Deconvolution Analysis of FMRI Time Series Data - Waisman ...

where C is an s P matrix, and P = total number of parameters in the full model. It canbe shown (see Ref. [2]) that the least squares estimate for which satises the above setof linear constraints is given by:b R = b F ; ; X t X ;1 C t C ; X t X ;1 C t ;1CbFwhere b F = least squares estimate of under the full model, and b R = least squaresestimate of under the reduced model. The error sum of squares for this reduced modelis then:SSE(R) =(Z ; Xb R) t (Z ; Xb R)Finally, the test statistic for the general linear test (GLT) is:F =SSE(R) ; SSE(F )df R ; df FSSE(F )df Fwhich has the F (df R ; df F df F ) distribution under the null hypothesis (Ref. [2]), wheredf R ; df F = s.Writing the s P matrix C in terms of row vectors:C =264c t 1c t 2.c t s375we can estimate C by:Cb F =2643c t 1b Fc t 2b F7.c t sb F25 643L 1L 27. 5L swhere L 1 , L 2 , :::, L s are the s linear combinations of the parameter vector b F speciedby the GLT.Program 3dDeconvolve calculates the s linear combinations L 1 , L 2 , :::, L s , and theGLT F statistic, for each operator specied GLT, and for each voxel, and appends thesevalues as sub-bricks of an AFNI \bucket" dataset. (See the Examples in Section 1.4.4.)1.2.12 Concatenation of RunsMany users choose to concatenate runs prior to time series analysis. This is a verydelicate operation, due to the implicit time dependence across runs. That is, consecutiveimage volumes are assumed to have been acquired at consecutive points in time but for16