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

Outline  What is ‘1st level analysis’?  The General Linear Model and 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? A B C D [ ]

Rebecca Knight Motion correction Smoothing kernel Spatial normalisation Standard template fMRI time-series Statistical Parametric Map General Linear Model Design matrix Parameter Estimates  Once the image has been reconstructed, realigned, spatially normalised and smoothed….  The next step is to statistically analyse the data Overview

 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 2 nd level analysis and 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 Matrix forms part of the General linear model, the majority of statistics at the analysis stage use the GLM Key concepts

Y Generic Model  Aim: To explain as much of the variance in Y by using X, and thus reducing ε Dependent Variable (What you are measuring) Independent Variable (What you are manipulating) Relative Contribution of X to the overall data (These need to be estimated) Error (The difference between the observed data and that which is predicted by the model) = X x β + ε Y = X 1 β 1 + X 2 β X n β n ε  More than 1 IV ? General Linear Model

Y Matrix of BOLD at 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) = X x β + ε  How does this equation translate to the 1 st level analysis ?  Each letter is replaced by a set of matrices (2D representations) Time (rows) Voxels (columns) Time (rows) Regressors (columns)Param. weights (columns) Voxels (rows) Time (rows) Voxels GLM continued

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

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

 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 Interest or 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 interest or 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 6 movement regressors (rotations x3 & translations x3 ) or physiological factors e.g. heart rate, breathing or others (e.g., scanner known linear drift) Regressors

Time (n) Regressors (m)  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  Grey represents intermediate values  The representation of each regressor column depends upon the type of variable specified Regress. of Inter. (Covariates)

 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 When you IV is presented When you IV is absent (implicit baseline) Changes in the bold activation associated with the presentation of a stimulus Fitted Box-Car  Red box plot of [0 1] doesn’t model the rise and falls Regress. of inter. (Indicators)

 E.g Movement 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 Regr. of no inter. (Covariate)

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

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

 The type of design and the type of variables used in your experiment will affect the construction of your design matrix  Another important consideration when designing your matrix is to make sure your regressors are separate  In other words, you should avoid 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 Separating regressors

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

V A C 1 C 2 C 3 M N h m l  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 N h m l MNhmlMNhml Example Con’t

h m l h m l M M M N N N  If you make each condition a 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 MNMN h m l MhMh NhNh MlMlMmMm NmNm NlNl h m l h m l M M M N N N hmlhml hmlhml MMMNNNMMMNNN Orthogonal Design Matrix

Y Matrix of BOLD signals Design matrixMatrix parameters = X x + ε Time Voxels Time Regressors Voxels Time Voxels Error matrix β  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 Interim Summary

BRIEF OVERVIEW: SPECIFY THE 1 ST MODEL IN SPM

Thank you!