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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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

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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.

In this module,

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

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

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

So inference is based on the fact that our estimate,

beta hat, is normally distributed with mean beta and

variance-covariance matrix X transpose V inverse X inverse.

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

F procedures to perform tests on effects of interest.

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

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

So the term c transpose beta

specifies a linear combination of the estimated parameters as follows.

So here, c is called a contrast vector.

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

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

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

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

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

So this is a very simplified GLM model.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

So how would we test this?

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

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

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

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

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

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

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

V, the variance-covariance matrix.

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

the null hypothesis and test this hypothesis.

We're often interested in trying to make simultaneous tests

of several contrasts at once.

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

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

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

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

both equal to zero.

What is an example?

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

So consider the model with a boxcar shaped activation and

drift using the discrete cosine basis.

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

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

the second column corresponds to the baseline, and

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

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

do these drift components add anything to the model?

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

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

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

whether they're simultaneously equal to zero.

So the drift won't contribute if beta 3 through

beta 9 are all simultaneously equal to 0.

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

And this is how we would formalize that, mathematically.

So this is equivalent to testing whether or

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

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

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

So to understand what this implies,

we split the the design matrix into two parts.

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

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

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

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

needed in the model.

So maybe the X1 part here is superfluous.

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

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

Does it contribute in a significant way to the model?

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

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

which removes the X1 terms.

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

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

which includes all the drift components.

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

because we can use the more parsimonious design matrix X0.

So how do we test that?

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

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

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

And then assuming the errors are normally distributed,

this F-statistic has an approximate

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

So basically, this is what we do.

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

some variant of those.

So for each voxel,

a separate hypothesis test is performed, and the statistic corresponding

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

So this image that I'm showing here is actually

an image of t-statistics across space.

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

So now I want to move, to show you,

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

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

We typically use what is called the massive univariate approach,

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

And here, these regression models,

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

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

there's the GLM analysis.

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

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

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

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

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

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

So this is a t-map here.

So this is nice and

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

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

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

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

us to say that that voxel is statistically significant?

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

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

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

However, this last step is trickier.

How do we actually determine the threshold?

How do we determine which voxels are actually active?

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

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

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

So how do we determine this threshold?

Well, this is a very problematic part, and

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

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

So if we have 100,000 voxels,

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

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

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

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

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

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

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

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

which is the true negative rate.

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

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

You'll see that to the left,

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

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

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

have a widespread activation.

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

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

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

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

But where?

Which one do we choose?

Well, that's a good question,

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

So that's the end of this module,

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

In the next couple of modules,

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

determining the appropriate threshold when we talk about multiple comparisons.

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

Bye.

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