Information Technology Reference
In-Depth Information
scores in each treatment condition can be summed as (
Y
i
1
,
Y
i
2
,
Y
i
3
,
Y
5
). We can also
represent these sums more simply with a capital letter
A
and a numerical
subscript, which yields
A
1
,
A
2
,
A
3
, and so forth (read as “big 'A' one,” “big
'A' two,” “big 'A' three”). These sums represent the aggregated total score
across all participants within a single treatment condition. By adding these
five sums (
A
1
+
Y
i
4
,
Y
i
5
), or more simply (
Y
1
,
Y
2
,
Y
3
,
Y
4
,
A
2
+
A
3
+
A
4
+
A
5
) or by summing all participants'
scores (
Y
ij
) we obtain the grand total, represented by
T
.
In addition to identifying the individual (
Y
ij
or simply
Y
)scores,
summing them, and generating a grand total, we also need to create a
sum of squared scores for each treatment group; this is designated
Y
aj
Y
a
5
.Thesenewsumsare
calculated by squaring the individual
Y
scores for each treatment and then
summing them. The sums of the squared scores are then added together
to produce a total sum of squared scores, designated
Y
a
1
,
Y
a
2
,
Y
a
3
,
Y
a
4
,and
or more simply
Y
2
.
Alowercase
n
with a subscripted number (read “little '
n
' one” or “little
'
n
' two”) is used to refer to the number of observations within a particular
treatment group. The total number of observations or cases is designated
with a capital
N
(read “big
N
”). For a one-way between-subjects design
with equal group sizes,
N
can be obtained by multiplying the number of
groups by the number of cases contained in each group; thus
N
Y
aj
or simply,
(
a
)(
n
).
In all the ANOVA examples in this text, sample size (
n
)willalwaysbeequal
across independent variable groups, in order to keep the computational
procedures and formulas as clear as possible.
As we noted previously, treatment group means are calculated by
dividing th
e t
reatment sum
A
1
by
n
1
and
A
2
by
n
2
, and so forth, and is
de
signated
Y
(read “
Y
bar”) with a subscripted number. A grand mean
Y
T
(read as “
Y
bar
T
”) can also be calculated by dividing the total number
of scores by the total number of observations in the study.
=
6.5.2 SOURCES OF VARIANCE
The summary table for this design is shown in Table 6.2. In a between-
subjects design, our focus is on between-subjects variance. We have
emphasized this point by showing a general heading for the summary
table as “between-subjects effects.” SPSS and SAS will use a similar head-
ing for the summary tables that they produce.
The first column of a summary table contains the sources of variance.
Total variance is always placed in the last row of the table. Above the total
variance are the only two sources we have in a one-way between-subjects
design. One source is a known source and represents the independent
variable. This is labeled as
between groups
in the summary table and refers,
generically, to Factor
A
. In the example study, this represents the five levels
of the independent variable: study preparation time in months (zero, two,
four, six, and eight). The other source is the
within groups
or
error
. This
represents variation within the groups that is unable to be linked to the
independent variable.
