Information Technology Reference
In-Depth Information
Statistically, nonparallel functions are indicative of an
interaction,
and
that is the case here. Because the interaction is indicative of the differences
in patterns of reactions to the treatment, and because differences in pat-
terns of reactions are indicative of error variance, the interaction effect is
defined as the error term in the ANOVA. The two variables that interact are
(a) time of measurement, which represents our within-subjects indepen-
dent variable - our treatment effect - and (b) the subjects in the study.
Because this interaction effect is the error variance, the mean square asso-
ciated with the Treatment
×
Subjects interaction is used in the ANOVA
as the denominator (it is used as the error term) in the
F
ratio assessing
the effect of the within-subjects independent variable.
10.9 COMPUTING THE OMNIBUS ANALYSIS BY HAND
The procedures for conducting a one-way within-subjects ANOVA have
both some parallels to the one-way between-subjects design and some
very real differences. Table 10.2 depicts the basic observations of our
hypothetical drug therapy study, which we will refer to as the
AS
data
matrix. As can be seen, the eight subjects (or participants) are measured on
the dependent variable (symptom intensity) on five separate occasions (or
treatments). Each subject is designated with a lower case
s
and subscript
(read “little '
s
' one,” etc.), and the levels of Factor
A
continue to be
designated
a
1
,
a
2
,
a
3
, and so forth. Sums (
A
j
), sums of squared scores
(
Y
aj
), and means (
Y
j
) are computed for each treatment condition, as
in the single-factor between-subjects case we discussed in Chapter 6. A
new computational procedure, required of within-subjects designs, is to
sum each subject's score on the dependent variable across all treatment
conditions. We will designate these subject sums as
S
(read “big '
S
'”), and
they can be seen on the right side of Table 10.2. The
A
treatment sums
and the
S
subject sums both sum to the grand sum
T
.
10.9.1 SUM OF SQUARES
There are four sums of squares that we need to calculate for a one-way
within-subjects ANOVA, all of which are produced by manipulating the
three types of sums found in this design:
A
,
S
, and
T
.
Sum of squares between groups or treatments (
SS
A
)inwhichthe
subscript
A
represents the variability among the treatments of inde-
pendent variable or Factor
A
.
Sum of squares subjects (
SS
S
)inwhichthesubscript
S
reflects the
average variability among the subjects.
Sum of squares interaction (
SS
A
×
S
) (read “sum of squares
A
by
S
”)
reflects the treatments by subjects variability or how the individual
subjects respond to the various treatments. The larger this sum of
squares, the greater is the variability among the subjects.
Sum of squares total (
SS
T
).
