•Design
analysis
→X’X invertible but cond(X’X) is huge à
linear system is sensitive à
difficult to obtain
accurate estimates of regressor weights
→Condition
number: a measure of system's sensitivity to numerical computation
•cond(M) = ratio of
maximum to minimum eigenvalues of matrix M
•note, 3dDeconvolve can generate both X and (X’X)-1, but not cond()
→Covariance
matrix estimate of regressor coefficients vector b:
•s2(b) = (X’X)-1MSE
•t test for a
contrast c’b (including
regressor coefficient):
Øt = c’b /sqrt(c’ (X’X)-1c MSE)
Øcontrast for condition A only: c = [0 0 1 0
0]
Øcontrast between conditions A and B: c = [0 0 1 -1 0]
Øsqrt(c’ (X’X)-1c) in the
denominator of the t test indicates the relative stability and statistical power of the experiment design
•sqrt(c’ (X’X)-1c) =
normalized standard deviation of a contrast c’b (including regressor
weight) à these values are output by 3dDeconvolve
•smaller sqrt(c’ (X’X)-1c) à stronger statistical power in t test, and
less sensitivity in solving
the normal equation of the general linear system
•RSFgen helps find out a good design with relative small sqrt(c’ (X’X)-1c)