Information Technology Reference
In-Depth Information
Ta b l e 7 . 4 .
Types of polynomial functions
Name
Power of
x
Algebraic equation
Characteristics of the function
Linear
1
y
=
a
+
bx
A straight line function. It requires at
least two data points.
Quadratic
2
y
=
a
+
cx
2
The function has one change in direction
(one “peak”). It requires at least three
data points.
Cubic
3
y
=
a
+
dx
3
The function has two changes in
direction (two “peaks”). It requires at
least four data points.
y
=
a
+
ex
4
Quartic
4
The function has three changes in
direction (three “peaks”). It requires at
least five data points.
Quadratic functions have a single bend in the shape of the function.
Three data points (the means of three groups) at minimum are required
to draw this function. In the function,
x
is raised to the second power.
Figures 7.20C and D show quadratic functions.
Sometimes the shape can be described by more than one function, that
is, some curves have components of two or more polynomial functions.
Figures 7.20E and F show functions composed of both a linear as well as
a quadratic trend. We will discuss this further shortly.
Cubic shapes contain two bends in the function. Four data points (the
means of four groups) at minimum are required to draw this function.
In the function,
x
is raised to the third power. Figures 7.20G and H show
cubic functions.
The general rule to determine the maximum number of contrasts in
an analysis is this: The number of possible polynomial contrasts is equal to
a
−
1where
a
is the number of groups. SPSS will not perform polynomial
contrasts in excess of the fifth order; that is almost never going to be a
problem for you because most of the time you will not wish to deal
with more than linear and quadratic contrasts anyway. This is because
few theories in the social and behavioral sciences are precise enough to
necessitate higher-order polynomials to describe them.
7.21.3 TYING TREND ANALYSIS TO ANOVA
The data points that are schematically plotted in Figure 7.20 are the group
means. These are the very same means that an omnibus ANOVA as well
as the post hoc or preset contrasts are analyzing. If we have a statistically
significant
F
ratio, then the between-groups variance - this represents the
effect of the independent variable - accounts for a significant amount
of the total variance of the dependent variable; that is, the statistically
significant
F
ratio indicates that we have mean differences.
A trend analysis begins when a significant effect of the independent
variable has been obtained. But now, instead of asking which means differ
