STAT 312 10.1 - Single-Factor ANOVA Chapter 10 - Analysis of Variance (ANOVA) Introduction 10.1 - Single-Factor ANOVA 10.2 - Multiple Comparisons in ANOVA 10.3 - More on Single-Factor ANOVA

Grand Mean (Estimator) For simplicity, take k = 3 treatment groups, independent, normal, equivariant: equivariant: 1 2 = H0: (True) Grand Mean ith group (row) jth entry (col) Group Samples Group Means i = 1 i = 2 i = 3 Grand Mean (Estimator)

Grand Mean (Estimator) For simplicity, take k = 3 treatment groups, independent, normal, equivariant: equivariant: 1 2 = H0: (True) Grand Mean ith group (row) jth value (col) Group Samples Group Means i = 1 i = 2 i = 3 Grand Mean (Estimator)

How far is each group mean i from the grand mean ? In general… k groups Moreover… If H0 is true, then each i = 0!!! So… (True) Group Means (True) Grand Mean How far is each group mean i from the grand mean ? Recall… The sum of deviations of any set of values from its mean is 0.

k groups In general… Moreover… If H0 is true, then each i = 0!!! So… (True) Grand Mean Recall… The sum of deviations of any set of values from its mean is 0.

How far is each sample value Yij from its group mean i? In general… k groups (True) Group Means (True) Grand Mean How far is each sample value Yij from its group mean i? Recall… The sum of deviations of any set of values from its mean is 0.

k groups In general… (True) Group Means (True) Grand Mean “residuals” ANOVA Model Recall… The sum of deviations of any set of values from its mean is 0.

k groups Moreover… In general… (True) Group Means “residuals” ANOVA Model Residuals are independent, normally distributed about 0, and equivariant.

X1 and X2 are called indicator or dummy variables. For simplicity, take k = 3 treatment groups, independent, normal, equivariant: 1 2 = H0: “reference group” ANOVA Model X1 and X2 are called indicator or dummy variables.

For simplicity, take k = 3 treatment groups, independent, normal, equivariant: 1 2 = H0: “reference group” ANOVA Model > days = c(4,5,4,3,...,,5,6,5,5) Cure A: 4 5 4 3 2 4 3 4 4 Cure B: 6 8 4 5 4 6 5 8 6 Cure C: 6 7 6 6 7 5 6 5 5 > mean(days[1:9]) 3.666667 > mean(days[10:18]) 5.777778 > mean(days[19:27]) 5.888889

For simplicity, take k = 3 treatment groups, independent, normal, equivariant: 1 2 = H0: “reference group” ANOVA Model cure = c(rep("A",9),..., rep("C",9)) concrete = data.frame(days,cure) Cure A: 4 5 4 3 2 4 3 4 4 Cure B: 6 8 4 5 4 6 5 8 6 Cure C: 6 7 6 6 7 5 6 5 5 results = aov(days ~ cure, data = concrete) summary(results) # gave full ANOVA table

For simplicity, take k = 3 treatment groups, independent, normal, equivariant: 1 2 = H0: “reference group” ANOVA Model cure = c(rep("A",9),..., rep("C",9)) concrete = data.frame(days,cure) Cure A: 4 5 4 3 2 4 3 4 4 Cure B: 6 8 4 5 4 6 5 8 6 Cure C: 6 7 6 6 7 5 6 5 5 results = aov(days ~ cure, data = concrete) results$coeff (Intercept) cureB cureC 3.666667 2.111111 2.222222

For simplicity, take k = 3 treatment groups, independent, normal, equivariant: 1 2 = H0: “reference group” ANOVA Model cure = c(rep("A",9),..., rep("C",9)) concrete = data.frame(days,cure) Cure A: 4 5 4 3 2 4 3 4 4 Cure B: 6 8 4 5 4 6 5 8 6 Cure C: 6 7 6 6 7 5 6 5 5 results = aov(days ~ cure, data = concrete) results$coeff (Intercept) cureB cureC 3.666667 2.111111 2.222222

= H0: ANOVA Model Linear Regression For simplicity, take k = 3 treatment groups, independent, normal, equivariant: 1 2 = H0: “reference group” ANOVA Model Linear Regression