Multiple Comparison Procedures - Analysis of Variance Designs - page 148

because the means would not differ significantly in magnitude. Second, to
meaningfully address the question of the shape of the function, the groups
must be aligned on the x axis in an interval manner (or a reasonable
approximation to it). If the independent variable is so measured, as it is in
our SAT study time example where the levels are “spaced” in intervals of
two months, we can then legitimately ask about the shape of the function.

7.21.4 PARTITIONING THE BETWEEN-GROUPS VARIANCE
In the omnibus ANOVA, we partitioned the total variance into between-
groups and within-groups variance. In a trend analysis, we further par-
tition the between-groups variance into its polynomial components by
performing polynomial contrasts and testing their significance by using
an F ratio. The total of all of the polynomial components is equal to the
between-groups variance. We can summarize this as follows, where SS is
the sum of squares:
For 2 groups: between-groups SS = linear SS only

( SS A = SS linear )

For 3 groups: between-groups SS = linear SS + quadratic SS

( SS A = SS linear + SS quadratic )

For 4 groups: between-groups SS = linear SS + quadratic SS +

cubic SS

( SS A = SS linear + SS quadratic + SS cubic )

For 5 groups: between-groups SS = linear SS + quadratic SS +

cubic SS + quartic SS .

( SS A = SS linear + SS quadratic + SS cubic + SS quartic ) .

Let's revisit Figure 7.17 with this information in mind. With only
two groups, Figures 7.20A and B show us pure linear trends. All of the
between-groups variance is accounted for by the linear component of the
polynomial contrast (there is no other polynomial component possible).
Figures 7.20C and D are examples of almost pure quadratic trends. If
we tried to fit a straight line through either function, it would be almost
parallel to the x axis; such a “flat” slope indicates no mean differences
from that linear perspective, and the linear component should not come
close to reaching statistical significance.
Figures 7.20E and F are substantially less pure. There is a sufficient bend
in each function to most likely generate a statistically significant F ratio
for the quadratic trend in the data. But there is a general increase in the
values in Figure 7.20E and a general decrease in the values in Figure 7.20F
thatcanbeapproximatedbystraightlinefunctions(wehavedrawndashed