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

2. 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)

3. 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)

4. 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.

5. 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.

6. 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.

7. 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.

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

9. 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.

10. 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:
Cure B:
Cure C:
> mean(days[1:9])
> mean(days[10:18])
> mean(days[19:27])

11. 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:
Cure B:
Cure C:
results = aov(days ~ cure, data = concrete)
summary(results) # gave full ANOVA table

12. 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:
Cure B:
Cure C:
results = aov(days ~ cure, data = concrete)
results$coeff
(Intercept) cureB cureC

13. 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:
Cure B:
Cure C:
results = aov(days ~ cure, data = concrete)
results$coeff
(Intercept) cureB cureC

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