The 29