Higher values in the off-diagonals means that the parameter estimates

are correlated.

I'm not sure which ones are actually driving the results or should be.

And then, finally, this is the intercept, just to note that.

So it also has its own variance in which determines the fundamental stability with

which I can estimate the baseline levels.

So what makes these diagonals' values smaller and designs more efficient?

Well, there are three factors we'll cover here, one is a large rise and

fall in the predictors themselves, that's predictor variance.

Secondly, a low covariance among predictors, or orthogonal predictors.

And this is a minimal multicolinearity problem, is one way of saying that.

And finally, large sample sizes, so

the variance is proportional to the square root of the number of observations.

And all of that is factored into that matrix.

So in fMRI setting, it's a little bit more complex but

the principle is exactly the same.

We have to factor in contrast,

high-pass filtering, and autocorrelated noise in the scanner.

So now we'll take our four column design matrix, and we'll multiply it by contrast

weights, 1, 1, -1, -1, or multiply the parameter estimates by those.

And that's equivalent to doing a main effect of factor 1 in this 2 by

2 factorial design.

So that's from the previous class.

So now let's look at the variance of that contrast estimate.

So all we have to do is take that X transpose as inverse and

sandwich it between contrast transpose times contrast.

Now the t statistic becomes contrast times beta hat divided by its standard error.