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
Module 28: FDR Correction
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Principles of fMRI 1
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從本節課中
Week 4
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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'll continue
talking about multiple comparisons problem.
So, in the last module we talked about familywise error rate correction.
Here, we'll probably talk about false discovery rate correction.
So, methods that control familywise error rates, such as Bonferroni,
random field theory, and permutation tests,
provide a strong control over the number of false positives.
While this is really an appealing property, the resulting thresholds
often tend to be very stringent and lead to tests that suffer from low power.
So power is very important in fMRI applications because most
interesting effects lie in the edge of a detection.
The false discovery rate, or the FDR, is a recent development in multiple
comparisons that are due to Benjamini and Hochberg.
So this comes from a paper around 1995.
So while the familywise error rate controls the probability of
any false positives, the false discovery rate
controls the proportion of false positives among all rejected tests.
So this is a slightly different criteria that we're controlling here.
So, to get some notation down,
let's suppose that we're performing tests on m different voxels.
Now we can make the following little table.
We can separate voxels into those that are truly inactive and
those that are truly active.
Of course we're never privy to this information but
in general there are truly active voxels and truly inactive voxels in this setting.
Now we can also declare voxels inactive or active and
that's something that we have control over.
So, for example, in this table we have V
as the number of truly inactive voxels that were declared active.
So, those would be things that we should've not rejected but did reject.
That's a false positive.
So, both U, V, T and S are unobservable random variables,
because we don't know how many false positives we're making in practice.
The only thing we do know is R, which is the number of voxels that were
declared active, because we know which ones were declared active, and weren't.
But we don't know what proportion of those were truly inactive or truly active.
And that's something that we want to be able to control and
that's the basis behind FDR.
So in this notation, a familywise error rate is simply the probability that V
is greater than or bigger than one.
That means that we have one or
more false positives because V is the number of false positives.
The false discovery rate is defined to be 0 if R=0 in this case.
So, if we didn't declare any voxels active then we can't have any false positives.
Then this is not a problem.
So a procedure controlling the FDR ensures that on average the FDR is
no bigger than some pre-specified rate q which usually lies between 0 and 1.
Let's say 0.05.
However, for any given data set, the FDR need not be below the bound, because
the FDR is just the expected number of false positives among all rejected tests.
So, we just know that on average we're going to to control it at a certain rate,
not what's going to happen in any given situation.
So basically an FDR-controlling technique guarantees control of the FDR in the sense
that it's less than or equal to q.
So on average we're going to control the FDR at the level q.
So how do we do this?
Well, the most popular way of doing this is the so-called Benjamini-Hochberg
procedure.
So here we begin by selecting a desired limit q on the FDR, let's say 0.05.
The next thing we do is we just rank all the p-values over
all the voxels from 1 to m, where m is the total number of voxels.
So we just rank them from smallest to biggest, and
then we just plot them from smallest to biggest in a plot.
Then we let r be the largest i, so the largest voxel in this
ranking such that p(i) is less than or equal it i/m x q.
So this is, i/m x q is just a straight line that goes from 0 to q/m.
Now, in this case, we're going to get a straight line there,
and this is the black line we see in this little cartoon here.
And then anything that's below that black line is going to be deemed active and
anything above is not active.
So, here we're seeing the active voxels are those hypotheses,
we're going to reject all the hypotheses whose corresponding
p-values are between p(1) to p(r).
If all of the null hypotheses are true,
then the FDR is equivalent to the familywise error rate.
Any procedure that controls the familywise error rate will also control the FDR.
So a procedure that only controls the FDR can only be less stringent and
lead to a gain in power.
And so since FDR controlling procedures work only on the p-values and
not on the actual test statistics, it can be applied to any valid statistical test.
In the last couple of years,
FDR-controlling procedures have become increasingly popular as they
are less conservative than the familywise error rate-controlling procedures.
And so they're getting a lot of use in the neuroimaging context and
other big data context as well.
In the next module, we'll talk a little about pitfalls with multiple comparisons.
Okay, see you then.
Bye.
