Information Technology Reference
In-Depth Information
7.26.3 THREE EXAMPLE ORTHOGONAL CONTRASTS
Table 7.5 illustrates a set of three orthogonal contrasts. The first contrast
comparesthezeromonthstudygrouptoalloftheothers.Wehaveassigned
a weight of 4 to the zero month group because we are comparing it to a
combination of four groups with each assigned a value of
1(weareusing
equal weights here). We could just have easily assigned the zero month
group a
−
4 and the groups in the linear combination each a 1. In truth,
we could have used any combination of values that summed to zero, but
this set of weights is clear and simple. It is also the case that we assigned
each group in the set of four an equal weight; this need not be the case as it
is also possible for theoretical reasons to differentially weight the groups.
For example, we could have weighted the two, four, six, and eight month
groups as
−
2. We would have then needed to weight the
zero-month group as 6 to “balance” the weights such that they added to
zero.
The second contrast compares the two month group to the four month
group. One arbitrarily is assigned a weight of 1 and the other a weight of
−
−
1,
−
1,
−
2, and
−
1, and all of the others are weighted as zero to remove them from the
comparison.
The third contrast compares groups that have studied relatively less as
a set (the two month and four month study groups combined) to those
as a set that have studied relatively more (the six month and eight month
study groups). One set has each of its groups equally weighted as 1 and
the other set has each of its groups equally weighted as
1. This is because
each set contains the same number of groups; had the sets been of unequal
size, we would have chosen other numbers that would add to zero. For
example, with three groups in one set and two groups in another set,
we could have chosen weights such as
−
−
2,
−
2, and
−
2versus3and3,
respectively.
To verify that these specified contrasts represent orthogonal contrasts,
we make sure that each column sums to zero. Then we multiply the weights
of pairs of contrasts and add these products to achieve a value of zero; this
ensures that each pair of contrasts whose weights we multiplied represents
an orthogonal contrast. As you can see from Table 7.5, all of the contrasts
are orthogonal to each other.
7.27 PERFORMING USER-DEFINED (PLANNED) COMPARISONS
BY HAND
Let's continue with our hypothetical study dealing with student study-
time preparation (in months) on SAT performance. Suppose we had the
following three hypotheses (or comparisons) that we wanted to assess.
Students who do not study versus those that do study.
Students studying for two months versus those studying for four
months.
