Print

---

Hi Stuart,

I have a slightly different take on this...

When ever you have one categorical variable (drug/no drug) and one
continuous variable (stimulus rating), it's invaluable to draw an
ANCOVA graph:  X-axis stimulus rating, Y-axis response.  Draw two
lines, one for drug, one for no drug.  In the SPM parameterization,
the first columns of the design matrix are the intercept for each line
(call them beta1 and beta2); the slope of each line are the next two
columns (beta3 and beta4).

What I would call the natural parameterization is this:  The
midpoint between the two intercepts is the grand mean, call it
alpha0.  The distance from the midpoint to each of the two intercepts
is the main effect of drug (alpha1).  Draw a dotted line that goes
through the y-axis at alpha0 and halfway between the two lines; this
is the main effect of the covariate, call it alpha 2.  The effect that
expresses the difference between the two (solid) lines is the
drug by stimulus interaction (alpha3).

OK, you can put down your pencil now.  The point of this is that
you obviously want this contrast:

        alpha0   alpha1   alpha2   alpha3
        [  0       0        0        1  ]

Note that this doesn't involve the intercepts (alpha0 and alpha1) at
all.

Now I can write down the relationship between the alphas and the
betas:

        alpha0 = 1/2*(beta1 + beta2)
        alpha1 = 1/2*(beta1 - beta2)
        alpha2 = 1/2*(beta3 + beta4)
        alpha3 = 1/2*(beta3 - beta4)

You can quickly see that the contrast above implies that you want a
contrast that tests Ho:  1/2*(beta3 - beta4) = 0.  Or simply

        beta1    beta2    beta3    beta4
        [ 0        0        1       -1  ]

Again, the intercepts don't come into play.


Also, note that the interpretation of beta1 and beta2 (and alpha0 and
alpha1) depend crucially upon whether the covariate was centered.  If
it was centered, then the intercepts are the response at the average
covariate value.  If it wasn't centered, then the intercepts are the
response at covariate value zero, which may, or may not make any
sense.

Of course, at the end of the day, it doesn't matter as the inferences
for estimable contrasts of the intercepts (t's and F's) are the same
no matter what.


So this answer wasn't one of your hypothesized ones... does this make
sense?

-Tom


    -- Thomas Nichols --------------------   Department of Biostatistics
       http://www.sph.umich.edu/~nichols     University of Michigan
       [log in to unmask]                     1420 Washington Heights
    --------------------------------------   Ann Arbor, MI 48109-2029