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2 ( Allowable Daly Intaes ADI )

4 . X- Kaplan-Meer 3. X X- 95% 4 4. X- 4

5 RFM X- 5. X X X- 7 : d =, d = 5, d =, d = 5, d = : = : = : =.7..3 : d =, d = 5, d = 8, d =, d = : = : = : =

6 ( Allowable Daly Intaes ADI ) ADI ADI ( no-obsorved-effect level NOEL ) ( Food and Drug Admnstraton FDA ) ( Safety Factor ) NOEL ADI :.. 3. NOEL ADI NOEL ADI

7 NOEL NOEL NOEL ADI NOEL NOEL ADI Crump (984 ) ( extra rs ) ( extra response ) ( Benchmar dose ; BD ) ( rght-censored survval tme ) ( Webull dstrbuton ( Proportonal hazards model ) ADI ( confdence lower

8 bound ) ADI ADI 3

9 . Benchmar dose n X extra rs P( d) P() P() d Crump (984) (-α)% 4

10 () m(d) d ( extra response ) m( d) m( o) m() d. ( Webull Dstrbuton ) T ( ) ( pdf ) f(t) F(t) = P(T>t ) T ( survval functon ) S(t) = P(T>t ) T ( hazard functon ) h(t) = lm t P( t T < t + t ) t = f ( t) S( t) 5

11 h(t) = λ t λ T ~W(λ ) T S(t) = exp(-λt ) f(t) = λ t exp(-λt ) H(t) = t h( s) ds ln H(t) = ln{-ln S(t)} = ln λ + ln t T > < = T ( log-lnear model ) ln T = µ + σw µ ( ntercept parameter ) σ ( scale parameter ) W ( extreme value dstrbuton ) f W w (w) = exp(w e ) 6

12 (t) S T S W = P(T t) w (w) = exp( e ) lnt µ = P(W ) σ lnt µ = exp[ exp{ }] σ λ λ = exp( µ σ) σ Y = ln T = ln T = [ lnλ + W] Y S Y y (y) = exp( λe ) Y f Y (y) = ( σ)exp[( y µ ) σ exp[( y µ ) σ]] S Y (y) = exp{ exp[( µ ) σ]} y.3 ( Proportonal hazards model ) ( ) 7

13 h (t Z) p Z = (Z Z p ) t Cox (97) ( Proportonal hazards model Cox regresson model ) t h (t Z) = h (t)c( β Z ) h (t) (baselne hazard functon) β = ( β β ) t h (t Z) p t c( β Z ) = exp( β t Z ) p = exp( = β Z ) t S(t) = exp{ exp( β Z) λt } T Z = (Z,,Z p ) t : ln T = µ + αz + α Z α P Z P + σw µ σ α, α,..., α W λ exp( µ σ) S T = σ (t) = P(T t) t ln t µ α Z = exp[ exp{ }] σ P Z,,Z p = β = α / σ j =,,,p j T j t ln T = [ ln λ β Z + W] 8

14 9

15 3. h(t d) = h (t)exp(βd) = λ t exp(βd) h (t) ; λ β d T W(λ ).3 λ = exp( µ σ) = σ ln T = µ+αd +σ ε (3.) β α σ (3.) = ε m(d) = µ+αd

16 m() m( d) m() αd = µ * c d α d * = c µ µ d * = c α d * = - β c ln λ ( relatve rs ) h(t d) h ( t) o = exp(βd * ) = exp[ ( α σ) *d * ] = λ c c c c.

17 ( Crump,984 ) d * ( Maxmum lelhood estmator ) ( α )% 3. n,n,, n ( ) j T j j =,,, n =,,, T j = t j ( Lelhood functon ) n L = h( t ; d ) S( t ; d ) = j= n j j = {[ λ exp( βd ) t ]exp[ λ exp( βd ) t ]} = j= n l= ln L = {ln[ λ exp( βd = j= j ) t j j ] λ exp( βd ) t j } l λ l β l = λ [ n ] = n - = j= [exp( βd ) t ] = [ n d ]- λ [exp( βd ) ] tjd = = [ ] n = n = j= n + j n [ lntj] -λ = j= = j= [exp( βd ) t j lnt j ] λ β λ β

18 λ = - λ [ n ] = l = n -λ ] [ exp( ) βd tjd = j= β = - [ n ]- λ [exp( βd ) (ln ) ] tj tj = n = j= λ β n = - = j= [exp( βd ) t j d ] λ n = - = j= [exp( βd ) t j lnt j ] β n = -λ [exp( βd = j= ) t d lnt ] j j β λ = λ β λ = λ β = β ( Informaton matrx ) I (λ, β, ) = -E λ β λ λ λ β β β λ β λ ( λ β, ) = β λ λ λ β β β λ β λ= λ, β = β, = 3

19 λ β ( covarance matrx ) Var( λ) = ( λ β, ) = Cov( λ, β ) Cov( λ, ) Cov( λ, β ) Var( β ) Cov( β, ) Cov( λ, ) Cov( β, ) Var( ) ( ) ( ) ( nverse matrx ) * c d = lnλ β δ Var ( * d ) = c [( ) λ β Var ( λ)+( ln λ ) Var ( ln λ β )-( 3 ) Cov( λ β )] β λ β * d (-α)% * d -z α / s.e.( * d ) * d ) = Var( d ) * 3.3 Tj Cj j =,,, n =,,, 4

20 X = mn{ T, C } j j j δ j = I < { Tj Cj } I( a) = a I( a) = a X j = x j n L = [ f ( t = j= n j ; d )] δj δ [ S( tj ; d )] j = [ h( t ; d )] S( t ; d ) = j = n j δ j δ j = {[ λ exp( βd ) x ] exp[ λ exp( βd ) x ]} = j= l= ln L = { δ n = j= j j ln[ λ exp( βd ) x j j ] λ exp( βd j ) x } j l λ l β l n = λ [δ j] - = j= n = = j= j n = j= [exp( βd ) x ] n [δ d ]- λ [exp( βd ) x d ] n = j= = [δ j] + = j= j n n [δ j ln xj] -λ = j= = j= j [exp( βd ) x j ln x ] j λ β λ β λ = - n λ = j= [δ j ] l = n -λ ] [ exp( ) βd xjd = j= β 5

21 = - λ β λ β = β λ n = - = j= = j= n n = - = j= = -λ λ β = j= [δ ]- λ [exp( βd ) x (ln x ) ] j [exp( βd [exp( βd n ) x ) x [exp( βd = λ n = j= j j d ] ln x ] j j ) x d ln x ] λ j = β j β j λ β ( λ β, ) = λ β λ λ λ β β β λ β λ= λ, β = β, = = Var( λ) Cov( λ, β) Cov( λ, ) Cov( λ, β) Var( β) Cov( β, ) Cov( λ, ) Cov( β, ) Var( ) * c d = lnλ β (-α)% d * L = * d -z α / s.e.( * d ) * * d ) = Var( d ) Var ( * d ) = c [( ) λβ Var ( λ)+( lnλ ) Var ( lnλ β)-( 3 ) Cov( λ β)] β λ β 6

22 : X- Patel and Hoel (973) Upton et al. (969) ( Leuemas ) X- ( X-radaton ) 535 RFM : ( rads ) X ( ) ; ( ) X- 4 Kaplan- Meer X- 7

23 95% 95% µ = 6.86 α = -.3 σ =.353 ( covarance matrx ) = λ = 3.48* 9 =.8386 β =.38 s.e.( β ) =.9 95% 3 c. 9% 4 8

24 5. ( ) 4 ( 3 4 ) λ λ = exp( βd ) =,,3,4 β * c β d = cln λ / λ c β U(,R) R ( ) p : 4 λ= = p=.3 R=39.6 λ= = p=.5 R=

25 IMSL RNWIB RNUN ( basedness ) 9% 5- ~5-4 6-~6-4 c λ β 5. λ c c λ c λ c β β =3 =

26 5-6- 9% 95% β β 9% β β λ λ λ 9%

27 Crump (984) Cox

28 . Survval Functon rads 5 rads rads 5 rads 3 rads Survval Tme ( Days ) X Kaplan-Meer Log-Cumulatve Hazard Functon rads 5 rads rads 5 rads 3 rads Log of Tme X 3

29 rads 5 rads rads 5 rads 3 rads Densty.3 3 x 4

30 9. RFM X- X * * * * * 49 4* * * 453* * 49* 496 5* * * * * * * 584* * * * 63 64* rads * * * 648* * 655* * * * * * * * 4* * 9 93* 38* 3 339* 34* * 358* 366* 379* 388* * * 45* * * * rads 56* * * * * * * * *

31 Contnue X * 93* * 6* 6* 67* 75* 76 77* 8* 85* * 37* 3* 3* 36 37* * 347* * 379* * 398* 44 48* * * 468* * * rads * * 55* * * * * * * * * 5 * 3* 4 44* 5* 65* 7 74* 76* 9* 98 39* 37* 39* * * 355* * * * 39* 399* 45* 45 4* * rads * * 58* 5* * * * * * 595* * * * * * 58* 6* 7* 73* 79* 95* 3 38* 33* 36* 39* 3* 33* * * 3 rads 377* * * 46 46* * * 58* * * 56* 576* * : * : Upton et al. (969) 6

32 . X- (rads) (λ) () 95% 5.966* (.9868,5.3) * (.9356,3.84).3* (.77,3.443) 5.37* (.935,3.455) 3.3* (.885,3.549) 3. X- (rads) (hazard rato) 95% 5.5 (.,.998).4653 (.49,.685) (.38,.663) (.7534,4.5386)

33 5-. d =, d = 5, d =, d = 5, d = c λ β true dose d E(d_hat) s.e(d_hat) P(d> d L ). * -4 * * -4 *

34 5-. d =, d = 5, d =, d = 5, d = : =.3 c λ β true dose d E(d_hat) s.e(d_hat ) P(d> d L ). * -4 * * -4 *

35 5-3. d =, d = 5, d =, d = 5, d = : =.5 c λ β true dose d E(d_hat) s.e(d_hat) P(d> d L ). * -4 * * -4 *

36 5-4. d =, d = 5, d =, d = 5, d = : =.7 c λ β true dose d E(d_hat) s.e(d_hat) P(d> d L ). * -4 * * -4 *

37 6-. d =, d = 5, d = 8, d =, d = c λ β true dose d E(d_hat) s.e(d_hat) P(d> d L ). * -4 * * -4 *

38 6-. d =, d = 5, d = 8, d =, d = : =.3 c λ β true dose d E(d_hat) s.e(d_hat) P(d> d L ). * -4 * * -4 *

39 6-3. d =, d = 5, d = 8, d =, d = : =.5 c λ β true dose d E(d_hat) s.e(d_hat) P(d> d L ). * -4 * * -4 *

40 6-4. d =, d = 5, d = 8, d =, d = : =.7 c λ β true dose d E(d_hat) s.e(d_hat) P(d> d L ). * -4 * * -4 *

41 [] Collett, D. (994). Modellng Survval Data n Medcal Research. London : Chapman & Hall. [] Cox, D.R. (97). Regresson Models and Lfe Tables (wth Dscusson ). Journal of the Royal Statstcal Socety, B, 74,87-. [3] John, P.K., and Melvn, L.M. (997). Survval Analyss Technques for Censored and Truncated Data. New Yor : Sprnger-Verlag. [4] Josef, S., and Gerald, J.H. (979). A Smple Method for Regresson Analyss Wth Censored Data. Technometrcs, [5] Kenny, S.C. (984). A New Method for Determnng Allowable Daly Intaes. Fundametal and Appled Toxcology 4, [6] Louse Ryan (99). Quanttatve Rs Assessment for Developmental Toxcty. Bometrcs 48, [7] NAS. Drnng Water and Heath. Safe Drnng Water Commttee, Natonal Academy of Scences, Washngton, D.C. (997) [8] Thoman, D.R. and Ban, L.J. (969). Two Sample Test n the Webull Dstrbuton. Technometrcs, [9] Upton, A.C., Allen, R.C., Brown, R.C., Clapp, N.K., Conln, J.W., Cosgrove, G.E., Darden, Jr., E.B., Kastenbaurm, M.A., Odell, Jr., T.T., Serrano, L.J., Tyndall, R.L. and Walburg, Jr., H.E. (969). Quanttatve Expermental Study of Low-Level Radaton Carcnogenss. Radaton- Induced Cancer, Internatonal Atomc Energy Agency, Venna,

) & ( +,! (# ) +. + / & 6!!!.! (!,! (! & 7 6!. 8 / ! (! & 0 6! (9 & 2 7 6!! 3 : ; 5 7 6! ) % (. ()

) & ( +,! (# ) +. + / & 6!!!.! (!,! (! & 7 6!. 8 / ! (! & 0 6! (9 & 2 7 6!! 3 : ; 5 7 6! ) % (. () ! # % & & &! # % &! ( &! # )! ) & ( +,! (# ) +. + / 0 1 2 3 4 4 5 & 6!!!.! (!,! (! & 7 6!. 8 / 6 7 6 8! (! & 0 6! (9 & 2 7 6!! 3 : ; 5 7 6! ) % (. () , 4 / 7!# + 6 7 1 1 1 0 7!.. 6 1 1 2 1 3