random field theory.ppt

Random Field TheoryMkael Symmonds, Bahador Bahrami

OverviewSpatial smoothing

Statistical inference

The multiple comparison problem

and what to do about it

Statistical inferenceAimto decide if the data represents convincing evidence of the effect we are interested in.

How perform a statistical test across the whole brain volume to tell us how likely our data are to have come about by chance (the null distribution).

Inference at a single voxela = p(t>t-value|H0)NULL hypothesis, H0: activation is zerot-value = 2.42t-distributionp-value: probability of getting a value of t at least as extreme as 2.42 from the t distribution (= 0.01). t-value = 2.42alpha = 0.025As p < , we reject the null hypothesist-value = 2.02

Sensitivity and SpecificityH0 True TN FP type I error

H0 False FN TPDontRejectRejectACTIONChanceNot by chance

Many statistical testsIn functional imaging, there are many voxels, therefore many statistical tests

If we do not know where in the brain our effect will occur, the hypothesis relates to the whole volume of statistics in the brain

We would reject H0 if the entire family of statistical values is unlikely to have arisen from a null distribution a family-wise hypothesis

The risk of error we are prepared to accept is called the Family-Wise Error (FWE) rate what is the likelihood that the family of voxel values could have arisen by chance

How to test a family-wise hypothesis?Height thresholdingThis can localise significant test results

How to set the threshold?Should we use the same alpha as when we perform inference at a single voxel?

How to set the threshold?LOTS OF SIGNIFICANT ACTIVATIONS OUTSIDE OF OUR SIGNAL BLOB!

How to set the threshold?So, if we see 1 t-value above our uncorrected threshold in the family of tests, this is not good evidence against the family-wise null hypothesis

If we are prepared to accept a false positive rate of 5%, we need a threshold such that, for the entire family of statistical tests, there is a 5% chance of there being one or more t values above that threshold.

Bonferroni CorrectionFor one voxel (all values from a null distribution)Probability of a result greater than the threshold = Probability of a result less than the threshold = 1-

For n voxels (all values from a null distribution)Probability of all n results being less than the threshold = (1-)nProbability of one (or more) tests being greater than the threshold: = 1-(1-)n ~= n. (as alpha is small)

FAMILY WISE ERROR RATE

Bonferroni CorrectionSo,

Set the PFWE < n. Gives a threshold = PFWE / n

Should we use the Bonferroni correction for imaging data?

NULL HYPOTHESIS TRUE10,000 tests 5% FWE rateApply Bonferroni correction to give threshold of 0.05/10000 = 0.000005This corresponds to a z-score of 4.42We expect only 5 out of 100 such images to have one or more z-scores > 4.42100 x 100 voxels normally distributed independent random numbers

100 x 100 voxels averagedNow only 10 x 10 independent numbers in our imageThe appropriate Bonferroni correction is 0.05/100= 0.0005This corresponds to z-score = 3.29Only 5/100 such images will have one or more z-scores > 3.29 by chance

Spatial correlationIndependent VoxelsSpatially Correlated VoxelsBonferroni is too conservative for brain images, but how to tell how manyindependent observations there are?Assumes Independent Voxels

Spatial smoothingIncreases signal-to-noise ratio

Enables averaging across subjects

Allows use of Gaussian Random Field Theory for thresholding

Why do you want to do it?

Spatial SmoothingReduces effect of high frequency variation in functional imaging data, blurring sharp edgesWhat does it do?

Spatial SmoothingTypically in functional imaging, a Gaussian smoothing kernel is usedShape similar to normal distribution bell curveWidth usually described using full width at half maximum (FWHM) measure e.g., for kernel at 10mm FWHM:How is it done?

Spatial SmoothingGaussian kernel defines shape of function used successively to calculate weighted average of each data point with respect to its neighbouring data pointsHow is it done?Raw dataGaussian functionSmoothed datax=

Spatial correlationIndependent VoxelsSpatially Correlated VoxelsBonferroni is too conservative for brain images, but how to tell how manyindependent observations there are?Assumes Independent Voxels

ReferencesPrevious MfD slidesAn Introduction to Random Field Theory, from Human Brain Mapping, Matthew Brett, Will Penny, Stefan KiebelStatistical Parametric Mapping short course lecture on RFT, Tom Nichols

Random Field Theory (ii)Methods for Dummies 2008Mkael SymmondsBahador Bahrami

What is a random field?A random field is a list of random numbers whose values are mapped onto a space (of n dimensions). Values in a random field are usually spatially correlated in one way or another, in its most basic form this might mean that adjacent values do not differ as much as values that are further apart.

Why random field? To characterise the properties our studys statistical parametric map under the NULL hypothesis NULL hypothesis = if all predictions were wrong all activations were merely driven by chanceeach voxel value was a random number

What would the probability of getting a certain z-score for a voxel in this situation be?

Random Field

Thresholded @ ZeroThresholded @ one

Average number of blobs = (4 + 0 + 1 + + 2)/1000000000

Therefore, for every z-score, the expected value of number of blobs = probability of rejecting the null hypothesis erroneously ()

The million-dollar question is:thresholding the random field at which Z-score produces average number of blobs < 0.05? Or, Which Z-score has a probability = 0.05 of rejecting the null hypothesis erroneously?Any z-scores above that will be significant!

So, it all comes down to estimating the average number of blobs (that you expect by chance) in your SPM Random field theory does that for you!

Expected number of blobs in a random field depends on Chosen threshold z-score

Volume of search regionRoughness (i.e.,1/smoothness) of the search region: Spatial extent of correlation among values in the field; it is described by FWHMVolume and Roughness are combined into RESELsWhere does SPM get R from: it is calculated from the residuals (RPV.img)Given the R and Z, RFT calculates the expected number of blobs for you: E(EC) = R (4 ln 2) (2) -3/2 z exp(-z2/2)

PFWE = average number of blobs under null hypothesisProbability of Family Wise Error = PFWE = R (4 ln 2) (2) -3/2 z exp(-z2/2)

Thank youReferences: Brett, Penny & Keibel. An introduction to Random Field Theory. Chapter from Human Brain MappingWill Pennys slides (http://www.fil.ion.ucl.ac.uk/spm/course/slides05/ppt/infer.ppt#324,1,Random Field Theory)Jean-Etienne Poirriers slides (http://www.poirrier.be/~jean-etienne/presentations/rft/spm-rft-slides-poirrier06.pdf)Tom Nichols lecture in SPM Short Course (2006)

False Discovery Rateu1o o o o o o o x x x o o x x x o x x x xEg. t-scoresfrom regionsthat truly do and do not activateFDR = FP/(# Reject) a = FP/(# H True)

False Discovery Rateu2o o o o o o o x x x o o x x x o x x x xEg. t-scoresfrom regionsthat truly do and do not activateFDR=1/8=13%a=1/10=10% At u2FDR = FP/(# Reject) a = FP/(# H True)

False Discovery Rate

Signal+NoiseNoise

Cluster Level Inference

We can increase sensitivity by trading off anatomical specificity

Given a voxel level threshold u, we can compute the likelihood (under the null hypothesis) of getting a cluster containing at least n voxels

CLUSTER-LEVEL INFERENCE

Similarly, we can compute the likelihood of getting c clusters each having at least n voxels

SET-LEVEL INFERENCE

Levels of inferenceset-levelP(c 3 | n 12, u 3.09) = 0.019cluster-levelP(c 1 | n 82, t 3.09) = 0.029 (corrected)

voxel-levelP(c 1 | n > 0, t 4.37) = 0.048 (corrected)At least onecluster withunspecifiednumber of voxels abovethresholdAt least one cluster with at least 82 voxels above thresholdAt least 3 clusters abovethreshold

We perform a statistical test to inform us how surprising our data are that is, how likely that the data is to have come about by chance. The null hypothesis is that the data are drawn from a purely random distribution of values, with a Gaussian distribution of noise. So our null hypothesis is that there is no activation and our data variance is pure noise.

Say we perform a t-test, and get a value of 2. We translate this into a p value, the probability of getting this t value from our distribution with our given degrees of freedom (p=0.01 for df=40 and t=2.42). We set the value of alpha at a level that we feel is reasonable (i.e. the proportion or rate of false positives that we are willing to accept). This can be anything we like in this case 2.5%, though in most social research it is set at 5% by convention.If p is less than alpha, we reject the null hypothesis it is unlikely that the data arose by chance.For the given example, a p value of 0.01 means that there is a 1% chance for making a type 1 error (falsely rejecting the null hypothesis).So from our threshold we can have 4 possibilities. Null hypothesis is true and we treat it is such a true -ve, Null hypothesis is true and we reject it - think its activation and get a False positive. Alternatively, the null hypothesis is false and we incorrectly reject it, a false negative. Lastly, we correctly reject the null hypothesis and so get a true positive..

We can define two terms. Firstly , the specificity that is given by the number of true negatives given that the null hypothesis is true (no activation) this is simply 1-alpha.

Secondarily, we get the sensitivity that is; given that there is an activation, the null hypothesis is wrong how much of the time do we get it correct.The family wise error is like the alpha for an individual test, but now we are considering a whole set of tests, one for each voxelTo test a family-wise hypothesis, we can look for any statistic values are larger than we expect, if they had all come from a null distribution. Therefore, we need to find a threshold that we can apply to every statistic value, in order to say that values above the threshold are unlikely to have arisen by chance.If we find voxels with statistic values above threshold, we can conclude that there is an effect at these voxel locations.What threshold should we set? Should we use the same value for alpha as we use in a single statistical test an uncorrected p valueIf alpha, our threshold, = 0.1, then 10% of our voxels will be falsely positive i.e. lead to false rejection of the null hypothesisIf we see one or more t-values above our threshold in the family of tests that we perform, this is not good evidence against the family-wise null hypothesis that all these values have been drawn from a null distribution.We want to set the threshold such that the probability of an individual test exceeding the threshold is 0.05, at a 5% significance level.So we want our family-wise error rate to be 5%, which the Bonferroni correction tells us is when our threshold alpha is 5%/nIf we smooth randomly generated data by averaging, we reduce the number of independent observations, so change the Bonferroni correction required.The Bonferroni assumes n independent statistical tests in fact the data is spatially correlatedIn imaging data, how can we tell how many independent observations there are? The data are spatially correlated before smoothing, and this is even more so after smoothing.

The Bonferroni assumes n independent statistical tests in fact the data is spatially correlatedIn imaging data, how can we tell how many independent observations there are? The data are spatially correlated before smoothing, and this is even more so after smoothing.

random field theory.ppt

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