7.
Product and Process Comparisons
7.4. Comparisons based on data from more than two processes 7.4.3. Are the means equal?


Basic Layout  
The balanced 2way factorial layout 
Factor A has 1, 2, ..., a levels. Factor B has 1, 2, ...,
b levels. There are ab treatment combinations (or
cells) in a complete factorial layout. Assume that each treatment
cell has r independent obsevations (known as replications).
When each cell has the same number of replications, the design is
a balanced factorial. In this case, the abrdata points
{y_{ijk}} can be shown pictorially as follows:


How to obtain sums of squares for the balanced factorial layout 
Next, we will calculate the sums of squares needed for the ANOVA
table.
Finally, the total number of observations n in the experiment is abr. With the help of these expressions we arrive (omitting derivations) at
These expressions are used to calculate the ANOVA table entries for the (fixed effects) 2way ANOVA. 

TwoWay ANOVA Example:  
Data 
An evaluation of a new coating applied to 3 different materials was
conducted at 2 different laboratories. Each laboratory tested 3
samples from each of the treated materials. The results are given
in the next table:


Row and column sums 
The preliminary part of the analysis yields a table of row and
column sums.


ANOVA table 
From this table we generate the ANOVA table.
