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\chapter{Summary and Future Work}
\label{ch:summary}
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Brain imaging applications often produce large data sets that pose challenges for statistical analyses due, in part, to the intricate neurophysiology. To date, there is a paucity of methods that incorporate spatial considerations present in functional neuroimaging data.
%Most of analyses are performed in a univariate setting, where each voxel is modeled independently of the others.
%(``massive univariate approach'').
The main objective of our research is to uncover aspects of the complex spatial relationships present in functional neuroimaging data and to develop statistical methods that either evaluate or leverage those correlations.

Our first proposed method is related to activation studies, which attempt to localize regions of the brain activity when performing an experimental task. We propose a two-stage, spatio-temporal, autoregressive model which simultaneously
accounts for spatial dependencies between voxels within the same anatomical region and for temporal dependencies between a
subject's estimates from multiple sessions  \citep{derado:10}. %We use Brodmann area templates to define neuroanatomic regions, but other maps are applicable.
Our approach provides a unified framework for voxel-level and region-level inferences.
An important contribution of our work is that we derive efficient computational solutions to
facilitate implementation. Using our proposed method, we analyze fMRI data from a study of inhibitory
control in cocaine addicts. In summary, our proposed spatio-temporal model provides an appealing, computationally
efficient alternative to standard GLM-based methods for analyzing fMRI data. Our model
is based on assumptions that are more neurophysiologically plausible, capturing correlations
between different brain locations and between estimates of neural activity at different scanning sessions.
These correlations lead to interpretive advantages over the GLM, e.g., revealing
information about the degree of coherence in brain activity within defined neuroanatomic
regions. Our simulation studies demonstrate that our model estimates are quite accurate
and that the standard errors associated with estimates of our mean model parameters are on
average smaller than those from a GLM.

Our next proposed methodology is related to connectivity studies, which seek to identify
what brain areas show similar patterns of activity over time, yielding distributed networks
of brain function (Derado et al., 2010a
).
We propose a method to evaluate the level of connectivity
within functionally defined neural networks. We introduce a global index to evaluate the validity of neural processing networks identified by functional clustering or by ICA.  Specifically, we propose the use of Moran's $I$ statistic \citep{cliff:73} to quantify and test the autocorrelation present within computed functional networks.  We tailor our definition of the global index to neuroimaging applications through the use of network-based neighborhoods, and we also compute network-specific contributions to the overall autocorrelation.
Consequently, our methods target autocorrelation exhibited by voxels within defined neural processing networks, rather than between voxels globally throughout the entire brain. We illustrate the use of our methodology with the data from two studies: a PET study of regional cerebral blood flow (rCBF) correlates of parametrically manipulated working memory among individuals with schizophrenia, and an fMRI resting-state study of depression.
We also provide empirical support for the use of our hypothesis testing framework using a bootstrap analysis.
\nocite{derado:10a}

One of our future research goals is to evaluate the accuracy of the normal approximation formulas under randomization assumption \citep{cliff:73} derived for neuroimaging data.	In particular, we think of the spatial correlations present in neuroimaging data. Inherent spatial correlations can be present in the data due to neurophysiology, data preprocessing, scanning acquisition etc. There are couple of different scenarios in our evaluation of the appropriateness of these approximations: to consider the data at the voxel level or to consider regional based clustering. Because the number of voxels or regions play the central role in the calculations, we may observe smaller or larger degrees of bias when we consider these two settings.
%Thus far, we have started investigating the first setting (voxel-level analysis) in our bootstrap analysis, but we plan to investigate both settings because we feel the answers may vary.
%	What gives the possibility: the large number of voxels really drives the story
%	What shall we do if we find the differences?
In case we find that the answers vary, we hope to propose an alternative hypothesis testing framework to the one presented in Chapter 3.
%In conclusion, our future work will be concentrated on the investigation into new testing framework (pending these investigations into appropriateness of the approximation
In our simulation study based on the PET data, the normal approximation seems to perform reasonably well. However, it is an open question whether or not the regional level analysis will need to do some adjustment for inherent spatial correlations in the data. We plan to investigate this in our future work.

Prediction studies try to use neuroimaging scans to predict some behavioral, psychological, or clinical outcomes, or to classify subgroups of individuals (e.g. for diagnostic purposes). Although functional brain imaging has become a powerful tool to investigate the physiological
basis of psychiatric or neurological disorders, it has only recently been used to
predict clinical outcomes in individual patients.

The third proposed method in this dissertation is a novel Bayesian hierarchical framework for predicting follow-up neural activity based on the baseline functional neuroimaging data. By borrowing strength from the spatial correlations present in the data, the prediction algorithm based on this model has improved prediction accuracy, compared to some competing models. The spatial correlations are incorporated in the model in two ways: the short-range correlations between neighboring voxels are incorporated through a multivariate conditional autoregressive (CAR)  prior of the spatial parameters, while the long-range correlations between anatomical brain regions are incorporated through the covariance matrix of the random effect parameters. We applied our proposed model to the data of 40 AD patients and 40 healthy controls to estimate the parameters in our model which we then used to predict 6 month brain activity, based on the baseline activity and some relevant subject's characteristics to a set of 33 test subjects (in each group).

In the future, we plan to apply the proposed prediction algorithm to the mild cognitive impairment (MCI) group of subjects' data to predict the 6 month, 12 month, and 24 month brain activity. We think that this method had a potential clinical application is a sense that would enable clinicians
to forecast future brain states, monitor the progression of dementia and to predict the onset of Alzheimer's disease. We also plan to apply the algorithm to the data from treatment studies, e.g. to forecast the effect of the treatment.
Also, we will investigate how to modify our prediction model to further improve the prediction accuracy and to improve the computation cost.

Our long-term future goal is to develop a framework for predicting an individual's clinical response to treatment.
To potentially improve classification (prediction) accuracy, we plan to investigate and develop methods that borrow
strength from the spatial correlations present in the data. We plan to predict an individual's clinical response to treatment using spatially correlated, region-specific neural processing characteristics, such as region trends in serial brain activity measurements.
