Two-Way ANOVA

[1] Two Way Classification or Two-Way Analysis of Variance
This is another special case of the GLM (General Linear Model). In the GLM, the response variable is continuous and the explanatory variable is categorical or continuous. For the GLM information, click here! :D So, the Two-Way ANOVA has two factors, each with at least two levels. The main question is weather the treatment variable have an effect or not.   

What is the factor? A factor is a categorical predictor variable consisting of different class levels like various types of treatments. For example, if you want to predict the grade of the course you are taking, then there are several factors such that the number of hours you are studying, the number of assignments or a female/male student etc. 

[2] Assumptions
The samples must be independent, and selected by randomization condition.
The equal variance assumption and normal error assumption should be satisfied.

[3] Model and the Expected Values Example  
Consider the model for a two-way analysis of variance with two levels of each factor (a 2x2) classification. $Y_{i}=\beta_{0}+\beta_{1}I_{factor 1,i}+ \beta_{2}I_{factor 2,i}+\beta_{3}I_{factor 1,i}I_{factor 2,i}+e_{i}$ where $I_{factor 1,i}$ if the ith observation is in the first group of factor 1 and is 0 otherwise.

**The expected values of $Y_{i}$ for each of the 4 groups means are following.
1) i : 1st level of the factor 1, and 1st level of factor 2 : $E[Y_{i}]=\beta_{0}+\beta_{1}+\beta_{2}+\beta_{3}$
2) i : 1st level of the factor 1 and 2nd level of the factor 2: $E[Y_{i}]=\beta_{0}+\beta_{1}$
3) i: 2nd level of the factor 1 and 1st level of the factor 2: $E[Y_{i}]=\beta_{0}+\beta_{2}$
4) i : 2nd level of the factor 1 and 2nd level of the factor 2: $E[Y_{i}]=\beta_{0}$  

[4] The Full Model & The Reduced Model and Test Statistics 
The full model is a model with all explanatory variables, whereas the reduced model (or additive model) is a model without variables whose coefficient you are testing.

The test statistics is following below.
$F_{obs}= \frac{(SStrt_{full}-SStrt_{reduced}) / \#of\beta's \ being \ tested }{MSE_{full}}$