Power es(ma(on for whole brain fMRI based on peaks Thomas E. Nichols University of Warwick NIHR Imaging Sta(s(cs Group 20 April, 2016

Overview •  fMRI spa(al sta(s(cs •  Detailed Univariate fMRI Power Analysis •  fMRI Power analysis based on peaks

fMRI Perspec(ve 1,000

•  4-Dimensional Data –  1,000 mul(variate observa(ons, each with 100,000 elements –  100,000 (me series, each with 1,000 observa(ons 2

•  Usual approach is the (me-series perspec(ve

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BOLD fMRI

•  Finger Tapping

Blood Oxygenation Level-Dependent Effect Red blood → Blue blood Diamagnetic → Paramagnetic

Tap fingers Rest Single fMRI Image

•  Learning task

Learn A

Learn B

Learn C Feedback

Recall A

Recall B

Recall C

Thresholded T statistic image overlaid on single fMRI image

•  Mass Univariate Model: Actually, just a simple regression model at each voxel

Fixation

Blue-sky inference: What we’d like •  Don’t threshold, model the signal! –  Signal location? •  Estimates and CI’s on (x,y,z) location

θˆMag.

–  Signal magnitude? •  CI’s on % change

–  Spatial extent?

θˆ

Loc.

θˆExt.

space

•  Estimates and CI’s on activation volume •  Robust to choice of cluster definition

•  ...but this requires an explicit spatial model –  We only have a univariate linear model at each voxel!

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Real-life inference: What we get •  Signal location –  Local maximum – no inference

•  Signal magnitude –  Local maximum intensity – P-values (& CI’s)

•  Spatial extent –  Cluster volume – P-value, no CI’s •  Sensitive to blob-defining-threshold

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Voxel-level Inference •  Retain voxels above α-level threshold uα •  Gives best spatial specificity –  The null hyp. at a single voxel can be rejected

uα space

Significant Voxels

No significant Voxels 7

Cluster-level Inference •  Two step-process –  Define clusters by arbitrary threshold uclus –  Retain clusters larger than α-level threshold kα

uclus space

Cluster not significant

kα

kα

Cluster significant 8

Cluster-level Inference •  Typically better sensitivity •  Worse spatial specificity –  The null hyp. of entire cluster is rejected –  Only means that one or more of voxels in cluster active uclus space

Cluster not significant

kα

kα

Cluster significant 9

Peak-level Inference •  Two step-process –  Find “peaks”, local maxima, above arbitrary threshold upeak –  Retain peaks larger than α-level threshold uα

•  Spatial specificity –  Relates to point spread function about peak Peak significant

uα

Peak not significant

uclus upeak space

Locations of local maxima

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Power: 1 Test •  Power: Probability of rejec(ng H0 when HA is true •  Must specify: •  Sample size n •  Level α (allowed false posi(ve rate) •  Standard devia(on σ (popula(on variability; not StdErr) •  Effect magnitude Δ •  Last two can be replaced with •  Effect size δ = Δ/σ

Alternative Distribution

Null Distribution

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Power

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0.05



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α 2

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Power: 100,000 Tests? •  Mul(ple tes(ng (easy part)

–  Set α to reflect mul(plicity –  If FWE corrected is typically t*=5, then α = 0.00036

•  Alterna(ve: δ1, δ2, δ3, …, δ99,999, δ100,000 (hard part) –  Must consider all an(cipated alterna(ves –  These 10 voxels ac(ve, and those other 20, and… –  Oh, and don’t forget to specify σ1, σ2, σ3 … too!

•  Cheap & cheerful

–  Base power on extracted summary “primary outcome” –  But doesn’t account for temporal or spa(al detail •  Can we do bener?

fMRI Modelling: A two-stage approach Intra-subject model for Subject k

Inter-subject group model

•  Previous result only for first level fMRI •  2nd level fMRI doesn’t depend on 1st level Pvalues •  Data quality also an issue 13

Level 1: Autocorrelation •  What’s your Vk (N×N matrix)?

–  Who has intuition on magnitude of autocorrelation? –  True Vk is very complicated •  SPM uses per-subject, global, cheap & cheerful AR(1) approximation (ρ≈0.2) •  FSL uses per-subject, local, tapered & spatially regularized arbitrary Autocorrelation function –  Different at every voxel!

•  Simple proposal: AR(1) + White Noise –  Estimated from residuals –  Specified by σWN, σAR & ρ

Mumford, J. A., & Nichols, T. E. (2008). Power calcula(on for group fMRI studies accoun(ng for arbitrary design and temporal autocorrela(on. NeuroImage, 39(1), 261–8. 14 Mumford, J. A. (2012). A power calcula(on guide for fMRI studies. Social Cogni1ve and Affec1ve Neuroscience, 7(6), 738–42.

Power as a function of run length and sample size

Block design 15s on 15s off TR=3s Pilot: reading study

δ=0.69% σg=0.433% ρ=0.73, σAR = 0.980 %, σWN=1.313% α= 0.005

Power as a function of cost •  Cost to achieve 80% power •  Cost=$300 per subject+$10 per each extra minute

fMRIpower tool hnp://fmripower.org for both SPM & FSL

Limita(ons to this detailed approach •  Requires knowledge of 4 separate noise parameters! –  Intrasubject •  σAR, σWN & ρ

–  Between subject •  σg

•  Tool es(mates these from data •  But with out pilot data in-had would be tough •  A univariate approach, no use of space

Power using spa(al sta(s(cs •  Set out to make power based imaging measure –  Voxel-wise •  Requires good model of spa(al dependence –  Data are ‘smooth’, but never know/es(mate spa(al ACF

–  Cluster-wise •  Clusters independent (yeah!) •  Parametric (RFT) cluster P-values awful (invalid, unstable)

–  Peak-wise •  Peaks independent •  Parametric RFT peak P-values good (i.e. U(0,1) for Ho) Durnez, J., Moerkerke, B., & Nichols, T. E. (2014). Post-hoc power es(ma(on for topological inference in fMRI. NeuroImage, 84, 45–64.

Power based on peaks •  Specify analysis solely by –  Propor(on of brain under H1 (π1) –  Effect size δ in H1 regions •  Equivalently, separately give μ and σ in H1

–  And the usual other stuff… •  α, desired power or N

•  From pilot data… –  Es(mate π1 from p-value distribu(on –  Given π1, es(mate mixture distribu(on for peaks •  Non-null mixture gives δ

Es(ma(ng π1: Propor(on Brain Ac(vated •  Peak heights have simple null distribu(on –  Exponen(al distribu(on with mean u+1/u

•  Use this to make P-values •  Many methods available to es(mate π1=1-π0 –  We use Pounds & Morris

Es(ma(ng μ1 & σ1 •  Null peaks: Exponen(al •  Alt. peaks: Normal(μ1,σ1) •  Propor(on of each –  Null π0 Alt π1 π0+π1=1

•  Mixture models are nasty to fit –  But if we fix π0, π1 not so nasty

Example: Es(ma(ng Parameters •  Pilot data’s sta(s(c image –  1620 peaks above u = 2 –  Es(mated Alt peak distn: N(3.26,1.5)

Power Predic(on •  Now need to set α –  Bonferroni FWE 5% (based on # voxels) –  Benjamini & Hochberg FDR 5% –  RFT voxel-wise 5% (based on # voxels & smoothness) –  Uncorrected α=0.001

•  FWE –  41-42 subjects

•  FDR –  27 subjects

•  Effect prevalence and effect size es(mated from peaks only •  Then computes power for given number of subjects, peak threshold

NeuroPower

http://neuropower.shinyapps.io/neuropower

Conclusions •  Precise fMRI power really hard –  Have to account for design, temporal autocorrela(on… –  But can play with experimental design, dura(on…

•  Peak power –  When a pilot sta(s(c image is available, –  Simple approach, just 3 parameters •  π1 & μ1 & σ1

–  But can’t generalise to other designs –  Peak inference not universally understood/used