deletions | additions       diff --git a/The_assumption_of_independence_between__.tex b/The_assumption_of_independence_between__.tex  index 8c55bce..1eb5566 100644  --- a/The_assumption_of_independence_between__.tex  +++ b/The_assumption_of_independence_between__.tex 
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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 In  dropping the covariance terms here, we power  would be overestimating power overestimated  because the denominator of the non-centrality parameter would be larger. This was a concern for us, which was why we We  scanned MS patients to see how different the scaling factors are, and if we could use that were  to estimate the covariance terms if needed. But, we found that the address this concern. The  scaling factors were very similar and in one case, identical, with the caveat thatwe needed more careful QC on  the images were QC'd more carefully  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{equation}  \begin{split} 
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\end{split}  \end{equation}  Sincewe could not detect  scaling factor differences were not detected  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 It  is necessary if other for  researchers believe the to investigate if this  independence assumption is strongly violated for the particular disease they are studying, and we have included the studying. The  point that this assumption is valid for MS only, only was discussed more thoroughly  in the discussion section.