Information Technology Reference
In-Depth Information
7.34 PERFORMING POLYNOMIAL CONTRASTS
(TREND ANALYSIS) IN SAS
Polynomial contrasts are performed in
SAS Enterprise Guide
by using the
same procedure as we just described for user-defined contrasts. That is,
the linear, quadratic, and other polynomial orders (polynomial partitions
of the variance) are specified by a set of contrast coefficients.
The various polynomial trends are defined in terms of the means of
certain groups being statistically different from the means of other groups.
Consider the quadratic function in Figure 7.20C. The second (middle)
data point must be significantly different from the other two (combined)
for there to be a quadratic trend. The coefficients corresponding to the
test of statistical significance would be 1,
−
2, and 1 for the groups ordered
first, second, and third.
It is possible for researchers to generate the appropriate coefficients for
various polynomial components (linear, quadratic, cubic, and so on). The
values chosen for these coefficients would be affected by the particular
polynomial component of interest as well as the number of groups in the
study. Rather than having to craft these anew with each study for each
research team, tables presenting sets of coefficients applicable to a wide
range of group set sizes and polynomial components (for interval level
spacing of the groups) were worked out more than half a century ago and
are currently widely disseminated. Technical explanations of how these
coefficients are derived and how to compute coefficients for unequal
intervals can found in several texts, including Keppel (1991) and Kirk
(1995).
Tables containing coefficients for testing orthogonal polynomial com-
ponents of the total variance can be found in Keppel and Wickens (2004,
Table A.3) and Kirk (1995, Table E.10) as well as in the public domain (e.g.,
university Web sites). As we discussed in the context of our hand com-
putations for polynomial contrasts, we present these coefficients from
a first-order (a linear function) through a fourth-order polynomial (a
quartic function) for up to eight groups in Appendix D. The coefficients
are to be read across for the specific case. For example, the contrast co-
efficients for evaluating a quadratic polynomial function for a set of 5
groupsorderedasGroups1,2,3,4,and5are2,
−
1,
−
2,
−
1, and 2,
respectively.
Readers can refer to the figures we presented for planned comparisons
if they wish, as we briefly describe the setup here. Select from the main
menu
Analyze
➜
ANOVA
➜
LinearModels
.Onthe
Task Roles
tab, spec-
ify
satscore
as the
Dependent variable
and
group
as the
Classification
variable
.Onthe
Model
tab, select
group
and click the
Main
bar in the
middle of the window to place
group
in the
Effects
panel. In the
Model
Options
window click only the
Type III sum of squares to show
.Inthe
Least Squares
window, specify
group
as the
Classeffecttouse
and select
All pairwise differences
for the
Show p-values for differences panel
.
