1 1 () () 1 random variable error 1-1 1-1
1 measurement data. cm (kg) (10 1 /L) / KPa count data ABABO 3 ordinal data ++++++ population sample probability A PA 0PA n A m m/n A n frequency PA= m/n
3 sampling error [ ] A B C D D [ ] A B CD D 10 9 / L
1 ( ) A C A B. D. B. C D. 3 A B. C D. 4 A B. C D. 5 A 4
B C D 50 10 1population sample random sampling variation 4 (ordinal data) (probability) A PA PA A 0PA n A m m/n A n (freqency) PA= m/n random error (systematic error) 5
6 1 10 9 / L 10
7 X max X min R =
X X X = n X f fx X = f lg X G = lg 1 n f lg X G = lg 1 f M = X n+ 1 1 M = X n + X n ( ) ( + 1) R = X max X min P x 8
x P 5 ix P = L + ( n x% f ) x x L f x Q 3 = P 75 Q 1 = P 5 QR = Q3 Q1 ( ) X µ S = n 1 ( ) X ( X X X ) S = = n n 1 n 1 S CV = 100 % X S X ± 9
10
11
1
X ( X X) X 13
R = X max X min X f fx fx 14
ix n.0 M = Lx + fl fm 17 S = X = ( X X ) n 1 fx f fx (( fx) f ) f 1 X f 15
16 = f fx X + = L M x x f n f i L M = + + + + = = 5 15 14 10 9 7 n X X ( ) X X = = n X X ( ) 1 = n X X S 100% S CV X = = = n X X ( ) 1 = n X X S 100% S CV X = = = = 4 65.70 lg lg lg 1 1 f X f G
() 1 () () X 1 ( X µ ) (σ ) f ( X ) = e < X < + (3-1) σ π X X N( µ, σ ) µ σ µ σ µ σ µ x = µ µ σ σ σ σ σ σ µ = 0 σ = 1u Z u N1 17
µ = X u X N( µ, σ ) u σ u Φ( u) X, ) ( X 1 X 1 ( X µ ) ( σ ) D = e dx = Φ( u ) Φ( u ) 1 X1 σ π X1 µ X µ u1 = u = σ σ 3- X σ µ ± σ µ ± 1. 64 σ µ ± 1. 96 µ ±. 58σ 1. 3- ( X, X ) 1 % 90 X ± 1. 64S X 1. 8S X 1. 8S 95 X ± 1. 96S X 1. 64S X 1. 64S 99 X ±. 58S X. 33S X 33S + P ~ P P P 5 95 10 90 + P ~ P P P.5 97. 5 5 95 +. P 0.5 ~ P99. 5 P 1 P 99 X ± S X ± 3S t 18
+ µ σ X = µ µ X µ σ X µ 1. 64σ µ + 1. 64σ µ + 1. 96σ µ +. 58σ µ ± 1. 64σ µ ± 1. 64σ µ + 1. 64σ X σ µ + 1. 64 X A. X ±. 58S BX +.33S Clog 1 ( Y ±.58 S Y ) D log 1 ( Y +.33S ) Y X X Y X ( 10 1 / L) 1 ( 10 / L) 1 ( 10 / L) X X X X X 19
4.00 4.18 X µ 4.50 4.18 P(4.00< X < 4.50) = P( < < ) 0.9 σ 0.9 = P ( 0.6 < u < 1.10 ) = 1 Φ( 1.10) Φ( 0.6) = 1 0.1357 0.676 = 0.5967 X 1 X 1.96σ = 4.18 1.96(0.9) = 3.61( 10 / L) 1 X + 1.96σ = 4.18+ 1.96(0.9) = 4.75( 10 / L) 1 3.61 ~ 4.75( 10 / L) 1P 0. 5 0.5 0.5 1 f L =0L X = 10 f X =7i X =1 1 P 0. 5 =10+ (4.98 0) =10.71 7 P 95.5 95.5 18 f L =985 L = 18 X f X =8i X =1 1 P 95.5 =18+ (991.0 985) =18.5 8 99% 10.71~18.5 µ σ µ µ σ σ 0
u X ± S µ σ µ g /100g ( µ g /) l 1
X = S 38 ( µ g / g) 1 ( X µ ) (σ ) f ( X ) = e, < X < + σ π X ) N( µ, σ µ σ 3 N (0,1 ) µ = X u X N( µ, σ ) u σ ± X
X 19.5 16.1 19.5 PX ( < 16.1) = P( < ) = Pu ( < 1.48) =Φ ( 1.48) = 0.0694.3.3 X -19.5.9-19.5 PX ( >.9) = 1 - PX (.9) = 1 - P( ) = 1 - Pu ( 1.48) = Φ( - 1.48) = 0.0694.3.3 (14.6 P X 3.9) = PX ( 3.9) P( X 14.6) X 19.5 3.9 19.5 X 19.5 14.6 19.5 = P( ) P( ).3.3.3.3 = Pu ( 1.91) Pu (.13) =1 Φ ( 1.91) Φ (.13) = 0.9719-0.0166= 0.9553 x x 1 x 3 x1 19.5 P ( X > x ) = 0.05 Pu ( ) = 0.95 1.3 x 19.5 P ( u 1.645) = 0. 95 1 = 1.645 x = 3.3 1.3 PX ( > x ) = 0.10 X 19.5 x 19.5 X 19.5 x 19.5 x 19.5 x 19.5 P P Pu.3.3.3.3.3.3 ( > ) = ( < ) = ( < ) =Φ ( ) = 0.10 x 19.5 = 1.8 x.3 =.4 x 3 = 4.0 µ x n fx fx fx 79.04 X = = = 1.15 µ n 00 3
µ fx ( fx) n 79.04 (30) /00 S = = = 0.703 n 1 00 1 log 1 1 ( X + 1.64SX ) = log (1.15+ 1.64 0.703) = 39 µ µ 3-300 X = 15.08 ( µ g / l) S ) ( µ g / l 90% X + 1. 8S ( µ g / l) 95%X + 1. 64S ) ( µ g / l 99%X +. 33S ) ( µ g / l 4 90% P90 = 8+ (300 90% 67) =9.33 ( µ g / l) 9 4 90% P95 = 36+ (300 95% 85) =36.00 ( µ g / l) 9 4 99% P99 = 5+ (300 99% 97) =5.00 ( µ g / l) 9 X ± 1. 96S ± X ±. 58S ± ( µ g/ g) 4
4 1 sampling error standard errorse S S P S b X 4-1 4-1 X σ X S X σ S σ σ = X n σ ( X = µ ) n S S X = n S= ( X X ) n 1 5
1 confidence intervalci CI 1-α CI CI 1-α α = 0.05 95% 95%CI α = 0.01 99% 99%CI 1-α () t t t t ν t (), 4-4- X S x X tα /, ν Sx X + tα /, ν Sx X uα / S x X + uα / S x X X 1 X tα /, ν S x 1 x X 1 X + tα /, ν S x 1 x () 1 t H µ = µ 0 0 H 1 µ µ 0 X µ t = 0 ν = n 1 4-1 S X H µ = 0 H 0 d 1 µ 0 d t d µ d = = n 1 Sd ν 4- H0 µ 1 = µ H 1 µ 1 µ 6
X X t = ν = n + n 4-3 1 S X 1 X 1 S ( ) ( ) c S = 1 1 n1 1 S1 + n 1 S X 1 X + S c = 4-4 n1 n n1 + n S X 1 X n>50 σ 0 X µ u = 0 n 4-5 S / n X µ 0 u = σ 0 σ / n 0 1 x 4-6 u = X 1 X 4-7 S x + S H 0 H0 H 1 α H 1 H1 H 0 β 4 β power of test α 1 A. B. C. D B [ ] S x = S / n n S x 7
X ±.58S x A99% B 99% C99% D 99% D 1 α CI 1 α BD A D 1 P=0.05 H P>0.05 H P=0.05 H 0 α P>0.05 α H 1 I II I II α 0 X S x 0 s 5.90 + 6.17 ν = 14 8
1 0 g/l 4-4 4-4 g/l 4 36 5 14 6 34 3 0 15 19 14 18 0 15 4 1 5 7 3 H : µ = µ 0 1 1 1 S x 1 x H : µ µ α = 0. 05 1 X X t = ν = n + n 1.7 t= = 1. 019 ν = 18.6485 P>0.05 0 mm/h 4-51 4-5 1 3 4 5 6 7 8 9 10 30 33 6 31 30 7 8 8 5 9 6 9 3 30 30 4 5 3 3 11 1 13 14 15 16 17 18 19 0 9 30 9 33 8 6 30 31 30 30 6 3 5 3 3 5 8 7 4 1 t t= d / S = 3./0.611=5.37 d t= d / S = 5.0/0.948=5.303 d v=9p<0.001α =0.05 H 0 H 1 9
d t= d 1 S d 1 d =-1.60v=180.>P>0.1α =0.05 H 0 3 90 10 4 /mm 3 418 9 1 95% 95% 1, X +1.96S=418+1.96 9=474.8410 4 /mm 3 X -1.96S=418-1.96 9=361.1610 4 /mm 3 361.16474.8410 4 /mm 3 n=90>50 X +1.96S =418+1.96 9/ 90 =43.9910 4 /mm 3 X X -1.96S X =418-1.96 9/ 90 =41.0110 4 /mm 3 41.0143.9910 4 /mm 3 6 AS BSE CS X DSD 7 A B C D 8 A B C t D 9 A B C D 10 A B t t C P 0.05 H 0 Du 6. Aα =0.05 Bα =0.01 Cα =0.10 Dα =0.0 7. 30
A B CAB DAB 8 At Bt Cu DF 9 30 X 1 S 1 X S AX 1 = X S 1 = S B t C F D 95% 10 A Bσ X Cσ X D 1 3 σ X 4 5 6 P 7I II 8 9 α 1. 10 mmhg 4-6 4-6 10 mmhg 1 3 4 5 6 7 8 9 10 31
117 17 141 107 110 114 115 138 17 1 13 108 10 107 100 98 10 15 104 107 0 4-7 4-7 0 1 170 150 155 145 3 140 105 4 115 100 5 35 6 15 115 7 130 10 8 145 105 9 105 15 10 145 135 11 155 150 1 110 15 13 140 150 14 145 140 15 10 90 16 130 10 17 105 100 18 95 100 19 100 90 0 105 15 11 mg% 4-8 4-8 mg% 170 155 140 115 35 15 130 145 105 145 150 15 150 140 90 10 100 100 90 15 mg% 4-9 3
4-9 mg% 11 106.49 9.09 106 95.93 6.63 116 103.91 7.96 10 97.93 8.71 1B 1 statistical inference sampling error σ X standard error of meansem confidence intervalci 1- α 1-α P H 0 I II I type I error H 0 I α II type II error H 0 II β 1- β power of test α H 0 H 1 level of a test α 1P P σ X 33
3 1P H 0 P 0.05 H 0 α H 0 α α α P H 0 1 t t=.484v=9p<0.05 α =0.05 H 0 t t=.157v=19p<0.05 α =0.05 H 0 α 95% 99% 95%99% 3.1018.0 0 95%99% 0 34
1 1 3 LSD-t Dunnett-t SNK-q () 1 analysis of varianceanova sum of squares of deviations from meanss SS F 1 variation among groups, MS MS / SS SS = k i= 1 n ( x x) i i = ν ν =k-1 k (variation within groups) MS ) k ni MS = SS / ν SS = i= 1 j= 1 ( ( xij xi ), ν = N k 3 (total variation) 35
(MS ) MS = SS / ν SS = k n i ( i= 1 j= 1 N x ij x), k n i i ν =N-1 1 (homoscedasticity) 1 1 (completely random design) () 5-1F F α H 0 : µ 1 = µ = LL = µ k SS ν MS SS ν = k 1 ν SS ν ν ν ν ν MS MS 5-F Fα H 0: µ = µ = LL = µ 1 k SS ν MS ν SS ν MS MS 36
ν ν ν ν 37 ν ν SS SS ν ν MS MS t X X A B = d AB S d AB 1 1 S MS ( + ) = n A nb t x x i 0 = x i x 0 S xi x0 1 1 S MS ( + ) n i n 0 ( XA XB) MS 1 1 q = S = ( n + ) d A nb S d 1.5 1 X = lg X X = lg( X + 1) Possion X = X X = X + 0. 5 3
X =1/ X 4 <30% >70% X = sin 1 X 5 6 4 a 0 b 0 a 1 b 0 a 0 b 1 a 1 b 1 A B A B AB 7 1 3 4 A B A AB 1 ASS <SS BMS <MS CMS =MS +MS D DSS =SS +SS [ ] SS =SS +SS C AB D H 1 H 1 38
t = F t = F 1 3 4 5 6 7 1 A Bt C D ν Aν -ν Bν -ν Cν -ν +ν Dν -ν -ν 3 5% t s X µ t s A1.96 1.96 σ C 0.05, ν D. 0.05, ν x x 4 ν 1,ν α A B C D 39
5 A B C D 6 A B C D 7 A B C D 8 A B C D 9 SS A B3 C4 D33 10 t A B C DAB α α 1 5-3 3 5-3 mg/l.6 19.1 18.9 19.0.8.8 13.6 16.9 1.0 4.5 17. 17.6 16.9 18.0 15.1 14.8 0.0 15. 16.6 13.1 1.9 18.4 14. 16.9 1.5 0.1 16.7 16. 1. 1. 19.6 14.8 X ij 167.9 159.3 131.9 19.3 588.40 40
n i 8 8 8 8 3 X i 0.99 19.91 16.49 16.16 18.39 X ij 3548.51 331.95 06.7 114.11 11100.84 s i 3.53 8.56 4.51 3.47 5-4 5-4 5 6 8 7 6 9 8 7 10 9 8 10 9 8 10 10 9 11 10 9 1 11 10 1 11 10 14 1 11 16 X ij 9 84 11 88 n i 10 10 10 30 X i 9. 8.4 11. 9.6 X ij 886 73 1306 94 s i 4.4.93 5.73 4Dunnett-t 5-5 % 41
8 78.86 0.43 0.5 5 79.65 0.68 3 5 79.77 0.66 6 8 80.94 0.75 9 79.61 0.66 n X i SS 16 SS = 8SS = 110 S = 5-6 10 4 /mm 3 1 3.8 6.3 8.0 4.6 6.3 11.9 3 7.6 10. 14.1 4 8.6 9. 14.7 5 6.4 8.1 13.0 6 6. 6. 9 13.4 7 0.3ml/kg 3 45 50% 5-7 3.80 3.88 1.85 1.94 3.90 3.84.01.5 4.06 3.96.10.03 3.85 3.9 1.9.10 3.84 3.80.04.08 1 MS 4
1C 1t t α P t α I H 0 α I H 1 α P H 1 P α H 0 H 0 C ij SS SS SS = ( X ) / n = 588.4 /3 = 10819.05 = X ij C = 11100.84 10819.05 = 81.635 = [( X ) / n ] C = (167.9 = 141.170 SS SS = ij i + 159.3 + 131.9 + 19.3 ) / 8 10819.05 = 81.635 141170 = 140.465 SS ν MS F 43
81.635 141.170 31 3 47.057 140.465 8 F F =. 95 FF 0.05,3,8 0.05,3, 8 P<0.05=0.05 H 0 H 1 SNK-q H 0 A = B H 1 A B =0.05 5.017 9.380 5-9 X i 0.99 19. 91 16.49 16.16 1 3 4 5-10 q q P 1, 4 4. 83 4 6. 099 <0.01 1, 3 4. 50 3 5. 68 <0.01 1, 1. 08 1.364 >0.05, 4 3. 30 3 4. 735 <0.01, 3 3. 4 4. 319 <0.01 3, 4 0. 33 0. 417 >0.05 P>0.05=0.05 H 0 4 P<0.01=0.05 H 0 H 1 H 0 H 1 =0.05 C ij SS = ( X ) / n = 88 / 30 = 764.8 = X ij C = 94 764.8 = 159. = [( ij) / i] = (9 + 84 + 11)/10 764.8 = 41.6 SS X n C 44
SS = SS SS = 159. 41.6 = 117.6 5-11 SS ν MS F 159. 41.6 117. 6 9 7 0.80 4.36 4.77 F F 3. 35 F 0.05,, 7 = 0.05,,7 F P <0.05=0.05 H 0 H 1 Dunnet -t H 0 H 1 =0.05 Dunnett-t t = X X 1 0 MS (1/ n + 1/ n ) 1 = (9.-11.)/ 4.36(1/10 + 1/10) = /0.93 =.14 ν =7Dunnett-t P<0.05=0.05 H 0 H 1 t = ( 8.4 11.) / 4.36(1/10+ 1/10) =.99 ν =7Dunnett-t P<0.05=0.05 H 0 H 1 3 H 0 H 1 =0.05 5-1 SS ν MS F 40 14 10 30 1 F F 3.88, P>0.05=0.05 H 0 = 0.05,,1 5.5 45
4 = ( n i 1) si SS = (8 1) 0.43 + (5 1) 0.68 = 1.3086 ν = N - k = 35-5 = 30 + (5 1) 0.66 MS = SS / ν = 1.3086/30= 0.4103 + (8 1) 0.75 + (9 1) 0.66 1 0.5 H 0 0.5 H 1 0.5 =0.05 79.65 78.86 t = 1 1 0.4103( + ) 8 5 =.16 ν = 30 =4 Dunnett -t.5t=.16<.5, P0.05 =0.05 H 0 0.5 3 H 0 3 H 1 3 =0.05 79.77 78.86 t = =.49 1 1 0.4103( + ) 8 5 t>.5 P<0.05=0.05 H 0 3 3 6 H 0 6 H 1 6 =0.05 80.94 78.86 t = = 6.49 1 1 0.4103( + ) 8 8 t>.5 P<0.05=0.05 H 0 6 4 H 0 H 1 =0.05 79.61 78.86 t = =.41 1 1 0.4103( + ) 8 9 t>.5 P<0.05=0.05 H 0 5 N=36 k=3 n=1 46
SS SS SS SS = 16 8 110 = 44 = v = N 1 = 36 1 = v = k v = n N v 35 1 = 3 1 = 1 = 1 1 = 11 = k n + 1 = 36 3 1 + 1 = 5-11 5-13 SS ν MS F P 8 4 0.8 >0.05 44 11 4 0.8 >0.05 110 5 16 35 6 1 H 0 H 1 =0.05 H 0 H 1 =0.05 3 5-14 SS ν MS F F F 0.05,,10 = 4.10, F 0.05,4,10 = 3.48, P<0.05=0.05 H 0 7A, B 5-15 5-15 SS ν MS F 47
* 19 F F 0.01,1,16 = 8.68, P<0.01=0.01 48
49 1 3 4 1 () 6-1 6-1 rate % proportion A 100% = 100% = B A =
ratio B A B =106.04100 () 50 =1.64 130.5% standardization method standardized rate adjusted rate (dynamic series) = n a n a0 = 1
51 1 4% A4% B4% C4% D4% B [ ] 100% =
11 10 114 6 95%5% A. B. C. D. 1 A. B. C. D. 13 A. B. C. D. 14 40/10 45/10 38/10 A. B. C. D. 15 a 0 a 1 a a n a + a +... + a A 0 1 n B n+ 1 a0 a1 n + 1 a a n n C n n D 1 a 16 1.5 a 0 A B 0 a n 5
C D 17 a 0 a 1 a a n a + a +... + a A 0 1 n B n+ 1 a0 a1 n + 1 a n C a n n D 1 n a 0 a 18 C B D 1 3 1.. 6-0 % 1/10 0~ 890 4.90 0~ 63 19.05 5.73 40~ 8161 17 4 60 3 167090 715 90 1.59 a n 3. ~1981 6-3 ~1981 1/10 53
1971 197 1973 1974 1975 1976 1977 1978 1979 1980 1981 0.5 6.31 1.87 3.07 1.08 1.38.9.31.47.76.94 45 35 77.77 300 15 71.67 710 450 68.38 83 4 50.60 755 485 64.4 383 57 67.10 1 1 relative number rate = 100% % 3 proportion = 100% 4ratio AB A B A = B 5 (standardization method) 6 (dynamic series) 7 8 9 54
55 10 1 1 100%1000 AB = / 100% 1 6-5 % =/ 1/10 =/ 100% = 100% = 1 0 1 = = n a n a
0~ 890 138 4.90 4.8 1.66 0~ 46638 63 1 19.05 5.73 1.35 40~ 8161 17 4 4.4 149.14 6.11 60~ 9371 34 3 9.36 341.48 36.50 167090 715 90 1.59 53.86 4.8 40~ 1/40~ 19.05%60~ 9.36%0~.90% 60~ 341.50/10 0 40 60 36.50 3 ~ % % 1/10 1971 0.5 100 100 197 6.31-14.1-14.1 30.75 30.75-69.5-69.5 1973 1.87-18.56-4.44 9.11 9.64-90.89-70.36 1974 3.07-17.45 1.0 14.96 164.17-85.04 64.17 1975 1.08-19.44-1.99 5.6 35.18-94.74-64.8 1976 1.38-19.14 0.30 6.73 17.78-93.7 7.78 1977.9-18.3 0.91 11.16 165.94-88.84 65.94 1978.31-18.1 0.0 11.6 100.87-88.74 0.87 1979.47-18.05 0.16 1.04 106.93-87.96 6.93 1980.76-17.76 0.9 13.45 111.74-86.55 11.74 1981.94-17.58 0.18 14.33 106.5-85.67 6.5 4 N i P i N i P i P i N i P i 77.77 71.67 68.38 50.60 56
N i 64.4 N ip i 67.10 N ip i N ' i Pi 771 p = 100 % = 100 % = 67.75% N 1138 N ' i Pi 648 p = 100% = 100 % = 56.94% N 1138 57
Poisson 1 Poisson Poisson 1 Poisson Poisson X 0,1,,n P( X = k) = ( π ) n k n k ) π (1 (7-1) k X n 0,1,,n k µ µ P( X = k) = e >0 (7-) k! X Poisson XP 58
1 1 X = X σ = X X σ = n π ( 1 π) X XP X = X σ = X X X = µ Poisson X 1 X X k n i pi=1,,,k X=X 1X X k n pn=n 1+n ++n k X 1X X k i=1,,,k Poisson X=X 1 X X k = 1 + ++ k Poisson Poisson n nπ ( 1 π) Poisson n nπ = λ Poisson Poisson P 1 n50 9599 n p 0 1 1- p-u p+u p ( 1 p) n 59
X Poisson S u p p 1 = S p p 1 60 X 1 + X X 1 + X 1 1 p = (1 )( ) n + n n + n n n p + 1 1 1 1 1 X50 9599 X50 Poisson 1- X u X X + u X α α X Poisson k µ µ P( X = k) = e k=0,1,, k! X X Poisson u u X u 0 = u 0 u 3 Poisson u u
u X 1 = X X 1 + X u u 1 1 X 1 / n1 X / n = X n X + n () 1 A p=x/n Bn, Bn X Bn, C Bn, D Bn, B [ ] An Cn D 61 Bn Dn n nπ ( 1 π) P
10000 ± 1.96 10000 10 ± 1.96 10 10 ± 1.96 10000 1000 10 ± 1.96 10000 Y = X 1000 S Y S = X 1000 = 10000 1000 14 14 1 13 P X 1 P = 1 (1 0.) + ( ) 0. (1 0.) = 1 0.044 + 0.154 0. [ ] [ ] 80 = = X 1 1 1. X 1X Bn 1p 1Bn p X 1X X= X 1 X AX 1=X B. n 1=n Cp 1 =p D. n 1 p 1 =n p 6
. Bnp An=50 B. p=0.5 Cnp=1 D. p=1 3. Poisson P µ N µ µ A µ µ B. =1 C µ =0 D. µ =0.5 4. 50% 3 A0.15 B. 0.375 C0.5 D. 0.5 5. 100 100 95 A5195 C95105 B80.4119.6 D74.15.8 1 Poisson Poisson 3 S p 1. 10% 10 3. 100 80 50 3. 100 4 1000 4. 10000 10 100 1. X 0,1,,n P( X π n k n k = k) = ( ) π (1 ) k X n Binomial Distribution 63
. 0,1,,n k µ µ P( X = k ) = e >0 k! X Poisson XP 3. Bernoulli A A Bernoulli Bernoulli Test 1 1.. n nπ ( 1 π) Poisson Poisson P 3. P n P S p 1. H 0 10% H 1 10% =0.05 10% 3 3 P(X3)=1-[P(X=0)+P(X=1)+P(X=)]=1-[0.9 10 +10*0.1*0.9 9 +45*0.1 *0.9 8 ]=0.070 P(X3)0.05=0.05 H 0. H 0 H 1 =0.05 S X + X 1 ( n 1 p 1 p = + n1 + n 1 1 ) n S p1 p = 80 + 50 1 1 ( + ) = 0.1095 100 + 100 100 100 64
u = p p 1 S p 1 p 0.8 0.5 u = =.631.58P0.01 0.114 =0.05 H 0 H 1 3. 4 µ =100*4=400 µ Poisson H 0 400H 1 400=0.05 u = X u u 0 0 1000 400 u = = 30.58P0.01 400 =0.05 0 1 4. 100 P( X = 0) P( X = 1) P( X = 10 ) 100 µ = 100 = 0. 1 10000 Poisson µ P ( X ) = e 0! 0 1 µ µ µ µ µ + e 1! + e! = 0.90484+ 0.09048+ 0.0045 = 0.99984 100 0.99984 65
χ 1. χ. χ 1 χ 3 χ 3. χ χ 1χ () χ χ Chi-square test 1 3 () χ 1χ χ H 0 H 0 π 1 = π χ χ H 0 H 0 H 1 π 1 π ( ) A = T χ A Actual Frequency,T T Theoretical Frequency χ χ () 1 σ p π ( 1 π) = π (8-1) n p( 1 p) S p = p (8-) n n p 1-p p 66
p uα / S p, p + uα / S p (8-3) () χ 8-1 8-1 χ H 0 H 1 H 0 H1 1 ( ad bc) n χ = ( a + b)( c + d)( a + c)( b + d ) n 40 1 T<5 ( ad bc n / ) n χ = ( a + b)( c + d)( a + c)( b + d ) ( b c 1) χ = b + c R C H0 ( H0 ) H1 ( H 0 ) (R-1)(C-1) A χ = n( 1) n R n C H 0 H1 ( A T ) T 1 n<40 χ χ 1 1<T<5n>40 T 1n 40 Fisher R C 1 <5 1/5 R C R C R C Kappa 1 χ A B C D 67
A [ ] χ χ 1 40 1 100 1 40 χ χ χ + 0 χ + χ 1 χ 0 χ χ χ χ u χ 1 χ χ u = χ 1 χ R C χ R C χ 1 1 8-8- 1 19 5 4 1 86 98 31 91 1 χ H 0 H 1 α = 0.05 ( b c 1) = b+ c χ =.11< χ 0.05, 1 5 1 1 ( ) χ = =. 1 5+ 1 ν = 1 P>0.05, 68
8-3 8-3 % 51 84 135 37.78 6 6 3 18.75 5 13 18 7.78 6 13 185 33.51 H 0 H 1 α = 0.05 A χ = n( 1) =185 51 84 6 6 5 13 + + + + + 1 =4 n r n c 6 135 13 135 6 3 13 3 6 18 13 18.498ν =-13-1= α = 0.05 H 0 % 100 41 100 94 100 89 100 7 χ 41 59 100 94 6 100 89 11 100 7 73 100 51 149 400 H 0 H 1 α = 0.05 A χ = n( 1) =400 n r n 41 59 73 + +... + c 51 100 149 100 149 100 1 =146.175 ν =4-1-1=3 α = 0.05 H0 H 1 69
1. p A B χ Cn p 1-p p D t. A u B t C u = χ D t = χ 3. p ( 1 p) A p( 1 p) B n p n 1 4. χ A ν = n B 3 4 ν = 11 C ν =4 Dχ > χ ν > η 0.05, ν 0.05, η 70 p ( 1 p) n n 5. 10 60% 50% 35% A B C D 6 A B χ C D 7 A u χ B u χ C u χ D χ u 1 χ 3 4 5 6 7 8Fisher
9McNemar 10Yates χ > χ 0.05(3) 111 8 11 39 1 8-6 8-6 0 340 11 6 357 I 73 13 6 9 II 97 18 18 133 III 3 1 6 513 44 31 588 38-7 10 8-7 + - + 4 18 60-30 30 60 7 48 10 4 8-8 8-8 % 9 5 86. 8 17 60.7 57 4 73.7 5 8-9 8-9 % 110 8 74.5 150 130 86.7 63 56 88.9 33 68 83.0 71
6 8-10 8-10 1 43 3 87 77 39 11 37 89 8 43 614 7 8-11 8-11 34 6 8 14 7 8 0 75 57 105 5 14 118 195 100 413 8. 8-1 8-1 A B O AB Eskdale 33 6 56 5 100 Annandale 54 14 5 5 15 87 0 108 10 5 1C χ χ α χ χ α χ 7 3 goodness of fit 4 A B n 8-1
8-1 A B + - + a b - c d 5. R C 6. σ P σ p π ( 1 π) = π n π P π n p( 1 p) S p = n 7. 8. Fisher 1 n<40 R.A.Fisher1934 H 0 9. McNemar McNemar s test for correlated proportions ( b c 1) χ = v=1 b + c 10Yates Yates F χ χ 1 P χ correction of continuity Yates Yates correction H 1 H 1 1 + - 11 71 8 1 38 39 1 109 11 = χ. 37 P>0.05,,χ H 0: H 1 : α = 0.05 χ =61.59χ < χ 0.01, 6 P<0.05, 73
α = 0. 05 H 0 H 1, 3 χ H 0 : H 1 : α = 0.05 χ =3.00α = 0. 05 H 0 4 u χ χ χ =4.774α = 0. 05 H 0 H 1, 5R C χ χ =8.539v=P<0.05α = 0. 05 H 0 H 1 6R C χ χ =443.456v=P<0.05α = 0. 05 H 0 H 1 7χ χ =4.00v=4P>0.05α = 0. 05 H 0 8 5 χ χ =5.710 v=3p>0.05 α = 0. 05 H 0 74
1 non-parametric statistics distribution-free statistics assumption free statistics ( ) 1 1 3 4 1 75
3 4 5 6 ( ) 7 (Wilcoxon ) H 0 M d H 1 M d α = 0.05 T + T-T + T- T + T- T n T P T P P n u u T n( n + 1)/ 4 0.5 u = n( n + 1)(n + 1) / 4 u = T n( n + 1) / 4 n( n + 1)(n + 1) 4 0.5 ( t 3 j 48 t j ) (Wilcoxon ) 1. (1)H 0 H 1 α = 0.05 () (3) n 1 T T (4) P T P T 76
P T P n 1 n -n 1 u u T n ( N + 1) / 0.5 1 u = n n ( N + 1) /1 1 u c = u C 3 3 C = 1 ( t t ) ( N N ) j j (Kruskal -Wallis ) 1H 0 H 1 α = 0.05 3 4 H 1 Ri H = ( ) 3( N + 1) N( N + 1) n H c i H c = H / C 3 3 C = 1 ( t t ) ( N N) j j 5 P H P T P T P (Nemenyi ) 1H 0 H 1 α = 0.05 D= R A -R B 3 P 1 D P 77
χ RA RB = C[ N ( N + 1)/1][1 n A + 1 n B ] 3 3 C = 1 ( t t ) ( N N ) j 1. j χ α,( k 1) (1) () R i (3) :R=b(k+1)/ b k (4) (R i -R); (5)M=(R ī R) (6)M M. Friedman (1) () R i χ k 1 = R bk( k + 1) j= 1 j 3b( k + 1) (3)χ α, ( k 1) P (1) R i () R R A B u = R A R B bk( k + 1)/ 6 u P c /c 78
A. B. C. D. D [ ] A.t B.u C. D.? C [ ] t H 0 A B C D B [ ] H H c C 79
[ ] Kruskal-wallis 3 3 Hc H c = H / C C = 1 ( t t ) ( N N) j C H CH P j j 1 3 4 At B C D A B C D A B C D A B C D X AH 0 BH 0 80
CH 0 DH 0 AH 0 BH 0 CH 0 D At B C D A B C D A B C D n n i i i A B C D t u At u Bu t Ct u Dt u A B C D A B C D P 81
A < < B < C < D P n An Bn Cn Dn n 1 n An 1 n Bn 1 n Cn 1 n Dn 1 n A B C D A B C D X 1 X AX X BXX CX X DX X t A B C D A B C D H 0 A B Cµ d? DM d n 1 n n n 1 An 1 n 1 Bn n Cn 1n n 1n Dn 1 n n 1 n A B C 8
D At B Ct D t n 1n A B C D u c A B C D A B C D u c H 0 A B C D? 83
1 mol/l 84
mol/l () () 85
? T - T + P n 1 T 1 n T P T - T + P t tp t n 1 T 1 n T P n 1 T 1 n T u c P 86
() 1. (linear regression) simple regression Y ˆ= a + bx ab 10-1 10-1 ab a Y intercept regression coefficient X Y X Y b >0 a>0 b>0 Y X <0 a<0 b<0 Y X =0 a=0 b=0 X Y X ( X X )( Y Y ) lxy b = = a = Y bx ( X X ) l XX 87
. b 1 t () coefficient of product -moment correlat ion r. ( X X )( Y Y) r = ( X X ) ( Y Y ) = 88 l l XX XY l r 1r1 1 YY 0 r r r 1 3. r 1r t 1. 1 Y X XY 3b X Y b r 4b= l xy / l xx r = l xy / l l xx yy 5b+ 1r1 6b r.
1 b r, r b t t b =t r l xy 3 r = = SS SS l l xx yy r 1 rank correlation 1 A r B t C F D D [ ] t r b r b t t b=t r Ab 1 =b Ct r 1=t r D Bt b1 =t b D r b b 1 =b r b t r1 = t b 1 t r =t b t b1 =t b t r 1 =t r r 3 r >r 0.05( n-) X Y A B. C D. D r r r >r 0.05( n- ) P<0.05 H 0 H 1 0 X Y 4 H 0 A=0 B. 0 89
C>0 A D. <0 r r =0 r Ar>0b<0 Cr<0b>0 B B. r>0b>0 D. r b b r, r>0b>0 19 ( ) AXX BY Y /n C(YY ) DXX YY 0Y=14+4X 1~7 kg ( ) A C 1 r=1 ( ) Ab=1 CS Y. X =0 B D Ba=1 DS Y. X = S Y ( ) A B C D 3 X Y ( ) A µ S = ( X X ) ( n ) $ xy, B. Sr = ( Y Y) ( n 1) ( ) C. $ S = ( Y Y) ( n ) yx, D. S µ b = Sxy X X 4 ( ) An Bn1 90
Cn Dn1 5 Y ( ) ASS =SS CSS =SS 6 SS ( ) Al YY l XY b Bl YY bl XX YY XY XX C. l l l D(1 r ) l YY 7 ( ) BSS =SS D Al XY lxx lyy Bb YX lxx lyy Cb YX b XY D 8 r=0 Y ˆ= a + bx ( ) Aa B. a X Ca D. a Y 1 SS 1 =SS r 1 r 3 r 1 3 n=100x Y r=0.1 X Y 4r r s 5 6 7 110 0 10- =0 95% cm 10-10 0 170 173 160 155 173 188 178 183 180 165 45 4 44 41 47 50 47 46 49 43 cm 10-3 91
900g 95% 900g Y 95% 10-3 1 3 4 5 6 7 8 g 800 780 70 867 690 787 934 750 g 185 158 130 180 134 167 186 133 3 a 10-4 a 1 3 4 5 6 7 8 1/10 5.60 18.50 16.3 11.40 13.80 8.13 18.00 1.10 ag/100m 3 0.05 1.17 1.05 0.10 0.75 0.50 0.65 1.0 4 10-5 1 1 3 4 5 6 7 8 9 10 11 1 10 130 160 310 40 540 740 1060 160 130 1440 000 10 9 /L ++ +++ + + ++ 1D 1 linear regression simple regression regression coefficient (slope) b b X Y b 3 residual sum of squaresss ( Y Yˆ ) X Y Y X 9
Y Yˆ 4 ( ) regr ession sum of squares SS ( Yˆ Y ) X Y Y X 5 linear correlation simple correlation 6 zerro correlation 9 coefficient of product -moment correlation r 30 coefficient of determination r SS l XY l XY l XX r = = = SS r l l l SS XX YY YY r 1 31 rectification 14. rank correlation 1 SS 1=SS SS SS r 1 r 3 r 1r1 r 1 0 1 Y X a X ( ) ( ˆ) ( ) s Y. X s Y. X = SS n = Y Y n X Y Y X : b=0 y s Y. X Y X Y b = s l s b Y. X X Yˆ s s XX 1 n+ ( X X) ( X X ˆ = ) Y Y. X Yˆ s Y. X X Y y y 93
s ˆ = sy. X 1 n + ( X X ) ( X X) Y Y 3n=100r=0.1 t =0.05 H 0 (=0) H 1 (0) r =0.1 =0.01 1% 4 r r s 5 t 6 X YX Y Y ˆ= a + bx P X Y X Y 1~7 X Y Y X Y X X Y 7 X Y XY X Y X Y 1 10-1 94
1 X = 175 X = 9855 X = Y = 454 Y = 0690Y = 45.4 lxx = X ( X ) n = 9855 175 l = Y ( Y ) n = 0690 454 YY lxy = XY ( X )( Y ) n = 78541 r = l l XY l XX YY = 6 = 0.87 96.5 78.4 =0 17.5 XY = 78541 10 = 96.5 10 = 78.4 175 454 10 = 6 H 0=0 H 1 0 r 0 t = = s r r ( 1 r ) ( n ) 0.87 10 = = 4.09 1 0.87 =0.05 ν = n = 10 = 8 t 0.00<P<0.005=0.05 H 0 H 1 0 95% r z 1 1+ r 1 1+ 0.87 z = ln = ln = 1.1651 1 r 1 0.87 z=tanh 1 0.87=1.1651 z 95% 95
( z u0.05 n 3 z + u0. 05 n 3) = ( 1.1651 1.96 10 3 1.1651 + 1.96 10 3) = ( 0.443 1.9059 ) r=tanhz z 0 95% 0.40050.9567 10- X=638X =5048814X = 791 Y=173Y =06619Y = 159. 15 XY=101863 l l l XX YY XY = = = l b = l XY XX Y X ( ( XY ( X ) Y ) X ) ( 1130 = = 0.61 43366 n = 5048814 638 n = 06619 173 Y) n = 101863 638 173 8 = 1130 a = Y bx = 159.15 0.61 791 = 47.36 8 = 43366 8 = 405.875 Y=17.94+0.4X Y=47.33+0.6X Y=78.9+0.10X 10-1 H 0 0 H 1 0 0.05 SS = l SS SS = l YY XY = SS = 405.875 l XX SS = 1130 10-6 43366 = 954.905 = 405.875 954.905 = 1097.97 96
10-6 SS MS F 405.875 7 954.905 1 954.905 16.147 1097.970 6 18.995 F=16.147F P<0.010.05 H 0 H 1 t H 0 0 H 1 0 0.05 SS SS SS s Y. X = l = l YY XY = SS = = 405.875 l XX b 0 t = = s s b SS SS = 1130 43366 = 954.905 = 405.875 954.905 = 1097.97 n = 1097.97 8 = 13.576 Y. X b l XX 0.61 = 13.576 43366 = 4.018 =6t 0.01>P>0.050.05 H 0 H 1 F = 16.147 = 4. 018 = t Y ˆ = a + bx = 47.36 + 0. 61X YYˆ 10-7 10-7 X Y Yˆ YYˆ 1 800 185 161.474 3.56 780 158 156.54 1.746 3 70 130 140.594 10.594 4 867 180 178.961 1.039 5 690 134 13.764 1.36 6 787 167 158.081 8.919 7 934 186 196.448 10.448 8 750 133 148.44 15.44 X=800Y=185 95% 95% 97
95% bt 0.05(n) S b bt 0.05(n ) S b =0.61.44713.510743366 0.61.44713.510743366 =0.100.4198 X 1=690 Yˆ=47.36+0.61XY 1=13.76X =934Y =196.45 69013.76934196.45 10-95% a = Y bx a 1 =78.85a = 17.937 95% 10- Yˆ=78.85+0.10X Yˆ=17.937+0.4198X 900g 95% s Y = s 1 n + ( X X ) ( X X ) Y. X = 13.576 1 8 + (900 791) 43366 = 8.5446 X=900 µ Ŷ 95% Yˆt 0.05(6) sy ˆYˆt 0.05(6) sy ˆ =187.574.4478.5446187.574.4478.5446=166.6708.48 900g 187.574g95% 166.67~08.48g 900g Y 95% s Y = s 1+ 1 n + ( X X ) ( X X ) Y. X = 13.576 1+ 1 8 + (900 791) 43366 = 16.000 X=900 Yˆ=47.36+0.61X=187.574 Y 95% Yˆt 0.05(6) S Y Yˆt 0.05(6) S Y =187.574.44716.000187.574.44716.000=148.46.73 900g 95% 148.4~6.73g 3 10-8 X 10-8 a 1/10 Y a d = 1 5.60 1 0.05 1 0 0 18.50 8 1.17 7 1 1 3 16.3 6 1.05 6 0 0 4 11.40 3 0.10 1 1 d 98
5 13.80 5 0.75 5 0 0 6 8.13 0.50 3 1 1 7 18.00 7 0.65 4 3 9 8 1.10 4 1.0 8 4 16 d =8 H 0 s 0 a H 1 s0 a 0.05 r s =16d [n ( n 1)]=168[8(8 1)]=0.6667 r s 0.10>P>0.050.05 H 0 a 4 10-9 X 10-9 10 9 /L Y d = 1 10 1 ++ 10.5 9.5 90.5 d 130 +++ 1.5 10.0 100.00 3 160 3 7.0 4.0 16.00 4 310 4 3.5 0.5 0.5 5 40 5 + 8.5 3.5 1.5 6 540 6 + 8.5.5 6.5 7 740 7 3.5 3.5 1.5 8 1060 8 3.5 4.5 0.5 9 160 10 3.5 6.5 4.5 10 130 9 3.5 5.5 30.5 11 1440 11 ++ 10.5 0.5 0.5 1 000 1 3.5 8.5 7.5 d =40.5 H 0 s 0 H 1 s 0 0.05 Y r s r s T X =0 T Y=t 3 t 1=[(6 3 6)+( 3 )+( 3 )]1=18.5 99
3 [( n n) 6] ( T + T ) X Y 3 3 [( n n) 6] T [( n n) 6] X 3 [( 1 1 ) 6] ( 0 + 18.5) 3 3 [( 1 1 ) 6] 0 [( 1 1 ) 6] d r = s T = = 0.5095 40.5 Y 18.5 r s 0.10>P>0.050.05 H 0 100
P 6logistic SPSS SAS () Y Yˆ = b + bx + bx + + bx 0 1 1 Yˆ X 1 X X k k b 0 b 1 b, bk partial regression coefficient b j X X j j Y () Yˆ X1 X X k Y b b 1, b k Yˆ Y e i = ( Y Y)ˆ b b 1, bk b1 b, bk normal equation k k 101
b l 1 b1l b1l 11 1 k1 + b l 1 + b l + b l k + Λ + b l + Λ k 1k + b l k k + Λ + b l l l ( X X )( X X ) XX l k kk = l 1 y = l = l y ky ( X )( X ) i = = = ij ji i i j j i j iy = ( X i X i )( Y Y ) = ( X Y b0 b = Y b X b X Λ 0 1 1 i X i )( n b X k k n Y ) j X1 X X k Y H 0 β 1 = β = β 3 = L = β k = 0, H1 β j 0 0 F MS lk F = = MS l( n k 1) n k l = b l + b l + Λ + b l 1 1y l = l l y ( Y Y ) l = l () logistic = yy X, X, Λ, X 1 k Y Y Y =1Y k ky Y =0P Q P + Q=1 Logistic Q P e 1+ e β0+ β1x1+ βx + Λ + βk X k = β + β X + β X + Λ + β X 0 = β + β X + β X + Λ + β 1+ e 0 1 1 1 1 1 β 0 β j ( j= 1 L k) X j P x β x β ( ) P Q 01 n i X, X, Λ, X Y i1 i ik i k k X k k 10
Y =1 Y i i =0 P Q i i P i + Q i =1P i Q i l P i β + β + β + L + β e = 1 + L e 1 = + L X X X 0 1 i1 i k ik β + β X + β X + + β X 0 1 i1 i k ik Q i X X X 0 1 i1 i k ik 1 e β + β + β + + β i odds P Q l l i i P Q OR odds ratio Pi Q i ln = 1( X i1 X l1 ) + ( X i X l ) + + k ( X ik X lk ) Pl Q β β Λ β l ( ) j = 1Λ k X X ( ) ij X lj i X ij X lj β j X j X j logistic logistic maximum likelihood estimate β b ( j= 1 L k) j j Y X, X, 1 Λ, X k 1i Y i = 0 i i l = P Q Yi 1 Yi i i i n L n n Y = = i 1 Yi L li Pi Qi i= 1 i= 1 i 1 n L Newton Raphson e β j 103
( j = 1Λ k ) b j logistic logistic likelihood ratiotest (score test)wald (Wald test) Λ = ln ' ' ( L L) = (ln L ln L) ln L m ( m < k) L ' m 1 X H χ Λ χ α(1) α X j j X j 0 1 Y D E F D D [ ] X Y j SS SS R R = SS SS 104 R p R Logistic logistic logistic SPSS SAS
11-1 0 X 1 /kg X /cm X 3 /cm Y /L 1 50.8 73. 36.3.96 49.0 84.1 34.5 3.13 3 4.8 78.3 31.0 1.91 4 55.0 77.1 31.0.63 5 45.3 81.7 30.0.86 6 45.3 74.8 3.0 1.91 7 51.4 73.7 36.5.98 8 53.8 79.4 37.0 3.8 9 49.0 7.6 30.1.5 10 53.9 79.5 37.1 3.7 11 48.8 83.8 33.9 3.10 1 5.6 88.4 38.0 3.8 13 4.7 78. 30.9 1.9 14 5.5 88.3 38.1 3.7 15 55.1 77. 31.1.64 16 45. 81.6 30..85 17 51.4 78.3 36.5 3.16 18 48.7 7.5 30.0.51 19 51.3 78. 36.4 3.15 0 45.8 75.0 3.5 1.94 SPSS EXAP11sav 4 0 Statistic Regression Linear DependentY Independent(s)X 1 X X 3 MethodEnter Variables Entered/Removed Model Variables Entered Variables 1 X3, X, Removed Method. Enter 105
X1 a All requested variables entered. b Dependent Variable: Y Model Summary Model R R Square Adjusted RSquare Std. Error of the Estimate 1.846.715.66.893 a Predictors: (Constant), X 3, X, X 1 ANOVA Model Sum of df Mean F Sig. Squares Square 1 Regression 3.367 3 1.1 13.413.000 Residual 1.339 16 8.368E- 0 Total 4.706 19 a Predic tors: (Constant), X, X, X 3 1 b Dependent Variable: Y Coefficients Unstandardized Standardized Model Coefficients Coefficients t Sig. B Std. Error Beta 1 (Constant) -4.676 1.31-3.541.003 X 3 X X 1 6.036E-0 3.508E-0 5.010E-0.01.015.09.474.333.307.899.7 1.735.010.037.10 a Dependent Variable: Y SAS DATA EXAP11INPUT x1 x x3 y@ @ PROC REG CARDS MODEL y=x1 x x3 50.8 73. 36.3.9645.8 75.03.5 1.94 RUN 106
Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 3.3673 1.144 13.41 0.0001 Error 16 1.33893 0.08368 Corrected Total 19 4.7066 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept 1-4.67553 1.3051-3.54 0.007 X1 1 0.06036 0.008.90 0.0105 X 1 0.03508 0.01544.7 0.037 X3 1 0.05010 0.0888 1.73 0.100 [ ] SPSS SAS 1 H 0 β 1 = β = β 3 = 0 F=13.413P=0.0001 H 0 b 1b b 3 Yµ = 4.68+ 0.06X + 0.04X + 0.05X R =0.715R a =0.66 1 3 3 11-11- SE t P b 0-4.675 1.31-3.54 0.00 b 1 0.060 0.01.90 0.01 b 0.035 0.015.7 0.04 b 3 0.050 0.09 1.73 0.10 3 A χ B F C U 107 D Ridit 33 A B C D
34 k B C 1/k D E 35 F A B C D 1..3. 4. 5. 6. logistic β i Y kg X 1 cm X 1 X1 =1611 X1 = 19631 X =106 X = 976 Y = 341 Y = 9883 X 1 X = 14454 X 1Y = 46439 X Y = 3079 1 Y ˆ= b0 + b1 X1 + b X b =.114 b = 0. 135 b = 0. 93 0 1 3 11-3 1 v SS MS F 1B A 3B 4A 1 Y X multiple linear regression multiple regression Yˆ = b 0 + bx 1 1 + bx + + bx b k k 1 b, bk partial regression coefficient b j X j X j Y 3 R coefficient of multiple correlation, R 108
4 coefficient of determination R SS SS R = SS / SS R y x 5logistic β0+ β1x1 + βx + Λ + βk X k P = β0+ β1x1 + βx + Λ + βk X 1+ e Q e = β + β X + β X + Λ + β 1+ e 0 1 1 1 i oddsp Q i i k k X k odds 6logistic OR odds ratio β j X j j X j 1 Y ˆ= b0 + b1 X1 + b X b1 l11 + bl1 = l1y b l + b l = l 1 1 y b =.114 0 b = 0. 1 135 b = 0. 93 Y ˆ=.114 + 0.135 X1 + 0. 93 X 3 11-4 1 v SS MS F 151.35 75.675 16.380 9 41.57 4.6 11 19.9 e β 109
110 1 (statistical graph) 1 1
111
1. 1-11
113
114 [ ] 1-
115
116 1-4 1949-1957 15 1-5
117
118 (1)
119
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(%) 11
1. 1
13 1-π () (experimental study) experiment clinical trial
14 error bias 1 random error systematic error selection bias measurement bias contamination intervention compliance noncompliance lost of follow-up disagreement blind method
confounding bias randomized block design π ( ) σ u α + u β N = 13-1 δ N u β δ N = uα 13- σ u β 1- β 15 β
N ( u u ) σ + α β d N = 13-3 δ u β N = δ uα 13-4 σ d 3 ( u α + u β ) σ N = 13-5 δ N uβ = δ uα 13-6 σ 4 N = ( u + u ) α β ( π π ) 1 4π 1 π c c ) 13-7 u β = N π π π c 1 ( 1 π ) c u α 13-8 π 1 π π c N 5 χ uα π c + uβ π + π+ N = π + π + b c = π = + a + b a + c N π + π + uα π u β = π π π π + + + c c π c π c π = π + + + 13-9 13-10 6 u = α σ N 13-11 δ 7 π( 1 π ) = u α N 13-1 δ 5. Kappa PA Pe Kappa = 13-13 1 P e Kappa Kappa 0.75 0.4 16
experimental effect [ ] 1. [ ]. [ ] 1. 10% 5% 0.05, α = 90% α u 0.05 = 1.64β u0.10 1 π π = 0. 05 δ π c 1 ( u + u ) 4 π (1 π ) α β c c (1.64+ 1.8) 4 0.075 0.95 N = = = 946 ( π π ) 0.05 946 473. 300g 467g 0g α =0.05 1 1- β =0.08 1- β 0.90 3 10 α =0.05 17
( ) µ α + µ β σ 1 ( ) 1.64+ 0.84 467 N = = =111 δ 0 11 ( ) µ α + µ β σ ( ) 1.64+ 1.8 467 N = = =154 δ 0 154 δ N 0 40 3 u β = uα = 1.64 =. 00 σ 467 µ β =.00 β =0.0power=1-0.0=0.98 3. 5 40~64 5.5% 5.0% 1α =0.05 80% 0.90 3 0.80 4 5000 α =0.05 (1) α =0.05,u α β uβ π1 π π c N = ( u + u ) 4π ( 1 π ) α β ( π π ) 1 c c = ( 1.96 + 0.84) 4 0.05 ( 1 0.05) ( 0.05 0.0) = 7588 α =0.05 80% 13794 α =0.05,u α β uβ N = ( u + u ) 4π ( 1 π ) α β ( π π ) 1 c c = ( 1.96 + 1.8) 4 0.05 ( 1 0.05) ( 0.05 0.0) 0.90 18471 3α =0.05, u α β u β N = ( u + u ) 4π ( 1 π ) α β ( π π ) 1 c c = ( 1.64 + 0.84) 4 0.05 ( 1 0.05) ( 0.05 0.0) 0.80 1081 4α =0.05, u α = 3694 N π 1 π 10000 0.05 0.0 u β = uα = 1.96 =0.1686 π c ( 1 π c ) 0.05( 1 0.05) β =0.435 1-0.435=0.5675 = 1644 4. 37.5 10% 10% 0% α β α u 0.05 = 1.96π 0 δ 18
uα 1.96 N = π0(1 π0)( ) = 0.1 (1 0.1) = 864 δ 0.0 864 α =0.05,β α u 0.05 = 1.96π 0 δ uα 1.96 N = π0( 1 π0) = 0.1478(1 0.1478) = 4838 δ 0.01 4838 1.. 3. 4. 5. 6. 7. 8. 9. 10. 1.. A. B. C. D. 50 A. B. C. D. A. B. C. D. A. B. C. D. A. B. C. A. C. D. B. D. 19
A. 130 B. C. D. A. B. C. D.. E A 8 E A A 1 A. B. C. D. A. t B. C. t D. 3 A. H0 A H A 1 B. H0 A H1 A C. H A 0 H1 A D. H 0 A H1 A 4 A. P0.05 A B. P0.05 A C.P0.05 A D.P0.05 A
1. A 1.8ml/minB.4ml/min 1.0ml/min =0.05=0.10. 37 0.1 0. III α β 0.05 3. kpa 10 5kPa α 5. 30 0 3 10% 6. 10 ABC 56 191547089100 116ABC placebo control randomization confounding factor systematic error bias (experimental study) 131
lost of follow-up randomized control trial. 1-1. D. D 3. C 4. A 5. D 6. D 7. D 8. BC, A 1B A 3D 4D =.4-1.8=0.6ml/min =1 =0.05 =0.1 u = u 0.01 = 1.8 t ( ) σ ( ) 0.05 1.96 u α + u β 1.96 + 1.8 1 N = = =10 δ 0.6 60 α u 0.05 = 1.64 β u 0.05 = 1.64 δ 10 0 Cσ 0 C t ( u + u ) α β (1.64 1.64) 0. N σ + = = = 43 δ 0.1 43 α u 0.05 = 1.64 β u 0.01 =.33 δ kpaσ 5kPa σ t ( u + u ) α β (1.64.33) 5 N σ + = = = 98 δ α u 0.05 = 1.96 β u 0.05 = 1.64δ 6σ 6 t ( uα + uβ) σ (1.96+ 1.64) 6 N = = = 51 δ 6 5 6 13
π = c 3 α u 0.05 = 1.96 β u 0.10 = 1.8π 1 = π = 3 30 43 60 43 43 ( u (1.96 1.8) 4 (1 ) α + uβ)4 πc(1 π ) + c N = = 60 60 = 85 ( π 3 1 π) ( ) 3 30 46 13-1 13-1 10 133
13-1 A B C 140 ABC 13-13- ABC χ χ A 14 16 30 B 6 48 C 0 4 56 64 10 0.05, 134
135 () 1 (1) () (3) (4) (5) () (survey) ( ) () 1
136
t / σ n = ( α ) δ δ σ n ν t 0.05 / u t 0.05 / 0.05 / n u α / p(1 p) n = δ δ p p p p p p p () 1 (probability sampling) Sampling frame) 1 simple random sampling) 137
n S = ( 1 ) 14-3 N n S X n p(1 p) S p = (1 ) N n 1 14-4 n / N (sampling fraction) ( 1 n / N) (finite population correction) (systematic sampling) 3 (stratified sampling) strata) (proportional allocation) ni Ni n = n = N 14-5 i i n N N (optimum allocation) Niσ i ni = n 14-6 N σ i i Ni πi (1 πi ) ni = n 14-7 N π (1 π ) i i i W = N N X i i / = i X W i X 14-8 n = i S X (1 ) Wi S N i p = W i pi n = i S p 1 ) Wi N X i 14-9 14-10 ( S pi i 14-11 4 (cluster sampling) K k 138
X K X m X i = i Nk 14-1 k K k 1 S X = ( 1 )( ) ( Ti T ) 14-13 N K k( k 1) i= 1 T i i T T i T = Ti / k K p = a 14-14 i Nk k K k 1 S p = ( 1 )( ) ( ai a ) 14-15 N K k ( k 1) i a a i= 1 (non-probability sampling) 1 (quota sampling) 30% 30% (snowballing) (judgement) () (case control study) (cases) (controls) (cohort study) (exposure) (outcome) 139
14-1 (odds ratioor) (cumulative incidenceci) OR RR OR RR (relative riskrr) (attributable risk proportionar%) (incidence densityid) (attributable riskar) (attributable risk proportionar%) ORRR () ( ) ( ) (standardization) 1 (direct standardization) ( ) (indirect standardization) ( standardized incidence ratiosir standardized mortality ratiosmr)sir(smr) 140
3 14-1 n 1 r 1 p 1 N 1 R 1 P 1 n r p N R P 3 n 3 r 3 p 3 N 3 R 3 P 3 i n i r i p i N i R i P i k n k r k p k N k R k P k n r p N R P N p i i p = 14-16 N r p = P n P 14-17 i i r SMR = n P 14-18 i i (survey) ( ) [ ] 1 A B C D 141
A [ ] A B C D D [ ] 3 A C [ ] 4 A B C D A [ ] 5 14
[ ] n k K k [ ] (kpa) 0.5 95%.(kPa) σ =.δ =0.5 u 05 =1.96 t σ n = ( α / ) n ν t 0.05 / δ u0.05/ t 0.05/ n u σ n = ( α / ) n δ n ν ν t t 0.05/,73 143
t / σ n = ( α ) δ n ν ν t t n = ( t α / σ ) δ 0.05/,75 t / σ n = ( α ) δ u σ n = ( α / ) δ 1 A C B D n A δ σ β B α σ µ C δ σ α D δ σ µ 3 (5000 ) 0.5mg/L 0.1mg/L A97 B95 C96 D94 4 A B C D 5S = S / n x ACD 6 144
7 8 9 A B C D 10 A B 1 13 C 1 D 11 1000 600 A B C 400 D 600 1 1 37.5 10% 0%α 0.05 0.5mg/L 0.1mg/L 314-3 14-3 ( ) ( ) 0~ 4 1900 1406 74.0 6 1 80.8 5~14 3100 186 6.0 30 6.7 15~44 9400 1786 19.0 17 7 1.3 45~64 4900 7350 150.0 5 4 168.0 65~ 000 17400 870.0 5 48 960.0 145
1300 818 13.1 13 140 65.7 4 50 50 14-4 OR 14-4 110( a ) 5(b ) 135( m 1 ) 140( c ) 5( d ) 365( m 0 ) 50( n 1) 50( n ) 0 500(n ) 1 (sampling survey) simple random sampling) 3 (systematic sampling) 4 (stratified sampling) strata) 5 (cluster sampling) K k 6 (probability sampling) 7 (non-probability sampling) 146
134B 56A 7B 1011A 8C 9D 1 1 p =0.1δ =0.p =0.0.1=0.0 u 05 =1.96 α / u p(1 p) n = =1.96 0.10.9/0.0.1 865 δ σ =0.5δ =0.1 u 0. 05 =1.96 σ t α n = ( / ) n ν t 0.05/ δ u t 0.05 / 0.05 / n u σ n = ( α / ) n δ n ν ν t t n = ( t α / σ ) δ 0.05 /, 95 147
3 14-5 0~4 1900000 80.8 1535 5~14 3100000 6.7 08 15~44 9400000 1.3 00 45~64 4900000 168.0 83 65~ 000000 960.0 1900 1300000 31177 N i pi p 31177 = = 100000 = 146.4 / 10 N 1300000 14-6 0~4 74.0 6000 19. 5~14 6.0 30000 1.8 15~44 19.0 17000 4.1 45~64 150.0 5000 37.5 65~ 870.0 5000 43.5 16.1 r p = P 140 n P = 13.1 = 146.5 / 10 i i 16.1 4OR 110 5 = = 7.07 5 140 OR Mantel-Haenszel χ H OR 1 0 H OR 1 1 ( ad bc) ( n 1) MH χ = =73.17ν =1 n n m m 1 0 1 0 148
73.17> χ 0.05, 1=7.88P <0.05 H 1 7 149
150 1 () (population) = 65 =
14 = (pyramid) = (general fertility rategfr) = 15~49 = = Σ( ) natural increase rate, NIR = 151
15 standardized mortality ratesmr IMR = 50% 100% 5 5 5 = 8 =
153 = () incidence rate n n n n
154 Kaplan-Meier infant mortality rate, IMR = [ ] 1 A B C D B [ ]
155 A B C D D = ( ) A B C D C A B CD A,
= 100 = 100 156
157
158
159 (pyramid) = = standardized mortality ratesmr
160 n n n n Kaplan-Meier
161
() current life table cohort life table 1 5 () 1 (exact age) age specific probability of dying X n d X n d X q X = n q X = 16-1 lx lx q X X n q X X n 16
d X n d X XX+n 3 number of survival person-years l X lx d X n d X qx n q X d = l q d = l q 16- X X X n X X n X lx+1 = lx d X l X+ n = lx n dx 16-3 4 number of survival person-years total number of survival person-years X n LX n L X l X X X+n L 0 = l1 + a0 d0 16-4 a 0 0 lw L w = 16-5 mw L w l w m w 5 life expectancy X T e = x 16-6 x l x p i n X p i n X r i x p i = ( p ) n (16-7) n X n X 1 lx n d X n q X e X X 163
1 3 4 life expectancy X X X 0 65.5 1 G 65.5 H 65.5 I 65.5 D D [ ] 0 1 1 X X+1 1q x X+1 X+ 1q x+1 X X+ ( ) A 1q x 1q x+1 B1-1q x1q x+1 C1-1 q x 1 q x +1 D1-1- 1 q x 1 q x+1 ( ) 164
A 0 B 0 C 0 D m85 ( + ) e 85 1 1998 16-1 1998 0~ 18753 46 40~ 56806 134 1~ 5435 60 45~ 65863 39 5~ 64063 46 50~ 5443 346 10~ 94683 64 55~ 43355 58 15~ 11433 90 60~ 3004 763 0~ 16941 13 65~ 4445 97 5~ 118930 17 70~ 1818 897 30~ 919 104 75~ 5813 647 35~ 690 9 80~ 685 517 a 0 =0.145 1 life table current life table 3 complete life table 4 abridged life table 5 5 cohort life table 6 age specific probability of dying X n 165
7 number of survivors X 1D D 1 m = D / p X n q = n m ) /( + n m ) n X ( n X n X n qx = mx /[ 1+ (1 ax ) mx ] a X XX+1 X m 85 ( + ) 85 85 e 85 85 85 () 1 n DX 1 n m X = 16- P n q n X 46 q 0 013118 q = = 0. 0 18753 1.000000 3 X 16-3 d = l q X l d n X l = 100000 0 = 100000 0.013118 1311 = 0 0 0 l 1 = l0 d0 = 100000 1311 = = l q = 98689 0.004406 = 1 1 1 l = l d = 98689 434 = 1 1 d n X 98689 434 9855 (4) L n X a = 0. 145 0 L = l + a d = 98698+ 0.1450 1311 98879 L 0 1 0 0 = l 41 = 0.19551 80 80 ( + ) = = m80( + ) 15743 n X 16-166
(5) T L X = n X n L X T L 15743 = = 80 80 ( + ) 75 = L75 + T80 = T 93473 T X (6) e X = lx T0 6994553 e 0 = = = 69.95 l 0 100000 T1 6895674 e 1 = = = 69.87 l 98689 16-1 X ~ 1 P n X D n X 3 16-1998 m n X 4 q n X 5 l X 6 d n X 7 L n X 8 T X 9 0~ 18753 46 0.013118 0.013118 100000 131 98878 6994553 69.95 1~ 5435 60 0.001104 0.004406 98689 434 393888 6895674 69.87 5~ 64063 46 0.000718 0.003584 9855 35 490395 6501786 66.17 10~ 94683 64 0.000676 0.003374 97903 330 488690 6011391 61.40 15~ 11433 90 0.000787 0.00398 97573 383 486907 55701 56.60 0~ 16941 13 0.000969 0.004833 97190 469 484777 5035794 51.81 5~ 118930 17 0.001068 0.00535 9671 515 48317 4551017 47.05 30~ 919 104 0.001131 0.005641 9606 54 479675 4068700 4.9 35~ 690 9 0.001477 0.007358 95664 703 47656 358905 37.5 40~ 56806 134 0.00359 0.01175 94961 1113 470 311463 3.78 45~ 65863 39 0.00369 0.017981 93848 1687 4650 640441 8.14 50~ 5443 346 0.006379 0.031393 9161 893 45357 175419 3.60 55~ 43355 58 0.01179 0.059093 8968 575 43315 171847 19.9 60~ 3004 763 0.03841 0.11499 83993 9449 39634 188695 15.34 65~ 4445 97 0.039763 0.180837 74544 13480 33900 89353 11.97 70~ 1818 897 0.069980 0.97799 61064 18184 59860 553333 9.06 75~ 5813 647 0.11130 0.435368 4880 18668 167730 93473 6.84 80~ 685 517 0.19551 1.000000 41 41 15743 15743 5.19 e X 10 167
1 Cox 1 survial time 11-1 t 360,990,1400,1800 1 complete data 11-1 3609901800 censored data 11-1 1400 1400 + =1 11-1 4 =1 1 0 1 11/9/80 11/04/85 1 360 1 1 06/13/8 06/08/83 1 990 3 1 0 03/0/83 1/31/86 0 1400 + 4 0 0 08/04/83 04/10/86 1 1800 1 mortality probability q q = 168
1 survival probability p p = 1 q = Kaplan-Meier 1 1 survival rate t ) S t k t S ( t k ) = P( T tk ) = k 11-1 T () p 1, p, Λ, p k S ( t k ) = P( T tk ) = p 1 p Λ pk 11- p, p, Λ, p 1 k 0 t k 1 product -limit method Kaplan-Meier 1958 Kaplan-Meier life-table method log-rank test m ( Ak Tk ) χ = υ = m 1 11-3 T k= 1 k A k T k k H 0 χ m 1 χ m χ H 0 Cox Cox ( k 169
Cox h t ) = h ( t ) exp( b X + b X + Λ + b X ) p 11-4 ) h(t ( 0 1 1 p h 0 ( t ) X, b Cox D R b Cox i 1 ( ) A B C B D B C D 3Cox ( ) A 170 B C D Cox Cox DCox h ( ) AB C 0 t 4 log-rank ( ) A B C D D
Kaplan-Meier k () t t n t d q = d n p =1 q k S ( t k ) SE( S ( t k ) ) 1 360 4 1 1 4 3 4 ( 3 4 )=0. 75 0.165 990 3 1 1 3 3 ( 3 4 )( 3)=0.50 0.500 3 1800 1 1 1 1 0 1 ( 3 4 )( 3)( 0 1)=0.00 0 11-1 k =1 1 t + 1400 ) 3 t n t 3 4 t d t 4 5 t 5 6 t 6 7 78 8 171
1Cox ( ) A B C D A B C D 3 ( ) A C B Db,c 1550 + 687079 + 83 + 91 + 114 + 114 + 11917153738587 + 73 1 survival analysis survival time 3 complete data 17
4 censored data 5 mortality rate 6 mortality probability 7 survival rate tk t k 8 survival probability 1 h 0 () t 1 11-3 11-3 96 k k () p = 1 q S ( t k ) SE( S ( t k ) ) t d c n 0 n c = - c n 0 q = d n (7) (8) (9) (10) 1 0~ 94 10 96 91.0 0.330 0.6770 0.6770 0.074 1~ 74 15 19 184.5 0.4011 0.5989 0.4055 0.094 3 ~ 10 103 98.0 0.45 0.7755 0.3144 0.085 4 3~ 6 71 68.0 0.335 0.6765 0.17 0.063 5 4~ 5 5 43 40.5 0.135 0.8765 0.1864 0.055 6 5~ 6 6 33 30.0 0.000 0.8000 0.149 0.045 7 6~ 4 1 1 0.5 0.1951 0.8049 0.101 0.037 8 7~ 1 16 15.5 0.190 0.8710 0.1046 0.030 9 8~ 3 13 1.0 0.500 0.7500 0.0784 0.017 10 9~ 0 8 8.0 0.500 0.7500 0.0588 0.00 173
11 10~ 6 5.0 0.4000 0.6000 0.0353 0.0177 1 11~.0 1.0000 0.0000 0.0000 0.0000 1 k =1 1 n c = n 0 - c 6 3 q = d n 7 4 p = 1 q 8 5 910 6 t a 11- log-rank 1 H 0 H 1 α = 0. 05 A =4A 0 1 =10T 0 =8.6694T 1 =5.3306. 3 m ( A ) (4 8.6694) (10 5.3306) k Tk χ = = + = 6. 605 T 8.6694 5.3306 k = 1 k 4 P 1 χ P <0.05α = 0. 05 H0 H 1 174
175 evaluation synthetical evaluation (pre-event evaluation) (medial evaluation) (after-event evaluation)
176 (system s analysis method) (specialist-scored method) Satty Topsis 1998 11 18-1 18-18-1 11
1 % 3 % 4 5 6 % 7 8 % 9 % 10 % 11 % 18-1998 11 11 34511 18-1 18-18-3 y = X M M y = X 18-1 18-18-1 18- X M 177
18-3 1998 11 1 3 4 5 6 7 8 9 10 11 10 1 1.06 1.00 0.96 1.4 1.07 1.06 1.00 0.98 1.0 1.0 1.01 0.91 1.00 0.97 1.06 1.0 0.93 1.00 0.98 1.01 1.0 1.03 3 1.00 1.01 1.49 1.06 1.08 1.00 0.95 1.03 1.00 1.01 1.07 4 0.96 1.0 1.3 1.4 0.86 0.94 1.05 1.03 1.01 0.98 1.06 5 1.06 0.97 0.68 1.06 0.95 0.98 0.99 1.00 1.00 0.99 1.01 6 0.98 0.99 1.18 0.4 1.1 0.98 1.0 1.00 1.00 0.98 1.07 7 0.95 0.99 1.01 0.85 0.84 0.98 0.98 0.99 0.99 0.99 1.01 8 1.01 1.00 1.01.13 1.00 1.03 1.00 0.98 1.00 0.98 0.95 9 1.0 0.99 1.31 1.06 1.00 1.03 1.0 1.00 1.01 1.00 1.01 11 1 0.99 1.07 0.98 1.01 1.00 1.00 0.99 0.87 0.85 1.07 1.01 0.97 0.99 1.0 1.01 0.98.1 1.13 1.08 1.0 1.00 1.0 1.01 0.87 0.81 0.85 0.94 0.97 1.00 1.01 0.91 1.01 0.97 18-3 I = m n y ij i= 1 j= 1 I = 1.06 + 1.00 0.96 1.4 + 1.07 1.06 1.00 0.98 + 1.0 1.0 = 5. 5851 I = 0.91+ 1.00 0.97 1.06 + 1.0 0.93 1.00 0.98 + 1.01 1.0 = 4. 9509 1 + + 18-3 1.01 1.03 1998 1 3 4 5 6 7 8 9 10 11 1 5.5851 4.9509 5.746 5.664 4.6833 4.6476 4.599 6.151 5.484 4.8934 6.0795 4.4838 1 A C B D 178
A B C D 3 0.6370 0.970 1.0Saaty A1.9340 B0.9340 C1.7636 D0.189 4 A C B DTopsis 5 m 1 I = y n 1 Am Cy Bn D 18-5 Topsis 1995~1997 18-5 1995~1997 % % % % % % % % 1995 0.97 113.81 18.73 99.4 99.80 97.8 96.08.57 94.53 4.60 1996 1.41 116.1 18.39 99.3 99.14 97.00 95.65.7 95.3 5.99 1997 19.13 10.85 17.44 99.49 99.11 96.0 96.50.0 96. 4.79 1.A.B 3.D 4.C 5.C 18-6 18-6 % % % % % % % % 1995 0.97 113.81 5.34 99.4 99.80 97.8 96.08 97.43 94.53 95.40 1996 1.41 116.1 5.44 99.3 99.14 97.00 95.65 97.8 95.3 94.01 1997 19.13 10.85 5.73 99.49 99.11 96.0 96.50 97.98 96. 95.1 179
18-6 18-4 18-7 X ij Z = 18-4 ij n ( X ij ) 1995 18-4 0.97 Z11 = = 0. 509 0.97 + 1.41 + 19.13 i= 1 18-7 1995 0.590 0.59 0.560 0.577 0.580 0.580 0.577 0.577 0.57 0.581 1996 0.60 0.604 0.570 0.577 0.576 0.578 0.575 0.576 0.577 0.57 1997 0.538 0.535 0.601 0.578 0.576 0.574 0.580 0.580 0.583 0.579 18-5 18-6 + Z = a a, Λ a ( i1 max, imax, immax) Z = ( a, a, Λ, a ) i1 min i min im min + Z = ( 0.60,0.604,0.601,0.578,0.580,0.580,0.580,0.580,0.583,0.581 ) Z = ( 0.538,0.535,0.560,0.577,0.576,0.574,0.575,0.576,0.57,0.57) + 18-7 18-8 D D 18 8 D D + i i 1997 D + D D + = = 0.094 = = m ( a ij aij ) i= 1 18-5 18-6 max 18-7 m ( aij a ij ) i= 1 min 18-8 ( 0.60 0.538) + ( 0.604 0.535) + Λ + ( 0.581 0. 579 ) ( 0.538 0.538) + ( 0.535 0.535) + Λ + ( 0.57 0. 579 ) D = = 0.044 18-9 C 18-8 1997 C i 0.044 Ci = = 0.319 0.094 + 0.044 i Di C i = 18-9 D + D + i i 18-8 180
+ D D C i 1995 0.045 0.078 0.634 1996 0.034 0.095 0.736 1 1997 0.094 0.044 0.319 3 18-8 1996 181