1. Weekend Workshop I PROC MIXED

2. Random or Fixed ?RANDOMFIXEDLevels: Selected at random from infinite population Finite number of possibilities Another Experiment Different selections from same population Same Levels Goal Estimate variance components Compare means Inference All levels in population Only levels used in the experiment.

3. Twins: One gets SAS training method 1, the other gets method 2 Response Y = programming times

4. PROC MIXED Model PROC MIXED Model Y = X  Z  + e Variance of  is G =,Variance of e is R = Model ; Random ; Repeated ;

5. PROC MIXED DATA=TWINS; CLASS FAMILY METHOD; MODEL TIME = METHOD; * fixed; RANDOM FAMILY; *<- family ~ N(0,  2 F ) ; Covariance Parameter Estimates Cov Parm Estimate family 21.2184 Residual 40.8338 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F method 1 19 9.60 0.0059 Intraclass correlation (related to heritability)  2 F /(  2 F +  2 ) Estimated as 21.2/62 or about 1/3. Q: Why not usual (Pearson) correlation?

6. DemoGet_Twins.sas Twins_MIXED.sas Twins_MIXED.sas

7. BLUP Y ij =  + F i + e ij D i = Family mean –  F i + e i. best estimate of F i = ? Variance of (F i – b D i ) is (1-b) 2  2 F + b 2  2 /2 Use b =  2 F /(  2 F +  2 /2) Estimate: b = 21.2/(21.2 + 40.8/2) = 0.510 Overall mean + 0.510(Family i mean – Overall mean) PROC MIXED DATA=TWINS; CLASS FAMILY METHOD; MODEL TIME = METHOD; RANDOM FAMILY; ESTIMATE "1 " intercept 1 | family 1; ESTIMATE "2 " intercept 1 | family 0 1; PROC GLM DATA=TWINS; CLASS FAMILY METHOD; MODEL TIME = FAMILY METHOD; LSMEANS FAMILY;

8. MEANS andBLUPs MEANS and BLUPs (MIXED)(GLM)

9. Demo Twins_BLUP.sas Twins_BLUP.sasTwins_TEST.sas

10. REML Estimation (1)Regress out fixed effects (2)Maximze likelihood of residuals (mean known: 0) (3)Variance estimates less biased (unbiased in some simple cases) (3)Variance estimates less biased (unbiased in some simple cases) ML Estimation Search over all (fixed and random) parameters Search over all (fixed and random) parameters Estimates of variances biased low! 

11. Unbalanced Data Unbalanced Data SUBJ  Ear plug ABCDEFG I 25 (L)19 (L)29 (R)16 (R)25 (L) II 8 (R)7 (L)23 (L)16 (R)24 (R) III 22 (R)7 (R)14 (L)12 (L) I vs. III free of subject effects for red data. Misses info in other data.

12. proc glm; class plug worker; model loss = worker plug; Random Worker; Estimate "I vs III - GLM" Plug -1 0 1; run; proc mixed; class plug worker; model Loss=Plug; Random Worker; Estimate "I vs III - Mixed" Plug -1 0 1; run; GLM Source DF Type III SS F Value Pr > F worker 6 451.9062500 12.21 0.0074 plug 2 62.6562500 5.08 0.0625 Standard Parameter Estimate Error t Value Pr > |t| I vs III - GLM -4.8125 1.9635 -2.45 0.0579 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F plug 2 5 5.79 0.0499 Estimates Standard Label Estimate Error DF t Value Pr > |t| I vs III - Mixed -5.2448 1.9347 5 -2.71 0.0422 Covariance Parameter Estimates Cov Parm Estimate worker 37.578 Residual 6.1674

13. Demo Earplugs.sas Earplugs.sas

14. Soil Variety 1 1 1 2 2 1 2 2 3 1 3 2 4 Aquariums, 2 aerated 2 not six dishes / aquarium one plant / dish soil x variety combinations ANOVA SourceAir Error A V S VA VS AS AVS Error B SPLIT PLOT

15. PROC MIXED; CLASS VAR AQUARIUM SOIL AIR; MODEL YIELD = AIR SOIL VAR SOIL*VAR AIR*SOIL AIR*VAR AIR*SOIL*VAR / DDFM=SATTERTHWAITE; RANDOM AQUARIUM(AIR); ESTIMATE "SOIL 1: AIR EFFECT" AIR -1 1 AIR*SOIL -1 1 0 0 0 0; RUN; Compare Air to No Air within soil 1 Variance of this contrast is hard to figure out: (1/3)[MS(A)+2 MS(B)] Need Satterthwaite df AUTOMATIC IN MIXED!!!

16. Covariance Parameter Estimates Cov Parm Estimate AQUARIUM(AIR) 2.1833 Residual 7.7333 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F AIR 1 2 16.20 0.0565 SOIL 2 10 7.87 0.0088 VAR 1 10 24.91 0.0005 VAR*SOIL 2 10 0.04 0.9631 SOIL*AIR 2 10 1.08 0.3752 VAR*AIR 1 10 4.22 0.0669 VAR*SOIL*AIR 2 10 0.23 0.7973 Standard Label Estimate Error DF t Value Pr > |t| SOIL 1: AIR EFFECT 5.2500 2.4597 5.47 2.13 0.0812

17. Demo Aquarium.sas Aquarium.sas

18. Random Coefficient Models Random Coefficient Models the basic idea the basic idea mistakes Program writing time Average programmer Dave Line for individual j: (a 0 + a j ) + ( b 0 + b j )t a 0 + b 0 t

19. Hierarchial Models (1)Same as split plot - almost (2)Whole and split level continuous predictor variables (typically) (1)Aquarium level (level i): pH i (2)Dish level: Soil nitrogen test (N ij ) Y ij = a i + b i N ij +e ij (3) Idea: a i =  0 +  1 pH i + a i * b i =  0 +  1 pH i + b i * b i =  0 +  1 pH i + b i * Y ij = a i + b i N ij +e ij Y ij =  0 +  1 pH i + a i * + b i N ij +e ij Y ij =  0 +  1 pH i + a i * + (  0 +  1 pH i + b i * ) N ij +e ij Y ij = [  0 +  1 pH i +  0 N ij +  1 pH i N ij ] + [a i * +b i * N ij +e ij ] fixed random fixed random PROC MIXED DATA = UNDERWATER; MODEL GROWTH = N P N*P; RANDOM INTERCEPT N / SUBJECT = TANK TYPE=UN;

20. p Num Den Effect DF DF F Value Pr > F N 1 2 3.50 0.2018 pH 1 2.05 6.76 0.1186 N*pH 1 2 1.31 0.3702 aquarium N pH growth 1 2.21 5.5 27.05 1 1.25 5.5 25.92 1 4.36 5.5 30.09 1 7.14 5.5 33.66 1 8.61 5.5 36.13 1 6.53 5.5 33.00 2 6.58 4.7 35.72 2 3.12 4.7 31.17 2 5.28 4.7 34.35 2 1.09 4.7 28.34 2 4.83 4.7 33.56 2 9.61 4.7 40.25 3 7.99 4.2 47.04 3 7.79 4.2 46.56 3 8.32 4.2 48.27 3 2.53 4.2 34.20 3 6.85 4.2 44.59 3 4.73 4.2 39.29 4 0.95 5.1 24.94 4 2.00 5.1 27.33 4 9.99 5.1 43.84 4 0.23 5.1 23.54 4 0.13 5.1 23.56 4 1.17 5.1 25.68 Num Den Effect DF DF F Value Pr > F N 1 3 50.19 0.0058 pH 1 2.03 14.68 0.0603 Cov Parm Estimate UN(1,1) 1.8976 UN(2,1) -0.5563 UN(2,2) 0.2596 Residual 0.0286 pH N

21. Demo Hierarchial.sas Hierarchial.sas

22. Next: Repeated Measures Notes in pdf from NCSU experimental design class (ST 711)

23. Demo SURGERY.sas SURGERY.sas