1. Brain Mapping Unit The General Linear Model A Basic Introduction Roger Tait (rt337@cam.ac.uk)

2. Brain Mapping Unit Overview What is imaging data What is imaging data How is data pre-processed How is data pre-processed Hypothesis testing Hypothesis testing GLM: simple linear regression GLM: simple linear regression Analysis software Analysis software How to process results How to process results

3. Brain Mapping Unit What is imaging data?

4. Brain Mapping Unit DataStructuralfMRI A stack of numbers Functional

5. Brain Mapping Unit Multiple Data subjIDvoxel1voxel2voxel 3voxel 4……..voxel n 11227.3085411472.7702491417.7456321701.2947581288.742729 21612.4615231934.9538271677.6619272013.1943121465.051592 31466.2647391759.5176871559.7695861871.7235031827.678127 41499.700721799.6408641842.4744182210.9693021316.392368 51598.1216921917.7460311510.8507571813.0209091740.286976 61408.0662431689.6794921399.3938151679.2725781534.459154 71555.9514871867.1417841588.529211 1516.464089 81397.7218311677.2661971523.825912 1340.814881 91333.6591181600.3909411384.217926 1461.281399 101453.149661743.7795921558.603977 1406.575083

6. Brain Mapping Unit Reorientation Native MNI152 Reoriented

7. Basic pre-processing (fmri) omprage.nii obrain.nii worest.nii omrest.nii nomrest.nii wnomrest.nii

8. Brain Mapping Unit Basic pre-processing (structural) omprage.nii gmomprage.nii wgmomprage.nii

9. Brain Mapping Unit How does standard space data help?

10. Brain Mapping Unit Hypothesis testing Statistical inference is commonly done with a test statistic (t, F,   …) which has a distribution under H 0 mathematically derived. For example NB: this assumes that the errors are independent and normally distributed. 5% Parametric Null Distribution t t =    –   SE(    –   ) ^^ ^^

11. Brain Mapping Unit Introducing The GLM Y = X  +  DATA = MODEL + ERROR DATA = KNOWN * UNKNOWN + ERROR Encapsulates: t-test (paired, un-paired), F-test, ANOVA (one-way, two-way, main effects, factorial) MANOVA, ANCOVA, MANCOVA, simple regression, linear regression, multiple regression, multivariate regression…… Encapsulates: t-test (paired, un-paired), F-test, ANOVA (one-way, two-way, main effects, factorial) MANOVA, ANCOVA, MANCOVA, simple regression, linear regression, multiple regression, multivariate regression……

12. Brain Mapping Unit GLM definition Y = X  +  Where Y is a matrix with a series of observed measurements Where Y is a matrix with a series of observed measurements Where X is a matrix that might be a design matrix Where X is a matrix that might be a design matrix Where  is a matrix containing parameters to be estimated Where  is a matrix containing parameters to be estimated And  is a matrix containing error or noise And  is a matrix containing error or noise

13. Brain Mapping Unit GLM: Simple Linear Regression Y =    X 1   +  Y X   : is the Y axis intercept   : is the gradient of slope Y: the black circles  : diff between predicted Y and observed Y

14. Brain Mapping Unit GLM: Simple Linear Regression This is done by choosing   and   so that the sum of the squares of the estimated errors  i 2 is as small as possible. This is done by choosing   and   so that the sum of the squares of the estimated errors  i 2 is as small as possible. This is called the Method of Least Squares. This is called the Method of Least Squares.  i 2 is called the Residual Sum of Squares (RSS)  i 2 is called the Residual Sum of Squares (RSS) Y =    X 1   +  ^ ^

15. Brain Mapping Unit GLM example = mean reaction time + GENDER + AGE Y =    X 1    X 2    X 3    X 4   +  DATA = KNOWN * UNKNOWN + ERROR

16. Brain Mapping Unit Dummy Variables Continuous variables Continuous variables measurements on a continuous scale (age, mRT) measurements on a continuous scale (age, mRT) (-4.01, -0.47, 6.35, -7.06, -7.69, -14.24) Dummy Variables Dummy Variables Code for group membership (disease, gender) Code for group membership (disease, gender) controls = 0, patients = 1 females = 1, males = -1

17. Brain Mapping Unit Usage Hypothesis tests with GLM can be multivariate or several independent univariate tests Hypothesis tests with GLM can be multivariate or several independent univariate tests In multivariate tests the columns of Y are tested together In multivariate tests the columns of Y are tested together In univariate tests the columns of Y are tested independently (multiple univariate tests with the same design matrix) In univariate tests the columns of Y are tested independently (multiple univariate tests with the same design matrix)

18. Brain Mapping Unit fMRI model specification silent naming task The model BOLD signal

19. Brain Mapping Unit Actual retrieved data

20. Brain Mapping Unit fmri analysis with FSL

21. Brain Mapping Unit Structural analysis with CamBA group sex weight

22. Brain Mapping Unit Structural analysis output

23. Brain Mapping Unit Where are my clusters? here is a big cluster

24. Brain Mapping Unit Where is the cluster I am interested in? position mouse cursor here cluster location information shown here

25. Brain Mapping Unit How do my clusters help me?

26. Brain Mapping Unit Statistical Testing Convert cluster into a binary mask Convert cluster into a binary mask Overlay mask on subject data Overlay mask on subject data Extract voxel intensities Extract voxel intensities Do some statistical analysis to get more information from your data Do some statistical analysis to get more information from your data

27. Brain Mapping Unit Correlation with behaviour p>0.05 close but cluster Pos_001 does not significantly correlate with behaviour HIT1 for cluster Pos_002

28. Brain Mapping Unit Other Analyses two-sample t-test one-sample t-test simple regression Difference between means different from 0 Linear relationship between 2 variables

29. Brain Mapping Unit What else can I do to find out more about my data?

30. Brain Mapping Unit Other types of analyses Factorial designs Factorial designs Permits analysis of multiple time data Permits analysis of multiple time data Shows Shows Main effects of Factor 1 (time) Main effects of Factor 1 (time) Main effects of Factor 2 (group) Main effects of Factor 2 (group) Interaction between Factor 1 and Factor 2 Interaction between Factor 1 and Factor 2

31. Brain Mapping Unit Useful software package CamBA – Cambridge CamBA – Cambridge  http://www-bmu.psychiatry.cam.ac.uk/software/ FSL Randomise – Oxford FSL Randomise – Oxford  http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/Randomise SPM8 – UCL SPM8 – UCL  http://www.fil.ion.ucl.ac.uk/spm/software/spm8/

32. Brain Mapping Unit In summary The GLM allows us to summarize a wide variety of research outcomes by specifying the exact equation that best summarizes the data for a study. If the model is wrongly specified, the estimates of the coefficients (the beta values) are likely to be biased (i.e. wrong) and the resulting equation will not describe the data accurately. The GLM allows us to summarize a wide variety of research outcomes by specifying the exact equation that best summarizes the data for a study. If the model is wrongly specified, the estimates of the coefficients (the beta values) are likely to be biased (i.e. wrong) and the resulting equation will not describe the data accurately. In complex situations (e.g. cognitive fMRI paradigms), this model specification problem can be a serious and difficult one In complex situations (e.g. cognitive fMRI paradigms), this model specification problem can be a serious and difficult one

33. Brain Mapping Unit Any questions?