It is important to appreciate that there are two different ways in which the z-statistic can be changed between [OLS] and [MCMC]. The first was demonstrated in (1), in that by taking into account lower-level covariances and their heterogeneity, a substantial increase in higher-level z-statistic is possible. This is due to the fact that the heterogeneity of the lower-level covariances is effectively used to weight the summary statistic data to give more efficient estimates (resulting in reduced top-level regression parameter variance). This is analogous to the way in which prewhitening is used in first-level analyses to weight the regression parameter estimation to give more efficient estimators (25).
(1) were unable to demonstrate the second way in which the z-statistic can be changed between [OLS] and [MCMC], because they assumed that variances were known. In this paper, when we estimate the higher-level variances, they are constrained to be positive. This overcomes the well-known ``negative variance'' problem in OLS (19), by forcing the total variance to be greater than it would be in the OLS case. This increased variance translates into lower z-statistics in voxels which would have suffered from this problem.
In summary, we have two ways in which z-statistics can change between [OLS] and [MCMC]. Firstly, they can increase due to increased efficiency from using lower-level variance heterogeneity. Secondly, they can decrease due to the higher-level variance being constrained to be positive.