Multiple Comparison Procedures - Analysis of Variance Designs - page 156

Ta b l e 7 . 5 . An orthogonal contrast coefficient table

Product of coefficients for
each contrast combination

Coefficients for Coefficients for Coefficients for

Group first contrast second contrast third contrast 1 vs. 2 1 vs. 3 2 vs. 3

Zero month 4 0 0 0 0 0

Two months − 1 1 1 − 1 − 1 1

Four months − 1 − 1 1 1 − 1 − 1

Six months − 1 0 − 1 0 1 0

Eight months − 1 0 − 1 0 1 0

Sum 0 0 0 0 0 0

orthogonal or nonorthogonal, but we will focus here on orthogonal con-
trasts; nonorthogonal contrasts simply involve following fewer rules.

7.26.1 COEFFICIENTS/WEIGHTS IN GENERAL
Contrasts in general call for weights or coefficients to be assigned to each
group mean. When we mentioned weighted linear composites of means
earlier, these coefficients are those weights. With preset contrasts, there
was no need to explicitly talk about these weights since the procedure
automatically assigned them; with user-defined contrasts, you as the user
assign them.
One very convenient structure for assigning weights is to lay out a
simple structure such as that shown in Table 7.5. We have built the table
to demonstrate three contrasts that are all orthogonal to each other. Each
group occupies a row in the table and each contrast is represented as a
column. You fill in the coefficients (weights) for a particular contrast down
that respective column.
Coefficients can take on any values - large numbers, small numbers,
fractions - as long as they conform to the few rules that we will outline
below. We recommend, and we will use, what we consider to be the easiest
values to work with: ones and multiples of one.

7.26.2 RULES FOR ASSIGNING COEFFICIENTS/WEIGHTS
There are five reasonably straightforward rules governing the generation
of orthogonal contrasts (see Field, 2005; Keppel & Zedeck, 1989; Pedhazur
& Schmelkin, 1991 for a more complete treatment of these rules). They
are as follows:

Rule 1: Groups coded with positive weights (coefficients) are com-
pared with groups given negative weights (coefficients) .Itdoes
not matter which groups are coded as positive and which are coded
as negative.
Rule 2: If a group is not involved in a comparison, its weight (coeffi-
cient) should be zero.