an overview of data processing tasks in the analysis of ... · fmri signal + β 0 × = β 1 × + β...

An Overview of Data Processing Tasks

in the Analysis of fMRI-based Experiments

Functional MRI WorkshopsIntroduction to the GLM in fMRI Data Analysis

Robert L. Savoy, Ph.D.The Athinoula A. Martinos Center for Biomedical Imaging

HyperVision, Inc.e-mail: Robert.L.Savoy@alum.mit.edu

OutlineConnecting two “famous” cartoons... &

Data Processing lectures at the fMRI VFP (MGH)! Optimizing Data Acquisition! Preprocessing! Modeling for First Level Estimation! Modeling for Second Level Estimation! Inference and Critical Thresholds! (Sixth Lecture: Connectivity and Multivariate Analysis)

Now to draw connections between above two cartoons...apologizing that they may make more sense after the course...

NMR

MRI ExperimentalTask

ComparingBrains

Psycho-physiological

Laboratory

Data Analysis

Practicalities

FunctionalNeuroanatomy

Hemodynamics

ComputationalNeuroanatomy

ExperimentalDesign

DataAnalysis

Overview of SPM

“SpatialNormalisation”

Statistical parametric map (SPM)

Imagetime-series

Parameterestimates

GLM: GeneralLinear Model

Realignment Smoothing

Designmatrix

StandardizedAnatomicalTemplate

SpatialSmoothing

Kernel

Gaussian field theory

p <0.05

Statisticalinference

Overview of SPM

“SpatialNormalisation”

Statistical parametric map (SPM)

Imagetime-series

Parameterestimates

GLM: GeneralLinear Model

Realignment Smoothing

Designmatrix

StandardizedAnatomicalTemplate

SpatialSmoothing

Kernel

Gaussian field theory

p <0.05

Statisticalinference

Preprocessing Modeling Inference

Lecture 1

Optimizing Data Acquisition

This is the attempt to get the best quality images that you can from your scanner, subject, and experiment.

Lecture 2

Preprocessing: Realignment

Primarily, this is to compensate, on a global level, for the fact that living subjects move. It is “realigning” images with

similarly collected images---images like themselves.

Lecture 2

Preprocessing: Spatial Smoothing

This is almost always done, because the size of the activated region is typically larger than a single voxel. It benefits the

statistical analysis in a number of ways...in most experiments.

Lecture 2

Preprocessing: Spatial Normalization

Putting brain data in a “standard” 3-dimensional coordinate space permits group analysis and reporting findings.

Realignment can be associated with “spatial normalization when going between imaging modalities (PET & fMRI; T1 & T2*)

Lecture 3

Modeling for First Level Estimation

This is the search for “task-related activity” in each individual subject.

Overview of SPM

“SpatialNormalisation”

Statistical parametric map (SPM)

Imagetime-series

Parameterestimates

GLM: GeneralLinear Model

Realignment Smoothing

Designmatrix

StandardizedAnatomicalTemplate

SpatialSmoothing

Kernel

Gaussian field theory

p <0.05

Statisticalinference

Preprocessing Modeling Inference

effects of interest error varianceeffects of no interest

effects of interesterror variancestatistic = -----------------------------

The Jewelwe seek!

ExplainedVariance

The UnexplainedVariance,

called “noise”

Lecture 4

Modeling for Second Level Estimation

ExperimentalDesign

“Second Level” modeling can refer to many things.Most commonly, it refers to group analyses of various kinds.

Lecture 5

Inference and Critical Thresholds

Inference and “critical thresholds” are a core part of the history of the SPM software package. “Gaussian Random Field” theory was an attempt to deal with the problem of multiple comparisons

without invoking the overly stringent Bonferonni correction.

No Statistics Here!

fMR Model Specification and Estimation

!! CEO & Chief Software Developer ! Brain Innovation B.V. ! Maastricht, The Netherlands

! Maastricht Brain Imaging Center! ! ! F.C. Donders Center for! Department of Cognitive Neuroscience! ! Cognitive Neuroimaging! Maastricht University! ! ! ! ! Nijmegen, The Netherlands! Maastricht, The Netherlands! ! ! !

! Rainer Goebel

Basic fMRI Course Human Brain Mapping Conference Florence, Italy, 11th June 2006

View 1:A series of volumes(scans, 3D “images”) Useful for seeking spatio-temporal patterns, e.g., via PCA, ICA, etc.

View 2:Multiple voxel time series Useful for seeking temporal patterns in each individual voxel of the volumes. Standard hypothesis-driven statistical analysis (e.g., GLM) goes with view 2 since it is applied independently for each voxel time course, i.e., “voxel-wise” statistical analysis.

Two Views on fMRI Data

Inference: Use Uncertainty of Parameter Estimates

Rest Stim

a)

Rest Stim

b)

Mean 2

Mean 1

• Same mean values in Rest and Stim condition in cases a) and b) Simple subtraction (difference = Mean 2 – Mean 1) yields same result ! Simple subtraction does NOT consider variance of means (noise)

noise

• Statistical analysis (e.g., t test) relates estimate (mean difference) to uncertainty of estimate = pooled variance of values within conditions, i.e.

noise

noise

noise

Neural pathway Hemodynamics MR scanner

Convolution kernel

From Experimental Timing to Expected fMRI Time Course

fMRI signal

+

β0 ×

β1 ×= +

β2 ×

residuals

design matrix

+

The General Linear Model (GLM)

The design matrix models expected signal time courses for individual conditions (effects of interest), but may also include additional predictor time courses (confounds, effects of no interest) helping to improve explaining the data. In this example, two main predictors are defined (X1 and X2). A “constant” predictor (X0) is required to fit the base level of the signal time course. The expected time courses are obtained by convolving the condition “box-car” functions (red / green) with the assumed impulse response (HRF, two-gamma function).

X0

X1

X2Time Time

Time

fMRI signal

+

β0 ×

β1 ×= +

β2 ×

residuals

design matrix

+

The General Linear Model (GLM)

...The expected time courses are obtained by convolving the condition “box-car” functions (red / green) with the assumed impulse response (HRF, two-gamma function).

X0

X1

X2Time Time

Time

fMRI signal

+β0 × β1 ×= +β2 ×

residualsdesign matrix

+

Time

The General Linear Model (GLM)

The time courses of the signal, predictors and residuals can be rearranged in column form with time running from top to bottom. This view is helpful to understand the matrix form of the General Linear Model.

Time

fMRI signal= data

+

β0 × β1 ×

= +

β2 ×

residuals= error

design matrix= model

(defined by YOU)

+

estimate!

Time

The General Linear Model (GLM)

fMRI signal= data

+

β0 × β1 ×

= +

β2 ×

residuals= error

design matrix= model

(defined by YOU)

+

estimate!

Time

The GLM time courses, especially the design matrix is often shown in a graphical view using a black-to-white colour range for coding (expected) signal intensities.

The General Linear Model (GLM)

fMRI signal= data

+=

+

residuals= error

design matrix= model

(defined by YOU)

+

Time

+= ++Time

fMRI signal= data

residuals= error

design matrix= model

(defined by YOU)

The General Linear Model (GLM)

SPM Course Zurich2009

Klaas Enno Stephan Laboratory for Social & Neural Systems ResearchInstitute for Empirical Research in EconomicsUniversity of Zurich

Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London

With many thanks for slides & images to:FIL Methods group

The response of a linear time-invariant (LTI) system is the convolution of the input with the system's response to an impulse (delta function).

Problem 1: Shape of BOLD responseSolution: Convolution model

hemodynamic response function (HRF)

expected BOLD response = input function ⊗ impulse response function (HRF)

Convolution model of the BOLD responseConvolve stimulus function with a canonical hemodynamic response function (HRF):

⊗ HRF

Stimulus timing shown in REDStimulus timing convolved with the “canonical” HRF Shown in GREENMR data shown in BLUE

Stimulus timing convolved with the

“canonical” HRF...

What is the use, in most analyses, of “canonical HRF” function?

It refers to the fact that the SAME “HRF” is used FOR ALL voxels in the brain, FOR ALL stimulus types, FOR ALL subjects!

Stimulus timing convolved with the

“canonical” HRF...

One can use the “Finite Impulse Response (FIR)” model to estimate the HRF individually: for each voxel in the brain, for each stimulus type, for each subject.In practice, full FIR analysis is rarely done. But estimating the dispersion and delay of the HRF individually on a voxel basis, and then treating those as “covariates of no interest”, is more commonly useful.

The General Linear Model (GLM)

The GLM in Matrix Notation

observed variable,fMRI time course

designmatrix

predictor,explanatory variable,regressor, covariate,

basis function

beta weights,coefficients

errors,residuals

another predictor...one of “k” functions to help fit the data in the

GLM

The GLM in Matrix Notation

Observed fMRI signal:

Predicted fMRI signal:

Prediction error:

The GLM in Matrix Notation

Fitting a GLM

GLM – Example, block design

b0 = 101.9

residuals

design matrix

Observed fMRI signal

Predicted fMRI signal

+

b0 = 101.6

residuals

design matrix

b1 = 1.8

GLM - Example

Observed fMRI signal

Predicted fMRI signal

+

b0 = 101.0

residuals

design matrix

b1 = 4.5

GLM - Example

Observed fMRI signal

Predicted fMRI signal

+

b0 = 100.2

residuals

design matrix

+

b1 = 3.4

b2 = 5.5

GLM - Example

Observed fMRI signal

Predicted fMRI signal

Lessons from the GLM Example

• To obtain optimal GLM fits, all known effects should be modelled in the design matrix.

• Non-modelled effects are moved to the residuals, substantially increasing the error variance. This leads to poor fits and reduced statistical power (the error variance is one component of the standard error of contrasts).

• Inspection of the residuals is a good diagnostic to assess the goodness-of-fit of a GLM: If one observes prominent, low-frequency fluctuations, it is likely that not all effects have been modelled or that the modelled time courses do not fit to the data. The latter might be for example, due to delayed fMRI responses with respect to predictors.

• Confound predictors may be added to improve the fit, for example, predictors capturing low-frequency drifts (Fourier basis functions or Discrete Cosine Transform, DCT, basis functions), if drifts are not removed already during data preprocessing.

An Overview of Data Processing Tasks

in the Analysis of fMRI-based Experiments

Functional MRI WorkshopsIntroduction to the GLM in fMRI Data Analysis

Robert L. Savoy, Ph.D.The Athinoula A. Martinos Center for Biomedical Imaging

HyperVision, Inc.e-mail: Robert.L.Savoy@alum.mit.edu