Methods

We now want to ascertain the difference in the the bias of the resulting statistical distributions that exists for the different approaches for estimating the autocorrelation. This is determined experimentally on real rest (null) FMRI data by computing the t-statistic at each voxel for a dummy design paradigm The t-statistic is given by $t = \mathbf{c\hat{B}}/\sqrt{Var\{\mathbf{c\hat{B}}\}}$ where $\mathbf{\hat{B}}$ and $Var\{\mathbf{c\hat{B}}\}$ are given by equations 2 and 3 respectively. The t-statistics are then probability transformed to z-statistics. The probability transform involves converting the t-statistic into its corresponding probability (by integrating the t-distribution from the t-statistic's value to infinity) and then calculating the z-statistic that corresponds to the same probability (by integrating the normal distribution from the z-statistic's value to infinity).

These z-statistics form what we refer to as the null distribution. A technique with low bias should give a null distribution that closely approximates the theoretical z-distribution (or Normal distribution).

For the theoretical, Normal probability density function, $f(z)$, we can obtain the z-statistic, $z_p$, for a chosen probability $p$ such that $p=\int_{z_p}^{\infty}f(z)dz$. This can then be compared to $p_{null} = \frac{1}{2}Prob(\vert z\vert>z_p)$ for the empirically obtained null distribution, $d(z)$. This is given by:

$$p_{null}=\frac{\sum_{\vert z\vert>z_p}d(z)}{2\sum_{z}d(z)}$$ (23)

Since for purposes of inference the tail is the most important part of the distribution, we examine $p_{null}$ as far into the tail as the sample size will allow. This is aided by using both tails of the empirically obtained null distribution.

We intend to study data taken at TR=3 and 1.5 secs. Six different rest/null datasets (3 normal volunteers, 2 datasets per volunteer) were obtained using TR=3 secs and 9 null datasets (3 normal volunteers, 3 datasets per volunteer) were obtained using TR=1.5 secs. For each dataset 204 echo planar images (EPI) were acquired using a 3 Tesla system with time to echo (TE) = 30ms, in-plane resolution 4mm and slice thickness 7mm. The first 8 scans were discarded to leave $N=196$ scans and the data was motion corrected, intensity normalised by subtracting the global mean time series from each voxel's time series, and non-linear high-pass filtered. We computed an empirical distribution based on either all of the TR=3 secs data or on all of the TR=1.5 secs data. The z-statistics for all of the brain voxels in the six or nine null datasets are all pooled together to give one empirical null distribution. The resulting distributions consisted of z-statistics from approximately $80000$ voxels. This allowed for examination into the tail to probabilities as low as $1e-5$. It is important that we examine this far into the tail of the distribution as this is approximately where inference needs to take place when multiple comparison corrections are taken into account (Worsley et al., 1992).

We will consider two different paradigms - the simple boxcar HRF convolved paradigm (on the TR=3 secs data) and the single-event with randomized ISI design (on the TR=1.5 and 3 secs data) as described earlier. Various autocorrelation estimation techniques will be compared on the calibration plots when performing prewhitening.