Idiot's guide to... General Linear Model & fMRI fMRI model, Linear Time Series, Design Matrices, Parameter estimation, *&%@! Elliot Freeman, ICN. General Linear Model & fMRI How does GLM apply to fMRI experiments? Y = X . + Observed = Predictors * Parameters + Error BOLD = Design Matrix * Betas + Error Observed data Y is a matrix of BOLD signals: Each column represents a single voxel sampled at successive time points.
Preprocessing ... Y Time Y= X. + Intensity Univariate analysis Each voxel considered as independent observation Analysis of individual voxels over time, not groups over space SPM would still work on an Amoeba! Y= X. + Continuous predictors Y Scan no Voxel 1 Y= X. + X 1 57.84 Task difficulty 5
X 57.5 57 0 1 Condition X can contain values distinguishing experimental conditions Parameters & error : slope of line relating X to Y how much of X is needed to approximate Y? the best estimate of minimises : deviations from line Y= X. + this line is a 'model' of the data slope = 0.23 intercept = 54.5 Y= X. + Design Matrix Scan no
11 56.89 5 1 12 55.69 2 1 (t1)t1) (t1)t2) (t1)tN)) X1 X2 Parameter estimation and stats Find betas (by least squares estimation) Y= X -> B = Y / X X -> B = Y / X X X -> B = Y / X -> X -> B = Y / X B X -> B = Y / X = X -> B = Y / X Y X -> B = Y / X / X -> B = Y / X X X -> B = Y / X (B= estimated ) Matlab magic: >> B = inv(X) * Y Now find error term: e = Y (X * B ) ...and use these results for statistics: t = betas / standard error Covariates vs. conditions Covariates: X -> B = Y / X parametric modulation of independent variable e.g. task-difficulty 1 to 6
-> regression: beta = slope Conditions: 'dummy' codes identify different levels of experimental factor specify time of onset and duration e.g. integers 0 or 1: 'off' or 'on' -> ANOVA: beta = effect mean on off off on Modelling haemodynamics Brain does not just switch on and off! -> Reshape (convolve) regressors to resemble HRF Original HRF Convolved HRF basic function Anatomy of a design matrix Example: 5 subjects 2 conditions per
subject 6 replications per condition 1 covariate conditions subjects covariates Interesting vs. uninteresting Important to model all known variables, even if not experimentally interesting: e.g. head movement, block and subject effects minimise residual error variance for better stats effects-of-interest means adjusted to eliminate effectsof-no-interest global conditions: activity or movement effects of subjects interest Selecting and comparing betas A beta value is estimated for each column in design matrix A contrast variable is used to select (groups of) conditions and compare with others e.g. mean (2 4 6 ...) - mean (1 3 5 ...)
t statistic = ( 1 2 3 ... ) . -1 / SE 1 -1 ... t-test: t > critical value ? Summary: Reverse Cookery You start with the finished product and want to know how it was made You specify which ingredients to add (design matrix variables) For each ingredient, GLM finds the quantities (betas) that produce the best reproduction (model) Now you can compare your recipe with others (null hypothesis) to see if they differ! (statistical tests) How dumb was that? Sources: http://www.fil.ion.ucl.ac.uk/spm/doc/papers/SPM_3/welcome.html http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch7.pdf http://www.mrc-cbu.cam.ac.uk/Imaging/Common/spmstats.shtml
Image courtesy of defenselink.mil. ... Always hold personnel accountable for substandard S&H performance. There is a mutual respect between leadership and labor. Solve problems at the true root cause, without a tendency to blame employees.
Claremont School Assessment Data. 2015 - 2016 . Context . All students are assessed against Individual Learning Plan targets (ILP's). The ILP's follow long term outcomes from Education, Health, Care Plans and objectives in Statements of SEN.
Akramul Azim Department of Electrical and Computer Engineering University of Waterloo, Canada Queueing Theory - Terminology and Notation State of the system Number of customers in the queueing system (includes customers in service) Queue length Number of customers waiting for...
Specific intensity. Often we want to know more… the full 3D distribution of photon properties. Specific intensity describes everything contained in the flux, plus how the photons are arranged in direction… and in frequency…
HTCIA 2014 Conf - Aug 26, 2014. HTCIA 2014 International ConfHyatt Lost Pines Resort, Austin TexasTuesday August 26, 2014 8:00am. Introduction to the Microsoft exFAT File System. Robert Shullich . CPP, CISSP, CRISC, GSEC, GCFA, CEH, CHFI, CCFP-US
In first few minutes, a series of bands appears, increasing in intensity during the experiment up to 1hr of nitration. The bands at 1548 and 1584 cm−1 can be attributed to N = O vibrations of the nitrate species and...
Ready to download the document? Go ahead and hit continue!