STAT 213 Two-Way ANOVA II

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1 STAT 213 Two-Way ANOVA II Colin Reimer Dawson Oberlin College May 2, / 21

2 Outline Two-Way ANOVA: Additive Model FIT: Estimating Parameters ASSESS: Variance Decomposition Pairwise Comparisons 2 / 21

3 Outline Two-Way ANOVA: Additive Model FIT: Estimating Parameters ASSESS: Variance Decomposition Pairwise Comparisons 3 / 21

4 Alfalfa sprouts (Ex. 6.25) Some students were interested in the effect of acidic environments on plant growth. They planted alfalfa seeds in fifteen cups and randomly chose five to get plain water, five to get a moderate amount of acid and five to get a stronger acid solution. The cups were arranged indoors near a window in five rows of three with one cup from each Acid level in each row (with row a nearest the window, and row e farthest away). The response variable was average Height of the alfalfa sprouts after four days. A model: Height i = µ + α Acidi + ε i Acid i {water, moderate, strong} Any concerns about the ANOVA/regression conditions? The residuals might not be independent within rows! 4 / 21

5 Alfalfa Data Treatment/Row a b c d e Trt. mean water moderate acid strong acid Row mean Since each treatment is applied to each row, we can include row as a predictor. 5 / 21

6 Means Plots library("stat2data"); library("mosaic"); library("gplots") data("alfalfa") ## Using factor() to reorder the categories plotmeans(ht4 ~ factor(acid, levels = c("water", "1.5HCl", "3.0HCl")), data = Alfalfa, xlab = "Solution", ylab = "Height (in.)") Height (in.) n=5 n=5 n=5 water 1.5HCl 3.0HCl Solution 6 / 21

7 Means Plots plotmeans(ht4 ~ factor(row), data = Alfalfa, xlab = "Row", ylab = "Height (in.)") Height (in.) n=3 n=3 n=3 n=3 n=3 a b c d e Row 7 / 21

8 The One-way ANOVA Population Model (X categorical) Y i = f(x i ) + ε i Y = µ + α Xi + ε i, ε i N (0, σ 2 ε) One α X for each level of X: group deviation from overall mean The Two-way ANOVA Additive Model (A, B categorical) Y i = f(a i, B i ) + ε i Y i = µ + α Ai + β Bi + ε i, ε i N (0, σ 2 ε) One α A for each level of A ( row deviation from overall mean) One β B for each level of B ( column deviation from overall mean) 8 / 21

9 Height i = Concretely: Alfalfa Sprouts µ + α Water + β a if Acid = Water and Row = a µ + α Water + β b if Acid = Water and Row = b µ + α Water + β e if Acid = Water and Row = e µ + α HCl1.5 + β a if Acid = HCl1.5 and Row = a µ + α HCl1.5 + β e if Acid = HCl1.5 and Row = e µ + α HCl3.0 + β a if Acid = HCl3.0 and Row = a µ + α HCl3.0 + β e if Acid = HCl3.0 and Row = e 9 / 21

10 Outline Two-Way ANOVA: Additive Model FIT: Estimating Parameters ASSESS: Variance Decomposition Pairwise Comparisons 10 / 21

11 Population model: FIT: Parameter Estimation y A,B,i = µ + α A + β B + ε A,B,i where we let i index observations within combinations of A and B Estimate terms by ˆµ = Ȳ ( grand mean) ˆα A ˆβ B Ŷ A,B,i ˆε A,B,i = ȲA Ȳ ( row deviation) = ȲB Ȳ ( column deviation) = ˆµ + ˆα A + ˆβ B (predicted value) = Y A,B,i ŶA,B,i (residual) 11 / 21

12 Practice: Alfalfa Data Treatment/Row a b c d e Trt. mean water moderate acid strong acid Row mean Find: ˆµ, ˆα Water, ˆα moderate, ˆα strong, ˆβ a,..., ˆβ e 12 / 21

13 Outline Two-Way ANOVA: Additive Model FIT: Estimating Parameters ASSESS: Variance Decomposition Pairwise Comparisons 13 / 21

14 Sums of Squares Y A,B,i = ˆµ + ˆα A + ˆβ B + ε A,B,i (Y A,B,i ˆµ) 2 = (ˆα A + ˆβ B + ε A,B,i ) 2 SS A = n A,B ˆα A 2 = n A ˆα A 2 A B i=1 A SS B = n A,B ˆβ B 2 = n B ˆβ2 B A B i=1 B SS Error = n A,B ˆε 2 A,B,i ( doesn t simplify) A B i=1 SS T otal = n A,B (Y A,B,i ˆµ) 2 = SS A + SS B + SS Error A B i=1 14 / 21

15 Alfalfa: Sums of Squares Treatment Row i Height ˆµ ˆα ˆβ ˆε water a water b water e moderate a moderate e strong a strong e SS A = ˆα 2 SS B = ˆβ2 SS E = ˆε 2 15 / 21

16 The Two-Way ANOVA Table Source df SS M S F P Factor A J 1 Factor B K 1 Residuals N J K + 1 Total N 1 Pairs: Factor A has J = 3 levels, factor B has K = 5 levels, with one observation per cell. How many degrees of freedom in each row of the table above? 16 / 21

17 Two-Way ANOVA Table library(mosaic); library(stat2data); data(alfalfa) alfalfa.model <- aov(ht4 ~ Acid + Row, data = Alfalfa) summary(alfalfa.model) Df Sum Sq Mean Sq F value Pr(>F) Acid * Row Residuals Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Caution: The F tests here amount to sequential nested F -tests, so order matters if there is any collinearity (here there is none, since the design is perfectly balanced) 17 / 21

18 Getting Means ## Note: this only works if you used aov(), not lm() model.tables(alfalfa.model, type = "means") Tables of means Grand mean 1.74 Acid Acid 1.5HCl 3.0HCl water Row Row a b c d e / 21

19 Getting Effects (αs and βs) ## Note: this only works if you used aov(), not lm() model.tables(alfalfa.model, type = "effects") Tables of effects Acid Acid 1.5HCl 3.0HCl water Row Row a b c d e ## Notice that the alphas and betas each sum to zero ## This will happen when the data is perfectly balanced ## since overall average is unweighted average of group means ## (Otherwise the weighted sum is zero) 19 / 21

20 Outline Two-Way ANOVA: Additive Model FIT: Estimating Parameters ASSESS: Variance Decomposition Pairwise Comparisons 20 / 21

21 Post-Hoc Pairwise Comparisons library(desctools) comparisons <- PostHocTest(alfalfa.model, method = "hsd", ordered = TRUE) comparisons$acid %>% round(3) diff lwr.ci upr.ci pval 1.5HCl-3.0HCl water-3.0hcl water-1.5hcl comparisons$row %>% round(3) diff lwr.ci upr.ci pval c-a b-a d-a e-a b-c d-c e-c d-b e-b e-d / 21

STAT 213 Interactions in Two-Way ANOVA

STAT 213 Interactions in Two-Way ANOVA Colin Reimer Dawson Oberlin College 14 April 2016 Outline Last Time: Two-Way ANOVA Interaction Terms Reading Quiz (Multiple Choice) If there is no interaction present,