Hi Hans,
> X2 = contribution of RT covariate (contribution of RT effect alone)
>
> (1) You correctly point out that gamma1 will correspond to the
> "intercept" i.e., the response predicted by the linear model for a
> subject with zero RT. I interpret it as asking the question "Is the
> mean response WITHOUT any contribution from RT effect = 0"? i.e., we
> are measuring gamma1 alone without any contribution from RT.
>
Sorry- but I am not sure what you mean by "the mean response WITHOUT
any contribution from RT effect=0".
If you meant "the mean response WITHOUT any contribution from RT
effect" then this sounds more like the group mean instead, i.e. the
gamma1 you get when you fit "beta = gamma1 * ones(m,1) + gamma2 *
demeaned(X2) + error" which is the same as the gamma1 you get when
you fit "beta = gamma1*ones(m,1)+error".
> (2) If we demean X2, then we are testing ( gamma1 + gamma2 *
> X2_bar ) = 0. [X2_bar = mean(X2)]
Indeed. If we have non-demeaned X2: we fit:
beta = gamma1 * ones(m,1) + gamma2 * X2 + error
giving gamma1 equal to the intercept, and gamma2 equal to the gradient
of the line fit.
If we then demean X2 and fit:
beta = gamma1_d * ones(m,1) + gamma2_d * demeaned(X2) + error
then we get:
gamma1_d=gamma1+gamma2*mean(X2)=intercept+gradient*mean(X2)
which it should be easy to see is equivalent to mean(beta). It is that
which we are testing.
> This is a MIXED test where we ask the question "Is the mean response
> + average contribution of RT covariate = 0"?
Can not see how you get from gamma1+gamma2*mean(X2) to "the mean
response + average contribution of RT covariate = 0"
As above, gamma1+gamma2*mean(X2) = intercept+gradient*mean(X2) =
mean(beta) = mean response
Cheers, Mark.
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