SPMhelp,
In the SPM course notes chapter, "Statistical Models and Experimental
Design," it is written that "a *contrast* is an estimable function with the
additional property c^T beta-hat = c'^T Y-hat = c'T Y ... Thus a contrast is
an estimable function whose c' vector is a linear combination of the columns
of X."
Note that being a contrast is a property of c, not c'. I claim that any
estimable function c (shorthand for the estimable function c^T beta) must be
a contrast. Why: c estimable means that there is a c' such that
c^T = c'^T X. Pick such a c'. Now define c'' to be the orthogonal
projection of c' onto the range of X. Since c' - c'' is orthogonal to the
range of X, (c' - c'')^T X = 0. Hence c^T = c''^T X. Moreover, c''^T
(Y-hat - Y) = 0, since c'' lies in the range of X, and Y-hat - Y is
orthogonal to it. Thus, c^T beta-hat = c''^T X beta-hat = c''^T Y-hat =
c''^T Y.
Is this correct?
Furthermore, a natural extension of this is that c is estimable (and hence a
contrast) if and only if it is orthogonal to the null space of X.
A more specific question about contrasts: The same section also states that
"For most designs, contrasts have weights that sum to zero over the levels
of each factor." If the most liberal definition of contrast is used, which
I claim above is equivalent to c being orthogonal to the null space of X,
then this would imply that the row sums of the design matrix vanish. (I
assume that the constraint that c_0 vanish is to insist that the contrast
not "see" the constant term; it's not made necessary by the definition
itself.)
Steve Fromm, NIDCD
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