Latent model

1. BOLD Activity Neural activity A A Assumed HRF (Basis functions) B B C C D D Columbia Psychology SCAN group Time (s) Time (s) http://www.scan.psych.columbia.edu/ Brain regions mediating a heat stimulus - pain report relationship Fig. 2. BOLD activity deconvolved into an HRF and neural activity. Mediation in the latent model then uses the neural activity instead of the BOLD. Heat-brain path N = 18 Brain-report path N = 18 Subjs 1,2,3…N Mediating brain regions Controlling for stim. intensity Basis set HRF Parameters Trial-level amplitude estimates Noxious stimulation (4 levels) Reported pain Amplitude Multi-level path diagram Fig. 3. Single-trial analysis. Each trial's response is fitted to a basis set (left), then key HRF parameters are computed (middle), and then the resulting timeseries of trial parameter estimates (e.g., amplitude - right) are used in the mediation instead of the BOLD signal. Power and False Positive Rate as a function of the number of subjects. N = 10, 20, or 40. M b a Bootstrapping Bootstrapping Bootstrapping Bootstrapping Sign perm Sign perm Sign perm Sign perm Total: c Power and FPR by permutation option (Bootstrapping or Sign Permutation) and number of iterations.1k = 1000 iterations, 5k = 5000 iterations, 10k = 10000 iterations. Direct: c' X Y Mo (Moderator) Fig.1. The basic unit of analysis in the mediation / moderation framework is a 3-variable system. Power and FPR by shift option. N = 20. Norm = normal, +/-1,+/- 2, and +/-3 are max shift amounts in the latency model. Power and FPR by permutation option (Bootstrapping or Sign Permutation) and number of subjects. N = 10, 20, 40, 60. The M3 Toolbox: the Multi-level Mediation/Moderation Framework for Connectivity Analyses in fMRI Data Matthew Davidson, Lauren Atlas, Martin Lindquist, Niall Bolger & Tor WagerDepartments of Psychology and Statistics, Columbia University, New York 324 Schermerhorn Hall Department of Psychology 1190 Amsterdam Ave. New York, NY 10027 Download this poster: http://www.columbia.edu/cu/psychology/tor/ Introduction Multilevel Mediation/Moderation, cont. Real-world Experiment • Since the path coefficients are computed on l: • lx = g-1(x, h(B, px)) • Then the equations become: • g-1(y, h(B, py)) = c * g-1(x, h(B, px)) + ey • g-1(m, h(B, pm)) = a * g-1(x, h(B, px)) + em • g-1(y, h(B, py)) = b * g-1(m, h(B, pm)) + c' * g-1(x, h(B, px)) + e'y • B consists of two gamma functions such that • HRFx = h(B, px) = Bp/∑Bp This experiment looked at the relationship between 4 different levels of applied heat and reported pain. In the mediation diagram, X is the level of heat applied to the subject, Y is the level of pain reported, and M is any brain voxel mediating the relationship between heat and pain. Below are the results of a search across the brain for any voxels mediating the heat-pain relationship. E.g., you can see that the ACC is a strong mediator. It correlates strongly with applied heat, and with reported pain (controlling for heat). The product of the two paths is significant at the group level, and hence, we can infer that the ACC is a mediator of the heat-pain relationship. • MODERATION • Test whether relationship between two variables depends on a third • In a multilevel analysis, moderators can be: • 1st level (within-subjects) • 2nd level (between-subjects) • BACKGROUND • Early imaging analyses focused on identifying regional neuronal correlates of psychological processes. However, this is an incomplete picture, providing little detail in terms of the interrelationship of various brain regions. As a result, interest has shifted towards identifying and describing related regions in terms of their pathways and circuits. • PRIOR WORK • Existing techniques/software focus on univariate methods (SPM, AFNI, FSL VoxBo and BrainVoyager) • Even the tools that are multivariate (ICA variants, PPI, DCM1, SEM, Granger causality models) typically lack certain key properties: • The ability to search for pathways rather than confirm a priori pathways - useful when the paths are not known • Identifying mediating brain regions • Adjust for differences in hemodynamic response between brain regions - differing HRFs can be problematic for between-region correlations2 • Multilevel modeling - to properly account for intersubject variance3 Variable latency model • Conduct a time-shifted search between data sources • Assumes HRF shape the same, up to a delay d • x and m are replaced by f(x, d1) and f(m, d2), where f() is a time-shifting function implemented by linear interpolation • Equations become: • y = c * f(x, d1) + ey • f(m, d2) = a * f(x, d1) + em • y = b * f(m, d2) + c' * f(x, d1) + e'y • d1 and d2, are estimated with a genetic algorithm that maximizes -log(SSET) Latent model • Deconvolves the hemodynamic response function using a small 2-param basis set (see Fig. 2) • Run the mediation/moderation analyses directly on computed neural activity • Similar in principle to DCM • If g() represents the convolution operation and h(B,[1,2…p]T) the HRF-generating function given p basis parameters and an n (time points) x p matrix of basis functions B, x is the observed timeseries, lx is the latent ‘metabolic’ signal, and pxis the vector of basis parameters, then: • x = g(lx, h(B, px)) Single-trial analysis • As an alternative to the complex and computationally intensive full deconvolution or latency models, a single-trial analysis can be used. • In the single-trial analysis, the response to each trial is fitted with a set of basis functions, and certain HRF parameters, such as height, delay, width, and area under the curve (AUC) are estimated. • Then, instead of using a BOLD signal, the mediation will use the trial-level parameters. This is illustrated below, in Fig. 3: Multilevel Mediation/Moderation • MEDIATION • Simple, three-variable form of SEM extended to the multilevel setting, making it feasible to treat linkages (i.e., connectivity between regions) as random effects. • Uses two key concepts: • Mediation/moderation in path analysis • Mixed-effects (or hierarchical) models • The M3 analysis merges the two approaches, building on recent developments in multi-level mediation analyses in psychology4 • Mediation provides tests of whether relationship between two variables is explained (mediated) by a third, thus establishing either a direct or indirect linkage5 • A test for mediation should satisfy the following criteria: • X should be related to M (the a pathway in Fig. 1 below) • b should be significant after controlling for X • The indirect relationship (a*b)should be significant • This is generally assessed with the Sobel test, or more efficiently, with a bootstrap test6 Software Simulations SPM TOOLBOX The M3 toolbox is currently available as a toolbox for SPM5, downloadable from http://www.columbia.edu/cu/psychology/tor/software.htm. It piggybacks off of the SPM job manager, and thus, presents no new learning curve to those familiar with SPM5. The M3 toolbox supports both single- and multi-level analyses, shifting and latent correlations, and contains built-in checks to prevent insertion of bad data. Summary • M3 provides tests of population inference on within-subject pathways and their moderators. • Tests are efficient and valid for unbalanced designs because the method is explicitly designed for multilevel connectivity. • The M3 framework provides the capability for hybrid exploratory and confirmatory approaches to identifying functional pathways when the exact voxels that comprise such a pathway are unknown. • Provisions are made for specific characteristics of fMRI data, such as HRF and inter-regional latency differences. • Three linear equations: • y = cx + ey • m = ax + em • y = bm + c'x + e'y • If the relationship between x and y can be accounted for by an indirect relationship through m as described by slope coefficients a and b, then c - c’(the product ab) will be statistically different from zero. • MULTILEVEL • Equations: • ci = c + u0i • ai = a + u1i • bi = b + u2i • c'i = c' + u3i • The u's are between-subjects error terms • Subject-level path coefficients are random effects, enabling population inference References 1. Friston, K.J., L. Harrison, and W. Penny, Dynamic causal modelling. Neuroimage, 2003. 19(4): p. 1273-302. 2. Gitelman, D.R., et al., Modeling regional and psychophysiologic interactions in fMRI: the importance of hemodynamic deconvolution. Neuroimage, 2003. 19(1): p. 200-7. 3. Raudenbush, S.W. and A.S. Bryk, Hierarchical Linear Models: Applications and Data Analysis Second ed. Methods. 2002, Newbury Park, CA: Sage. 4. Kenny, D.A., J.D. Korchmaros, and N. Bolger, Lower level mediation in multilevel models. Psychol Methods, 2003. 8(2): p. 115-28. 5. Baron, R.M. and D.A. Kenny, The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. J Pers Soc Psychol, 1986. 51(6): p. 1173-82. 6. Shrout, P.E. and N. Bolger, Mediation in experimental and nonexperimental studies: new procedures and recommendations. Psychol Methods, 2002. 7(4): p. 422-45.