The assumption of independence between the unobserved difference between groups and the scaling factor of a site is certainly invalid if people that have more disease have tissue with different properties that affect regional contrast. It would mean that the scaling factor is correlated with the unobserved effect because different patient groups have different scaling factors. By dropping the covariance terms here, we would be overestimating power because the denominator of the non-centrality parameter would be larger. This was a concern for us, which was why we scanned MS patients to see how different the scaling factors are, and if we could use that to estimate the covariance terms if needed. But, we found that the scaling factors were very similar and in one case, identical, with the caveat that we needed more careful QC on the images to check and correct for gross segmentation missclassifications. The full equation for the calculation of variance has been added to the appendix, which includes the covariance terms before they are dropped:

\[\begin{split} var[XY] = \mu_Y^2var[X] + \mu_X^2var[Y] + 2\mu_X\mu_Ycov[X,Y] \\ - (cov[X,Y])^2 + E[(X-\mu_X)^2(Y-\mu_Y)^2] \\ + 2\mu_YE[(X-\mu_X)^2(Y-\mu_Y)] + 2\mu_XE[(X-\mu_X)(Y-\mu_Y)^2] \end{split}\]

Since we could not detect scaling factor differences between MS patients and healthy controls, we do not think an approximation of covariance is within the scope of this paper. We do think it is necessary if other researchers believe the independence assumption is strongly violated for the particular disease they are studying, and we have included this point in the discussion section.