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table of Coefficients of Orthogonal Polynomials in Appendix E. Our new
set of quadratic coefficients is as follows:
2
1
2
12
.
Hence,
ˆ
ψ quadratic =
.
+
.
(2)(412
86)
(
1)(474
29)
+
(
2)(552
.
86)(
1)(614
.
29)(2)(623
.
86)
=
825
.
72
474
.
29
1, 105
.
72
614
.
29
+
1, 247
.
72
= −
120.876
.
Thus,
n ( ˆ
ψ quadratic ) 2
86) 2
(7)(
120
.
SS A quadratic =
=
=
7,303.57
.
c 2
14
(7.11)
Next we calculate degrees of freedom, mean squares, and the F ratio.
The SS A quadratic has 1 df associated with it ( df A quadratic =
1); thus, MS A quadratic =
SS A quadratic /
1or MS A quadratic =
SS A quadratic .Thus,our F ratio becomes
MS A quadratic
MS S / A
7, 303
.
57
F A quadratic =
=
73 =
5.51
.
(7.12)
.
1, 325
This F value is evaluated with the same degrees of freedom as the
F A linear (1, 30) and is statistically significant at alpha
.05. This result
indicates a statistically significant quadratic trend is present in the data.
SAT scores improve progressively as the number of months of study time
increases; however, these scores appear to asymptote by six months of
study.
=
7.23 PERFORMING POLYNOMIAL CONTRASTS
(TREND ANALYSIS) IN SPSS
SPSS permits us to perform polynomial contrasts in point-and-click
mode, whereas SAS requires that this procedure be accomplished by spec-
ifying user-defined contrasts using SAS code (syntax). We will therefore
present the SAS procedure to perform a trend analysis in Section 7.34
(after discussing user-defined contrasts).
Open your data file. From the main menu, select Analyze Compare
Means One-Way ANOVA . That will bring you to the main dialog
window for the procedure as shown earlier in Figure 7.2. Configure it
with group as the Factor and satscore in the Dependent List panel.
Click the Contrasts pushbutton to reach the Contrasts dialog window
shown in Figure 7.22. This dialog window is somewhat different from the
onewejustworkedwithinthe GLM procedure,butitallowsustodo
both polynomial as well as user-defined contrasts very conveniently. Check
Polynomial. This action makes the Degree drop-down menu available.
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