Scott: /* Combining values from multiple subjects */

2011-01-29T00:07:45Z

Combining values from multiple subjects

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==Design of fMRI experiments==

* '''block design'''
** stimulus is presented in blocks of fixed length (eg 30s), alternated with blocks of fixation (control condition)
** useful for locating voxels in which activity is significantly different from control baseline
* '''event-related design'''
** stimulus is presented in a brief flash, alternated with blocks of fixation for an '''interstimulus interval''' (ISI, eg 30s)
** can be more-flexible, and even subject-driven
** long ISI allows blood flow to return to baseline, allowing you to study the BOLD/HRF at a single voxel

==Basic block analysis==

* t-test: we are interested in the difference in activation between task and control stimulus
* For voxel i, <math>t_i = \frac{\bar{X}_{task} - \bar{X}_{control}}{se}</math> where
** <math>\bar{X}_{task}</math> is the average level of activation in voxel i over all times during which the task stimulus was displayed, and <math>\bar{X}_{control}</math> is the average over control
** <math>se = s_p \sqrt{\frac{1}{n_{task}} + \frac{1}{n_{control}}}</math> where
** pooled variance <math>s_p^2 = \frac{(n_{task} - 1)s^2_{task} + (n_{control} - 1)s^2_{control}}{n_{task} + n_{control} - 2}</math>where
** <math>n_{task}</math> and <math>n_{control}</math> are the number of observations under each condition
** <math>s^2_{task}</math> and <math>s^2_{control}</math> are the sample variances of the activations of the two conditions
* The result is called a '''statistical parametric map'''.
* If there is more than one task condition, you can generate an F statistic at each voxel.
* An alternative to the t-test is '''correlation analysis'''. For voxel i calculate
** <math>r_i = r(S, X_i)</math> where
** S is the pattern of zeros and ones describing the block design stimulus, and X is the activation time course of voxel i
** r is Pearson's correlation coefficient:
** <math>r(Y, Z) = \frac{\sum_{j=1}^n (Y_j - \bar{Y})(Z_j-\bar{Z})}{\sqrt{\sum_{j=1}^n (Y_j - \bar{Y})^2\sum_{j=1}^n (Z_j - \bar{Z})^2}}</math>
** in the case where there are exactly n/2 task trials and n/2 control trials, the correlation analysis reduces to the t-test:
*** <math>t_i = r_i\sqrt{\frac{n-2}{1-r_i^2}}</math>

==Basic event-related analysis==

* '''trial averaging''' - trials are sorted by condition type and then standard t-test or correlation analysis is performed
* '''function estimation''' - trying to estimate the HRF function at each voxel (why?)
** parametric - we assume a certain functional form for the HRF and then estimate the parameters with the data
** nonparametric - may map with knotted spline curves, lots of Bayesian estimation

==General Linear Model==

* <math>Y = X\beta +\epsilon</math>, where
** Y is the measured responses, a matrix with one row for each time, one column for each voxel
** X is design matrix and includes
*** binary or categorical variables reflecting the stimuli
*** predicted hemodynamic response
** <math>\beta</math> the unknown coefficients of the design matrix, and
** <math>\epsilon</math> the error, usually assumed to be normal with mean zero and variance <math>\sigma^2</math>
* simplest approach assumes each voxel and time point independent and <math>\sigma^2</math> constant, so <math>\beta</math> can be estimated with least squares
* predicted hemodynamic response obtained by convolving the stimulus time course with a model for the HRF (e.g. gamma function model)
** <math>x_t = \int^\infty_0 h(u)s(t-u)du</math>
** [[Image:convolution.png]]
* The general linear model makes strong assumptions about independence that in practice that probably aren't realistic.

==Mapping to a common brain (coregistration)==

* Talaraich - a single postmortem brain with voxels defined at 1mm per side
* Montreal Neurological Institute (MNI) - average over 300 healthy brain scans
* nonlinear methods outperform simple affine transformations

==Combining values from multiple subjects==

We want to combine values at matching voxels over all subjects. There are many approaches, included simple average of the t statistic. A popular approach is to calculate the Fisher T statistic based on the p-value at each voxel:

<math>T_F = -2\sum_{i=1}^k \log{p_i}</math> where i sums over the k subjects

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Scott: /* Combining values from multiple subjects */