Weekend Workshop IWeekend Workshop I

PROC MIXED

Random or Fixed ?

RANDOMRANDOM FIXEDFIXED

Levels:Levels: Selected at random Selected at random from infinite populationfrom infinite population

Finite number of Finite number of possibilitiespossibilities

Another ExperimentAnother Experiment Different selections Different selections from same populationfrom same population

Same LevelsSame Levels

GoalGoal Estimate variance Estimate variance componentscomponents

Compare meansCompare means

InferenceInference All levels in populationAll levels in population Only levels used in the Only levels used in the experiment.experiment.

Twins: One gets SAS training method 1, the other gets method 2

Response Y = programming times

PROC MIXED ModelPROC MIXED Model

38.25

33.00

28.75

30.00

46.50

55.25

1

2

1 0 1

1 1 0

1 0 1

1 1 0

1 1 0

1 0 1

M

M

1

2

3

1 0 0

1 0 0

0 1 0

0 1 0

0 0 1

0 0 1

F

F

F

1

2

3

4

5

6

e

e

e

e

e

e

Y = X Z + e

Variance of is G = ,Variance of e is R = 2

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

2

1 0 0

0 1 0

0 0 1F

Model ; Random ; Repeated ;

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 /(2F + 2)

Estimated as 21.2/62 or about 1/3. Q: Why not usual (Pearson) correlation?

DemoDemoGet_Twins.sasGet_Twins.sas

Twins_MIXED.sasTwins_MIXED.sas

BLUPBLUP

Yij = + Fi + eij

Di = Family mean – Fi + ei.best estimate of Fi = ?

Variance of (Fi – b Di) is (1-b)22F + b2 2/2

Use b = 2F /(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;

MEANS andMEANS and BLUPsBLUPs

(MIXED)(GLM)

DemoDemo Twins_BLUP.sasTwins_BLUP.sasTwins_TEST.sasTwins_TEST.sas

REML EstimationREML Estimation(1)(1)Regress out fixed effectsRegress out fixed effects(2)(2)Maximze likelihood of residuals (mean known: 0)Maximze likelihood of residuals (mean known: 0)(3)(3)Variance estimates less biased (unbiased in some Variance estimates less biased (unbiased in some simple cases) simple cases)

ML EstimationML Estimation Search over all (fixed Search over all (fixed andand random) parameters random) parameters

Estimates of variances biased low! Estimates of variances biased low!

Unbalanced DataUnbalanced Data

SUBJ Ear plug

A B C D E F G

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.

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;

GLMSource DF Type III SS F Value Pr > F worker 6 451.9062500 12.21 0.0074 plug 2 62.6562500 5.08 0.0625 StandardParameter 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 StandardLabel 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

DemoDemo Earplugs.sasEarplugs.sas

Soil Variety 1 1 1 2 2 1 2 2 3 1 3 2

4 Aquariums, 2 aerated2 notsix dishes / aquarium

one plant / dish soil x variety combinations

ANOVA

SourceAir Air Error AError A V S VA VS AS AVSError B

SPLIT PLOT

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 1Variance of this contrast is hard to figure out:

(1/3)[MS(A)+2 MS(B)]

Need Satterthwaite df AUTOMATIC IN MIXED!!!

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

DemoDemo Aquarium.sasAquarium.sas

Random Coefficient ModelsRandom Coefficient Models the basic ideathe basic idea

mistakesmistakes

Program writing time

Average programmer

Dave

Line for individual j: (a0 + aj) + ( b0 + bj )t

2

2

,0

0~

BAB

ABA

j

j Nb

a

a0 + b0 t

Hierarchial ModelsHierarchial Models(1)(1)Same as split plot - Same as split plot - almost almost (2)(2)Whole and split level Whole and split level continuouscontinuous predictor predictor variables (typically)variables (typically)

(1)(1)Aquarium level (level i): pHAquarium level (level i): pHii

(2)(2)Dish level: Soil nitrogen test (NDish level: Soil nitrogen test (Nijij))

YYijij = a = aii + b + bii N Nijij+e+eijij

(3) Idea: a(3) Idea: aii = = 00 + + 11 pH pHii + a + aii**

bbii = = 00 + + 11 pH pHii + b + bii* *

YYijij = a = aii + b + bii N Nijij+e+eijijYYijij = = 00 + + 11 pH pHii + a + aii* * + b+ bii N Nijij+e+eijijYYijij = = 00 + + 11 pH pHii + a + aii

* * + + ((00 + + 11 pH pHii + b + bii* * ) ) NNijij+e+eijij

YYijij = = [[00 + + 11 pH pHii + + 00 N Nij ij + + 11 pH pHii N Nijij] ] + + [a[aii** +b +bii

* * NNijij+e+eijij]]

fixed fixed randomrandom

PROC MIXED DATA = UNDERWATER; MODEL GROWTH = N P N*P; RANDOM INTERCEPT N / SUBJECT = TANK TYPE=UN;

p

Num DenEffect DF DF F Value Pr > FN 1 2 3.50 0.2018pH 1 2.05 6.76 0.1186N*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 DenEffect DF DF F Value Pr > FN 1 3 50.19 0.0058pH 1 2.03 14.68 0.0603

Cov Parm EstimateUN(1,1) 1.8976UN(2,1) -0.5563UN(2,2) 0.2596Residual 0.0286

pHN

DemoDemo Hierarchial.sasHierarchial.sas

Next: Repeated Measures

Notes in pdf from NCSU experimental design class(ST 711)

DemoDemo SURGERY.sasSURGERY.sas