Statistical Methods in functional MRI. False Discovery Rate. Issues with FWER. Lecture 7.2: Multiple Comparisons ( ) 04/25/13

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1 Statistical Methods in functional MRI Lecture 7.2: Multiple Comparisons 04/25/13 Martin Lindquist Department of iostatistics Johns Hopkins University Issues with FWER Methods that control the FWER (onferroni, RF, Permutation ests) provide a strong control over the number of false positives. While this is appealing, the resulting thresholds often lead to tests that suffer from low power. Power is critical in fmri applications because the most interesting effects are usually at the edge of detection. False Discovery Rate he false discovery rate (FDR) is a recent development in multiple comparison problems due to enjamini and Hochberg (1995). While the FWER controls the probability of any false positives, the FDR controls the proportion of false positives among all rejected tests. Notation Suppose we perform tests on m voxels. Declared Inactive Declared ctive ruly inactive U V m 0 ruly active S m-m 0 m-r R m In this notation: Definitions ( ) FWER = P V 1 False discovery rate: V FDR = E R U, V, and S are unobservable random variables. R is an observable random variable. he FDR is defined to be 0 if R=0. 1

2 Properties procedure controlling the FDR ensures that on average the FDR is no bigger than a prespecified rate q which lies between 0 and 1. H Procedure 1. Select desired limit q on FDR (e.g., 0.05) 2. Rank p-values, p (1) p (2)... p (m) However, for any given data set the FDR need not be below the bound. n FDR-controlling technique guarantee controls of the FDR in the sense that FDR q. 3. Let r be largest i such that p (i) i/m q 4. Reject all hypotheses corresponding to p (1),..., p (r). p-value 0 1 p (i) i/m q 0 1 Comments he H procedure is adaptive in the sense that the larger the signal, the lower the threshold. If all null hypothesis are true, the FDR is equivalent to the FWER. Low signal High signal ny procedure that controls the FWER also controls the FDR. procedure that controls the FDR only can be less stringent and lead to a gain in power. q m q Since FDR controlling procedures work only on the p-values and not on the actual test statistics, it can be applied to any valid statistical test. Example Signal + Noise = Signal + Noise α=0.10, No correction Percentage of false positives FWER control at 10% FDR control at 10% FWER Occurrence of false positive Percentage of active voxels that are false positives 2

3 Uncorrected hresholds Most published PE and fmri studies use arbitrary uncorrected thresholds (e.g., p<0.001). With available sample sizes, corrected thresholds are so stringent that power is extremely low. Using uncorrected thresholds is problematic when interpreting conclusions from individual studies, as many activated regions may be false positives. Null findings are hard to disseminate, hence it is difficult to refute false positives established in the literature. Extent hreshold Sometimes an arbitrary extent threshold is used when reporting results. Here a voxel is only deemed truly active if it belongs to a cluster of k contiguous active voxels (e.g., p<0.001, 10 contingent voxels). Unfortunately, this does not necessarily correct the problem because imaging data are spatially smooth and therefore false positives may appear in clusters. Example ctivation maps with spatially correlated noise thresholded at three different significance levels. Due to the smoothness, the false-positive activation are contiguous regions of multiple voxels. α=0.10 α=0.01 α=0.001 Example Similar activation maps using null data. α=0.10 α=0.01 α=0.001 Note: ll images smoothed with FWHM=12mm Note: ll images smoothed with FWHM=12mm Lecture 8: Functional Connectivity 04/25/13 3

4 Data Processing Pipeline rain Networks Experimental Design It has become common practice to talk about brain networks, i.e. sets of interconnected brain regions with information transfer among regions. Data cquisition Reconstruction Preprocessing Slice-time Correction Motion Correction, Co-registration & Normalization Data nalysis Localizing rain ctivity Connectivity o construct a network: Define a set of nodes (e.g., ROIs) Estimate the set of connections, or edges, between the nodes. Spatial Smoothing Prediction C C C Network Methods number of methods have been suggested in the neuroimaging literature to quantify the relationship between nodes/regions. heir appropriateness depend upon: what type of conclusions one is interested in making; what type of assumptions one is willing to make; and the level of the analysis and modality. rain Connectivity Functional Connectivity Undirected association between two or more fmri time series. Makes statements about the structure of relationships among brain regions. VMPFC DLPFC dcc MG rain Connectivity Effective Connectivity Directed influence of one brain region on the physiological activity recorded in other brain regions. Makes statements about causal effects among tasks and regions. V5 Methods: Functional Connectivity Seed analysis Inverse covariance methods Multivariate decomposition methods Principle Components nalysis Independent Components nalysis Partial Least Squares V1 PPC Mediation analysis Psychophysiological interaction (PPI) analysis 4

5 Methods: Effective Connectivity Structural Equation Modeling Granger Causality Dynamic Causal Modeling ayes Net Mediation analysis Psychophysiological interaction (PPI) analysis Levels of nalysis Functional connectivity can be applied at different levels of analysis, with different interpretations at each. Connectivity across time can reveal networks that are dynamically activated across time. Connectivity across trials can identify coherent networks of task related activations. Levels of nalysis ivariate Connectivity Connectivity across subjects can reveal patterns of coherent individual differences. Connectivity across studies can reveal tendencies for studies to co-activate within sets of regions. Simple functional connectivity Region is correlated with Region. Provides information about relationships among regions. Can be performed on time series data within a subject, or individual differences (contrast maps, one per subject). ime Series Connectivity Calculate the cross-correlation between time series from two separate brain regions. Region 1 Region 2 Seed nalysis In seed analysis the cross-correlation is computed between the time course from a predetermined region (seed region) and all other voxels. Subject 1 Subject 2 Subject n Group nalysis his allows researchers to find regions correlated with the activity in the seed region. he seed time course can also be a performance or physiological variable 5

6 Correlations between brain activity and heart-rate VMPFC Issues One of the main problems with time series connectivity is the fact that there may be different hemodynamic lags in different regions: ime series from different regions may not match up, even if neural activity patterns match up. ime (Rs, 2 s) hreshold: p <.005 If lags are estimated from data, temporal order may be caused by vascular (uninteresting) or neural (interesting) response. verage within-subject correlation (r) eta Series eta Series he beta series approach can be used to minimize issues of inter-region neurovascular coupling. Region 1 Region 2 Procedure: Fit a GLM to obtain separate parameter estimates for each individual trial. Compute the correlation between these estimates across voxels. Subject 1 Subject 2 Subject n Group nalysis Individual Differences Partial Correlation Subject Contrast Image Seed Value Group Results 1 x 1 C 2.. x 2 Partial Correlation Correlation between two regions, after the effect of all other regions have been removed. Helps protect against illusory correlations between regions (e.g., and C uncorrelated after controlling for ). N x N 6

7 Inverse Covariance Methods For multivariate normal data there exists a duality between the inverse covariance (precision) matrix and the graph representing relationships between regions. Conditional independence between variables (regions) corresponds to zero entries in the precision matrix. Graphical lasso (GLSSO) can be used to estimate sparse precision matrices and graphs. 0 Σ 1 = C 0 Mediation Mediation he relationship between regions and is mediated by M Can identify functional pathways spanning > 2 regions Can be performed on time series data within a subject, or individual differences (contrast maps, one per subject) lso: est of whether task-related activations in are mediated, or explained, by M. ask M M Demonstrating Mediation Decomposition of Effects Full model, with mediator x a m c b m = i m + ax + e m y = i y + bm + c'x + e y y Reduced model, without mediator x y c y = i y + cx + e y he mediation framework allows us to decompose the total effect of x on y as follows: c = c' + ab otal effect = Direct effect + Mediated effect Does m explain some of the x-y relationship? est c c, which is equivalent to significance of the ab product. Sobel test or bootstrap test. Functional Mediation Pain Data X ( m t), m ( t), m ( )) 1( 2 n t M α (t) β (t) otal: γ Direct: γ ' Y ( x, x, 1 2 x n ) ( y, y, 1 2 y n ) y = γ x + ε ( ) i m ( t) = α ( t) x + ε, ( t) i i i m yi = α ( s) mi ( s) ds + γ ' xi + εi, y ( t) 0 i i, x t γ = γ ' + α( s) β ( s) ds 0 ctivation in the right anterior insula mediates the relationship between temperature and pain rating. α pathway function he key time interval driving the mediation is between seconds following activation. emp rain Response αβ pathway function β pathway function Rating 7

8 Moderation PPI M Moderation he relationship between regions and is moderated by M Connectivity between and depends on state (level) of M Can be performed on time series data within a subject, or individual differences (contrast maps, one per subject) M can be task state or other variable In the psychophysiological interaction (PPI) approach, the standard GLM model is supplemented with additional regressors that model the interaction between the task and the time course in a seed region. Y = β + β X + β R + β R* X + ε In SPM, on time series data: Psychophysiological interaction (PPI). ask ime course from seed region Interaction term PPI PPI can be used to determine whether the correlation between two brain areas is altered by different psychological contexts. he interaction term reflects the modulation of the slope of the linear relationship with the seed voxel depending on the variable used to create the interaction. Decomposition Methods We often use multivariate decomposition methods to study functional connectivity. Provides a decomposition of the data into separate components. Can be used to find coherent brain networks. Provides information on how different brain regions interact with one another. he most common decomposition methods are principal components analysis and independent components analysis. Data Organization Principal Components nalysis hroughout we organize the fmri data in a N matrix X. he row dimension is the number of time points and the column dimension the number of voxels. ime Voxels X Principal components analysis involves finding spatial modes, or eigenimages, in the data. hese are the patterns that account for most of the variance-covariance structure in the data. hey are ranked in order of the amount of variation they explain. he eigenimages can be obtained using singular value decomposition (SVD), which decomposes the data into two sets of orthogonal vectors that correspond to patterns in space and time. 8

9 Using SVD, we can decompose the matrix X as: X = USV where U and V are unitary orthogonal matrices and S is a diagonal matrix consisting of ranked singular values. ime Voxels = ime courses Eigenimages Each column of V defines a distributed brain region that can be displayed as an image (eigenimages). X = USV Each column of U correspond to the time-dependent profiles associated with each eigenimage. X = s1u 1v1 + s2u2v2 + + s N u N v N v 1 v 2 PPROX. = s 1 u 1 OF Y + s PPROX. 2 u OF Y Worsley Independent Components nalysis Independent Components nalysis (IC) is a family of techniques used to extract independent signals from some source signal. IC provides a method to blindly separate the data into spatially independent components. Cocktail Party Problem wo people are talking simultaneously in a room with two microphones. Speakers: s 1 (t) and s 2 (t). Microphones: x 1 (t) and x 2 (t) he key assumption is that the data set consists of p spatially independent components, which are linearly mixed and spatially fixed. x ( t) = a 1 x ( t) = a 2 s ( t) + a 11 1 s ( t) + a 21 1 s ( t) 12 2 s ( t) 22 2 X = S Mixing matrix Source matrix 9

10 ssumptions If the mixing matrix is known, the problem is straight forward. However, IC solves this problem without knowing the mixing parameters. IC Estimation We can find the independent components using a variety of different approaches. Maximizing non-gaussianity Minimizing the mutual information Maximum likelihood estimation Projection pursuit Instead it exploits some key assumptions: Linear mixing of sources. he components s i are statistically independent. he components s i are non-gaussian. IC for fmri IC for fmri It is assumed that the fmri data can be modeled by identifying sets of voxels whose activity both vary together over time and are different from the activity in other sets. Decompose the data set into a set of spatially independent component maps with a set of corresponding time-courses. fmri data Source 1 s = [ s1 s2] + ime course 1 fmri data is assumed to be a linear mixture of statistically independent sources, s. Source 2 ime course 2 Vince Calhoun IC for fmri Overview We seek to decompose X as follows: Voxels ime Courses Spatially independent Components X = S ime = where the matrix S contains statistically independent maps in its rows each with an internally consistent time-course contained in the associated column of the mixing matrix. Data X = S Mixing Matrix Components Use an IC algorithm to find and S. 10

11 Comments Unlike PC which assumes an orthonormality constraint, IC assumes statistical independence among a collection of spatial patterns. Independence is a stronger requirement than orthonormality. However, in IC the spatially independent components are not ranked in order of importance as they are when performing PC. ypes of IC n IC that decomposes the original data into spatially statistically independent components is called spatial IC (sic). It is possible to switch the order and make the temporal dynamics independent. his is called temporal IC (tic). Spatial IC is more common in fmri data analysis. Multi-subject nalysis Using IC to analyze fmri data from multiple subjects raises several questions. How should components be combined across subjects? How should the final results be thresholded and/or presented? here are several approaches: Stack time courses (forces time courses to be the same) Stack images and back-reconstruct (allows time courses to vary, allows some flexibility in images) Stack into a cube (forces images and time courses to be the same) McKeown, et. al. Group IC Group IC Group IC is based on temporal concatenation. It decomposes the group matrix, and estimates through back-reconstruction the spatial weights for each subject for a component of interest. For each subject the spatial weights at each voxel are treated as random variables, and a t- test is used to test whether that voxel loaded significantly on that component in the group. Data Subject 1 X Subject N IC = 1 S_agg N ack-reconstruction 1-1 Subject i i = S i 11