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Chapter 7 Factorial Treatment Designs: Random and Mixed Models 1 Random Effects for Factorial Treatment Designs Section 7.1 pp 232-237 This is really an RCBD with days as blocks. When we have random variables our interest is in the variance components rather than the means. In this example days and machines are both random. data new; do day = 1, 2, 3, 4; do machine = 1, 2, 3, 4; do obs = 1 to 2; input TriLevel @@; output; end; end; end; cards; 142.3 144.0 148.6 146.9 142.9 147.4 133.8 133.2 134.9 146.3 145.2 146.3 125.9 127.6 108.9 107.5 148.6 156.5 148.6 153.1 135.5 138.9 132.1 149.7 152.0 151.4 149.7 152.0 142.9 142.3 141.7 141.2 ;;;;

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Chapter 7 Factorial Treatment Designs: Random and Mixed Models 2 Proc Mixed data=new Class Day Machine; Model Trilevel=; Random Day Machine Day*Machine; run; The Mixed Procedure Class Level Information Class Levels Values day 4 1 2 3 4 machine 4 1 2 3 4 Covariance Parameter Estimates Cov Parm Estimate day 44.6855 machine 57.7194 day*machine 34.7210 Residual 17.8953 E =155.0112
Chapter 7 Factorial Treatment Designs: Random and Mixed Models 3 Proc Mixed data=new Method=Type3; Class Day Machine; Model Trilevel=; Random Day Machine Day*Machine; run; Type 3 Analysis of Variance Sum of Source DF Squares Mean Square day 3 1334.463437 444.821146 machine 3 1647.278438 549.092813 day*machine 9 786.035312 87.337257 Residual 16 286.325000 17.895313 Type 3 Analysis of Variance Note the F values differ from book Source F Value Pr > F due to rounding. day 5.09 0.0248 machine 6.29 0.0137 day*machine 4.88 0.0029 Residual . .

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Chapter 7 Factorial Treatment Designs: Random and Mixed Models 4 Source Expected Mean Square day Var(Residual) + 2 Var(day*machine) + 8 Var(day) machine Var(Residual) + 2 Var(day*machine) + 8 Var(machine) day*machine Var(Residual) + 2 Var(day*machine) Residual Var(Residual) Covariance Parameter Estimates Cov Parm Estimate ( books estimates are rounded ) day 44.6855 machine 57.7194 day*machine 34.7210 Residual 17.8953 E =155.0112 GLM and VARCOMP also work.
Chapter 7 Factorial Treatment Designs: Random and Mixed Models 5 Covariance Parameter Estimates Cov Parm Estimate day 44.6855/155.0112 = .288 machine 57.7194/155.0112 = .372 day*machine 34.7210/155.0112 = .224 Residual 17.8953/155.0112 = .115 . E =155.0112 E . 1 What does these tell us? Residual - Day - Machine - Day*Machine -

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Chapter 7 Factorial Treatment Designs: Random and Mixed Models 6 Three Factor Studies (All Random) Proc Mixed Data=a method=type3; Class D M T; Model y =; Random D M D*M T D*T M*T D*M*T; run; Term df Expected Mean Squares D 2 Var(Residual) + 2 Var(DMT) + 6 Var(DT) + 6 Var(DM) + 18 Var(D) M 2 Var(Residual) + 2 Var(DMT) + 6 Var(MT) + 6 Var(DM) + 18 Var(M) T 2 Var(Residual) + 2 Var(DMT) + 6 Var(MT) + 6 Var(DT) + 18 Var(T) DM 4 Var(Residual) + 2 Var(DMT) + 6 Var(DM) DT 4 Var(Residual) + 2 Var(DMT) + 6 Var(DT) MT 4 Var(Residual) + 2 Var(DMT) + 6 Var(MT) DMT 8 Var(Residual) + 2 Var(DMT) Residual 27 Var(Residual) What is the test for 3 fi? 2 fi? ME?
Chapter 7 Factorial Treatment Designs: Random and Mixed Models 7 0 Approximate F Tests (F or F N ) Two schools of thought for testing Way 1 (Allows for negative MS, but only requires Satterthwaite procedure for the denominator df.

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Chapter 7 Factorial Treatment Designs: Random and Mixed Models 8 Way 2 (This is better because it doesn’t allow for negative MS)
Chapter 7 Factorial Treatment Designs: Random and Mixed Models 9 Mixed Models Section 7.2 pp 237-243 Inference random factors - to the variation in a population of effects.

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