<commit_before>% tex file for hypothesis 
\par \indent Our simple linear regression model was created to better 
understand the relationship between the voxels in a given subject's brain and 
the convolved time course. In order to measure the strength of the association 
between these two measurements, we ran a hypothesis test on the coefficients of 
the simple linear regression model for each subject. 

\par There is an individual linear model associated with each voxel in a
subject’s image (and a total of $64 \times 64 \times 34$ voxels per subject).
Thus we ran a t-test on each voxel's $\beta$ coefficient that is associated
with the HRF response. The null hypothesis for each test was that $ \beta = 0$,
with the alternative hypothesis that $\beta \neq 0$. Once we had obtained each
t-statistic, we compared this value across voxels in two ways. First, we simply
compared this t-values with voxels within a subject. In this case, we took into
account the sign of the t-statistic in our analysis. Second, we converted this
t-statistic to a "p-value", in which case the sign of the t-value will become
irrelevant and we compared across voxels without taking into account this sign.
Later, we also run a multiple comparison test using a Benjamini\–Hochberg in
order to find the voxels that are most significant.


\par Having implemented a method to compare the voxels within a single subject, 
we next examined our results for the same voxels across subjects. Our initial 
approach was to aggregate the t-statistic data between all subjects for each 
voxel. This allowed us to decrease the variability of the fit on each voxel and 
detect a more clear signal. 

\par In order to do this, we ran the hypothesis test as stated above on all 24 
subjects of the study. Then for each voxel, we took the average of the t-
statistics across the subjects. An issue with our data was the presence of 
empty space detected by the scanner that is not directly part of the brain. To 
account for this, we took the masked data of the brain and ``cut out'' the parts 
of the images that were not relevant to our analysis. Ultimately, we were left with 
a single 3-d image with each voxel representing the average t-statistics across 
all subjects in the study. This image will later be used in our clustering step in 
order to pinpoint the regions of the brain that have the strongest relationship 
with the convolved time course. <commit_msg>Tiny typo ruining lives.<commit_after>% tex file for hypothesis 
\par \indent Our simple linear regression model was created to better 
understand the relationship between the voxels in a given subject's brain and 
the convolved time course. In order to measure the strength of the association 
between these two measurements, we ran a hypothesis test on the coefficients of 
the simple linear regression model for each subject. 

\par There is an individual linear model associated with each voxel in a
subject’s image (and a total of $64 \times 64 \times 34$ voxels per subject).
Thus we ran a t-test on each voxel's $\beta$ coefficient that is associated
with the HRF response. The null hypothesis for each test was that $ \beta = 0$,
with the alternative hypothesis that $\beta \neq 0$. Once we had obtained each
t-statistic, we compared this value across voxels in two ways. First, we simply
compared this t-values with voxels within a subject. In this case, we took into
account the sign of the t-statistic in our analysis. Second, we converted this
t-statistic to a "p-value", in which case the sign of the t-value will become
irrelevant and we compared across voxels without taking into account this sign.
Later, we also run a multiple comparison test using a Benjamini–Hochberg in
order to find the voxels that are most significant.


\par Having implemented a method to compare the voxels within a single subject, 
we next examined our results for the same voxels across subjects. Our initial 
approach was to aggregate the t-statistic data between all subjects for each 
voxel. This allowed us to decrease the variability of the fit on each voxel and 
detect a more clear signal. 

\par In order to do this, we ran the hypothesis test as stated above on all 24 
subjects of the study. Then for each voxel, we took the average of the t-
statistics across the subjects. An issue with our data was the presence of 
empty space detected by the scanner that is not directly part of the brain. To 
account for this, we took the masked data of the brain and ``cut out'' the parts 
of the images that were not relevant to our analysis. Ultimately, we were left with 
a single 3-d image with each voxel representing the average t-statistics across 
all subjects in the study. This image will later be used in our clustering step in 
order to pinpoint the regions of the brain that have the strongest relationship 
with the convolved time course. 
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