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Chapter 11: Finding Your Way through Two-Way ANOVA 193

If Factor A has i levels and Factor B has j levels, you have i * j different combi-
nations of treatments in your two-way ANOVA model.

Stepping through the sums of squares

The two-way ANOVA model contains the following three terms:

✓ The main effect A: Term for the effect of Factor A on the response
✓ The main effect B: Term for the effect of Factor B on the response
✓ The interaction of A and B: The effect of the combination of Factors A
and B (denoted AB)

The sums of squares equation for the one-way ANOVA (which I cover in
Chapter 9) is SSTO = SST + SSE, where SSTO is the total variability in the
response variable, y; SST is the variability explained by the treatment vari-
able (call it factor A); and SSE is the variability left over as error.

The purpose of a one-way ANOVA model is to test to see whether the differ-
ent levels of Factor A produce different responses in the y variable. The way
you do it is by using Ho: μ = μ = . . . = μ , where i is the number of levels of
1 2 i
Factor A (the treatment variable). If you reject Ho, Factor A (which separates
the data into the groups being compared) is significant. If you can’t reject Ho,
you can’t conclude that Factor A is significant.
In the two-way ANOVA, you add another factor to the mix (B) plus an interac-
tion term (AB). The sums of squares equation for the two-way ANOVA model
is SSTO = SSA + SSB + SSAB + SSE. Here, SSTO is the total variability in the
y-values; SSA is the sums of squares due to Factor A (representing the variabil-
ity in the y-values explained by Factor A); and similarly for SSB and Factor B.
SSAB is the sums of squares due to the interaction of Factors A and B, and SSE
is the amount of variability left unexplained, and deemed error.

Although the mathematical details of all the formulas for these terms are
unwieldy and beyond the focus of this book, they just extend the formulas for
one-way ANOVA found in Chapter 9. ANOVA handles the calculations for you,
so you don’t have to worry about that part.
To carry out a two-way ANOVA in Minitab, enter your data in three columns.

✓ Column 1 contains the responses (the actual data).
✓ Column 2 represents the level of Factor A (Minitab calls it the row factor).
✓ Column 3 represents the level of Factor B (Minitab calls it the column
factor).

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