Datasets

To avoid unnecessary consideration of first-level design matrices, and because we are only looking to validate the inference on equation 14, we do not generate artificial first-level data $Y$. Instead, we directly generate second-level summary ``data'', $\mu_{\beta_k}$, via equation 14. To do this we specify that we want null data by setting $\beta_g=\mathbf{0}$, and then choose values for $\sigma_{\beta_k}$ and $\sigma_g$. As a result, $\mu_{\beta_k}$, $\nu_k$ and $\sigma_{\beta_k}$ form the summary statistic data we then use in the second-level inference.

To generate artificial data we need to decide on our values for $\sigma_{\beta_k}$ (for k=1...K) and $\sigma_g$. Our choice is governed by the variance ratios we want between the top level and the lower levels. In section 6.1.3 we discussed two ways in which we would expect differences between [OLS] and [MCMC] inference. However, we would expect this difference in z-statistics to be less and less substantial as the top-level variance dominates over the lower-level variance. (1) demonstrated that at a 10:1 ratio of between-session/subject variance to within-session variance, the increase in higher-level z-statistic (due to taking into account variance heterogeneity) is negligible. One of our datasets ([Dataset 4]) utilises a 10:1 variance ratio to explore if the combination of the two possible effects discussed in section 6.1.3 shows any difference in z-statistics between [OLS] and [MCMC].

However, we also consider variance ratios of the order of 1:1. The widely reported existence of the negative variance problem in FMRI (19,28) along with the effects seen in the real FMRI data later in this paper, demonstrate that such low group to first-level variance ratios do exist in FMRI data. We need such a ratio to reproduce data which will suffer from the well reported ``negative variance'' problem when using traditional OLS estimation (19). Furthermore, we need to consider the case of three level hierarchies, which are popular in neuro-imaging studies (e.g. hierarchies containing within session levels, session levels and subject levels). When one uses the summary statistics from the output of the second level to infer on the third level, the variance ratio we are concerned with is between session variance to between subject variance, for which a ratio of the order of 1:1 is realistic.

The four datasets are:

• [Dataset 1] A group mean design, with $N_K=8$ subjects and $\sigma_{\beta_k}^2\approx 0$, $\sigma_g^2=1$. The second level design matrix is: $X_g=[1,1,1,1,1,1,1,1]^T$ with t-contrast $c=[1]$.
• [Dataset 2] A group mean design, with $N_K=8$ subjects and $\sigma_{\beta_k}^2\sim$   Uniform$(0.1,1.9)$, $v_k=8$ and random effects variance $\sigma_g^2=1$. The design matrix and t-contrast is the same as for dataset 1.
• [Dataset 3] A paired t-test design with 5 subjects under two conditions (giving $N_K=10$) and $\sigma_{\beta_k}^2\sim Uniform(0.1,1.9)$, $v_k=8$ and random effects variance $\sigma_g^2=0.5$. The second level design matrix is:
  $$X_g\! =\! \left[\! \begin{array}{cccccc} 1& 1& 0& 0& 0& 0\\ ... ...1& 0& 0& 0& 1& 0\\ -1& 0& 0& 0& 0& 1\\ \end{array}\! \right]$$
  with t-contrast $c=[1,0,0,0,0,0]^T$.
• [Dataset 4] A group mean design, with $N_K=8$ subjects and $\sigma_{\beta_k}^2\approx \sim$   Uniform$(0.1,1.9)$, $\sigma_g^2=10$. The second level design matrix is: $X_g=[1,1,1,1,1,1,1,1]^T$ with t-contrast $c=[1]$.