Contents. Introduction The General Linear Model. General Linear Linear Model Model. The General Linear Model, Part I. «Take home» message

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1 DISCOS SPM course, CRC, Liège, 2009 Contents The General Linear Model, Part I Introduction The General Linear Model Data & model Design matrix Parameter estimates & interpretation Simple contrast «Take home» message C. Phillips, Centre de Recherches du Cyclotron, ULg, Belgium Based on slides from: JB. Poline image data parameter estimates Statistical Parametric Map corrected p-values parameter estimates Statistical Parametric Map realignment & motion correction General Linear Linear Model Model model model fitting fitting statistic statistic image image correction for for multiple comparisons General Linear Linear Model Model model model fitting fitting statistic statistic image image normalisation smoothing Random effect effectanalysis anatomical reference kernel design matrix Dynamic causal causal modelling, Functional & effective connectivity, PPI, PPI, Realigned, normalised, smoothed image data design matrix

2 General Linear Model General Linear Model What does it mean? General Linear Model What does it mean? General Linear The model can be The model uses used to answer a simple linear wide variety of relationships questions. between the variables. Model A set of equations are used to describe the data. Questions about the data can then be stated as mathematical expressions. GLM is is the basic model or general framework underlying the analysis of variance and multiple regression. SPM key concepts... a a voxel by voxel hypothesis testing approach reliably identify regions showing a significant experimental effect of interest Key concepts Type I error significance test at each voxel Parametric statistics parametric model for voxel data, test model parameters o exact prior anatomical hypothesis multiple comparisons? Functional neuroimaging signal metabolic change cerebral blood flow (CBF physiological effects blood oxygenation physical effects MR properties glucose and oxygen metabolism Blood oxygen-level dependent (BOLD signal - H 5 2 O PET neuronal activity FDG PET cerebral blood volume (CBV magnetic field uniformity (microscopic Decay time (T2* - opening/closing of ionic channels generation of microscopic current sources combination in space and time of the current sources - Measurable electromagnetic fields on/outside the head fmri (T2*-weighted image EEG/MEG Source: Doug oll s primer

3 Hemodynamic Response Function fmri/pet Setup % signal change = (point baseline/baseline usually 0.5-3% initial dip -more focal and potentially a better measure -somewhat elusive so far, not everyone can find it time to rise signal begins to rise soon after stimulus start time to peak signal peaks 4-6 sec after stimulus begins post stimulus undershoot signal suppressed after stimulation ends Contents Introduction The General Linear Model Data & model Design matrix Parameter estimates & interpretation Simple contrast «Take home» message A simple experiment Stimuli: passive word listening versus rest BOLD response in the primary auditory cortex

4 Looking at 2 individual scans Looking at 2 individual scans O-OFF, just one scan per condition Simple fmri example dataset: can we do better? Voxel by voxel statistics One One session, one one subject Passive word listening versus rest rest 7 cycles of of rest restand listening series seriesof of BOLD BOLD responses in in one one voxel voxel Test your hypothesis visually do it voxel by voxel Each epoch 6 scans with 7 sec sec TR TR Question: Is Isthere therea change in in the thebold response between listening and and rest? rest? Stimulus function single voxel time series statistic image or SPM

5 Why do we need stats? Clearly voxel surfing isn t t a viable option. We d d have to do it,000 of times and it would require a lot of subjective decisions about whether activation was real Statistics: tell us where to look for activation that is related to our paradigm help us decide how likely it is that activation is real Voxel by voxel statistics model specification parameter estimation hypothesis test statistic single voxel time series statistic image or SPM parametric Classical statistics parametric one sample t-testtest two sample t-testtest paired t-testtest Anova AnCova correlation linear regression multiple regression F-tests etc non-parametric parametric? all cases of the General Linear Model assume normality to account for serial correlations: Generalised Linear Model SnPM Statistics Formal statistics are just doing what your eyeball test of significance did Estimate how likely it is that the signal is real given how noisy the data is p p value = probability value of the null hypothesis. ull hypothesis = o activation! confidence: : how likely is it that the results could occur purely due to chance? If p p =.03,, that means there is a 3% chance that the results are bogus By convention, if the probability that a result could be due to chance is less than 5% (p <.05, we say that result is statistically significant Significance depends on signal (differences between conditions noise (other variability sample size (more time points are more convincing

6 e.g. two-sample t-test? test? t distribution voxel time series μ 0 μ t = μ μ 2 σ n n0 0 Depends on set of degrees of freedom! ν 95% voxel intensity Question: Is Isthere a change in in the thebold response between listening and and rest? standard t-test assumes independence ignores temporal autocorrelation! t-statistic image SPM{t} compares size of effect to its error standard deviation Inf Regression example Regression example = β β 2 error ε Ν(0, σ 2 Ι (error is normal and independently and identically distributed = β β 2 error ε Ν(0, σ 2 Ι (error is normal and independently and identically distributed x x 2 Question: Is Isthere a change in in the BOLD response between listening and rest? ε Hypothesis test: β = 0? (using t-statistic correlation: test H 0 : ρ = 0 equivalent to test H 0 : β = 0 x x 2 ε two-sample t-test: test H 0 : μ 0 = μ equivalent to test H 0 : β = 0

7 Regression example Model as basis functions Data Boxcar constant = β β 2 error stimulus function can accomodate BOLD response model can be extended to account for temporal autocorrelation! time = β β 2 Error x x 2 ε What s wrong with this model?. Stimulus function is not expecting BOLD response 2. Data is serially correlated X 2 = β X β 2 ε time Data vector Design Matrix Design Matrix Parameter vector β = Error vector General Linear Model p = Xβ ε = X β p ε This Thisisisfor fora SIGLE voxel!! Design matrix X is isthe thesame for for ALL ALL voxels!! β 2 = X β ε : : number of of scans scans p: p: number of of regressors Model is isspecified by by.. Design matrix X Assumptions about ε

8 General Linear Model fmri time series:,, s,, acquired at times t,,t s s,,t Model: Linear combination of basis functions Model: Linear combination of basis functions β l l s = β f (t s β l f l (t s β L f L (t s ε s f l l (.: basis functions reference waveforms dummy variables parameters (fixed effects amplitudes of basis functions (regression slopes β l l : parameters (fixed effects ε (0,σ s : residual errors: ε s ~ 2 identically distributed independent, or serially correlated (Generalised Linear Model GLM Parameter estimation = Xβ ε ˆ β = ˆ β2 X Estimate parameters such that that t= ˆε 2 t εˆ minimal ˆ ε = X ˆ β residuals Assume iid iiderror ˆ β = ( X X T X T Least squares parameter estimate Mass univariate approach Design Matrix and contrast K p K = X p K βˆ β = β 2 = Xβ ε εˆ = X β ε Question: Is Isthere a change Hypothesis test: in in the BOLD response β = 0? between listening and rest? (using t-statistic

9 Model contrasts SPM{t} t distribution Contrast : specifies linear combination of parameter vector: c β c = 0 box-car amplitude > 0? = β > 0? (β : estimation of β = xβ 0xβ 2 > 0? = test H 0 : c β > 0? SPM{t} Depends on set of degrees of freedom! ν % contrast of estimated parameters T = variance estimate T = c β s 2 c (X X X c Inf = X β ε How is this computed? (t-test test Estimation [, X] [b, s] β= = (X X X X (β = estimation of e = - X β (e = ε ~ σ 2 (0,I ( : at one position estimation of β beta??? images e = estimation of ε s 2 = (e e/(n( e/(n - p (s s = estimation of σ, n: scans,, p: parameters image ResMS Test [b, s 2, c] [c b, t] Var(c β = s 2 c (X X X c compute for each contrast c t = c c β / sqrt(s 2 c (X X X c c β images spm_con??? compute the t images images spm_t??? under the null hypothesis under the null hypothesis H 0 : t ~ Student( df df = n-pn Example: a line through 3 points simple linear regression i = β x i β 2 ε i i =,2,3 = β x β 2 ε 2 = β x 2 β 2 ε 2 3 = β x 3 β 2 ε 3 x β ε 2 = x 2 ε 2 3 x 3 β 2 ε 3 = X β ε dummy variables β 2 (x, (x 2, 2 (x 2, 2 (x, parameter estimates β & β 2 fitted values, 2, 3 residuals e, e 2, e 3 (x 3, 3 (x 3, 3 x β

10 = Geometrical perspective x β ε 2 = x 2 ε 2 3 x 3 β 2 ε 3 (, 2, 3 β X β 2 X 2 ε = Geometrical perspective x β ε 2 = x 2 ε 2 3 x 3 β 2 ε 3 (, 2, 3 β X β 2 X 2 ε (0,0, X (x, x 2, x 3 (0,0, X (x, x 2, x 3 O (,0,0 (0,,0 X 2 (,, O (,0,0 (0,,0 X 2 (,, design space Estimation, geometrically Estimation, geometrically Question: Does X explain anything? (, 2, 3 e = (e,e 2,e 3 T Model: i = β X β 2 X 2 ε i test the null hypothesis: e.g. H 0 : β = 0 (zero slope i.e. (zero slope does X explain anything? (, 2, 3 e = (e,e 2,e 3 T (after X 2 X β2 X 2 β X (, 2, 3 X β2 X 2 β X (, 2, 3 X 2 design space X 2 design space

11 Contents Why modelling? Introduction The General Linear Model Data & model Design matrix Parameter estimates & interpretation Simple contrast «Take home» message Why? How? Model? Make inferences about effects of of interest.. Decompose data into intoeffects and and error Form statistic using estimates of of effects and and error data data Use Useany anyavailable knowledge Stimulus function model model effects estimate error error estimate Contrast: e.g. e.g. [ [- -]] statistic Way to proceed Prepare your questions. ALL the questions! Find a model which allows contrasts that translates these questions. takes into account ALL the effects (interaction, sessions,etc Devise task & stimulus presentation. Acquire the data & analyse. ot the other way round!!!