7.4.3.8. Models and calculations for the two-way ANOVA

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

7.4.3.8. Models and calculations for the two-way ANOVA

Basic Layout

The balanced 2-way 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:

Factor B

1 2 ... b

1 y₁₁₁, y₁₁₂, ..., y_11r y₁₂₁, y₁₂₂, ..., y_12r ... y_1b1, y_1b2, ..., y_1br

2 y₂₁₁, y₂₁₂, ..., y_21r y₂₂₁, y₂₂₂, ..., y_22r ... y_2b1, y_2b2, ..., y_2br

Factor
A .
. ... .... ...

a y_a11, y_a12, ..., y_a1r y_a21, y_a22, ..., y_a2r ... y_ab1, y_ab2, ..., y_abr

How to obtain sums of squares for the balanced factorial layout

Next, we will calculate the sums of squares needed for the ANOVA table.

Let A_i be the sum of all observations of level i of factor A, i = 1, ... ,a. The A_i are the row sums.
Let B_j be the sum of all observations of level j of factor B, j = 1, ...,b. The B_j are the column sums.
Let (AB)_ij be the sum of all observations of level i of A and level j of B. These are cell sums.
Let r be the number of replicates in the experiment; that is: the number of times each factorial treatment combination appears in the experiment.

Then the total number of observations for each level of factor A is rb and the total number of observations for each level of factor B is raand the total number of observations for each interaction is r.

Finally, the total number of observations n in the experiment is abr.

With the help of these expressions we arrive (omitting derivations) at

SS(total) = SUM(each observation)**2 - CM

SS(A) = {SUM[i=1 to a][A(i)**2]/(r*b)} - CM

SS(B) = {SUM[i=1 to b][B(i)**2]/(r*a)} - CM

SS(AB) = {SUM[i=1 to a]SUM[i=1 to b][(A*B)(ij)**2]/r} -
CM - SS(A) - SS(B)

SSE = SS(total) - SS(A) - SS(B) - SS(AB)

These expressions are used to calculate the ANOVA table entries for the (fixed effects) 2-way ANOVA.

Two-Way 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:


	Materials (B)
LABS (A)	1	2	3

	4.1	3.1	3.5
1	3.9	2.8	3.2
	4.3	3.3	3.6

	2.7	1.9	2.7
2	3.1	2.2	2.3
	2.6	2.3	2.5

Row and column sums

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


Material (B)

Lab (A)	1	2	3	Total (A_i)

1	12.3	9.2	10.3	31.8
2	8.4	6.4	7.5	22.3

Total (B_j)	20.7	15.6	17.8	54.1

ANOVA table

From this table we generate the ANOVA table.


Source	SS	df	MS	F	p-value

A	5.0139	1	5.0139	100.28	0
B	2.1811	2	1.0906	21.81	.0001
AB	0.1344	2	0.0672	1.34	.298
Error	0.6000	12	0.0500

Total (Corr)	7.9294	17