Information Technology Reference
In-Depth Information
The linear coefficients are found in a Table of Coefficients of Orthog-
onal Polynomials found in Appendix E. Coefficients are provided for
various components of trend (i.e., linear, quadratic) and are organized by
the number of treatment groups or means under scrutiny. In the present
example, we need the coefficients that correspond to
a
=
5 treatment
levels in the “linear” row:
−
1012
Applying the formula for
ˆ
2
−
ψ
,wefind
ˆ
ψ
linear
=
(
c
1
)(
Y
1
)
+
(
c
2
)(
Y
2
)
+
(
c
3
)(
Y
3
)
+
(
c
4
)(
Y
4
)
+
(
c
5
)(
Y
5
)
=
(
−
2)(412
.
86)
+
(
−
1)(474
.
29)
+
(0)(552
.
86)
+
(1)(614
.
29)
+
(2)(623
.
86)
=−
825
.
72
−
474
.
29
+
0
+
614
.
29
+
1247
.
72
=
562.00
.
Additionally, we need to compute the sum of the squared coefficients:
c
2
2)
2
1)
2
(0)
2
(1)
2
(2)
2
=
−
+
−
+
+
+
=
.
(
(
10
Thus,
00)
2
(7)(562
.
SS
A
linear
=
=
221,090.80
.
10
The next steps are to calculate our degrees of freedom, mean squares,
and
F
ratio. Each orthogonal (independent) trend component is associated
with 1
df
;hence,
df
A
linear
=
1.
Thus,
MS
A
linear
=
.
/
=
.
221, 090
80
1
221,090.80
Our final step is to calculate the
F
ratio:
MS
A
linear
MS
S
/
A
=
221,090.80
1,325.73
F
A
linear
=
=
166.77
.
This
F
value is evaluated at 1 and 30
df
, and is found to be statistically
significant at alpha
.05. We can then reject the null hypothesis and
conclude that a statistically significant linear relationship exists between
the amount of study time and SAT performance.
Again, we note that the hand calculated
F
value of 166.77 differs
slightly from the SPSS calculated value of 165.586 in Figure 7.20 as a result
of rounding error.
=
7.22.2 CALCULATING A NONLINEAR OR QUADRATIC TREND
The most common nonlinear trend component that social and behavioral
scientists test for is a
quadratic trend
that indicates a change of direction
commonly found in U-shaped or inverted U-shaped functions. To assess
for this quadratic trend component, we will again calculate a new sum of
squares and subsequent
F
ratio. We begin this process by consulting our
