## Systematic Detection of Change

The most obvious and simple statistical test that can be used in fMRI data analysis is student's t test. This test assumes that each number in each group is independent, and that the underlying distribution of numbers is Gaussian (i.e., it is a parametric test). In fact, both of these assumptions are often violated in actual fMRI data. Nonetheless, parametric statistics like the t test are the most widely used measures of the difference between the groups of numbers collected in fMRI images across conditions. Other statistical tests are possible and sometimes used.

The most commonly used approach to the detection of systematic effects in fMRI data is the general linear model, which uses correlational analysis. Here, the fMRI data are compared with some kind of reference temporal function to determine where in the brain there are high correlations between the reference function and the MR data. The reference function is obtained from the experimental design. For example, because the brain's hemodynamic response follows a fairly consistent profile, a boxcar function defining the experimental paradigm is often convolved with an estimated hemodynamic response function to yield the reference function. The resulting reference function is smoother than a boxcar and better takes into account the shape of the hemodynamic response, generally resulting in better correlation between the MR signal time courses and the regressor time course. Several functions have been historically used to model the hemodynamic response, including a Poisson function and, recently, a gamma function. Often, a single canonical hemodynamic response function is used across the entire brain and across subjects, though there is evidence for variation in hemodynamic response shape across subjects and brain regions. Some software packages make provisions for this, allowing for independent modeling of the hemody-namic response function on a voxelwise basis.

There are a number of variations on this general scheme. For multiple experimental conditions, the previously mentioned scheme can be easily extended using multiple regression. In addition, it is also possible, though less common, to perform nonlinear regression on fMRI data, given some nonlinear prior model of expected brain response.

As with any statistical test, one must exercise some caution when using correlational analysis to ensure that incorrect inferences are not made due to violation of the assumptions inherent in the statistical test. In particular, the assumption of independence of consecutive samples is sometimes badly violated in fMRI data, inflating estimates of significance.

All of the preceding approaches make the assumption that the variations of interest in the data are those that occur in temporal synchrony with the experimental variations built in to the design. These tests cannot detect novel temporal variations triggered by the experiment but not part of the design. For instance, if a change was triggered at stimulus onset and stimulus offset, most standard data analytic packages, as they are typically employed, could not detect that response. In contrast, various multivariate approaches (such as PCA) seek regularities in the spatiotemporal structure of the fMRI data that are not specified beforehand. Such techniques typically detect the experimental variation that was designed by the experimenter as well as some physiological variations (such as those due to breathing or heartbeat). The challenge is to refine these tests so that it is easy to interpret the regularities that are detected.

Principal component analysis is not really a statistical test. Rather, it is a re-representation of the data that condenses as much ofthe variability in that data as possible into a small number of eigenimages, each of which is associated with an eigenfunction that specifies a temporal fluctuation for the entire image. Thus, instead of one temporal variation for each voxel in a brain volume, there are a small number of volumetric images, each of which varies as a single relative image according to some time course. The key virtue of PCA is that it has the power to pick out particular areas in the brain that exhibit a time course similar to that in the experimental design without the experimenter ever having specified that design to the analysis procedure. Similarly, it can detect temporal changes that are different from the ones built into the experiment. On the other hand, there is no obvious way to know which eigenimages and eigenfunctions actually correspond to an important or interpretable variation. PCA and related techniques have great theoretical appeal, but they have been rarely used in practical fMRI data analysis. A related multivariate technique, called independent component analysis (ICA), is designed to help with the interpretation of the data. Instead of projecting data into a lower dimensional space that accounts for the most variation (as PCA does), ICA finds a space in which the dimensions are as independent as possible, thus facilitating interpretation of the components.

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