Page 208 - Intermediate Statistics for Dummies

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Chapter 11: Getting a Little Interaction with Two-Way ANOVA
The main effect A: A term for the effect of factor A on the response
The main effect B: A 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 (see 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 variable (call it factor A);
and SSE is the variability left over as error. The purpose of this model is to test
to see whether the different levels of factor A produce different responses in
the y variable. The way you do it is by using Ho: µ 1 = µ 2 = . . . = µ i , where i is the
number of levels of factor A (the treatment variable). If you reject Ho, then
factor A (which separates the data into the groups being compared) is signifi-
cant. 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 187
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. (While 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). Go to Stat>Anova>Two-
way. Click on Column 1 in the left-hand box and it appears in the Response
box on the right-hand side. Click on Column 2 and it appears in the row factor
box; click on Column 3 and it appears in the column factor box. Click OK.
For example, suppose you have six data values in Column 1: 11, 21, 38, 14, 15,
62. Suppose Column 2 contains 1, 1, 1, 2, 2, 2, and Column 3 contains 1, 2, 3, 1,
2, 3. This means that factor A has two levels (1, 2), and factor B has three
levels (1, 2, 3). The number 11 was the response when Level 1 of factor A and
Level 1 of factor B were applied. The second data value, 21, came from Level
1 of A and Level 2 of B. The third value, 38, came from Level 1 of A and Level 3
of B. The fourth number, 14, came from Level 2 of A and Level 1 of B. The
number 15 is the response from Level 2 of A and Level 2 of B, and finally, the
number 62 corresponds to the result of Level 2 of A and Level 3 of B.