Multi-Factor Experiments
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1 ETH p. 1/31 Multi-Factor Experiments Twoway anova Interactions More than two factors
2 ETH p. 2/31 Hypertension: Effect of biofeedback Biofeedback Biofeedback Medication Control + Medication
3 ETH p. 3/31 Main effects Treatment means: Medication no yes Biofeedback Total no yes Total main effect of biofeedback: = 10 mmhg
4 ETH p. 4/31 Interaction plot Effect of biofeedback with medi: 18 mmhg Effect of biofeedback without medi: 2 mmhg Interaction 190 Biofeedback 190 Medication 185 without with 185 without with mean of bp mean of bp without with without with Medication Biofeedback
5 ETH p. 5/31 Model for two factors Y ijk = µ+a i +B j +(AB) ij +ǫ ijk È Ai = 0, È B j = 0, È i (AB) ij = È j (AB) ij = 0. i = 1,...,I;j = 1,...,J;k = 1,...,n. A i : ith effect of factor A B j : jth effect of factor B µ+a i +B j : overall mean + effect of factor A on level i + effect of factor B on level j (AB) ij : deviation from additive model
6 ETH p. 6/31 Parameter estimation ˆµ = y...,  i = y i.. y... and ˆB j = y.j. y... ÂB ij = y ij. (ˆµ+Âi + ˆB j ) = y ij. y i.. y.j. +y... Medication A Biofeedback B no yes Total no y.1. = 188 yes y.2. = 178 Total y 1.. = 189 y 2.. = 177 y... = 183 ˆµ = 183,  1 = Â2 = 6, ˆB1 = ˆB 2 = 5
7 ETH p. 7/31 Predicted values of the additive model Treatment means: (ˆµ+Âi + ˆB j ) Medication no yes Biofeedback no yes ÂB 11 = ÂB 22 = ÂB 12 = ÂB 21 = 4.
8 ETH p. 8/31 Decomposition of Variability SS tot = SS A +SS B +SS AB +SS res SS tot = SS A = SS B = SS AB = (y ijk y... ) 2 (y i.. y... ) 2 (y.j. y... ) 2 (y ij. y i.. y.j. +y... ) 2 SS res = «Difference» degrees of freedom: main effect with I levels: I 1 df, interaction between two factors with I and J levels: (I 1)(J 1) df.
9 ETH p. 9/31 Anova table Source SS df MS F P value medi bio medi:bio residual total
10 ETH p. 10/31 Treatment effects medi without bio: medi with bio: bio without medi: bio with medi: effect size C.I. y 21. y 11. = = 4 ( 18,10) y 22. y 12. = = 20 ( 34, 6) y 12. y 11. = = 2 ( 16,12) y 22. y 21. = = 18 ( 32, 4) (standard error: 2 MS res /5 = 5)
11 ETH p. 11/31 Efficiency of factorials factorial medication biofeedback no yes no 6 6 yes 6 6 Two separate studies medication biofeedback no yes and no yes or: control medi bio 8 8 8
12 ETH p. 12/31 More than two factors bio/medi bio medi control without diet with diet
13 ETH p. 13/31 Treatment means without diet(c) medication (A) bio (B) no yes total no yes total with diet (C) medication (A) bio (B) no yes total no yes total
14 ETH p. 14/31 Main effects and interactions main effects: medication (A): = 9.5 biofeedback (B): = 8.5 diet(c): = way interactions AB, AC, BC: medi (A) Bio (B) no yes total no yes total way interaction ABC
15 Model and Anova table Y ijkl = µ+a i +B j +C k +(AB) ij +(AC) ik +(BC) jk +(ABC) ijk +ǫ ijkl È Ai = 0,..., È k (ABC) ijk = 0 Source df MS=SS/df F A I 1 MS A /MS res B J 1 MS B /MS res C K 1 MS C /MS res AB (I 1)(J 1) MS AB /MS res AC (I 1)(K 1) MS AC /MS res BC (J 1)(K 1) MS BC /MS res ABC (I 1)(J 1)(K 1) MS ABC /MS res Residual «Difference» MS res = ˆσ 2 Total IJKn 1 ETH p. 15/31
16 ETH p. 16/31 Anova table Source SS df MS F P value medi bio diet medi:bio medi:diet bio:diet medi:bio:diet Residual Total
17 Half normal plot Sorted Data bio2/diet2 + medi Half normal quantiles ETH p. 17/31
18 ETH p. 18/31 Unbalanced Factorials uncorrelated estimators: SS tot = SS A +SS B +SS AB + correlated estimators: SS res }{{} SS C +...+SS res SS tot = SS A +SS B +SS AB +SS C +...+SS res SS Typ I: SS A ignores all other SS SS Typ II: SS A takes into account all other main effects, ignores all interactions SS Typ III: SS A takes into account all other effects
19 ETH p. 19/31 Calculation of SS s by model comparison For SS Typ I: model 1: Y ijk = µ+ǫ ijk model 2: Y ijk = µ+a i +ǫ ijk model 3: Y ijk = µ+a i +B j +ǫ ijk SS e1 = SS T SS e2 SS e3 model 4: Y ijk = µ+a i +B j +AB ij +ǫ ijk SS e4 = SS res
20 ETH p. 20/31 Rat genotype Litters of rats are separated from their natural mother and given to another female to raise. 2 factors: mother s genotype (A, B, I, J) and litter s genotype (A, B, I, J) response: average weight gain of the litter.
21 ETH p. 21/31 Full model > summary(aov(y mother*genotype,data=gen)) Df Sum Sq Mean Sq F value Pr(>F) mother * genotype mother:genotype Residuals > summary(aov(y genotype*mother,data=gen)) Df Sum Sq Mean Sq F value Pr(>F) genotype mother * genotype:mother Residuals
22 ETH p. 22/31 SS Typ III in R > drop1(mod1,test="f") Model: y mother * genotype Df Sum Sq RSS AIC F value Pr(F) <none> mother:geno > drop1(mod1,..,test="f") Model: y mother * genotype Df Sum Sq RSS AIC F value Pr(F) <none> mother * geno * mother:geno
23 ETH p. 23/31 Offer for a 6-year old car Planned experiment to see whether the offered cash for the same medium-priced car depends on gender or age (young, middle, elderly) of the seller. 6 factor combinations with 6 replications each. Response variable y is offer made by a car dealer (in $ 100) Covariable: overall sales volume of the dealer
24 ETH p. 24/31 Analysis of Covariance Covariates can reduce MS res, thereby increasing power for testing. Baseline or pretest values are often used as covariates. A covariate can adjust for differences in characteristics of subjects in the treatment groups. It should be related only to the response variable and not to the treatment variables (factors). We assume that the covariate will be linearly related to the response and that the relationship will be the same for all levels of the factor (no interaction between covariate and factors).
25 ETH p. 25/31 Model for two-way ANCOVA Y ijk = µ+θx ijk +A i +B j +(AB) ij +ǫ ijk Ai = B j = (AB)ij = 0, ǫ ijk N(0,σ 2 )
26 ETH p. 26/31 Effect of Age and Gender y y young middle elderly male female
27 ETH p. 27/31 Interaction effect of Age and Gender mean of cash$y cash$gende male female mean of cash$y cash$age middle young elderly young middle elderly cash$age male cash$gender female
28 ETH p. 28/31 Two-way Anova > mod1=aov(y age*gender,data=cash) > summary(mod1) Df Sum Sq MeanSq Fvalue Pr(>F) age e-12*** gender age:gender Residuals
29 ETH p. 29/31 Sales and Cash Offer cash$y cash$sales
30 ETH p. 30/31 Sales and Cash Offer by Group female young female middle female elderly y male young male middle male elderly sales
31 ETH p. 31/31 Two-way Ancova > mod2=aov(y sales+age*gender,data=cash) > summary(mod2) Df SumSq MeanSq Fvalue Pr(>F) sales < 2e-16*** age < 2e-16*** gender * age:gender Residuals
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