This course covers the design, acquisition, and analysis of Functional Magnetic Resonance Imaging (fMRI) data. A book related to the class can be found here: https://leanpub.com/principlesoffmri

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From the course by Johns Hopkins University

Principles of fMRI 1

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This course covers the design, acquisition, and analysis of Functional Magnetic Resonance Imaging (fMRI) data. A book related to the class can be found here: https://leanpub.com/principlesoffmri

From the lesson

Week 3

This week we will discuss the General Linear Model (GLM).

- Martin Lindquist, PhD, MScProfessor, Biostatistics

Bloomberg School of Public Health | Johns Hopkins University - Tor WagerPhD

Department of Psychology and Neuroscience, The Institute of Cognitive Science | University of Colorado at Boulder

Hi.

Â In this module,

Â we're going to talk about performing inference within the GLM framework.

Â So after fitting the GLM model, we use the estimated parameters

Â to determine whether significant activation present in a voxel or not.

Â So inference is based on the fact that our estimate,

Â beta hat, is normally distributed with mean beta and

Â variance-covariance matrix X transpose V inverse X inverse.

Â So, we use this result, and we can derive t and

Â F procedures to perform tests on effects of interest.

Â So, as we talked about in earlier slides, we often use linear combinations of

Â the parameters and test whether they're significant, so these are contrasts.

Â So the term c transpose beta

Â specifies a linear combination of the estimated parameters as follows.

Â So here, c is called a contrast vector.

Â So to illustrate, let's consider the following event-related

Â experiment with two types of stimulus, let's say condition A and condition B.

Â So here we might specify the following GLM model, we would have beta 1

Â times the baseline, beta 2 times the condition A convolved with an HRF,

Â and beta 3 times condition B convolved with a HRF, and then plus noise.

Â So this is a very simplified GLM model.

Â We probably would have more repetitions of condition A and condition B in practice,

Â but this just gives you the flavor of what we're interested in estimating.

Â So what type of inference might we want to try to do?

Â Well, we might be interested in trying to find areas of the brain where

Â there's a difference between condition A and condition B.

Â So in statistical terms what we want to do is we want to test the null hypothesis

Â that beta 2 is equal to beta 3 against the alternative that they're not equal, or

Â say, that beta 2 is bigger than beta 3 if we believe that to be the case.

Â So in terms of contrast, we can write this as c transpose beta = 0,

Â where c transpose is now the contrast vector, which in this case is 0,

Â because we gave 0 weight to beta 1, 1 because we gave weight 1 to beta 2,

Â and -1 because we gave a -1 weight to beta 3.

Â So if we take c transpose beta, you would get beta 2 minus beta 3 is

Â equal to 0 which is the same thing as beta 2 is equal to beta 3.

Â So how would we test this?

Â Well the test, the null hypothesis that c transpose beta us equal to zero against

Â the alternative that it's not equal to zero, we would use a t-statistic, and

Â a t-statistic is computed in the normal way with an estimate of c transpose beta,

Â using c transpose beta hat divided by its variance, the square root of its variance.

Â And so the only thing that we have to think about here is that under H0,

Â T is approximately t distribution with degrees of freedom that depend on r,

Â the residual inducing matrix that we talked about a few modules ago, and

Â V, the variance-covariance matrix.

Â So if we compute this, we can calculate the distribution of t under

Â the null hypothesis and test this hypothesis.

Â We're often interested in trying to make simultaneous tests

Â of several contrasts at once.

Â In this case, c becomes what's called a contrast matrix.

Â So suppose for example, we have c defined in the following way, then c transpose

Â beta is simply going to be equal to beta 1 beta 2, the vector beta 1 beta 2.

Â So here we might want to test whether these two are simultaneously or

Â both equal to zero.

Â What is an example?

Â Well, let's take a look at an example when we might want to do such a test.

Â So consider the model with a boxcar shaped activation and

Â drift using the discrete cosine basis.

Â So this is the model that we introduced a few modules ago.

Â So we have the first column, corresponds to the boxcar shaped activation,

Â the second column corresponds to the baseline, and

Â columns 3 through 9 correspond to the discrete cosine basis set.

Â So here we might want to ask the following question, it's a very simple question,

Â do these drift components add anything to the model?

Â Well, in this case we might want to test the following, c transpose beta = 0,

Â where c is just simply an indicator for each of the drift components here.

Â So here, each of the rows here indicates which of the components we want to test,

Â whether they're simultaneously equal to zero.

Â So the drift won't contribute if beta 3 through

Â beta 9 are all simultaneously equal to 0.

Â If that's true, then there's no contribution to drift.

Â And this is how we would formalize that, mathematically.

Â So this is equivalent to testing whether or

Â not beta 3 to beta 9 are simultaneously equal to zero.

Â If that's true, then none of the discrete cosine basis sets have

Â a significant beta associated with them, and thus there's no drift.

Â So to understand what this implies,

Â we split the the design matrix into two parts.

Â One is X0, which corresponds to the first two columns, which in this little

Â cartoon is just the baseline, and it will also be equal to the boxcar shape thing.

Â So those are things that we think are important as signal components.

Â And we let X1 be all these discrete cosine basis things that we think maybe are not

Â needed in the model.

Â So maybe the X1 part here is superfluous.

Â So if we want to ask whether or not the drift components add anything to

Â the model, we have to ask, how much does this term X1 actually contribute?

Â Does it contribute in a significant way to the model?

Â So in that case, we typically compare a full model that includes X1, so

Â this is the full design matrix X with a reduced model that only includes X0,

Â which removes the X1 terms.

Â So the idea here is if X0, which is our reduced design matrix,

Â does just as good a job of modeling the data as our full design matrix,

Â which includes all the drift components.

Â Well, then in that case it's probably not necessary to include the drift components

Â because we can use the more parsimonious design matrix X0.

Â So how do we test that?

Â Well we test that using an F-statistic, and I'm not going to go over the details

Â here, but basically this just involves the residuals from the reduced model and

Â the full model, and using that to construct the F-tests.

Â And then assuming the errors are normally distributed,

Â this F-statistic has an approximate

Â F-distribution with the degrees of freedom calculated in the following manner.

Â So basically, this is what we do.

Â At each voxel of the brain, we perform either a t-test or an F-test or

Â some variant of those.

Â So for each voxel,

Â a separate hypothesis test is performed, and the statistic corresponding

Â to the test is used to create a statistical image over all the voxels.

Â So this image that I'm showing here is actually

Â an image of t-statistics across space.

Â And we see how they vary from, say, -5 to around 7.

Â So now I want to move, to show you,

Â illustrate how the GLM is used in an actual setting to analyze fMRI data.

Â So, the first step is to construct a model for each voxel of the brain.

Â We typically use what is called the massive univariate approach,

Â where a separate model is fit to each voxel of the brain.

Â And here, these regression models,

Â such as the GLM that we've been talking about, are commonly used.

Â And so here's how we would set up the design matrix, and

Â there's the GLM analysis.

Â Next we would perform a statistical test, as spoken about earlier in this module,

Â to determine whether there's task-related activation present in the voxels.

Â So for each voxel, we might test the new hypothesis that c transpose beta = 0.

Â And then we would get a t-test for each voxel of the brain.

Â And using those t-tests, we would put them in into the voxel location and

Â we get a statistical image, which is a map of t-tests across all voxels.

Â So this is a t-map here.

Â So this is nice and

Â this shows the results of the hypothesis test across the entire brain.

Â However, we often want to threshold these, and so the last step that we want to do is

Â we want to choose an appropriate threshold for determining statistical significance.

Â So how high does the t-statistic need to be for

Â us to say that that voxel is statistically significant?

Â After choosing a threshold, we can now color-code the significant voxels as

Â follows, and then this is what's called the statistical parametric map.

Â So each significant voxel is color coded according to the size of its p-value.

Â However, this last step is trickier.

Â How do we actually determine the threshold?

Â How do we determine which voxels are actually active?

Â because the implication here is that the color-coded voxels are active

Â while the non-color coded voxels are non-active.

Â But this is all dependent on this threshold that we choose.

Â So how do we determine this threshold?

Â Well, this is a very problematic part, and

Â this is a big thing in fMRI data analysis, because here are some of the problems.

Â The statistics are obtained by performing a large number of hypothesis tests.

Â So if we have 100,000 voxels,

Â we're actually performing 100,000 hypothesis tests simultaneously.

Â And because of this, many of the test statistics will be artificially inflated

Â due to noise, and this in turn will lead to many false positives.

Â So, if we choose an alpha equal to 0.05 level threshold, then in this case,

Â we're going to expect to actually get 5,000 false positive voxels.

Â So this could lead to entire areas of the brain to be falsely activated.

Â So in general, choosing a threshold is always going to be a balance between

Â sensitivity, so having the true positive rate, and specificity,

Â which is the true negative rate.

Â And here's an illustration for this little cartoon data about what happens if we

Â threshold at t bigger than 1, 2, 3, 4, and 5.

Â You'll see that to the left,

Â where we had a very lenient threshold, we have a lot of activation.

Â So we're probably capturing all the true activation there, but

Â we can't shake the feeling that we also have a lot of false positive because we

Â have a widespread activation.

Â If we go all the way to the right and we threshold at t equal to 5,

Â then we're pretty sure that the activation that we see is truly active, but

Â we can't shake the feeling that we've missed some activation.

Â So ideally we want to find a threshold somewhere in between these extremes.

Â But where?

Â Which one do we choose?

Â Well, that's a good question,

Â and this is going to be the topic of many of the future modules.

Â So that's the end of this module,

Â and the end of sort of the single-subject GLM analysis.

Â In the next couple of modules,

Â we'll talk about group analysis, and then we'll return to this question of

Â determining the appropriate threshold when we talk about multiple comparisons.

Â Okay. I'll see you in the next module.

Â Bye.

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