Hi Kaitlin,

CONN implements mixed within- between- subject second-level models, using a "partitioned variance" approach, which means that for any (subjects by conditions) matrix of connectivity measures Y, and a (subjects by between-subject effects) design matrix X, CONN will fit a multivariate general linear model of the form:

 Y = X*B + noise

and it will allow you to test any hypothesis of the form:

 C*B*D' = 0

where C is your between-subjects contrast (matrix or vector), and D is your between-conditions contrast (matrix or vector again), using the appropriate multivariate likelihood ratio test. 

In your case (I am going to assume in this example that you have 25 subjects), Y will be a 25 x 7 matrix of connectivity values, and (assuming you have a single group of subjects) X will be a 25 x 1 vector (all 1's, the 'AllSubjcets' second-level covariate). The between-subjects contrast is just 1, and the within-subjects contrast is your [1.28 ... 0.28] vector. The resulting likelihood ratio test in this case will be exactly equivalent to:

 a) first computing Y*D' for each subject (i.e. computing a linear combination of the connectivity measures in your 7 timepoints using your contrast vector D as weights)
 b) then performing a one-sample t-test on the resulting measures across all subjects

One way to think about this is in terms of the "growth curves" technique for longitudinal analyses. In growth curve analyses, you first posit a family of curves modeling the expected evolution of your longitudinal measures (for example, this family could be the set of all possible quadratic curves). Then you fit, individually for each subject, the curve that best approximates your observed longitudinal measures (in the quadratic example above, that will give you three parameters for each subject, modling the constant, linear, and quadratic terms of the best quadratic fit to each subject's data). Last you look at your parameter estimates across subjects to make inferences (e.g. you could test whether there is evidence of hyperlinear or hypolinear longitudinal changes by testing the third -quadratic- parameter estimate across subjects). Coming back to your case, your analyses can be interpreted as a simple growth-curve analyses where you are positing a family of longitudinal trajectories defined by a constant term plus some non-linear change in the direction specified by your contrast [1.28 ... 0.28] (i.e. stable during the first two time points, followed by decreases in connectivity during 3-5th time points, and a reversal to higher connectivity in the last 6-7th timepoints). Your second-level model is then testing the parameter estimates associated for each subject with this non-linear term. Note that, while this test will in fact highlight regions that show the expected connectivity trajectory, it will also show regions that might follow other similar trajectories that your growth-curve model is not explicitly modeling (e.g. a simple linear change in connectivity measures across your 7 timepoints). If you wish to rule-out these other potential trajectory shapes you simply need to orthogonalize your between-conditions contrast with respect of these other effects (e.g. using gram-schmidt to orthogonalize your contrast vector with respect to [-3 -2 -1 0 1 2 3] will lead to this alternative contrast [0.3214 0.6428 -0.03573 -0.7143 -1.393 -0.07144 1.25], which will allow you to rule-out simpler linear longitudinal changes in connectivity). 

Hope this helps
Alfonso 

 


 
Originally posted by Kaitlin Cassady: