Inference via Spatial Mixture Modelling

Commonly in FMRI, null hypothesis testing is used on a GLM to label voxels, or clusters of voxels, as being ``active'' if they reject the null hypothesis at a given false positive rate (FPR). This depends on knowing the null distribution for the relevant statistic (e.g., regression parameter, t-statistic or pseudo-t) under the null hypothesis. An alternative approach for inference is to use mixture modelling on the statistic of interest [16]. This involves fitting a mixture of distributions to the histogram of the statistic of interest; in our case we use a Gaussian for the central non-activation part of the data, and a gamma for the activation (and possibly another for ``deactivation'').

A well-known problem in null hypothesis testing of FMRI is that if enough observations are made, then every voxel in the brain will reject the null hypothesis. This is because in practice all voxels will show some response to the stimulus, if only due to modelling inadequacies such as unmodelled stimulus-correlated motion or the point spread function of the scanner. By using mixture modelling, instead of asking the question ``Is the activation zero or not?'', we ask the question ``Is the activation bigger than the overall background `signal'?''. Adaptability in modelling the non-activating (``null'') part of the distribution can also help to protect against violations of the modelling assumptions, such as poorly modelled noise structure. Mixture modelling also provides us with far more inference flexibility compared with null hypothesis testing. We can either control FPRs or TPRs (true positive rates) by using the ``non-activating'' or ``activating'' distributions respectively. Controlling the TPR may be of real importance when using FMRI for pre-surgery planning. Non-spatial mixture modelling is already used in FSL for statistically inferring areas of ``activation'' in MELODIC (see section 2.5).

In null hypothesis testing one can incorporate spatial information by using smoothing in combination with Gaussian Random Field Theory [44]. However, this requires the arbitrary setting of parameters (the t/z threshold for forming clusters, and the amount of spatial smoothing) and makes the resulting inference hard to interpret. In contradistinction, we can incorporate spatial information into mixture modelling by encoding the prior belief that we expect to find areas of activation next to other areas of activation. In [40] we describe a novel way to do this with the amount of spatial regularisation determined adaptively from the data using a discrete-labels Markov Random Field (MRF) prior on the classification. Heuristic tuning of control parameters is no longer required. All parameters in the model are adaptively determined from the data, and hence we can infer regions of brain activity completely objectively. Figure 7 shows the spatial mixture model applied to a single-event pain stimulus experiment.

Figure 7: Results of applying the spatial mixture modelling on a single-event pain stimulus experiment. [top left] z-statistic parametric maps. [bottom left] Probability of being in the activation class. [right] Mixture model fit to histogram of pseudo-z-statistics.