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Conjunction Inference Using the Bayesian
Interpretation of the Positive
False Discovery Rate
(pFDR)
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Thomas E. Nichols1 ,
Tor D. Wager,2
1Department of Biostatistics,
University Michigan, Ann Arbor, MI 48109, 2Department of Psychology, Columbia
University, New York, NY 10027
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Introduction
Functional neuroimaging often requires a test for the
conjunction of several effects. For example, task A
and B may each involve working memory but use
different modalities; it is of interest to find brain
regions where both task A and B are significantly
activated.
In separate work (see poster by Brett etal), we show
that conjunction inference based on the minimum
statistic test (Worsley & Friston, 2000; SPM99; SPM2)
does not control the relevant false positive rate.
That is, a significant P-value for a minimum statistic
only means that one or more of the effects are
significant, not that all effects are
significant.
In this work we propose an approach to conjunction
inference which overcomes this fundamental limitations
of the minimum statistic test. We use the Bayesian
interpretation of the Positive False Discovery Rate
(pFDR) (Storey, 2003).
A pFDR "q-value" is also the posterior probability of
the (random) null hypothesis given the extremity of
the data. It is precisely the Bayesian complement of
the P-value (a P-value is the probability of the
extremity of the data given the null hypothesis).
Using posterior probabilities, it is easy to control
the relevant conjunction false positive rate, the
probability of "not all effects true".
We introduce pFDR, describe our method, and apply it
to a real dataset.
Gentle Methods
1. Create images of pFDR q-values. 2. Sum the q-value
images to be conjoined. 3. Reject the conjunction
null where sum image is less than 0.05.
Statistical Methods
pFDR. For one statistic image and a given
threshold u, the FDR is the expected proportion
of false positives among suprathreshold voxels. The
pFDR is same except that it conditions on there
being at least one suprathreshold voxel. pFDR has
been described as "the rate at which discoveries are
false".
Bayes & pFDR. Let Ti be the
statistic value at voxel i and let
Hi be the null hypothesis;
Hi = 0 if the null is true or 1
otherwise. Let Hi be random.
Then for a threshold u, pFDR(u) =
P(Hi = 0 | Ti
≥ u) (Storey 2003). This is the posterior
probability of the null given that voxel i is
as or more extreme than u.
Conjunction. Now consider conjoining K
statistic images. The conjunction null is
Uk{Hik=0},
the state of one or more nulls being true. We control the
posterior conjunction null probability:
P(Uk{Hik=0}|Πk{Tik>ti})
≤ ΣP(Hik=0|Πk{Tik>ti})
= Σk P(Hik=0|Tik>ti)
The inequality uses Bonferroni and the equality uses
independence over the K statistics at each voxel
(not spatial independence). The last summation is just
a summation of pFDR q-values.
Results
We applied this method to a study of response
inhibition. We considered K=2 tasks, a
go-no-go task and a flanker task. See Figure.
Conclusion
Using pFDR we have proposed a new method for
conjunction inference. Our method is easy to apply,
yet controls the appropriate conjunction false
positive rate.
References
Storey JD. (2003) The positive false discovery rate: A
Bayesian interpretation and the q-value. Annals of
Statistics, 31: 2013-2035
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Conjunction of two tasks involving
response inhibition. Top two panels show t statistic images.
Bottom shows -log10 posterior probability of either or both
effects being null. Yellow regions correspond to posterior
probability less than 0.1; 3 voxels are less than 0.05.
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