1 st level analysis: Design matrix, contrasts, and inference Stephane De Brito & Fiona McNabe

1. 1st level analysis: Design matrix, contrasts, and inferenceStephane De Brito & Fiona McNabe

2. Outline A B C D [1 -1 -1 1] • What is ‘1st level analysis’? • The General Linear Modeland how this relates to the Design Matrix • Design matrix • What are we testing for? • What do all the black lines mean? • What do we need to include? • Contrasts • What are they for? • t and F contrasts • Inferences • How do we do that in SPM5?

3. Overview StatisticalParametricMap Designmatrix fMRItime-series kernel Motion correction Smoothing General Linear Model ParameterEstimates Spatial normalisation Standard template • Once the image has been reconstructed, realigned, spatially normalised and smoothed…. • The next step is to statistically analyse the data Rebecca Knight

4. Key concepts • 1st level analysis – A within subjects analysis where activation is averaged across scans for an individual subject • The Between- subject analysis is referred to as a 2nd level analysisand will be described later on in this course • Design Matrix–The set of regressors that attempts to explain the experimental data using the GLM A dark-light colour map is used to show the value of each variable at specific time points – 2D, m = regressors, n = time. • The Design Matrixforms part of theGeneral linear model, the majority of statistics at the analysis stage use the GLM

5. General Linear Model Generic Model β ε + Y X x = Dependent Variable (What you are measuring) Independent Variable (What you are manipulating) Relative Contributionof X to the overalldata (These need tobe estimated) Error (The difference between the observed data and that which is predicted by the model) • Aim: To explain as much of the variance in Y by using X, and thus reducing ε Y = X1β1 + X2β2 + ....X nβn.... + ε • More than 1 IV ?

6. GLM continued • How does this equation translate to the1st level analysis ? • Each letter is replaced by a set ofmatrices(2D representations) β X Y + ε x = Matrix of BOLDat various time points in a single voxel (What you collect) Design matrix (This is your model specification in SPM) Parameters matrix (These need to be estimated) Error matrix (residual error for each voxel) Time (rows) Time (rows) Time(rows) Voxels (rows) Voxels (columns) Regressors (columns) Param. weights (columns) Voxels

7. ‘Y’ in the GLM Y Y = Matrix of Bold signals Voxel time course fMRI brain scans Time Time (scan every 3 seconds) 1 voxel = ~ 3mm³ Amplitude/Intensity Rebecca Knight

8. ‘X’ in the GLM X = Design Matrix Time (n) Regressors (m)

9. Regressors • Regressors– represent hypothesised contributors to the fMRI time course in your experiment. They are represented by columns in the design matrix (1column = 1 regressor) • Regressors of Interestor Experimental Regressors– represent those variables which you intentionally manipulated. The type of variable used affects how it will be represented in the design matrix (2 types: Covariates and Indicators, next slides) • Regressors of no interestor nuisance regressors – represent those variables which you did not manipulate but you suspect may have an effect. By including nuisance regressors in your design matrix you decrease the amount of error. • E.g. -The 6movement regressors (rotations x3 & translations x3 ) or physiological factors e.g. heart rate, breathing or others (e.g., scanner known linear drift)

10. Regress. of Inter. (Covariates) • Covariates = Regressors that can take any of a continuous range of values (e.g, task difficulty) • A dark-light colour map is used to show the value of each regressor within a specific time point • Black = 0 and illustrates when the regressor is at its smallest value • White = 1 and illustrates when the regressor is at its largest value • Greyrepresents intermediate values • The representation of each regressor column depends upon the type of variable specified Time (n) Regressors (m)

11. Regress. of inter. (Indicators) • As they indicate conditions they are referred to as indicator variables • Type of dummy code is used to identify the levels of each variable • E.g. Two levels of one variable is on/off, represented as ON = 1 OFF = 0 Changes in the bold activation associated with the presentation of a stimulus When you IV is presented When you IV is absent (implicit baseline) • Red box plot of [0 1] doesn’t model the rise and falls Fitted Box-Car

12. Regr. of no inter. (Covariate) • E.gMovement regressors–not simply just one state or another • The value can take any place along the X,Y,Z continuum for both rotations and translations

13. E.g., Regress. of no inter. Scanner Drift Artifact and t-test

14. Ways to improve your model: modelling haemodynamics The brain does not just switch on and off. Reshape (convolve) regressors to resemble HRF Modelling haemodynamic More on this next week! HRF basic function Original HRF Convolved

15. Separating regressors • Thetype of designand thetype of variablesused in your experiment will affect the construction of yourdesign matrix • Another important consideration when designing your matrix is to make sure yourregressors are separate • In other words, you shouldavoid correlations between regressors(collinear regressors) –because correlations in regressors means that variance explained by one regressor could be confused with another regressor • This is illustrated by an example using a 2 x 3 factorial design

16. Example Design Motion No Motion High Medium Low High Medium Low • IV 1 = Movement, 2 levels (Motion and No Motion) • IV 2 = Attentional Load, 3 levels (High, Medium or Low)

17. Example Con’t V A C1 C2 C3 • If you made each level of the variables a regressor you could get 5 columns and this would enable you to test main effects • BUT what about interactions? How can you test differences between Mh and Nl • This design matrix is flawed – regressors are correlated and therefore a presence of overlapping variance (Grey) M Nh m l M Nh m l MNh ml

18. Orthogonal Design Matrix M M M N N N • If you make each conditiona regressor you create 6 columns and this would enable you to test main effects • AND it enable you to test interactions! You can test differences between Mh and Nl • This design matrix is orthogonal – regressors are NOT correlated and therefore each regressor explains separate variance h m l h m l M M M N N N h m l h m l h m l M M M N N N h m l h m l Mh Mm Ml M N Nl Nh Nm

19. Interim Summary β Y X ε + x = Matrix of BOLD signals Design matrix Matrix parameters Error matrix Time Time Time Regressors Voxels Voxels Regressors Voxels • Aim: To explain as much of the variance in Y by using X, and thus reducing ε • β = relative contribution that each regressor has, the larger the β value = the greater the contribution • Next: Examine the effect of regressors and the contrasts

20. BRIEF OVERVIEW:SPECIFY THE 1ST MODEL IN SPM

29. Thank you!