one way anova ss total ss between ss within logic of two way anova
TRANSCRIPT
One Way ANOVA SStotal
SSbetween SSwithin
Logic of Two Way ANOVA
Two Way ANOVA SStotal
SSbetween SSwithin
Logic of Two Way ANOVA
SSfactorA SSfactorB SSinteraction
Main Effect 1
2
2
w
factorA
s
sF
Main Effect 2
2
2
w
factorB
s
sF
Interaction
2
2
w
ninteractio
s
sF
Logic of Two Way ANOVA
This is no surprise
2)( XXSStot 2)( cellw XXSS
2)( XXnSS cellcellB
This is news2)( XXnSS rowrowrowfactor
2)( XXnSS colcolcolfactor colfrowfwithintoteraction SSSSSSSSSS int
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Step 1: Get the cell means, row means, column means, and grand mean
2X 5X 2X
2X 1X 0X
3X
1X
2X 3X 1X 2X
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Step 1: Get the SStotal
2X
= 56
2)( XXSStot
2X 5X 2X
2X 1X 0X
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Step 2: Get the SSwithin
2X
= 14
2)( cellw XXSS
2X 5X 2X
2X 1X 0X
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Step 3: Get the SStime
2X
= 18
2)( XXnSS rowrowrowfactor
3X
1X
2X 5X 2X
2X 1X 0X
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Step 4: Get the SStemp
2X
= 12
2)( XXnSS colcolcolfactor
2X 3X 1X
2X 5X 2X
2X 1X 0X
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Step 5: Get the SSinteraction
= 12
SStotal
SSbetween SSwithin
SStime SStemp SSinteraction
= 12= 18
= 14
= 56
2X 5X 2X
2X 1X 0X
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Step 6: Finish the table
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Source SS df s2 F
Between
SStime 18
SStemp 12
SStimXtemp 12
Within 14
Total 56
Step 6: Finish the table
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Source SS df s2 F
Between
SStime 18 1
SStemp 12 2
SStimXtemp 12 2
Within 14 12
Total 56 17
Step 6: Finish the table
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Source SS df s2 F
Between
SStime 18 1 18
SStemp 12 2 6
SStimXtemp 12 2 6
Within 14 12 1.17
Total 56 17
Step 6: Finish the table
Steps for Two Way ANOVA
Morning
Afternoon
70o50o 90o
1 2 3 4 5 6 3 1 2
1 4 1 0 1 2 0 0 0
Source SS df s2 F
Between
SStime 18 1 18 15.38
SStemp 12 2 6 5.12
SStimXtemp 12 2 6 5.12
Within 14 12 1.17
Total 56 17
Step 6: Finish the table
p < .01
p < .05
p < .05
Steps for Two Way ANOVA
Since there are only two levels of time, you know they are significantly different from one another
Since there are three levels of temp, you will need to use post-hoc multiple comparisons (e.g. Tukey’s HSD) to determine which levels differ from one another.
Steps for Two Way ANOVA
To understand the interaction more fully, you might want to look at the “simple effects”
You could compute the effect of temp just in the morning, for example.
You would simply compute SStemp at AM
Assumptions for the Two Factor ANOVA
1. Observations within each sample are independent.
2. Populations are normally distributed.
3. Populations from which the samples are selected must have equal variances (homogeneity of variance).
4. Equal cell Ns (if calculating by hand)