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Rule 3: The sum of weights (coefficients) for a given comparison
should be zero . Thus, if one group is compared to a combination
of groups and if those groups in the combination are to be weighted
equally, then each group in the combination can be assigned a
weight of
1 and the group in isolation is assigned a weight
of opposite valence equal to the total of the individual weights. For
example, if one group is compared to a combined set of 3 groups,
each group in the set could be assigned a coefficient of
+
1or
+
1; the
3. But,
thegroupsinthecompositecanbeassigneddifferentweightsso
long as their sum balances to the total of the other group or groups
to which the composite is being compared.
Rule 4: Once a group or set of groups is involved in a contrast, that
group or set of groups cannot be used in the same way again .
Rule 5: The sum of the products of the coefficients for each combi-
nation of contrasts must sum to zero .
single comparison group is then assigned a coefficient of
Rule 1 holds that any one contrast in a set of orthogonal contrasts
must be “balanced.” That is, the groups being compared can be thought
of as being on either side of a seesaw. The total weight of one set must
cancel the total weight of the other. Because of this, one set of weights
must be positive and the other must be negative such that their sum is
zero.
Rule 2 indicates that not all of the groups must be involved in any one
contrast. Any group not involved in a particular contrast is assigned a zero
weight for that contrast; this is not counted as a “use” (see Rule 4).
Rule 3 synthesizes the first two rules by giving you a quick check to
make sure that the contrast is balanced by adding to zero. Note that the
groups in a combination are not required to be weighted equally, although
that is certainly the most convenient and easiest way to combine them.
But it is possible that in some research contexts there are good theoretical
reasons to weight the groups in an unequal manner. You are free to use
any weighting scheme to form a linear combination that makes sense -
just make sure that the group or combination of groups to which you are
comparing this unequally weighted combination gives you a total of the
coefficients that exactly balances the total of the other set of coefficients.
Rule 4 states that you can use a group or a set of groups in an isolated
manner in a contrast only one time. Zeros do not count. You must therefore
select your contrasts with care.
Rule 5 is probably the key rule in defining orthogonal contrasts (Ped-
hazur & Schmelkin, 1991), and it is the reason we included the last set
of columns in Table 7.5. Multiply the two coefficients on each row of
the contrasts on which you are checking for orthogonality to get your
products. With five groups, we have five products for each combination
of contrasts. Then add these products and obtain zero to verify that the
contrasts specified in the table are orthogonal.
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