Dear Steve,


>
> 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.

I think what was meant here is that an estimable function is defined by the
property above. (which is not really, this is just a corollary of the usual
definition that c^T beta is estimable if there exists c' such that E(c'^T Y) = c^T beta
for any beta. I guess it could be used as a definition ...)
I do agree that the sentence in that text is misleading ....

> Why: c estimable means that there is a c' such that
> c^T = c'^T X.

yes, and your "means" is an "if and only if" ...

> 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.

yes, more simply put, c' needs not be unique, but its ortho projec. onto
the range of X is unique. Another way of seeing c' is to think of it as
a constraint on the model X ... (so it must lies in its space !)

> 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?

absolutly, and it is used in the spm routines.

>
> 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.

yes, or more simply that the contrast lies in the range of X^T, and that's
how we check estimability ... (have a look at spm_SpUtil, spm_FcUtil,
and spm_sp if you want more details on how this is handled.
In fact, because as you point out c'' is in the range of X, we use the
coordinates of it in an orthonormal basis of X to save memory and computation
time and work in the parameter dimension rather than the temporal dimension.

>
> 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.

not quite sure what you mean here, what was meant is that in simple factorial
designs, valid contrasts usually have weights that sum up to zero.

> (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.)

yes. But in any case, the spm interface would not allow a non valid contrast.

best,
jb



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