The General Linear Model and Statistical Parametric Mapping Stefan Kiebel Andrew Holmes SPM short course, May 2002 Stefan Kiebel Andrew Holmes SPM short.

The General Linear Model and Statistical Parametric Mapping Stefan Kiebel Andrew Holmes SPM short course, May 2002 Stefan Kiebel Andrew Holmes SPM short course, May 2002

…a voxel by voxel hypothesis testing approach  reliably identify regions showing a significant experimental effect of interest Key conceptsKey concepts TypeType I error – –significance test at each voxel Parametric statisticsParametric statistics –parametric model for voxel data, test model parameters No exact prior anatomical hypothesisNo exact prior anatomical hypothesis –multiple comparisons Statistical Parametric MappingStatistical Parametric Mapping General Linear ModelGeneral Linear Model Theory of continuous random fieldsTheory of continuous random fields …a voxel by voxel hypothesis testing approach  reliably identify regions showing a significant experimental effect of interest Key conceptsKey concepts TypeType I error – –significance test at each voxel Parametric statisticsParametric statistics –parametric model for voxel data, test model parameters No exact prior anatomical hypothesisNo exact prior anatomical hypothesis –multiple comparisons Statistical Parametric MappingStatistical Parametric Mapping General Linear ModelGeneral Linear Model Theory of continuous random fieldsTheory of continuous random fields SPM key concepts... ?

…a voxel by voxel hypothesis testing approach  reliably identify regions showing a significant experimental effect of interest The General linear model & Statistical Parametric MappingThe General linear model & Statistical Parametric Mapping General Linear Model – –models & design matrices – –model estimation Generalised Linear Model – –serial correlations – –variance components Contrasts & Statistic imagesContrasts & Statistic images –Statistical Parametric Maps – SPMs Global effectsGlobal effects Model selectionModel selection …a voxel by voxel hypothesis testing approach  reliably identify regions showing a significant experimental effect of interest The General linear model & Statistical Parametric MappingThe General linear model & Statistical Parametric Mapping General Linear Model – –models & design matrices – –model estimation Generalised Linear Model – –serial correlations – –variance components Contrasts & Statistic imagesContrasts & Statistic images –Statistical Parametric Maps – SPMs Global effectsGlobal effects Model selectionModel selection Overview…Overview…

realignment & motion correction smoothing normalisation General Linear Model Ümodel fitting Üstatistic image corrected p-values image data parameter estimates design matrix anatomical reference kernel Statistical Parametric Map random field theory

Epoch fMRI –7s RT – interscan interval –84 scans – images 16–99 –1 session –1 condition/trial: words –Deterministic design: Fixed SOA of 12 scans (42s) –Epoch design –First trial at time 6 scans, 6 scans / words epoch SPM parameters (event terminology) Single subjectSingle subject –RH male ConditionsConditions Passive word listeningPassive word listening –Bisyllabic nouns –60wpm –against rest Epoch fMRIEpoch fMRI –rest & words –epochs of 6 scans  42 second epochs –7 BA cycles experiment was 8 cycles: first pair of blocks dropped  BABABABABABABA  last 84 scans of experiment images 16–99  ~10 minutes scanning time Single subjectSingle subject –RH male ConditionsConditions Passive word listeningPassive word listening –Bisyllabic nouns –60wpm –against rest Epoch fMRIEpoch fMRI –rest & words –epochs of 6 scans  42 second epochs –7 BA cycles experiment was 8 cycles: first pair of blocks dropped  BABABABABABABA  last 84 scans of experiment images 16–99  ~10 minutes scanning time Example epoch fMRI activation dataset: Auditory stimulation (50, -40, 8) — Primary Auditory Cortex seconds

model specification f MRI time series statistic image or SPM Voxel by voxel statistics… voxel time series parameter estimation hypothesis statistic

Voxel statistics… parametricparametric one sample t-testone sample t-test two sample t-testtwo sample t-test paired t-testpaired t-test AnovaAnova AnCovaAnCova correlationcorrelation linear regressionlinear regression multiple regressionmultiple regression F-testsF-tests etc…etc… non-parametric ?  SnPMnon-parametric ?  SnPM parametricparametric one sample t-testone sample t-test two sample t-testtwo sample t-test paired t-testpaired t-test AnovaAnova AnCovaAnCova correlationcorrelation linear regressionlinear regression multiple regressionmultiple regression F-testsF-tests etc…etc… non-parametric ?  SnPMnon-parametric ?  SnPM all cases of the General Linear Model assume normality to account for serial correlations: Generalised Linear Model

…e.g. two-sample t-test? ¢ standard t-test assumes independence  ignores temporal autocorrelation! voxel time series t-statistic image SPM{t} compares size of effect to its error standard deviation Image intensity

Regression example…  correlation: test H 0 :  = 0 equivalent to test H 0 :  = 0 Xtwo-sample t-test: test H 0 :  0 =  1 equivalent to test H 0 :  = 0 box-car reference function voxel time series Y s =  +  f(t s ) +  s =  ++ error  s ~ N(0,  2 ) f(t s ) = 0 or 1  t-statistic for H 0 :  = 0 4can extend to account for temporal autocorrelation!

…revisited…revisited =  + + ss =  ++ f(t s )1YsYs error 

General Linear Model… fMRI time series: Y 1,…,Y s,…,Y NfMRI time series: Y 1,…,Y s,…,Y N –acquired at times t 1,…,t s,…,t N Model: Linear combination of basis functionsModel: Linear combination of basis functions Y s =  1 f 1 (t s ) + …+  l f l (t s ) + … +  L f L (t s ) +  s f l (.): basis functionsf l (.): basis functions –“reference waveforms” –dummy variables  l : parameters (fixed effects)  l : parameters (fixed effects) –amplitudes of basis functions (regression slopes)  s : residual errors:  s ~ N(0,  2 )  s : residual errors:  s ~ N(0,  2 ) –identically distributed –independent, or serially correlated (Generalised Linear Model  GLM) fMRI time series: Y 1,…,Y s,…,Y NfMRI time series: Y 1,…,Y s,…,Y N –acquired at times t 1,…,t s,…,t N Model: Linear combination of basis functionsModel: Linear combination of basis functions Y s =  1 f 1 (t s ) + …+  l f l (t s ) + … +  L f L (t s ) +  s f l (.): basis functionsf l (.): basis functions –“reference waveforms” –dummy variables  l : parameters (fixed effects)  l : parameters (fixed effects) –amplitudes of basis functions (regression slopes)  s : residual errors:  s ~ N(0,  2 )  s : residual errors:  s ~ N(0,  2 ) –identically distributed –independent, or serially correlated (Generalised Linear Model  GLM)

Design matrix formulation… Y s =  1 f 1 (t s ) + …+  l f l (t s ) + … +  L f L (t s ) +  s Y s =  1 f 1 (t s ) + …+  l f l (t s ) + … +  L f L (t s ) +  s s = 1, …,N Y 1 =  1 f 1 (t 1 )+…+  l f l (t 1 )+…+  L f L (t 1 )+  1 : : : : : : : : : : : : : : : : Y s =  1 f 1 (t s )+…+  l f l (t s )+…+  L f L (t s )+  s : : : : : : : : : : : : : : : : Y N =  1 f 1 (t N )+…+  l f l (t N )+…+  L f L (t N )+  N Y 1 f 1 (t 1 ) … f l (t 1 )… f L (t 1 )  1  1 : : … :… : : : : : … :… : : : Y s = f 1 (t s ) … f l (t s )… f L (t s )  l +  s : : … :… : : : : : … :… : : : Y N f 1 (t N )… f l (t N ) … f L (t N )  L  N Y= X×  +  Y = X ×  +  N  1 N  L L  1 N  1 data vector design matrix vector of parameters error vector

Box-car regression… (revisited) =  + + ss =  ++ f(t s )1YsYs  error

Box car regression: design matrix…   =+  =  +YX data vector (voxel time series) design matrix parameters error vector  

Low frequency nuisance effects… DriftsDrifts –physical –physiological Aliased high frequency effectsAliased high frequency effects –cardiac (~1 Hz) –respiratory (~0.25 Hz) Discrete cosine transform basis functionsDiscrete cosine transform basis functions –r = 1,…,R –R  cut-off period DriftsDrifts –physical –physiological Aliased high frequency effectsAliased high frequency effects –cardiac (~1 Hz) –respiratory (~0.25 Hz) Discrete cosine transform basis functionsDiscrete cosine transform basis functions –r = 1,…,R –R  cut-off period

…design matrix =  +  =  + YX data vector design matrix parameters error vector 

Example: a line through 3 points… Y i =  x i +  +  i i = 1,2,3 Y 1 =  x 1 +  × 1 +  1 Y 2 =  x 2 +  × 1 +  2 Y 3 =  x 3 +  × 1 +  3 Y 1 x 1 1  1 Y 2 =x 2 1+  2 Y 3 x 3 1  3 Y= X  +  simple linear regression parameter estimates  &  fitted valuesY 1, Y 2, Y 3 residualse 1, e 2, e 3 parameter estimates  &  fitted valuesY 1, Y 2, Y 3 residualse 1, e 2, e 3 x Y  (x 1, Y 1 )   (x 2, Y 2 ) (x 3, Y 3 )  ^ ^  1 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ dummy variables

Geometrical perspective… (1,1,1) (x 1, x 2, x 3 ) (Y 1, Y 2, Y 3 ) design space Y O xx xx x 1 1 Y=   x 2 +   1+  x 3 1 Y=   x    x  

Estimation, geometrically… (Y 1, Y 2, Y 3 ) design space Y (Y1,Y2,Y3)(Y1,Y2,Y3) ^^^ e = (e 1,e 2,e 3 ) T Y ^ ^   x  x   x  x ^ xx xx

Estimation, formally… Residuals Residual sum of squares Minimised when …but this is the l th row of Consider parameter estimates Giving fitted values so the least squares estimates satisfy the normal equations giving (obviously! )

Inference, geometrically… (Y 1, Y 2, Y 3 ) design space Y (Y1,Y2,Y3)(Y1,Y2,Y3) ^^^ e = (e 1,e 2,e 3 ) T ^   x  x   x  x ^ xx xx model: Y i =  x i +  +  i null hypothesis: e.g. H 0 :  = 0 (zero slope…) i.e.does x  explain anything ? ( after x  ) model: Y i =  x i +  +  i null hypothesis: e.g. H 0 :  = 0 (zero slope…) i.e.does x  explain anything ? ( after x  )

Inference, formally… For any linear compound of the parameter estimates: So hypotheses can be assessed using: Further (independently): a Student’s t statistic, giving an SPM{t} Suppose the model can be partitioned… “Extra sum-of-squares”

Multivariate perspective…  =  +YX data matrix design matrix parameters errors + ? =  ? voxels scans Üestimate  ^  residuals estimated component fields parameter estimates “Image regression” variance  estimated variance   =

fMRI box car example… =  +  =  +YX data vector design matrix parameters error vector

…fitted…fitted raw fMRI time series scaled for global changes adjusted for global & low Hz effects residuals fitted “high-pass filter” fitted box-car

hæmodynamic response power spectrum DriftsDrifts –physical –physiological Aliased high frequency effectsAliased high frequency effectse.g. –cardiac (~1 Hz) –respiratory (~0.25 Hz) DriftsDrifts –physical –physiological Aliased high frequency effectsAliased high frequency effectse.g. –cardiac (~1 Hz) –respiratory (~0.25 Hz)

Serial correlations... Y = X  +  ~ N(0,  2 V) –intrinsic autocorrelation V Y = X  +  ~ N(0,  2 V) –intrinsic autocorrelation V Problem: Estimate  2 V at each voxel and make inference about  Problem: Estimate  2 V at each voxel and make inference about c T  Model: V as linear combination of m variance components Model V as linear combination of m variance components V = 1 Q 1 + 2 Q 2 + … + m Q m Assumptions: V is the same at each voxel  2 is different at each voxel  2 is different at each voxel Y = X  +  ~ N(0,  2 V) –intrinsic autocorrelation V Y = X  +  ~ N(0,  2 V) –intrinsic autocorrelation V Problem: Estimate  2 V at each voxel and make inference about  Problem: Estimate  2 V at each voxel and make inference about c T  Model: V as linear combination of m variance components Model V as linear combination of m variance components V = 1 Q 1 + 2 Q 2 + … + m Q m Assumptions: V is the same at each voxel  2 is different at each voxel  2 is different at each voxel Q1Q1 Q2Q2 Example: For one fMRI session, use 2 variance components. Choice of Q 1 and Q 2 motivated by autoregressive model of order 1 plus white noise (AR(1)+wn)

Serial correlations…estimation Estimation:   = (X T X) – X T Y – unbiased, ordinary least squares estimate Compute sample covariance matrix of data at all activated voxels: C Y =  k  Y k Y k T /K Y k Important: Data Y k must be high-pass filtered. C Y C Y = X   X  i Q i i Model C Y as C Y = X   T X T +  i Q i and estimate hyperparameters i using Restricted Maximum Likelihood (ReML) V V = N  i Q i /trace(  i Q i ) Estimate V by V = N  i Q i /trace(  i Q i )  2 Estimate  2 at each voxel in the usual way by  2T V  2 = (RY) T (RY) / trace(R V) – unbiased where R = I – X(X T X) – X Estimation:   = (X T X) – X T Y – unbiased, ordinary least squares estimate Compute sample covariance matrix of data at all activated voxels: C Y =  k  Y k Y k T /K Y k Important: Data Y k must be high-pass filtered. C Y C Y = X   X  i Q i i Model C Y as C Y = X   T X T +  i Q i and estimate hyperparameters i using Restricted Maximum Likelihood (ReML) V V = N  i Q i /trace(  i Q i ) Estimate V by V = N  i Q i /trace(  i Q i )  2 Estimate  2 at each voxel in the usual way by  2T V  2 = (RY) T (RY) / trace(R V) – unbiased where R = I – X(X T X) – X CYCY CYCY ^ ^ ^ ^ ^

Serial correlations…inference Inference:  To test null hypothesis c T  = 0, compute t-value by dividing size of effect by its  standard deviation:t = c T  / std[c T  ]  V where std[c T  sqrt(   c ' (X T X) – X T V X (X T X) – c )  V … but … std[c T  is not a    variable because of V Find approximating    distribution using Satterthwaite approximation:  2 V V V Var[  2 ] = 2   trace(R V R V ) / trace(R V ) 2  2  2 VVV = 2 E [  2 ] 2 /Var[  2 ] = trace(RV) 2 / trace(RVRV) – effective degrees of freedon –Worsley & Friston (1995) “Analysis of fMRI time series revisited – again” Use t-distribution with degrees of freedom to compute p-value for t Inference:  To test null hypothesis c T  = 0, compute t-value by dividing size of effect by its  standard deviation:t = c T  / std[c T  ]  V where std[c T  sqrt(   c ' (X T X) – X T V X (X T X) – c )  V … but … std[c T  is not a    variable because of V Find approximating    distribution using Satterthwaite approximation:  2 V V V Var[  2 ] = 2   trace(R V R V ) / trace(R V ) 2  2  2 VVV = 2 E [  2 ] 2 /Var[  2 ] = trace(RV) 2 / trace(RVRV) – effective degrees of freedon –Worsley & Friston (1995) “Analysis of fMRI time series revisited – again” Use t-distribution with degrees of freedom to compute p-value for t ^ ^ ^ ^ ^

Inference — contrasts — SPM{t} SPM{t} contrast — linear combination of parameters: c T  c = +1 0 0 0 0 0 0 0 T = contrast of estimated parameters variance estimate t-test H 0 : c T  = 0 e.g. activation: box-car amplitude > 0 ? correct variance estimate & degrees of freedom for temporal autocorrelation is there an effect of interest after other modelled effects have been taken into account

Inference — F-tests — SPM{F} SPM{F} H 0 :  2 = 0 F = error variance estimate e.g. Does HPF basis set model anything ? multiple linear hypotheses additional variance accounted for by effects of interest X 2 (   ) correct variance estimate & degrees of freedom for temporal autocorrelation 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 H 0 : c T  = 0 c = is there an effect of interest after other modelled effects have been taken into account

Conclusions…Conclusions… General Linear Model General Linear Model 4(simple) standard statistical technique temporal autocorrelation – a Generalised Linear Modeltemporal autocorrelation – a Generalised Linear Model 4single general framework for many statistical analyses flexible modelling  basis functionsflexible modelling  basis functions 4design matrix visually characterizes model fit data with combinations of columns of design matrixfit data with combinations of columns of design matrix 4statistical inference: contrasts… t–tests: planned comparisons of the parameterst–tests: planned comparisons of the parameters F–tests: general linear hypotheses, model comparisonF–tests: general linear hypotheses, model comparison General Linear Model General Linear Model 4(simple) standard statistical technique temporal autocorrelation – a Generalised Linear Modeltemporal autocorrelation – a Generalised Linear Model 4single general framework for many statistical analyses flexible modelling  basis functionsflexible modelling  basis functions 4design matrix visually characterizes model fit data with combinations of columns of design matrixfit data with combinations of columns of design matrix 4statistical inference: contrasts… t–tests: planned comparisons of the parameterst–tests: planned comparisons of the parameters F–tests: general linear hypotheses, model comparisonF–tests: general linear hypotheses, model comparison

Friston KJ, Holmes AP, Worsley KJ, Poline J-B, Frith CD, Frackowiak RSJ (1995) “Statistical parametric maps in functional imaging: A general linear approach” Human Brain Mapping 2:189-210 Worsley KJ & Friston KJ (1995) “Analysis of fMRI time series revisited — again” NeuroImage 2:173-181 Friston KJ, Josephs O, Zarahn E, Holmes AP, Poline J-B (1999) “To smooth or not to smooth” NeuroImage 12: 196-208 Zarahn E, Aguirre GK, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5: 179-197 Aguirre GK, Zarahn E, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5: 199-212 Holmes AP, Friston KJ (1998) “Generalisability, Random Effects & Population Inference” NeuroImage 7(4-2/3):S754 Friston KJ, Holmes AP, Worsley KJ, Poline J-B, Frith CD, Frackowiak RSJ (1995) “Statistical parametric maps in functional imaging: A general linear approach” Human Brain Mapping 2:189-210 Worsley KJ & Friston KJ (1995) “Analysis of fMRI time series revisited — again” NeuroImage 2:173-181 Friston KJ, Josephs O, Zarahn E, Holmes AP, Poline J-B (1999) “To smooth or not to smooth” NeuroImage 12: 196-208 Zarahn E, Aguirre GK, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5: 179-197 Aguirre GK, Zarahn E, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5: 199-212 Holmes AP, Friston KJ (1998) “Generalisability, Random Effects & Population Inference” NeuroImage 7(4-2/3):S754 The General Linear Model and Statistical Parametric Mapping Ch4Ch3

The General Linear Model and Statistical Parametric Mapping Stefan Kiebel Andrew Holmes SPM short course, May 2002 Stefan Kiebel Andrew Holmes SPM short.

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The General Linear Model and Statistical Parametric Mapping Stefan Kiebel Andrew Holmes SPM short course, May 2002 Stefan Kiebel Andrew Holmes SPM short.

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Presentation on theme: "The General Linear Model and Statistical Parametric Mapping Stefan Kiebel Andrew Holmes SPM short course, May 2002 Stefan Kiebel Andrew Holmes SPM short."— Presentation transcript:

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