5 Armitage. x,, x n y i = 0x i + 3 y i = log x i x i y i.2 n i i x ij i j y ij, z ij i j 2 y = a x + b 2 2. ( cm) x ij (i j ) (i) x, x 2 σ 2 x,, σ 2 x,2 σ x,, σ x,2 t t x * (ii) (i) m y ij = x ij /00 y ij ȳ, ȳ 2 σ 2 y,, σ 2 y,2 σ y,, σ y,2 t t y (iii) (i) (ii) (i = ) 65, 72, 60, 55 (i = 2) 70, 50, 60 * t Student t
( ) (i) x = n x j = (65 + 72 + 60 + 55) = 63 n 4 j= x 2 = x 2j = (70 + 50 + 60) = 60 n 2 3 j= σ 2 x, = n σ 2 x,2 = n 2 n j= j= (x j x ) 2 = 3 (65 63) 2 + (72 63) 2 + (60 63) 2 + (55 63) 2 52.7 (x 2j x 2 ) 2 = 2 (70 60) 2 + (50 60) 2 + (60 60) 2 = 00 σ x, = t σ 2 x = σ 2 x, = 56.7 7.3, σ x,2 = σ 2 x,2 = 00 = 0 (n ) + (n 2 ) (n ) σ 2 x, + (n 2 ) σ 2 x,2 7.6 t x = x x 2 63 60 ( ) ( n + n σ2 2 x 4 + ) 0.46 3 7.6 (ii) m y ij (i = ).65,.72,.60,.55 (i = 2).70,.50,.60 2 m (i) σ 2 y, = n σ 2 y,2 = n 2 n j= j= ȳ = n y j = (.65 +.72 +.60 +.55) =.63 n 4 j= ȳ 2 = y 2j = (.70 +.50 +.60) =.60 n 2 3 j= (y j ȳ ) 2 = 3 (.65.63) 2 + (.72.63) 2 + (.60.63) 2 + (.55.63) 2 0.00527 (y 2j ȳ 2 ) 2 = 2 (.70.60) 2 + (.50.60) 2 + (.60.60) 2 = 0.0 σ y, = σ 2 y, = 0.00527 0.073, σ y,2 = σ 2 y,2 = 0.0 = 0. 2
t σ 2 y = (n ) σ (n ) + (n 2 ) 2 y, + (n 2 ) σ 2 y,2 0.0076 t y = ȳ ȳ 2.63.60 ( ) ( n + n σ2 2 y 4 + ) 0.46 3 0.0076 (iii) (i), (ii) x ij y ij y ij = 00 x ij x = 63, x 2 = 60 ȳ =.63, ȳ 2 =.60 ȳ = 00 x, ȳ 2 = 00 x 2 σ2 x, = 52.7, σ2 x,2 = 00 σ2 y, = 0.00527, σ2 y,2 = 0.0 σ2 y, = 0,000 σ 2 x,, σ2 y,2 = 0,000 σ 2 x,2 σ x, = 7.3, σ x,2 = 0 σ y, = 0.073, σ y,2 = 0. σ y, = 00 σ x,, σ y,2 = 00 σ x,2 t t x = 0.46 t y = 0.46 t x = t y /00 /00 ( ) (/00) 2 ( 2 ) /00 ( ) t = t p 2 t 2.2 2 2 ( cm) (i = ) 65, 72, 60, 55 (i = 2) 70, 50, 60 3 x ij 3cm (i) t (ii) (i) t 3
( ) (i) z ij 3cm x ij z ij = x ij + 3 (i = ) 68, 75, 63, 58 (i = 2) 73, 53, 63 4 (z ij ) σ 2 z, = n σ 2 z,2 = n 2 n j= j= z = n z j = (68 + 75 + 63 + 58) = 66 n 4 j= z 2 = z 2j = (73 + 53 + 63) = 63 n 2 3 j= (z j z ) 2 = 3 (68 66) 2 + (75 66) 2 + (63 66) 2 + (58 66) 2 52.7 (z 2j z 2 ) 2 = 2 (73 63) 2 + (53 63) 2 + (63 63) 2 = 00 σ z, = t σ 2 z = σ 2 z, = 56.7 7.3, σ z,2 = σ 2 z,2 = 00 = 0 (n ) + (n 2 ) t z = (ii) (n ) σ 2 z, + (n 2 ) σ 2 z,2 7.6 z z 2 66 63 ( ) ( n + n σ2 2 z 4 + ) 0.46 3 7.6 x ij z ij z ij = x ij + 3 x = 63, x 2 = 60 z = 66, z 2 = 63 z = x + 3, ȳ 2 = x 2 + 3 σ2 x, = 52.7, σ2 x,2 = 00 σ2 z, = 52.7, σ2 z,2 = 00 σ2 z, = σ 2 x,, σ2 z,2 = σ 2 x,2 σ x, = 7.3, σ x,2 = 0 σ z, = 7.3, σ z,2 = 0 σ z, = σ x,, σ z,2 = σ x,2 t t x = 0.46 t z = 0.46 t x = t z +3 t t 4
2.3 x,, x n, x 2,, x 2n2 a, b (a 0) y ij = a x ij + b y,, y n, y 2,, y 2n2 2.3. y ij ȳ = n y j = n (a x j + b) = a n n n n + b = a x + b σ 2 y, = n = a 2 σ 2 y,2 = n 2 = a 2 j= j= j= ȳ 2 = y 2j = (a x 2j + b) = a n n 2 n n n n j= j= (y j ȳ ) 2 = n n j= n j= j= (x j x ) 2 = a 2 σ 2 x, (y 2j ȳ 2 ) 2 = n 2 j= n 2 j= j= (x 2j x 2 ) 2 = a 2 σ 2 x,2 j= x j x 2j + b = a x 2 + b (a x j + b) (a x + b) 2 = n (a x 2j + b) (a x 2 + b) 2 = n 2 n j= j= a(x j x ) 2 a(x 2j x 2 ) 2 σ y, = σ y, = a 2 σ 2 x, = a σ 2 x, σ y,2 = σ y,2 = a 2 σ 2 x,2 = a σ 2 x,2 t t y = σ 2 y = (n ) σ (n ) + (n 2 ) 2 y, + (n 2 ) σ 2 y,2 = (n )a 2 (n ) + (n 2 ) σ 2 x, + (n 2 )a 2 σ 2 x,2 = a 2 (n ) σ (n ) + (n 2 ) 2 x, + (n 2 ) σ 2 x,2 = a 2 σ 2 x ȳ 2 ȳ ( ) = n + n 2 σ y (a x 2 + b) (a x + b) ( ) = n + n 2 a 2 σ x a a x 2 x ( ) = sign(a) t x n + n 2 σ x 5
sign(a) a > 0 a = 0 0 a < 0 *2 y ij = a x ij + b ȳ = a x + b, ȳ 2 = a x 2 + b σ2 y, = a 2 σ 2 x,, σ2 y,2 = a 2 σ 2 x,2 σ y, = a σ x,, σ y,2 = a σ x,2 t t y = sign(a) t x t *3 2.3.2 *4 E[x ij ] = µ x,i, V [x ij ] = σx,i 2 E[y ij] = µ y,i, V [y ij ] = σy,i 2 µ y,i = E[y ij ] = E[a x ij + b] = a E[x ij ] + b = aµ x,i + b σy,i 2 = V [y ij ] = V [a x ij + b] = a 2 V [x ij ] = a 2 σx,i 2 σ y,i = σ 2y,i = a 2 σ 2x,i = a σ 2 x,i = a σ2 x,i y ij = a x ij + b µ y, = a µ x, + b, µ y,2 = a µ x,2 + b (E[y j ] = a E[x j ] + b, E[y 2j ] = a E[x 2j ] + b) σ 2 y, = a 2 σ 2 x,, σ 2 y,2 = a 2 σ 2 x,2 (V [y j ] = a 2 V [x j ], V [y 2j ] = a 2 V [x 2j ]) σ y, = a σ x,, σ y,2 = a σ x,2 *2 a 0 0 a > 0 a a = a a = a < 0 a a = a = a a = sign(a) *3 a t t a < 0 *4 6
3 3. 3 3 5 6 (i = ) 0, 3, 20, 5, 7 (i = 2), 8, 5, 2, 5 (i = ) 0.00, 0.077, 0.050, 0.067, 0.059 (i = 2) 0.09, 0.25, 0.200, 0.500, 0.200 5 6 5 t t 5 ( ) 2 SAS x y data d; input group x; y=/x; cards; 0 3 20 5 7 2 2 8 2 5 2 2 2 5 ; run; proc means data=d mean var std; var x y; by group; run; 7
group= x 5.0000000 4.5000000 3.8078866 y 0.0704827 0.000370697 0.092535 group=2 x 6.2000000.7000000 3.4205263 y 0.22388 0.0262097 0.6894 t proc ttest data=d; class group; var x y; run; group N x 5 0.272 5 9.728 2.284 3.8079 0.942.7029 0 20 x 2 5.9529 6.2 0.447 2.0494 3.4205 9.829.5297 2 x Diff (-2) 3.523 8.8 4.079 2.4447 3.694 6.9339 2.289 y 5 0.0466 0.0705 0.0944 0.05 0.093 0.0553 0.0086 0.05 0. y 2 5 0.0222 0.2232 0.4242 0.097 0.69 0.4652 0.0724 0.0909 0.5 y Diff (-2) -0.32-0.53 0.054 0.0779 0.53 0.2209 0.0729 t t Pr > t x Pooled Equal 8 3.84 0.0049 y Pooled Equal 8-2.09 0.0696 t 8
x ij /x ij y ij x = 5, x 2 = 6.2 / x = 0.067, / x 2 = 0.6 ȳ = 0.075, ȳ 2 = 0.223 σ2 x, = 4.5, σ2 x,2 =.7 = 0.06897, = 0.08547 σ2 y, = 0.00037, σ2 y,2 = 0.0262 cσ 2 x, cσ 2 x,2 σ x, = 3.8079, σ x,2 = 3.4205 bσ x, = 0.2626, bσ x,2 = 0.2924 σ y, = 0.093, σ y,2 = 0.69 t t x = 3.84 t y = 2.09 p p x = 0.0049 ( ) p y = 0.0696 ( ) x (y ) 3.2 Delta x y Delta Delta f(x) y = f(x) µ x = E[x] y E[y] = E[f(x)] f(µ x ), V [y] = V [y] (f (µ x )) 2 V [x] E[y], V [y] f(x) = x f (x) = x 2 µ x = E[x] x = 5, x 2 = 6.2 Delta µ x x, x 2 E[y] f(µ x ) = µ x E[y j ] f(µ x, ) x = 5 = 0.067, E[y 2j] f(µ x,2 ) x 2 = 6.2 = 0.6 ȳ = 0.075, ȳ 2 = 0.223 *5 *5 ȳ x, ȳ 2 x 2 Jensen ( 2 Delta ) f(x) = /x 9
Delta V [y] (f (µ x )) 2 V [x] = ( ) 2 (µ x ) 2 V [x] = (µ x ) 4 V [x] µ x x, x 2 σ 2 x, = 4.5, σ 2 x,2 =.7 V [y j ] (f (µ x, )) 2 V [x ] ( x ) 4 σ 2 x, = 4.5 = 0.000286 54 V [y 2 ] (f (µ x,2 )) 2 V [x 2 ] ( x 2 ) 4 σ 2 x,2 =.7 = 0.00798 6.24 y x Delta x Delta y 0.0690 0.000286 0.00037 2 0.0855 0.00798 0.0262 Delta *6 3.3 Armitage [] 5 3 (i) (ii) (iii) (iv) (v) 3.4 Box Cox Box Cox λ x λ λ (λ 0) f(x λ) = log x (λ = 0) x x lim λ λ 0 λ = log x Box Cox log Delta *6 Delta 2 Delta 0
(i) λ 0 f (x λ) = x λ E[f(x λ)] f(µ x ) = (µ x) λ λ V [f(x λ)] (f (µ x )) 2 V [x] = (µ x ) 2(λ ) V [x] (ii) λ = 0 f (x 0) = x E[f(x 0)] log(µ x ) V [f(x 0)] (µ x ) 2 V [x] [] P.Armitage, G.Berry (, ) (Statistical Methods in Medical Research, 3rd ed.)., 200.