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Question 562:

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The two key advantages to a 2-way ANOVA over say running 2 1-Way ANOVAs are you can reduce your probability of a Type I error and you can test for interactions between the two-levels of your variable. A common example is there might not be any side-effects for drug A or drug B, but drug A and B together cause a side-effect.

 

Testing for main-effects in a 2-Way ANOVA is similar to the 1-Way ANOVA, we divide the Sum-of Squares by the degrees of freedom (levels-1). To get the F-ratio 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 no-longer 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 2-Way 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.

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