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table shows that 219,520.000 of the 230,497.143 is associated with the
linear component or trend. That leaves 10,977.143 remaining for all of the
other polynomial contrasts; this leftover is called
Deviation
in the
Linear
Te rm
portion of the summary table.
Of this remaining 10,977.143, we see that 7,546.939 is associated with
the quadratic trend. That leaves 3,430.204 for everything else. The cubic
trend handles 3,430.000 of that and only the quartic trend is left; it is
associated with a sum of squares of .204.
As for the degrees of freedom, we know that with five groups we have
4
df
associated with the between-groups source of variance. One degree of
freedom is associated with each of the polynomial contrasts; with four pos-
sible functions (linear, quadratic, cubic, and quartic), we have 4
df
in total.
Each polynomial trend is associated with an
F
ratio,whichinturnis
associated with a probability that we evaluate against our alpha level to
determine statistical significance. As can be seen from the summary table,
both the linear and quadratic components meet a .05 alpha level.
We can now compute eta squared values for our trends, and we will do
this with respect to the between-groups variance. The eta squared value
for the linear component is .95 (219, 520
95) and
the eta squared value for the quadratic component is .03 (7, 546
.
000
÷
230, 497
.
143
=
.
.
939
÷
230, 497
03). It does appear that the linear trend is dominant with
just a hint of some quadratic component to round out the picture.
.
143
=
.
7.25 COMMUNICATING THE RESULTS OF THE TREND ANALYSIS
In reporting the results of the trend analysis, we would construct a graph
exactly as shown in Figure 7.18 to show the function. Given that, one way
to report the results is as follows:
The amount of preparation for the SAT in which students engaged appeared
to significantly affect their performance on the test,
F
(4, 30)
=
43
.
47,
p
<
2
.
85. The function relating study time to performance is presented
in Figure 7.21. A trend analysis revealed that both the linear,
F
(1, 30)
05,
η
=
.
=
165
05, components were
statistically significant. The linear trend was the dominant one, accounting for
95 percent of the between-groups variance; it largely reflects the increases in
SAT scores from zero months of study to six months of study. The quadratic
component accounted for 3 percent of the between-groups variance and
reflects the relative lack of improvement in SAT scores between six months
and eight months of study.
.
59,
p
<.
05, and quadratic,
F
(1, 30)
=
5
.
69,
p
<.
7.26 USER-DEFINED CONTRASTS
User-defined contrasts are planned comparisons, and they ordinarily rep-
resent a small subset of all possible comparisons. They can be either
