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## Random Field Theory

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**Random Field Theory**Will Penny SPM short course, London, May 2005**image data**parameter estimates designmatrix kernel • General Linear Model • model fitting • statistic image realignment &motioncorrection Random Field Theory smoothing normalisation StatisticalParametric Map anatomicalreference corrected p-values**Overview**1. Terminology 2. Random Field Theory 3. Imaging Data 4. Cluster level inference • SPM Results • FDR**Overview**1. Terminology 2. Random Field Theory 3. Imaging Data 4. Cluster level inference • SPM Results • FDR**Inference at a single voxel**NULL hypothesis, H: activation is zero a = p(t>u|H) p-value: probability of getting a value of t at least as extreme as u. If a is small we reject the null hypothesis. u=2 t-distribution u=(effect size)/std(effect size)**Sensitivity and Specificity**ACTION Don’t Reject Reject H True TN FP H False FN TP TRUTH Specificity = TN/(# H True) = TN/(TN+FP) = 1 - a Sensitivity = TP/(# H False) = TP/(TP+FN) = b = power a = FP/(# H True) = FP/(TN+FP) = p-value/FP rate/sig level**Sensitivity and Specificity**ACTION At u1 Don’t Reject Reject Spec=7/10=70% Sens=10/10=100% H True (o) TN=7 FP=3 H False (x) FN=0 TP=10 TRUTH Specificity = TN/(# H True) Sensitivity = TP/(# H False) Eg. t-scores from regions that truly do and do not activate o o o o o o o x x x o o x x x o x x x x u1**Sensitivity and Specificity**ACTION Don’t Reject Reject At u2 H True (o) TN=9 FP=1 H False (x) FN=3 TP=7 TRUTH Spec=9/10=90% Sens=7/10=70% Specificity = TN/(# H True) Sensitivity = TP/(# H False) Eg. t-scores from regions that truly do and do not activate o o o o o o o x x x o o x x x o x x x x u2**Inference at a single voxel**NULL hypothesis, H: activation is zero a = p(t>u|H) We can choose u to ensure a voxel-wise significance level of a. This is called an ‘uncorrected’ p-value, for reasons we’ll see later. We can then plot a map of above threshold voxels. u=2 t-distribution**Signal**Inference for Images Noise Signal+Noise**Use of ‘uncorrected’ p-value, a=0.1**11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% Percentage of Null Pixels that are False Positives Using an ‘uncorrected’ p-value of 0.1 will lead us to conclude on average that 10% of voxels are active when they are not. This is clearly undesirable. To correct for this we can define a null hypothesis for images of statistics.**FAMILY-WISE NULL HYPOTHESIS:**Activation is zero everywhere If we reject a voxel null hypothesis at any voxel, we reject the family-wise Null hypothesis A FP anywhere in the image gives a Family Wise Error (FWE) Family-wise Null Hypothesis Family-Wise Error (FWE) rate = ‘corrected’ p-value**Use of ‘uncorrected’ p-value, a=0.1**Use of ‘corrected’ p-value, a=0.1 FWE**The Bonferroni correction**The Family-Wise Error rate (FWE), a, fora family of N independent voxels is α = Nv where v is the voxel-wise error rate. Therefore, to ensure a particular FWE set v = α / N BUT ...**The Bonferroni correction**Independent Voxels Spatially Correlated Voxels Bonferroni is too conservative for brain images**Overview**1. Terminology 2. Random Field Theory 3. Imaging Data 4. Cluster level inference • SPM Results • FDR**Consider a statistic image as a discretisation of a**continuous underlying random field Use results from continuous random field theory Random Field Theory Discretisation**Topological measure**threshold an image at u EC=# blobs at high u: Prob blob = avg (EC) So FWE, a = avg (EC) Euler Characteristic (EC)**Example – 2D Gaussian images**• α = R (4 ln 2) (2π) -3/2 u exp (-u2/2) Voxel-wise threshold, u Number of Resolution Elements (RESELS), R N=100x100 voxels, Smoothness FWHM=10, gives R=10x10=100**Example – 2D Gaussian images**• α = R (4 ln 2) (2π) -3/2 u exp (-u2/2) For R=100 and α=0.05 RFT gives u=3.8**Resel Counts for Brain Structures**(1) Threshold depends on Search Volume (2) Surface area makes a large contribution FWHM=20mm**Overview**1. Terminology 2. Theory 3. Imaging Data 4. Levels of Inference 5. SPM Results**Functional Imaging Data**• The Random Fields are the component fields, Y = Xw +E, e=E/σ • We can only estimate the component fields, using estimates of w and σ • To apply RFT we need the RESEL count which requires smoothness estimates**^** Estimated component fields voxels ? ? = + parameters design matrix errors data matrix scans parameterestimates • estimate residuals estimated variance = Each row is an estimated component field estimatedcomponentfields**Smoothness**smoothness » voxel size practically FWHM 3 VoxDim Typical applied smoothing: Single Subj fMRI: 6mm PET: 12mm Multi Subj fMRI: 8-12mm PET: 16mm Applied Smoothing**Overview**1. Terminology 2. Theory 3. Imaging Data 4. Levels of Inference 5. SPM Results**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**n=12**n=82 n=32 Levels of inference voxel-level P(c 1 | n > 0, t 4.37) = 0.048 (corrected) At least one cluster with unspecified number of voxels above threshold set-level P(c 3 | n 12, u 3.09) = 0.019 At least 3 clusters above threshold cluster-level P(c 1 | n 82, t 3.09) = 0.029 (corrected) At least one cluster with at least 82 voxels above threshold**Overview**1. Terminology 2. Theory 3. Imaging Data 4. Levels of Inference 5. SPM Results**SPM results I**Activations Significant at Cluster level But not at Voxel Level**SPM results II**Activations Significant at Voxel and Cluster level**False Discovery Rate**ACTION At u1 Don’t Reject Reject FDR=3/13=23% a=3/10=30% H True (o) TN=7 FP=3 H False (x) FN=0 TP=10 TRUTH Eg. t-scores from regions that truly do and do not activate FDR = FP/(# Reject) a = FP/(# H True) o o o o o o o x x x o o x x x o x x x x u1**False Discovery Rate**ACTION Don’t Reject Reject H True (o) TN=9 FP=1 H False (x) FN=3 TP=7 TRUTH At u2 FDR=1/8=13% a=1/10=10% Eg. t-scores from regions that truly do and do not activate FDR = FP/(# Reject) a = FP/(# H True) o o o o o o o x x x o o x x x o x x x x u2**Signal**False Discovery Rate Noise Signal+Noise**Control of Familywise Error Rate at 10%**FWE Occurrence of Familywise Error Control of False Discovery Rate at 10% 6.7% 10.5% 12.2% 8.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% Percentage of Activated Pixels that are False Positives**Summary**• We should not use uncorrected p-values • We can use Random Field Theory (RFT) to ‘correct’ p-values • RFT requires FWHM > 3 voxels • We only need to correct for the volume of interest • Cluster-level inference • False Discovery Rate is a viable alternative