Workshop 7.4a: Single factor ANOVA

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-1- Workshop 7.4a: Single factor ANOVA Murray Logan November 23, 2016 Table of contents 1 Revision 1 2 Anova Parameterization 2 3 Partitioning of variance (ANOVA) 10 4 Worked Examples 13 1. Revision 1.1. Estimation Y X 3 0 2.5 1 6 2 5.5 3 9 4 8.6 5 12 6 3.0 = β 0 1 + β 1 0 + ε 1 2.5 = β 0 1 + β 1 1 + ε 1 6.0 = β 0 1 + β 1 2 + ε 2 5.5 = β 0 1 + β 1 3 + ε 3 9.0 = β 0 1 + β 1 4 + ε 4 8.6 = β 0 1 + β 1 5 + ε 5 12.0 = β 0 1 + β 1 6 + ε 6 1.2. Estimation 3.0 = β 0 1 + β 1 0 + ε 1 2.5 = β 0 1 + β 1 1 + ε 1 6.0 = β 0 1 + β 1 2 + ε 2 5.5 = β 0 1 + β 1 3 + ε 3 9.0 = β 0 1 + β 1 4 + ε 4 8.6 = β 0 1 + β 1 5 + ε 5 12.0 = β 0 1 + β 1 6 + ε 6

-2-3.0 1 0 2.5 1 1 6.0 1 2 5.5 = 1 3 9.0 1 4 8.6 1 5 12.0 1 6 }{{}}{{} Response values Model matrix ( ) β0 + β 1 }{{} Parameter vector ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 }{{} Residual vector 1.3. Matrix algebra 3.0 1 0 2.5 1 1 ε 1 6.0 1 2 ( ) ε 2 5.5 = 1 3 β0 + ε 3 9.0 1 4 β 1 }{{} ε 4 8.6 1 5 ε Parameter vector 5 ε 12.0 1 6 6 }{{}}{{}}{{} Residual vector Response values Model matrix Y = X β + ϵ 2. Anova Parameterization ˆβ = (X X ) 1 X Y 2.1. Simple ANOVA Three treatments (One factor - 3 levels), three replicates

-3-2.2. Simple ANOVA Two treatments, three replicates

-4-2.3. Categorical predictor Y A dummy1 dummy2 dummy3 --- --- -------- -------- -------- 2 G1 1 0 0 3 G1 1 0 0 4 G1 1 0 0 6 G2 0 1 0 7 G2 0 1 0 8 G2 0 1 0 10 G3 0 0 1 11 G3 0 0 1 12 G3 0 0 1 y ij = µ + β 1 (dummy 1 ) ij + β 2 (dummy 2 ) ij + β 3 (dummy 3 ) ij + ε ij 2.4. Overparameterized y ij = µ + β 1 (dummy 1 ) ij + β 2 (dummy 2 ) ij + β 3 (dummy 3 ) ij + ε ij

-5- Y A Intercept dummy1 dummy2 dummy3 --- --- ----------- -------- -------- -------- 2 G1 1 1 0 0 3 G1 1 1 0 0 4 G1 1 1 0 0 6 G2 1 0 1 0 7 G2 1 0 1 0 8 G2 1 0 1 0 10 G3 1 0 0 1 11 G3 1 0 0 1 12 G3 1 0 0 1 2.5. Overparameterized y ij = µ + β 1 (dummy 1 ) ij + β 2 (dummy 2 ) ij + β 3 (dummy 3 ) ij + ε ij three treatment groups four parameters to estimate need to re-parameterize 2.6. Categorical predictor y i = µ + β 1 (dummy 1 ) i + β 2 (dummy 2 ) + β 3 (dummy 3 ) i + ε i 2.6.1. Means parameterization y i = β 1 (dummy 1 ) i + β 2 (dummy 2 ) i + β 3 (dummy 3 ) i + ε ij y ij = α i + ε ij i = p 2.7. Categorical predictor 2.7.1. Means parameterization y i = β 1 (dummy 1 ) i + β 2 (dummy 2 ) i + β 3 (dummy 3 ) i + ε i Y A dummy1 dummy2 dummy3 --- --- -------- -------- -------- 2 G1 1 0 0 3 G1 1 0 0 4 G1 1 0 0 6 G2 0 1 0 7 G2 0 1 0

-6-8 G2 0 1 0 10 G3 0 0 1 11 G3 0 0 1 12 G3 0 0 1 2.8. Categorical predictordd 2.8.1. Means parameterization Y A 1 2.00 G1 2 3.00 G1 3 4.00 G1 4 6.00 G2 5 7.00 G2 6 8.00 G2 7 10.00 G3 8 11.00 G3 9 12.00 G3 y i = α 1 D 1i + α 2 D 2i + α 3 D 3i + ε i y i = α p + ε i, where p = number of levels of the factor and D = dummy variables 2 1 0 0 ε 1 3 1 0 0 ε 2 4 1 0 0 ε 3 6 0 1 0 α 1 ε 4 7 = 0 1 0 α 2 + ε 5 8 0 1 0 α 3 ε 6 10 0 0 1 ε 7 11 0 0 1 ε 8 12 0 0 1 ε 9 2.9. Categorical predictor 2.9.1. Means parameterization Parameter Estimates Null Hypothesis α1 mean of group 1 H 0 : α 1 = α 1 = 0 α2 mean of group 2 H 0 : α 2 = α 2 = 0 α3 mean of group 3 H 0 : α 3 = α 3 = 0 > summary(lm(y~-1+a))$coef Estimate Std. Error t value Pr(> t ) AG1 3 0.5773503 5.196152 2.022368e-03 AG2 7 0.5773503 12.124356 1.913030e-05 AG3 11 0.5773503 19.052559 1.351732e-06 but typically interested exploring effects 2.10. Categorical predictor y i = µ + β 1 (dummy 1 ) i + β 2 (dummy 2 ) i + β 3 (dummy 3 ) i + ε i

-7-2.10.1. Effects parameterization y ij = µ + β 2 (dummy 2 ) ij + β 3 (dummy 3 ) ij + ε ij 2.11. Categorical predictor 2.11.1. Effects parameterization y ij = µ + α i + ε ij i = p 1 y i = α + β 2 (dummy 2 ) i + β 3 (dummy 3 ) i + ε i Y A alpha dummy2 dummy3 --- --- ------- -------- -------- 2 G1 1 0 0 3 G1 1 0 0 4 G1 1 0 0 6 G2 1 1 0 7 G2 1 1 0 8 G2 1 1 0 10 G3 1 0 1 11 G3 1 0 1 12 G3 1 0 1 2.12. Categorical predictor 2.12.1. Effects parameterization Y A 1 2.00 G1 2 3.00 G1 3 4.00 G1 4 6.00 G2 5 7.00 G2 6 8.00 G2 7 10.00 G3 8 11.00 G3 9 12.00 G3 y i = α + β 2D 2i + β 3D 3i + ε i y i = α p + ε i, where p = number of levels of the factor minus 1 and D = dummy variables 2 1 0 0 ε 1 3 1 0 0 ε 2 4 1 0 0 ε 3 6 1 1 0 µ ε 4 7 = 1 1 0 α 2 + ε 5 8 1 1 0 α 3 ε 6 10 11 12 1 0 1 1 0 1 1 0 1 ε 7 ε 8 ε 9 2.13. Categorical predictor 2.13.1. Treatment contrasts Parameter Estimates Null Hypothesis Intercept mean of control group H 0 : µ = µ 1 = 0

-8- Parameter Estimates Null Hypothesis α 2 α 3 mean of group 2 minus mean of control group mean of group 3 minus mean of control group H 0 : α 2 = α 2 = 0 H 0 : α 3 = α 3 = 0 > contrasts(a) <-contr.treatment > contrasts(a) 2 3 G1 0 0 G2 1 0 G3 0 1 > summary(lm(y~a))$coef Estimate Std. Error t value Pr(> t ) (Intercept) 3 0.5773503 5.196152 2.022368e-03 A2 4 0.8164966 4.898979 2.713682e-03 A3 8 0.8164966 9.797959 6.506149e-05 2.14. Categorical predictor 2.14.1. Treatment contrasts Parameter Estimates Null Hypothesis Intercept mean of control group H 0 : µ = µ 1 = 0 α 2 α 3 mean of group 2 minus mean of control group mean of group 3 minus mean of control group H 0 : α 2 = α 2 = 0 H 0 : α 3 = α 3 = 0 > summary(lm(y~a))$coef Estimate Std. Error t value Pr(> t ) (Intercept) 3 0.5773503 5.196152 2.022368e-03 A2 4 0.8164966 4.898979 2.713682e-03 A3 8 0.8164966 9.797959 6.506149e-05 2.15. Categorical predictor 2.15.1. User defined contrasts User defined contrasts Grp2 vs Grp3 Grp1 vs (Grp2 & Grp3)

-9- Grp1 Grp2 Grp3 α 2 0 1-1 α 3 1-0.5-0.5 > contrasts(a) <- cbind(c(0,1,-1),c(1, -0.5, -0.5)) > contrasts(a) [,1] [,2] G1 0 1.0 G2 1-0.5 G3-1 -0.5 2.16. Categorical predictor 2.16.1. User defined contrasts p 1 comparisons (contrasts) all contrasts must be orthogonal 2.17. Categorical predictor 2.17.1. Orthogonality Four groups (A, B, C, D) p 1 = 3 comparisons 1. A vs B :: A > B 2. B vs C :: B > C 3. A vs C :: 2.18. Categorical predictor 2.18.1. User defined contrasts > contrasts(a) <- cbind(c(0,1,-1),c(1, -0.5, -0.5)) > contrasts(a) [,1] [,2] G1 0 1.0 G2 1-0.5 G3-1 -0.5 0 1.0 = 0 1 0.5 = 0.5 1 0.5 = 0.5 sum = 0

-10-2.19. Categorical predictor 2.19.1. User defined contrasts > contrasts(a) <- cbind(c(0,1,-1),c(1, -0.5, -0.5)) > contrasts(a) [,1] [,2] G1 0 1.0 G2 1-0.5 G3-1 -0.5 > crossprod(contrasts(a)) [,1] [,2] [1,] 2 0.0 [2,] 0 1.5 > summary(lm(y~a))$coef Estimate Std. Error t value Pr(> t ) (Intercept) 7 0.3333333 21.000000 7.595904e-07 A1-2 0.4082483-4.898979 2.713682e-03 A2-4 0.4714045-8.485281 1.465426e-04 2.20. Categorical predictor 2.20.1. User defined contrasts > contrasts(a) <- cbind(c(1, -0.5, -0.5),c(1,-1,0)) > contrasts(a) [,1] [,2] G1 1.0 1 G2-0.5-1 G3-0.5 0 > crossprod(contrasts(a)) [,1] [,2] [1,] 1.5 1.5 [2,] 1.5 2.0 3. Partitioning of variance (ANOVA) 3.1. ANOVA

-11-3.1.1. Partitioning variance 3.2. ANOVA 3.2.1. Partitioning variance > anova(lm(y~a)) Analysis of Variance Table Response: Y Df Sum Sq Mean Sq F value Pr(>F) A 2 96 48 48 0.0002035 *** Residuals 6 6 1 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 3.3. Categorical predictor 3.3.1. Post-hoc comparisons No. of Groups No. of comparisons Familywise Type I error probability 3 3 0.14 5 10 0.40 10 45 0.90

-12-3.4. Categorical predictor 3.4.1. Post-hoc comparisons Bonferoni > summary(lm(y~a))$coef Estimate Std. Error t value Pr(> t ) (Intercept) 7 0.3333333 21.000000 7.595904e-07 A1-8 0.9428090-8.485281 1.465426e-04 A2 4 0.8164966 4.898979 2.713682e-03 > 0.05/3 [1] 0.01666667 3.5. Categorical predictor 3.5.1. Post-hoc comparisons Tukey s test > library(multcomp) > data.lm<-lm(y~a) > summary(glht(data.lm, linfct=mcp(a="tukey"))) Simultaneous Tests for General Linear Hypotheses Multiple Comparisons of Means: Tukey Contrasts Fit: lm(formula = Y ~ A) Linear Hypotheses: Estimate Std. Error t value Pr(> t ) G2 - G1 == 0 4.0000 0.8165 4.899 0.00653 ** G3 - G1 == 0 8.0000 0.8165 9.798 < 0.001 *** G3 - G2 == 0 4.0000 0.8165 4.899 0.00679 ** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Adjusted p values reported -- single-step method) 3.6. Assumptions Normality Homogeneity of variance Independence As for regression

-13-4. Worked Examples 4.1. Worked Examples > day <- read.csv('../data/day.csv', strip.white=t) > head(day) TREAT BARNACLE 1 ALG1 27 2 ALG1 19 3 ALG1 18 4 ALG1 23 5 ALG1 25 6 ALG2 24 4.2. Worked Examples Question: what effects do different substrate types have on barnacle recruitment Linear model: 4.3. Worked Examples Barnacle i = µ + α j + ε i ε N (0, σ 2 ) > partridge <- read.csv('../data/partridge.csv', strip.white=t) > head(partridge) GROUP LONGEVITY 1 PREG8 35 2 PREG8 37 3 PREG8 49 4 PREG8 46 5 PREG8 63 6 PREG8 39 > str(partridge) 'data.frame': 125 obs. of 2 variables: $ GROUP : Factor w/ 5 levels "NONE0","PREG1",..: 3 3 3 3 3 3 3 3 3 3... $ LONGEVITY: int 35 37 49 46 63 39 46 56 63 65... 4.4. Worked Examples Question: what effects does mating have on the longevity of male fruitflies Linear model: Longevity i = µ + α j + ε i ε N (0, σ 2 )

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