1. The Analysis of Variance

2. The Analysis of Variance (ANOVA)
Fisher’s technique for partitioning the sum of squares
More generally, ANOVA refers to a class of sampling or experimental designs with a continuous response variable and categorical predictor(s)
Ronald Aylmer Fisher ( )

3. Goal
The comparison of means among 2 or more groups that have been sampled randomly
Both regression and ANOVA are special cases of a more generalized linear model

4. ANOVA & Partitioning the Sum of Squares
Remember: total variation is the sum of the difference between each observation and the overall sample mean
Using ANOVA, we can partition the sum of squares among the different components in the model (the treatments, the error term, etc.)
Finally, we can use the results to test statistical hypotheses about the strength of particular effects

5. Symbols Y= measured response variable
= grand mean (for all observations)
= mean that is calculated for a particular subgroup (i)
= a particular datum (the jth observation of the ith subgroup)

6. EXAMPLE: Effects of early snowmelt on alpine plant growth
Three treatment groups (a = 3) and four replicate plots per treatment (n = 4):
Unmanipulated
Control: fitted with heating coils that are never activated
Treatment: warmed with permanent solar-powered heating coils that melt spring snow pack earlier in the year than normal

7. Effects of early snowmelt on alpine plant growth
After 3 years of treatment application, you measure the length of the flowering period, in weeks, for larkspur (Delphinium nuttallianum) in each plot

8. Data
Unmanipulated
Control
Treatment 10 9 12 11 13 15 16

9. Partitioning of the sum of squares in a one-way ANOVA9

10. SStotal=
SSag
SSwg
41.66 =
19.50

11. The Assumptions of ANOVA
The samples are randomly selected and independent of each other
The variance within each group is approximately equal to the variance within all the other groups
The residuals are normally distributed
The samples are classified correctly
The main effects are additive

12. Hypothesis tests with ANOVA
If the assumptions are met (or not severely violated), we can test hypotheses based on an underlying model that is fit to the data.
For the one way ANOVA, that model is:

13. The null hypothesis is
If the null hypothesis is true, any variation that occurs among the treatment groups reflects random error and nothing else.

14. ANOVA table for one-way layout
Source df
Sum of squares
Mean square
Expected mean square
F-ratio
Among groups
a-1
Within groups
a(n-1)
Total
an-1
P-value = tail probability from an F-distribution with (a-1) and a(n-1) degrees of freedom

15. Partitioning of the sum of squares in a one-way ANOVA15

16. ANOVA table for larkspur data
Source df
Sum of squares
Mean square
F-ratio
P-value
Among groups 2
22.16
11.08
5.11
0.033
Within groups 9
19.50
2.17
Total 11
41.67

17. Constructing F-ratios
Use the mean squares associated with the particular ANOVA model that matches your sampling or experimental design.
Find the expected mean square that includes the particular effect you are trying to measure and use it as the numerator of the F-ratio.

18. Constructing F-ratios (cont.’d)
Find a second expected mean square that includes all of the statistical terms in the numerator except for the single term you are trying to estimate and use it as the denominator of the F-ratio.
Divide the numerator by the denominator to get your F-ratio.

19. Constructing F-ratios (cont.’d)
Using statistical tables or the output from statistical software, determine the P-value associated with the F-ratio.
WARNING: The default settings used by many software packages will not generate the correct F-ratios for many common experimental designs.
Repeat steps 2 through 5 for other factors that you are testing.

20. ANOVA as linear regression
treatment
data X1 X2
unmanipulated 10 12 13
control 9 1 11
Treatment 15 16

21. EXAMPLE X1 X2
Expected
Unmanipulated
11.75
Control 1
10.75
Treatment
14.0
Coefficients
Value
Unmanipulated
Intercept
11.75
Control -1
Treatment
2.25

22. Regression Source of variation SS df MS Regression p-1 Residual n-p
Total
n-1

23. ANOVA table Source df Sum of squares Mean square F-ratio P-value
Regression 2
22.16
11.08
5.11
0.033
Residual 9
19.50
2.17
Total 11
41.67 23