HST 583 fMRI DATA ANALYSIS AND ACQUISITIONOutlinePowerPoint PresentationSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35ConclusionsHST 583 fMRI DATA ANALYSIS AND ACQUISITIONA Review of Statistics forfMRI Data AnalysisEmery N. BrownMassachusetts General HospitalHarvard Medical School/MIT Division of Health, Sciences and TechnologyDecember 2, 2002Outline•What Makes Up an fMRI Signal?•Statistical Modeling of an fMRI Signal•Maxmimum Likelihoood Estimation for fMRI•Data Analysis•ConclusionsTHE STATISTICAL PARADIGM (Box, Tukey)QuestionPreliminary Data (Exploration Data Analysis)ModelsExperiment (Confirmatory Analysis)Model FitGoodness-of-Fit not satisfactoryAssessment SatisfactoryMake an InferenceMake a DecisionCase 3: fMRI Data Analysis Question: Can we construct an accurate statistical model to describe the spatial temporal patterns of activation in fMRI images from visual and motor cortices during combined motor and visual tasks? (Purdon et al., 2001; Solo et al., 2001)A STIMULUS-RESPONSE EXPERIMENTAcknowledgements: Chris Long and Brenda MarshallWhat Makes Up An fMRI Signal?Hemodynamic Response/MR Physics i) stimulus paradigma) event-relatedb) block ii) blood flow iii) blood volume iv) hemoglobin and deoxy hemoglobin contentNoise Stochastic i) physiologic ii) scanner noiseSystematic i) motion artifact ii) drift iii) [distortion] iv) [registration], [susceptibility]Physiologic Response Model: Block DesignGamma Hemodynamic Response ModelPhysiologic Model: Event-Related Design0 20 40 60 80 100 12000.51Flow Term0 20 40 60 80 100 12000.51Volume Term0 20 40 60 80 100 12000.51Interaction Term0 20 40 60 80 100 120-0.200.20.40.6Modeled BOLD Signalfa=1 fb=-0.5fc=0.2Physiologic Model: Flow, Volume and Interaction TermsScanner and Physiologic Noise ModelsDATA:, …,1 Ty yThe sequence of image intensity measurements on a singlepixel.fMRI Signal and Noise Model= ( ) + vt ty h tMeasurement on a single pixel at timettyPhysiologic response( )h tActivation coefficientPhysiologic and Scanner Noisevtfor = , …,t 1 TWe assume the vtare independent, identically distributedGaussian random variables.fMRI Signal ModelPhysiologic Response( ) = ( ) ( - )h t g u c t u du( )g themodynamic response ( )c tinput stimulusGamma model of the hemodynamic response -( ) =1 - tg t t eAssume we know the parameters of g(t). MAXIMUM LIKELIHOODDefine the likelihood function( ) = ( )L | y f y | , the joint probability density viewed as a function of the parameter with the data yfixed. The maximum likelihood estimateof isˆMLˆ( ) = arg max ( ) = arg max ( ). MLy L | y logL | yThat is,ˆ( )MLyis a parameter value for which ( )L | yattains a maximum as a function offor fixed y.ESTIMATIONJoint DistributionT22Tt tt 12 2y h1 1f y22 =( - )( | ) = exp -Log Likelihood=log ( | ) = log( ) - ( - )= ( ) 2 2 2Tt tt 12T 1f y 2 y h /2 2Maximum Likelihood ˆˆˆ-= =-ε=== ( - ) 12T Tt t tt 1 t 12 1 2Tt tt 1h h yT y h GOODNESS-OF-FIT/MODEL SELECTIONAn essential step, if not the most essential step in a data analysis,is to measures how well the model describes the data. This should be assessed before the model is used to make inferencesabout that data. Akaike’s Information CriterionˆML2 f | 2p- log (y ) +For maximum likelihood estimates it measures the trade-off between maximizing the likelihood (minimizingˆML2 f |- log (y )and the numbers of parametersp, the model requires.)GOODNESS-OF-FIT• Residual Plots:ˆˆ= -t t ty h • KS Plots:ˆ( )2tΝ 0, We can check the Gaussian assumption with our K-S plots.Measure correlation in the residuals to assess independence.EVALUATION OF ESTIMATORSGiven w( ),yan estimator ofbased on= ( , …, )1 ny y yMean-Squared Error:[ ( ) - ] = Variance + bias2 2E w yBias=[ ( )]- ; unbiasedness [ ( )] =E w y E w y Consistency:( ) as (sample size) w y n Efficiency: Achieves a minimum variance (Cramer-Rao Lower Bound)FACTOIDS ABOUT MAXIMUM LIKELIHOOD ESTIMATES•Generally biased.•Consistent, hence asymptotically unbiased.•Asymptotically efficient.•Variance can be approximated by minus the inverse of the Fisher information matrix.•If ˆis the MLestimate of,thenˆ( )his the MLestimate of( ).hCramer-Rao Lower Bound2dE w ydw yf y-E [ ( )]Var[ ( )][ log ( | ]CRLB gives the lowest bound on the variance of an estimate.CONFIDENCE INTERVALSThe approximate probability density of the maximum likelihood estimates is the Gaussian probability density withmean and variance-- ( )1Iwhere ( )Iis the Fisherinformation matrixlog (y )( ) = - 22f |I E An approximate confidence interval for a component of isˆ-± ( ) 121i,ML |z iiz I THE INFORMATION MATRIX-=- -( )( ) = -( 2 1 2Ttt 12 2 1h 0I0 ) T 2CONFIDENCE INTERVAL ˆˆˆ ˆ-=-±± 12122tTT 12 22 h2 2 TKolmogorov-Smirnov Test White Noise Model2 2 White Noise ModelPixelwise Confidence Intervals for the SlicefMRI Signal and Noise Model 2= ( ) + vt ty h tMeasurement on a single pixel at timettyPhysiologic response( )h tActivation coefficientPhysiologic and Scanner Noisetv v t t-1for = , …,t 1 TWe assume the vtare correlated noise AR(1)Gaussian random variables.Simple Convolution Plus Correlated NoiseKolmogorov-Smirnov Test Correlated Noise Model2 2 Correlated Noise ModelPixelwise Confidence Intervals for the SliceAIC Difference = AIC Colored Noise-AIC White NoisefMRI Signal and Noise Model 3t ty t= s( ) + vMeasurement on a single pixel at timettyPhysiologic response1cos( ) sin( )qrs t rt B rt r r( ) = APhysiologic and Scanner Noisevtfor = , …,t 1 TWe assume the vtare independent, identically distributedGaussian random variables.Harmonic Regression Plus White Noise
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