Armitage 1 1.1 2 t 1.2 SAS Proc GLM 2 2.1 1 1 2.1.1 50 1 1
t sex N y 50 116.45 119.6 122.75 11.071 1.5657 93.906 154.32 y 50 127.27 130.7 134.13 12.072 1.7073 102.68 163.37 y Diff (1-2) -15.7-11.1-6.504 11.582 2.3165 t t Pr > t y Pooled Equal 98-4.79 <.0001 130.7 119.6 11.1 t p 0.0001 0.05 2.1.2 2 t 2
sex N x 50 48.883 50.214 51.546 4.6862 0.6627 41.734 66.356 x 50 53.785 55.262 56.74 5.1979 0.7351 44.028 65.512 x Diff (1-2) -7.012-5.048-3.084 4.9486 0.9897 t t Pr > t x Pooled Equal 98-5.10 <.0001 5kg p 0.0001 0.05 *1 2.1.3 3 *2 *1 *2 3
F Pr > F Model 2 12613.94771 6306.97385 169.29 <.0001 Error 97 3613.73757 37.25503 Corrected Total 99 16227.68528 Type III F Pr > F sex 1 21.372367 21.372367 0.57 0.4506 x 1 9533.205941 9533.205941 255.89 <.0001 sex p 0.4506 x 3 model / solution clparm *3 t Pr > t 95% Intercept 20.55955993 B 6.93919926 2.96 0.0038 6.78717027 34.33194959 sex -1.04010430 B 1.37322916-0.76 0.4506-3.76558405 1.68537544 sex 0.00000000 B..... x 1.99305978 0.12459289 16.00 <.0001 1.74577737 2.24034219 Note X X B *4 x 1.04 1.04 *5 100 *3 solution clparm 95 *4 Note 0 B *5 (better ) PPK 4
2.1.4 t t t 11.1 11.6 1.0 6.1 t 11.1 1.0 *6 Error 37.3 = 6.1 *7 2.1.5 1 2.2 2 2 y 1 50 2.2.1 t 4 y *6 *7 Error 5
t dose N y 50 22.957 25.474 27.991 8.8567 1.2525 8.7731 48.316 y 50 23.48 26.181 28.881 9.5022 1.3438 0.147 47.26 y Diff (1-2) -4.352-0.706 2.9391 9.1851 1.837 t t Pr > t y Pooled Equal 98-0.38 0.7014 0.7 1.8 *8 t p 0.7014 0.05 2.2.2 1 x y x x 5 x x t *8 0.7 6
dose N x 50 4.4943 5.3135 6.1328 2.8828 0.4077-0.383 11.332 x 50 6.8843 7.7772 8.6702 3.1419 0.4443-1.424 14.269 x Diff (1-2) -3.66-2.464-1.267 3.0151 0.603 t t Pr > t x Pooled Equal 98-4.09 <.0001 x 0.05 2.2.3 y x 6 x y 4 x F Pr > F Model 2 7533.384392 3766.692196 489.13 <.0001 Error 97 746.977786 7.700802 Corrected Total 99 8280.362178 7
Type III F Pr > F dose 1 889.212565 889.212565 115.47 <.0001 x 1 7520.909275 7520.909275 976.64 <.0001 dose x 1 model /solution clparm t Pr > t 95% Intercept 3.584008459 B 0.82270482 4.36 <.0001 1.951167133 5.216849786 dose 6.451886876 B 0.60041554 10.75 <.0001 5.260228231 7.643545521 dose 0.000000000 B..... x 2.905497330 0.09297224 31.25 <.0001 2.720973175 3.090021485 Note X X B y x y = 3.58 + 6.45 + 2.91x (= 10.03 + 2.91x) y = 3.58 + 2.91x (= 3.58 + 2.91x) *9 x 6.45 x 1 y 2.91 x x x 1 * 10 2 x Proc MEANS : x x N 100 6.5453926 3.2452716-1.4236279 14.2692448 x = 6.55 2 ŷ = 3.58 + 6.45 + 2.91 x = 3.58 + 6.45 + 2.91 6.55 29.1 ŷ = 3.58 + 2.91 x = 3.58 + 2.91 6.55 22.6 Proc GLM model lsmeans lsmeans dose; * 11 2 dose y 2 29.0535160 22.6016291 *9 2 *10 *11 8
2.2.4 t t t 2.5 3.0 6.5 2.8 1 t 2.5 x x 6.5 1 Error 7.7 = 2.8 2.2.5 2 y x x y x 2.3 2 19BioS (2) * 12 (1) (2) (3) (1), (2) 1 2 (3) * 13 (1) (3) *12 Greenland, S. and Robins, J. M. Identifiability, exchangeability, and epidemiological confounding. International Journal of Epidemiology 1986;15:413-419. Rothman, K. J., Greenland, S. Modern Epidemiology, 2nd ed. Philadelphia: Lippincott and Raven, 1998. *13 9
3 3 * 14 * 15 (2) x y 60 y 7 y y t dose N y 60 23.937 29.459 34.98 21.374 2.7594-10.42 67.793 y 60 30.817 36.236 41.654 20.975 2.7078-5.529 74.552 y Diff (1-2) -14.43-6.777 0.8789 21.176 3.8661 t t Pr > t y Pooled Equal 118-1.75 0.0822 6.8 p 0.0822 0.05 y x *14 *15 10
8 x 1 2 t dose N x 60 3.962 4.994 6.026 3.9949 0.5157-3.765 14.197 x 60 4.2014 5.1546 6.1078 3.6898 0.4763-1.273 15.766 x Diff (1-2) -1.551-0.161 1.2297 3.8454 0.7021 t t Pr > t x Pooled Equal 118-0.23 0.8194 0.16 y x 11
9 x y * 16 2 * 17 F Pr > F Model 2 35916.53973 17958.26986 114.36 <.0001 Error 117 18373.02321 157.03439 Corrected Total 119 54289.56294 Type III F Pr > F dose 1 1102.09815 1102.09815 7.02 0.0092 x 1 34538.70408 34538.70408 219.94 <.0001 dose p 0.0092 0.05 t Pr > t Intercept 13.30234710 B 2.23796616 5.94 <.0001 dose -6.06241693 B 2.28840583-2.65 0.0092 dose 0.00000000 B... x 4.44910146 0.29999683 14.83 <.0001 Note X X B *16 y *17 2 12
y = 13.30 6.06 + 4.45x (= 7.24 + 4.45x) y = 13.30 + 4.45x (= 13.30 + 4.45x) x 6.06 x Proc MEANS : x x N 120 5.0743056 3.8300419-3.7646217 15.7657095 x = 5.07 * 18 2 ŷ = 13.30 6.06 + 4.45 x = 13.30 6.06 + 4.45 5.07 29.80 ŷ = 13.30 + 4.45 x = 13.30 + 4.45 5.07 35.86 SAS model lsmeans dose; 2 dose y 2 29.8160306 35.8784476 3.1 t t t 6.8 21.2 6.1 12.5 t Error 157.0 = 12.5 3.2 y y x x 3.3 *18 13
= * 19 1 2 3 7 x * 20 x 4 * 21 *19 1 t *20 t 6.8 6.8 x 6.1 =0 *21 14
1 SAS SAS y x dose proc glm data=d1; class dose; model y= dose x / solution clparm ; lsmeans dose; run; quit; model solution clparm dose 2 lsmeans 2 t t t y 11, y 12,, y 1n N(µ 1, σ 2 ) y 21, y 22,, y 2n N(µ 2, σ 2 ) y ij = µ i + ɛ ij ( ɛij N(0, σ 2 ) ) i = 1, 2, j = 1,, n i j x ij y ij = µ i + βx ij + ɛ ij ( ɛij N(0, σ 2 ) ) i = 1, 2, j = 1,, n ( ) x ij x y x 15
3 1 2 x 10 x * 22 2 x *22 y x = 10 x y 2 x x x 11 12 x y 16