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This error component represents the errors made driving a sports car and
an SUV on a student by student basis. If all of the students showed exactly
the same pattern for driving the car and the SUV (averaging over the
number of drinks), the functions would be completely parallel and the
value of the sum of squares for this interaction would be zero.
11.5.2 THE MAIN EFFECT OF DRINKS
The error term for the main effect of Drinks is represented by the
Drinks
×
×
S
) in the summary
table. This error term focuses on the driving errors made as a function
of alcohol consumed on a student by student basis. Again, if all students
showed exactly the same pattern of errors (averaging across the vehicles
they drove), then there would be no unpredictability and the sum of
squares for the interaction would be zero.
Subjects interaction, shown as Error (
B
11.5.3 THE TWO-WAY INTERACTION
The error term for the interaction follows the same logic that we just
discussedforthemaineffects.Itisreallytheinteractionoftheeffect(which
is in this case is the Vehicle
×
Drinks interaction itself) with subjects. The
Vehicle
B
) interaction is based on the cells of the design.
We had laid out the matrix for the design earlier in Figure 11.1. Imagine
one student's driving scores in that matrix. Those scores would reflect a
particular pattern. If all of the other students had scores that fit that pattern
exactly, the Vehicle
×
Drinks (
A
×
×
Drinks
×
Subjects interaction - the
A
×
B
×
S
-
would yield a sum of squares of zero.
11.6 COMPUTING THE OMNIBUS TWO-FACTOR
WITHIN-SUBJECTS ANOVA BY HAND
The computational process for a two-factor within-subjects ANOVA is
both a direct extension of the computational procedures we covered in
Chapter 10 for the single-factor within-subjects design, and also parallels
some of the computational procedures we encountered with the two-factor
between-subjects ANOVA design of Chapter 8. We will continue with
the hypothetical example introduced in this chapter. The independent
variables are type of vehicle (Factor
A
) and amount of alcohol consumed
(Factor
B
). The type of vehicle independent variable has two levels (
a
1
=
car,
a
2
=
SUV) and amount of alcohol consumed has three levels (
b
1
=
0
drinks,
b
2
=
3 drinks). The dependent variable is the number
of driving errors. This combination of independent variables produces a
2
1 drink,
b
3
=
3 factorial and is displayed with the raw data in the first matrix (
ABS
Matrix) of Table 11.2. The
ABS
Matrix is accompanied by the usual sums
of scores and sums of squared scores, needed for future calculations. The
AB
matrix of means can be used to develop a plot of the treatment effects
as in Figure 11.2.
Three additional matrices (
AB
matrix,
AS
matrix, and
BS
matrix)
also need to be created from the original
ABS
matrix and can be seen
×
