1. ANOVA: Analysis of Variance

2. An ANOVA situation Subjects: 25 patients with blisters
Treatments: Treatment A, Treatment B, Placebo
(randomly assigned)
Measurement: # of days until blisters heal
Data [and means]:
A: 5,6,6,7,7,8,9, [7.25]
B: 7,7,8,9,9,10,10, [8.875]
P: 7,9,9,10,10,10,11,12,13 [10.11]
Are these differences in the means significant?

3. H0: The means of all the groups are equal.
What does ANOVA do?
At its simplest (there are extensions) ANOVA tests the following hypotheses:
H0: The means of all the groups are equal.
Ha: Not all the means are equal
Doesn’t say how or which ones differ.
Can follow up with “multiple comparisons”

4. Standard deviations of each group are approximately equal
Conditions of ANOVA
Random
Normal/Large Sample
Standard deviations of each group are approximately equal
Rule of thumb: ratio of largest to smallest sample st. dev. must be less than 2:1
(largest sd/ smallest sd < 2)

5. Normality Check

6. Standard Deviation Check
Variable treatment N Mean Median StDev
days A B P
Compare largest and smallest standard deviations:
largest: 1.764
smallest: 1.458
1.764/1.458 = 1.21 < 2

7. How ANOVA works (outline)
ANOVA measures two sources of variation in the data and compares their relative sizes
variation BETWEEN groups
for each data value look at the difference between its group mean and the overall mean
variation WITHIN groups
for each data value we look at the difference between that value and the mean of its group

8. The ANOVA F-statistic is a ratio of the Between Group Variation divided by the Within Group Variation:
A large F is evidence against H0, since it indicates that there is more difference between groups than within groups.

10. Degrees of Freedom
The between group df (numerator) is one less than the number of groups
We have three groups, so df(numerator) = 3 – 1 = 2
The within group df is the sum of the individual df’s of each group or Total number of observations – number of groups
The sample sizes are 8, 8, and 9
df(denominator) = = 22 or 25 – 3 = 22

11. Our Data Variable treatment N Mean Median StDev
days A B P

12. Calculations 1. Calculate the overall mean for our data.
2. Calculate MSG
Calculate MSE
4. Calculate the F-statistic. F = MSG
MSE

13. Minitab ANOVA Output Analysis of Variance for days Source DF SS MS F P
treatment
Error
Total

14. So How big is F? Mean Square Between / Mean Square Within = MSG / MSE
Since F is
Mean Square Between / Mean Square Within
= MSG / MSE
A large value of F indicates relatively more
difference between groups than within groups (evidence against H0)
To get the P-value, we will need to use the F-distribution. Be thankful for technology!!!

15. Our Original Situation
Subjects: 25 patients with blisters
Treatments: Treatment A, Treatment B, Placebo
(randomly assigned)
Measurement: # of days until blisters heal
Data [and means]:
A: 5,6,6,7,7,8,9,10
B: 7,7,8,9,9,10,10,11
P: 7,9,9,10,10,10,11,12,13
Put each treatment into a list. Run an ANOVA test on your calculator.