H BioS (i) I treat II treat data d; input group patno treat treat; cards; 8 7 4 8 8 5 5 6 ; run; I II sum data d; set d; sum treat + treat; run; sum proc gplot data d; plot sum * group ; symbol c black v dot; run;
I II (ii) t proc ttest data d; class group; var sum; run;
TTEST group N sum. 9. 47.644.878 7.7 46.6 4.557 5 sum 7.8. 45.459.966 5.686 5.77.8 5 6 sum Diff (-) -6.9 -.9.944 6.588 8.96 5.748 t t Pr > t sum Pooled Equal 4 -.7.787 sum Satterthwaite Unequal.76 -.7.798 F Pr > F sum Folded F.68.746
(iii) I II I II (P) (A) (P) (A) 8(P) 7(A) 5 4 8(A) 8(P) 6 5 5(A) (P) 5 6 (A) (P) t µ, µ σ σ µ ( + + 5) 9., µ (6 + 5 + ). σ 4 {( 9.) + ( 9.) + (5 9.) σ +(6.) + (5.) + (.) } 4. σ 4. 6.588 (ii) proc ttest t µ µ t ( + ) σ 9...7 4. proc ttest p data p; p value *( - cdf( t, abs(-.7), 4); * t 4; run; p.787 proc ttest 4
(i) data d; input group patno period treat y; label group patno period ; cards; 8 7 4 8 4 8 5 5 5 6 6 ; run; proc glm data d; class group treat patno period ; model y group treat patno(group) period / solution p ss; test h group e patno(group); run; quit; 5
GLM : y F Pr > F Model 7 4. 7.76948..66 Error 4 5. 8.8 Corrected Total 59.6666667 R y.77876 9.596.979 5.6667 Type III F Pr > F group...4.59 treat...77.4 patno(group) 4 86.66666667.66666667.45. period...5.774 Type III patno(group) Type III F Pr > F group...4.787 6
t Pr > t Intercept 7.8 B.467 7.5.8 group. B.9794.4.754 group. B... treat -. B.75986 -.94.4 treat. B... patno(group) -7. B.9794 -.6.78 patno(group) -.5 B.9794 -.5.64 patno(group). B... patno(group) 4.5 B.9794.5.64 patno(group) 5-4. B.9794 -.5.496 patno(group) 6. B... period.66666667 B.75986.9.774 period. B... Note:X X B patno(fixed) + period GLM. 9.6666667.8..8 -.8. 4.66666667 -.66666667 4. 7..66666667 5 8. 6.6666667.8 6 7. 8.8 -.8 7 8.. -. 8 8. 6.. 9 5. 4.5.5..5 -.5. 8.5.5. 4.5 -.5 TypeIII patno(group) p.787 t p 7
(ii) g i i (i, ) d j (j : j : ) p k k (k,, 6) t l (l : I l : II ) y ijkl µ + g i + d j + p k + t l + ɛ ijkl y µ + g + d + p + t + ɛ y µ + g + d + p + t + ɛ y µ + g + d + p + t + ɛ y µ + g + d + p + t + ɛ y µ + g + d + p + t + ɛ y µ + g + d + p + t + ɛ y 4 µ + g + d + p 4 + t + ɛ 4 y 4 µ + g + d + p 4 + t + ɛ 4 y 5 µ + g + d + p 5 + t + ɛ 5 y 5 µ + g + d + p 5 + t + ɛ 5 y 6 µ + g + d + p 6 + t + ɛ 6 y 6 µ + g + d + p 6 + t + ɛ 6 () X, β µ g g d d p p p p 4 p 5 p 6 t t () () y Xβ + ɛ () SAS g, d, p, p 6, t p,, 4, 5, 6 8
* X, β () y X β + ɛ µ g d p p p 4 p 5 t (iii) () β β β (X X ) X y (4) X X 6 6 6 6 6 6 6 6 6 * X,, 4, 5, 6 9
(X X ) 6 6 6 6 X y 8 7 8 8 5 + + + + 8 + 7 + 8 + 8 + 5 + + + + + + + 8 + 7 + + 8 + 8 + + + + 8 + 7 5 + + + 8 + 8 + 5 +
8 88 8 5 5 9 (4) β (X X ) X y 6 6 6 6 8 88 8 6 5 9 8 88 6 8 6 5 6 9 8 + 88 + 6 + 5 6 8 + 8 88 + + 88 + + 8 + 88 + 6 + 5 8 + 88 + 6 + 5 6 8 + 9 7 6 7 4
7.8. 7.5.5 4.6667 β µ ĝ d p p p 4 p 5 t ĝ, d, p, p 6, t β µ 7.8 ĝ ĝ d. d p 7 β p.5 p p 4.5 p 5 4 p 6 t.6667 t Intercept group group treat treat patno(group) patno(group) patno(group) patno(group) 4 patno(group) 5 patno(group) 6 period period proc glm ŷ
ŷ X β 7.8. 7.5.5 4.6667 7.8 +. 7 +.6667 7.8 + 7 7.8 +..5 +.6667 7.8 +.5 7.8 +. +.6667 7.8 + 7.8 +.5 +.6667 7.8. +.5 7.8 4 +.6667 7.8. 4 7.8 +.6667 7.8. 9.667.8 4.6667 7. 6.667 8.8 6 4.5.5 8.5 4.5
e e y ŷ 8 7 8 8 5 9.667.8 4.6667 7. 6.667 8.8 6 4.5.5 8.5 4.5.8.8.6667.6667.8.8.5.5.5.5 proc glm (iv)..... F.. p + p + p, p 4 + p 5 + p 6 p, p 6 X g, β g () y X g β g + ɛ µ d p p p 4 p 5 t 4
β g β g (X gx g ) X gy (5) X gx g 6 6 6 6 4 4 4 4 6 6 (X gx g ) 4 4 4 4 4 4 X gy 8 7 8 8 5 5
+ + + + 8 + 7 + 8 + 8 + 5 + + + + + 8 + 8 + + + 8 7 + 8 7 8 + 8 5 + + + 8 + 8 + 5 + 8 8 4 8 9 β g (5) β g (X gx g ) X gy 4 4 4 4 4 4 8 8 9 8 + 4 8 4 ( 4) ( ) ( 4) + 4 ( ) 4 ( 8) + 4 ( 8) 8 + 4 9 6.5. 4.667...667.6667 8 8 4 8 9 ŷ g 6
ŷ g X g βg 6.5. 4.667...667.6667 6.5. 4.667 +.6667 6.5 4.667 6.5. +. +.6667 6.5 +. 6.5. + 4.667. +.6667 6.5 + 4.667. 6.5 +. +.6667 6.5. +. 6.5.667 +.6667 6.5..667 6.5. +.667 +.6667 6.5.. +.667 9.6667. 5.667 7.8 6.6667 9. 9.5 5.5 4 8 4 7
ŷ ŷ g 9.667.8 4.6667 7. 6.667 8.8 6 4.5.5 8.5 4.5 9.6667. 5.667 7.8 6.6667 9. 9.5 5.5 4 8 4.5.5.5.5.5.5.5.5.5.5.5.5 (.5) + (.5) + (.5) + (.5) + (.5) + (.5) +.5 +.5 +.5 +.5 +.5 +.5 proc glm group TypeIII. () X p, β p µ g g d d t t () y X p β p + ɛ 8
g, d, t * X p, β p β p µ g d t β p (X px p ) X py (6) X px p 6 6 6 6 6 6 6 6 6 ( ) X p X p 6 6 6 6 6 6 * 9
X py 8 7 8 8 5 + + + + 8 + 7 + 8 + 8 + 5 + + + + + + + 8 + 7 + + + 7 + 8 + + + + + 8 + 5 + 8 88 8 9 β p (6) β p ( X px p ) X p y 6 6 6 6 6 6 8 88 8 9 7 7..6667
ĝ, d, t β p µ ĝ ĝ d d t t 7..6667 ŷ p ŷ p X p βp 7..6667 7. +.6667 7 7. +.6667 7 7. +.6667 7 7 +.6667 7. 7 +.6667 7. 7 +.6667 7.
. 6. 6. 6 7.6667.6667 7.6667.6667 7.6667.6667 ŷ ŷ p 9.667.8 4.6667 7. 6.667 8.8 6 4.5.5 8.5 4.5. 6. 6. 6 7.6667.6667 7.6667.6667 7.6667.6667 4.666 4.667.4..84.8...667.667.8.8 ( 4.666) + ( 4.667) +.4 +. +.84 +.8 +. +. + (.667) + (.667) +.8 +.8 86.6667 proc glm patno(group) TypeIII. ŷ ŷ g, ŷ ŷ p ŷ : µ, g, d, p, p, p 4, p 5, t 8 ŷ g : µ, d p, p, p 4, p 5, t 7 ŷ p : µ, g, d, t 4 ŷ ŷ p : 8 4 4 ŷ ŷ g : 8 7 F F ŷ ŷ g 4 ŷ ŷ p
.85 4 86.6667 p data p value; p - cdf( F,.85,, 4) run; p.787 proc glm TypeIII patno(group)