Lecture 8: Statistical Analysis of fMRI data,Yin Yao Jan. 12, 2017 Email: Yin.yaonih.gov,Outlines,Overview T-test Examples ANOVA Example Linear Regression Examples Reference,How Statistical Inference relevant to fMRI?,How Statistical Inference relevant to fMRI?,Many methods are available for the statistical analysis of fMRI data T-test to compare the means of blood oxygen level dependent (BOLD) for two groups (or conditions) ANOVA to compare the means of BOLD for three or more groups (or conditions) Linear model to investigate the relationship between the response (BOLD) and predictors (conditions) Autoregressive model to adjust for the temporal errors,T-test: Example 1,Is there different activation of the fusiform gyrus (FFG) for faces vs. objects? Within-subjects design: Condition 1: Presented with face stimuli Condition 2: Presented with object stimuli Hypotheses H0 = There is no difference in activation of the FFG during face vs. object stimuli HA =There is a significant difference in activation of the FFG during face vs. object stimuli,Compare the mean between 2 conditions (Faces vs. Objects) H0: A = B (null hypothesis) No difference in brain activation between these 2 groups/conditions HA: A B (alternative hypothesis) There is a difference in brain activation between these 2 groups/conditions If 2 samples are taken from the same population, then they should have fairly similar means If 2 means are statistically different, then the samples are likely to be drawn from 2 different populations, i.e they really are different,T-test: Example 1 (cont.),T = differences between sample means / standard error of sample means,Calculating T,BOLD response,The exact equation varies depending on which type of T-test used,T-test: Example 2,T = (hot warm effect) / SD,fMRI Experiment on pain perception (Chen et al., 2000) After 9 s of rest, a subject was given a painful heat stimulus (49C) to the left forearm for 9 s, followed by 9 s of rest, then a warm stimulus (35C) for 9 s, repeated 10 times for 6 min in total. During this time the subject was scanned every TR=3 s (120 scans) using a Siemens 1.5 T machine 13 slices of 128 x 128 pixel BOLD images were obtained.,Compare the differences in means between three or more groups (or conditions) Use the variance of data to calculate if means are significantly different Tests the null hypothesis (the means are the same) by the F- test F-Statistics = Variability between groups/ Variability within groups,ANOVA: Analysis of Variance,Between-group variance is smaller relative to the within-group variance; F critical values, therefore statistically not significant,Between-group variance is large relative to the within-group variance; F critical value, therefore statistically significant,Is there different activation of the fusiform gyrus (FFG) for faces vs. objects vs. bodies? Null Hypothesis: H0 : m1 = m2 = m3 i.e. stimuli (face, object, body) has no effect on BOLD signal change Alternative Hypothesis: Ha: not all m are equal i.e. at least one stimulus had an effect on BOLD signal change,ANOVA: Example 1,How Statistical Inference relevant to fMRI?,Linear Regression: Example 1,How does Linear Model (LM) apply to fMRI experiments? Y = X * + Observed = Predictors * Parameters + Error BOLD = Design Matrix * Betas + Error,Linear Regression (cont.),Y,Observed Data Y is a matrix of BOLD signals Each column represents a single voxel sampled at successive time points. Each voxel is considered as independent observation Univariate analysis of individual voxels over time,Linear Regression: Example 1,X can contain values quantifying experimental variable,Y X,Continuous predictors,Parameters & error,This line is a model of the data,slope = 0.23,intercept = 54.5,: slope of line relating X to Y How much of X is needed to approximate Y? - The best estimate of minimizes : deviations from line,Linear Regression (cont.),Design Matrix,Matrix represents values of X Different columns = different predictors,Y X1 X2,X1 X2,Linear Regression (cont.),Matrix formulation,(t) Y X1 X2,Y1 = (5 * 1) + (1 * 2),Y2 = (4 * 1) + (1 * 2) .,YN = (X1(tN) * 1) + (X2(tN) * 2),X1 X2,Linear Regression (cont.),Parameter estimation Least Squares Estimation: What are we trying to estimate? , the slope How? Minimize the squared distance between the observation values and the predicted values Find the that gives the minimum sum of the squared difference Resulting formulas:,Linear Regression (cont.),How Statistical Inference relevant to fMRI?,fMRI Experiment on pain perception (T-test Exemple 2 in Slide 8) Modelling the Noise: The errors are not independent in time Using least squares model fitting to estimate each b, and neglecting to take the correlation structure into account, or using the wrong correlation model, can cause biases in the estimated error of the estimates of b The simplest model of the temporal