Special
A user-defined contrast. Allows entry of special contrasts
in the form of square matrices with as many rows and columns as there
are categories of the given independent variable. For MANOVA and LOGLINEAR,
the first row entered is always the mean, or constant, effect and
represents the set of weights indicating how to average other independent
variables, if any, over the given variable. Generally, this contrast
is a vector of ones.
The remaining rows of the matrix contain the special contrasts indicating the comparisons between categories of the variable. Usually, orthogonal contrasts are the most useful. Orthogonal contrasts are statistically independent and are nonredundant. Contrasts are orthogonal if:
- For each row, contrast coefficients sum to 0.
- The products of corresponding coefficients for all pairs of disjoint rows also sum to 0.
For example, suppose that treatment has four levels and that you want to compare the various levels of treatment with each other. An appropriate special contrast is
(1 1 1 1) weights for mean calculation
(3 -1 -1 -1) compare 1st with 2nd through 4th
(0 2 -1 -1) compare 2nd with 3rd and 4th
(0 0 1 -1) compare 3rd with 4thwhich you specify by means of the following CONTRAST subcommand
for MANOVA, LOGISTIC REGRESSION,
and COXREG:
/CONTRAST(TREATMNT)=SPECIAL( 1 1 1 1
3 -1 -1 -1
0 2 -1 -1
0 0 1 -1 ) For LOGLINEAR, you need to specify:
/CONTRAST(TREATMNT)=BASIS SPECIAL( 1 1 1 1
3 -1 -1 -1
0 2 -1 -1
0 0 1 -1 ) Each row except the means row sums to 0. Products of each pair of disjoint rows sum to 0 as well:
Rows 2 and 3: (3)(0) + (–1)(2) + (–1)(–1) + (–1)(–1) = 0
Rows 2 and 4: (3)(0) + (–1)(0) + (–1)(1) + (–1)(–1) = 0
Rows 3 and 4: (0)(0) + (2)(0) + (–1)(1) + (–1)(–1) = 0The special contrasts need not be orthogonal. However, they must not be linear combinations of each other. If they are, the procedure reports the linear dependency and ceases processing. Helmert, difference, and polynomial contrasts are all orthogonal contrasts.