what would you have to say about the results? A linear or quadratic component would make
some sense, but a quartic component could not be explained by any theory I know.Unequal Intervals
In the preceding section we assumed that the levels of the independent variable are equally
spaced along some continuum. In fact, I actually transformed the independent variable into
a scale called Composite to fulfill that requirement. It is possible to run a trend analysis
when we do not have equal intervals, and the arithmetic is the same. The only problem
comes when we try to obtain the trend coefficients, because we cannot take our coefficients
from Appendix Polynomial unless the intervals are equal.
Calculating quadratic coefficients is not too difficult, and a good explanation can be
found in Keppel (1973). For higher-order polynomials the calculations are more laborious,
but a description of the process can be found in Robson (1959). For most people, their
analyses will be carried out with standard statistical software, and that software will often
handle the problem of unequal spacing. Without diving deeply into the manuals, it is often
difficult to determine how your software handles the spacing problem. The simplest thing
to do, using the attractiveness data as an example, would be to code the independent vari-
able as 1, 2, 3, 4, and 5, and then recode it as 2, 4, 8, 16, 32. If the software is making
appropriate use of the levels of the independent variable, you should get different answers.
Then the problem is left up to you to decide which answer you want, when both methods
of coding make sense. For example, if you use SPSS ONEWAY procedure and ask for
polynomial contrasts, where the independent variable is coded 1, 2, 3, 4, 5, you will obtain
the same results as above. If you code the variable 2, 4, 8, 16, 32, you will obtain slightly
different results. However, if you use SPSS General Linear Model/Univariateprocedure,
the way in which you code the independent variable will not make any difference—both
will produce results as if the coding were 1, 2, 3, 4, 5. It always pays to check.
An example containing both a quadratic and a cubic component can be found in
Exercise 12.25 Working through that exercise can teach you a lot about trend analysis.408 Chapter 12 Multiple Comparisons Among Treatment Means
Key Terms
Error rate per comparison (PC) (12.1)
Familywise error rate (FW) (12.1)
A priori comparisons (12.1)
Post hoc comparisons (12.1)
Contrasts (12.3)
Linear combination (12.3)
Linear contrast (12.3)
Partitioned (12.3)
Standard set (12.3)
Orthogonal contrasts (12.3)
Dunn’s test (12.3)
Bonferroni t(12.3)
Bonferroni inequality (12.3)
Dunn-Sidák test (12.3)ˇ
Fisher’s least significance difference
(LSD) (12.6)
Fisher’s protected t(12.6)
Studentized range statistic (q) (12.6)
Tukey test (12.6)
Tukey HSD (honestly significant
difference) test (12.6)
WSD (wholly significant difference)
test (12.6)Newman–Keuls test (12.6)
Ryan procedure (REGWQ) (12.6)
Scheffé test (12.6)
Dunnett’s test (12.6)
False Discovery Rate (FDR) (12.6)
Benjamini and Hochberg’s Linear
Step Up (LSU) procedure (12.6)
Trend (12.10)
Quadratic functions (12.10)
Polynomial trend coefficients (12.10)