SASによる共分散分析

1 SAS .. ancova using SAS. Chikuma Hamada Tokyo University of Science .. SAS . 2.. STATISTICS IN MEDICINE 2002;21:2917- 2930. Subgroup analysis , covariate adjustment and baseline comparisons in clinical trial reporting: current practice and problems Stuart J. Pocock, Susan E. Assmann, Laura E. Enos and Linda E. Kasten 3.. 1 50 . 1997 7 9 50 . British Medical Journal (BMJ)5 . New England Journal of Medicine (NEJM)24.. Journal of the American Medical Association (JAMA)6 . Lancet 15 . 4. 5. 6. 7.. N Of VAriance . C N of COVA riance.

2 C( . ANOVA . 8.. N Of VAriance (variance) .. (two-way) .. (6,7,8). 9.. A B C. 6 6 6. 10.. covariates : adjust .. 11.. bias) (precision). 12.. 13. (confounding) . (drug) (response).. (confounding factor) . 1) . 2) . 3) . 14. 15. (confounding).. 16.. 1) . 2) . 3) .. 17.. 1) .. 2) . a) .. b) . MH . c) . Cox 18.. (drug) (response). (C). C C .. 19.. (adjusted analysis ).. : .. :1) .. 2) .. Cox 20. ( mmHg). ( . 39 41 42 43 43 44 45 46 47 49 95 99 106 111 115 116 101 117 104 119 ( . 36 37 38 39 39 40 41 42 44 47 88 94 96 92 98 89 103 97 110 105 p= **.

3 P= *.. 21.. 110. b . p 100. 90. 1 P 2 A. g p= ** 22. GLM . data data;. do g=1 to 2;. do i=1 to 10;. input age bp @ end;end;. cards;. 39 95 41 99 42 106 43 111 43 115. 44 116 45 101 46 117 47 104 49 119. 36 88 37 94 38 96 39 92 39 98. 40 89 41 103 42 97 44 110 47 105. ;. proc glm;. class g;. model bp=g; 23. lsmeans g/pdiff tdiff;. ( ) . Source DF Sum of Squares Mean Square F Value Pr > F. Model 1 Error 18 Corrected Total 19 2. g bp LSMEAN 95% Confidence Limits . 1 2 Least Squares Means for Effect g . i j Difference Between 95% Confidence Limits for LSMean(i)- Means LSMean(j).

4 1 2 24.. 48. 46.. a 44.. g 42. e 40. 38. 36. 1 2. P g A. p= * 25.. p< . 26. P A.. 1) . 2) . 3) .. 27.. 28. 1) .. 39 41 42 43 43 44 45 46 47 49 95 99 106 111 115 116 101 117 104 119 . 36 37 38 39 39 40 41 42 44 47 88 94 96 92 98 89 103 97 110 105 .. 29. 2) .. 39 41 42 43 43 44 45 46 47 49 95 99 106 111 115 116 101 117 104 119 95 . 36 37 38 39 39 40 41 42 44 47 88 94 96 92 98 89 103 97 110 105 105.. N . 2 2. ni1ni 2. V [ i 2 i1 ] , wi . ni 2 ni1 ni1 ni 2 30. 2) . data data;set data;. if age <= 39 then agec=1;. if 40<=age <=44 then agec=2.

5 If 45<=age then agec=3;. proc glm;. class agec g;. model bp=agec g ;. lsmeans g/pdiff tdiff;. 31. 2) .. (N) (N). 30 1 5. (1) (5) 1 5. 40 6 4. (6) (4) 6 4. 40 105 3 1. (3) (1) 3 1. (10) (10) p= 32. 3) .. 39 41 42 43 43 44 45 46 47 49 95 99 106 111 115 116 101 117 104 119 . 36 37 38 39 39 40 41 42 44 47 88 94 96 92 98 89 103 97 110 105 =a b b= . (p= ). 33. 3) . =a . P. A.. 34. P A. 3) . proc glm;. class g;. model bp=age g;. lsmeans g/pdiff tdiff;. 35.. CL(Confidence . Limit). 1 Y1 2 Y2 Y .. 2 1 Y2 Y1 Y1. 2. 2 Y2. V [ 2 1 ].

6 N2 n1. 2. 2 Z . CL 2 1 t DF . n2 n1. 36.. 2 1 Y2 Y1 2. 2. V [ 2 1 ] . n2 n1 10 10.. SE[ 2 1 ]2. 2. 2 2. CL 2 1 t18 . n2 n1. , p 37.. P . Y1.. Y2 . A .. 38.. y 1 z . y 2 z .. 1. : . 2 0. 39.. Yij i zij ij i : . Y z .. ij N(0, ). 2.. 40.. SS (Yij i zij ) 2. i j e12. dSS. zij (Yij i zij ). d i j e 22. dSS. (Yij i zij ) 0. d i i e32.. i Yi z i 41. 2 .. Y 0 1 z .. 0 42.. SS (Yij i zij )) 2. i j (Yij Yi ( zij z i )) 2. i j dSS. ( zij z i )(Yij Yi ( zij z i )) 0. d i j ( z z i )(Yij Yi ) ( zij z i ) 0. 2. ij i j i j . ( z ij z i )(Yij Yi ).

7 S zy1 S zy 2. i j . ( z i j ij z i. ) 2. S zz1 S zz 2 43.. S zy1 S zy1. S zz1 S zz1. S zy1 S zy 2 S zz1 S zz1.. S zz1 S zz 2 S zz1 S zz 2.. 1 1.. 1 . 2 . V [ 1 ] V [ 2 ] w1 1 w2 2.. 1.. 1 w1 w2.. V [ 1 ] V [ 2 ].. wi . 1.. 1 2. V [ i ]. 44.. 1 Y1 z1 2 Y2 z 2.. 2 1 Y2 Y1 ( z 2 z1 ). 2. 2 . V [ 2 1 ] ( z 2 z1 ) V [ ]. adj adj 2. n2 n1. 45.. 2 1 Y2 Y1. 2. 2. V [ 2 1 ] . n2 n1.. 2 1 Y2 Y1 ( z 2 z1 ). adj 2. adj 2.. V [ 2 1 ] ( z 2 z1 ) V [ ]. 2. n2 n1 46.. 1) z 2 z1 .. 2) 2> adj2. z 2 z1 .. 47. ( ) . Source DF Sum of Squares Mean Square F Value Pr > F.

8 Model 1 Error 18 Corrected Total 19 2. g bp LSMEAN 95% Confidence Limits . 1 2 Least Squares Means for Effect g . i j Difference Between 95% Confidence Limits for LSMean(i)- Means LSMean(j). 1 2 48.. Source DF Sum of Squares Mean Square F Value Pr > F. Model 2 <.0001. Error 17 Corrected Total 19 adj2. g bp LSMEAN 95% Confidence Limits . 1 2 2 Least Squares Means for Effect g . i j Difference Between 95% Confidence Limits for LSMean(i)-LSMean(j). Means 1 2 49.. , z1 , z 2 . 2 1 Y 2 Y1 ( z 2 z1 ). ( ) adj 2. adj 2.

9 V [ 2 1 ] ( z 2 z1 ) V [ ]. 2. n2 n1. 2 2 . ( ) 2 V [ ]. 10 10. 2. 50.. 2 1 Y2 Y1 CL , p .. 2 1 CL , p 51.. Yij i i zij ij i : . Y z .. ij N(0, ) 2. 52.. SS (Yij i i zij )) 2. i j dSS. (Yij i i zij )) 0. d i i . i Yi i z i 53.. SS (Yij i i zij )) 2. i j (Yij Yi i ( zij z i )) 2. i j dSS. ( zij z i )(Yij Yi ( zij z i )) 0. d i j ij i ij i ij i 0.. j ( z z )(Y Y ) ( z z ) 2. j . (z ij z i )(Yij Yi ). S zyi i j . (zj ij zi ) 2. S zzi 54.. (z ij z i )(Yij Yi ) (z ij z i )(Yij ). i j . j ( zij z i ). j 2. ij i ( z j z ) 2.

10 Ij i . ( z z ) 2 2. 2 2. V [ i ] . j . ( zij z i ) 2 . 2. ( z ij z i). 2. S zzi j j . 55. P. A. 56.. 1) .. 2) .. 3) . Yij i zij ij , . ij N(0, ). 2. 57.. 1) .. 58.. Y1 . (post Y2) .. ancova . 59.. post(Y2) 2. Y2 Y1 2+ 2 2. ancova ( 2 1) 2 2 2. =0 = = .0. post(Y2 2 2 2. Y2 Y1 2 2 2 . ancova 2 2 0 60.. Y2 Y1 . post(Y2 . ancova . 61.. ancova ( 2 1). 0 =0 post(Y2) . 1 =0 Y2 Y1 . 0< <1 . Y2 Y1 > . post(Y2) 62.. (mg). 0 351 361. 1 319 343 3 .. 10 261 288 266 63.. 0 1 3 ** 10 ** 64.. 65.. proc glm data=ow;. class dose;. model liver=dose/ss2.