Chapter 10 Analysis of Variance 411for an appliance manufacturer. Assembly line and shift are the two factors;
the assembly line factor has four levels, and the shift factor has two levels.
Each combination of the factors line and shift is called a cell, so there are
43258 cells. The response variable is the total number of microwaves
assembled in a week for one assembly line operating on one particular shift.
For each of the eight combinations of assembly line and shift, six separate
weeks’ worth of data are collected.
You can describe the mean number of microwaves created per week with
the effects model where
Mean number of microwaves 5 overall mean 1 assembly line effect
1 shift effect 1 interaction 1 error
Now let’s examine a possible model of how the mean number of micro-
waves produced could vary between shifts and assembly lines. Let the over-
all mean number of microwaves produced for all shifts and assembly lines
be 240 per week. Now let the four assembly line effects be A, +66 (that is,
assembly line A produces on average 66 more microwaves than the overall
mean); B, 22 ; C, 2100 ; and D, +36. Let the two shift effects be p.m., 26 , and
a.m., +6. Notice that the four assembly line effects add up to zero, as do the
two shift effects. This follows from the need to constrain the values of the
effect terms to avoid overparametrization, as was discussed with the one-
way effects model earlier in this chapter.
If you exclude the interaction term from the model, the population cell
means (the mean number of microwaves produced) look like this.
A B C D
p.m. 300 232 134 270
a.m. 312 244 146 282
These values are obtained by adding the overall mean + the assembly line
effect + the shift effect for each of the eight cells. For example, the mean for
the p.m. shift on assembly line A is
Overall mean 1 assembly line effect 1 shift effect 5240166265300Without interaction, the difference between the a.m. and the p.m. shifts
is the same (12) for each assembly line. You can say that the difference be-
tween a.m. and p.m. is 12 no matter which assembly line you are talking
about. This works the other way, too. For example, the difference between
line A and line C is the same (166) for both the p.m. shift 130021342 and
the a.m. shift^131221462. You might understand these relationships better
from a graph. Figure 10-13 shows a plot of the eight means with no interac-
tion (you don’t have to produce this plot).