Question 562:
1Answer:
No answer provided yet.The two key advantages to a 2way ANOVA over say running 2 1Way ANOVAs are you can reduce your probability of a Type I error and you can test for interactions between the twolevels of your variable. A common example is there might not be any sideeffects for drug A or drug B, but drug A and B together cause a sideeffect.
Testing for maineffects in a 2Way ANOVA is similar to the 1Way ANOVA, we divide the Sumof Squares by the degrees of freedom (levels1). To get the Fratio we divide the MS for both factors by the MSerror and then test the significance of F.
With interaction effects we do the same thing. The major thing to consider with an interaction term however is that if we have a significant interaction effect, the main effects nolonger are meaningful to us. This is what the interaction term identified: there is a statistically significant change that occurs but only in one level of the variable, not both.
In the table below I have an example of a typical 2Way ANOVA table with the relevant values. You can see both main effects and interaction are significant, so we'd look at an interaction plot of the data to see what levels interact with A and B.
In the computations, there will always be an Ms Interaction value, but as this figure approaches 1 it no longer becomes statistically significant.
Source 
SS 
df 
MS 
F 
p 
A 
192.2 
1 
192.2 
40.68 
£.05 
B 
57.8 
1 
57.8 
12.23 
£.05 
AxB 
168.2 
1 
168.2 
35.60 
£.05 
Error 
75.6 
16 
4.725 
 
Total 
493.8 
19 

The best intuitive explanation I can get from reading all that algebra is that basically when you have random factors (model II), if they are correlated you get pieces of one with the main effect of the other one. If they are uncorrelated then the terms fall away. This doesnt happen in a fixed model (Model I) since you've already covered each combination of factors.