Spatial Smoothing and Multiple Comparisons Correction for Dummies Alexa Morcom, Matthew Brett Acknowledgements

Spatial Smoothing andSpatial Smoothing andMultiple Comparisons Multiple Comparisons

Correction for DummiesCorrection for Dummies

Alexa Morcom, Matthew BrettAcAcknowledgementsknowledgements

Spatial Smoothing andSpatial Smoothing andMultiple Comparisons Multiple Comparisons

Correction for DummiesCorrection for Dummies

(i.e., no equations)(i.e., no equations)

Alexa Morcom, Matthew BrettAcAcknowledgementsknowledgements

OverviewOverview• Spatial Smoothing

– What does it do?

– How is it done?

– Why do you want to do it?

• Correction for Multiple Comparisons– Bonferroni correction

– Random field theory

– Uncorrected thresholds

– False discovery rate

– Which correction method to use?



– How is it done?







Spatial SmoothingSpatial Smoothing

• Reduces effect of high frequency variation in functional imaging data, “blurring sharp edges”

WhWhat does it do?at does it do?


• Typically in functional imaging, a Gaussian smoothing kernel is used– Shape similar to normal distribution

bell curve

– Width usually described using “full width at half maximum” (FWHM) measure

e.g., for kernel at 10mm FWHM:

How is it doneHow is it done??

0 5-5


• Gaussian kernel defines shape of function used successively to calculate weighted average of each data point with respect to its neighbouring data points


Raw dataRaw data Gaussian fuGaussian functionnction SSmoothedmoothed data dataxx ==


• Gaussian kernel defines shape of function used successively to calculate weighted average of each data point with respect to its neighbouring data points


Raw dataRaw data Gaussian fuGaussian functionnction SSmoothedmoothed data dataxx ==

http://www.mrc-cbu.cam.ac.uk/Imaging/rnd_figures.pdf



• Increases signal-to-noise ratio– Depends on relative size of smoothing kernel and effects to

be detected

– Matched filter theorem: smoothing kernel = expected signal

– Practically, rule of thumb: FWHM ≥ 3 x voxel size

– May consider varying kernel size if interested in different brain regions, e.g. hippocampus vs. parietal cortex

WhWhy do you want to do it?y do you want to do it?


• Enables averaging across subjects– Reduces influence of functional and/or anatomical

differences between subjects

– Even after realignment and normalisation, residual between-subject variability may remain

– Smoothing data improves probability of identifying commonalities in activation between subjects, but trade-off with anatomical specificity



• Allows use of Gaussian Field Theory for thresholding– Assumes error terms are roughly Gaussian in form

– Requires FWHM to be substantially greater than voxel size

– Enables hypothesis testing and dealing with multiple comparison problem in functional imaging …




– How is it done?







Correction for Multiple ComparisonsCorrection for Multiple Comparisons

• Typically in hypothesis testing, the null hypothesis is rejected if ≤ 5% probability effect emerged by chance– If 100 tests are done, this means that on average, 5 will be

significant by chance

• In the brain, there are many thousands of voxels– Unless this is corrected for, may have many false positive

voxels and may reject the null hypothesis incorrectly

– Evidence against null hypothesis relates to volume of values, so need to calculate family-wise error rate (FWE)

WhWhat is the problem?at is the problem?


• In behavioural experiments, might adjust threshold for rejecting the null hypothesis depending on the number of independent tests: ' = / n

• But in functional imaging, there is typically correlation between signal in neighbouring voxels– This means there is likely to be fewer independent values in the

brain volume than there are voxels– Therefore Bonferroni correction is way too conservative

WhWhy not do Bonferroni correction?y not do Bonferroni correction?


• Allows thresholds to be determined for smooth statistical maps, such as those found in functional imaging

• Procedure:– Estimate smoothness (spatial correlation) of data

– Calculate expected Euler characteristic (EC) at different thresholds

– Calculate threshold for required control of false positives

RRandom field theoryandom field theory


• Calculate number of resels in image– Resels = resolution elements

– Number of resels similar to number of independent observations (but not identical)

– Defined as a block of pixels of same size as FWHM of smoothing kernel

Estimating smoothnessEstimating smoothness

0 5-510

10

Gaussian kerneGaussian kernell RReselesel


• Property of an image after it has been thresholded– Can be thought of as number of “blobs” in image

• At high thresholds, EC drops towards zero– expected EC corresponds to probability of finding an above threshold

blob in image

CaCalculating expected Euler characteristiclculating expected Euler characteristic

Thresholded at Z > 2.75,Thresholded at Z > 2.75,EC = 2EC = 2

Thresholded at Z > 3.5,Thresholded at Z > 3.5,EEC = 1C = 1

SSmoothedmoothed data data





• If number of resels is known, it is possible to calculate the expected EC at any given threshold– If x is the Z score threshold that gives expected EC of 0.05

– Threshold image at x

– There is ≤ 5% probability that any remaining blobs have occurred by chance

– Note that this threshold x depends only on number of resels

CaCalculating expected EC and thresholdlculating expected EC and threshold


• Expected EC for peak of each cluster shown in ‘PFWE-corr’ column

ResultsResults in SPM2 in SPM2


• Expected EC for peak of each cluster shown in ‘PFWE-corr’ column

• Values in ‘Puncorrected’ column refer to uncorrected values ...



• Many research groups use uncorrected thresholds, such as P < 0.001, considering clusters of ≥ 5 voxels as significant– Tends to be more ‘sensitive’ than RFT correction

– Officially considered bad practice because not clear how this threshold relates to FWE

– But often used on basis of control studies finding no false positive activations during visual fixation using these conventions (e.g., Zarahn et al., 1997 NeuroImage)

Uncorrected threshold conventionsUncorrected threshold conventions


• Values in ‘PFDR-corr’ column refer to False discovery rate ...



• Alternative method of inference– Based on observation that subjects vary within an experiment

in degree of overall signal exhibited

– Suggests different thresholds may be appropriate for different subjects

• FDR is proportion of false positive voxels amongst voxels declared positive– cf. FWE which is proportion of false positive voxels amongst

all voxels in volume whether or not declared positive

False discovery rateFalse discovery rate


• Bonferroni, FWE, uncorrected, FDR ...– FWE is most “correct” method, but FDR may be more

sensitive in some cases

– May be a good idea to use whatever method is employed in previous related studies, to increase comparability

• Most important point is to decide on correction method a priori, rather than subjectively adjusting thresholds to give desirable results!

Which correction method to use?Which correction method to use?

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Spatial Smoothing and Multiple Comparisons Correction for Dummies Alexa Morcom, Matthew Brett Acknowledgements