The Joy of Path Diagrams Doing Normal Statistics

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1 Intro to SEM SEM and fmri DBN Effective Connectivit Theor driven process Theor is specified as a model Alternative theories can be tested Specified as models Theor A Theor B Data The Jo of Path Diagrams Variable Correlation Causal Arrow Correlational Arrow T-Test One wa ANOVA (Dumm coding) 1 2 3

2 Two- wa ANOVA (Dumm coding) Regression * 2 MANOVA ANCOVA z Sampling Variation and χ 2 Equations and numbers Eas to determine if its correct Sample data ma var from the model Even if the model is correct in the population Use the χ 2 test to measure difference between the data and the model Some difference is OK Too much difference is not OK R ab = 0.3, N = 100 a Eample 1 b Estimate = 0.3, SE = 0.105, C.R. = The correlation is significantl different from 0

3 Model a b Tests the hpothesis that the correlation in the population is equal to zero It will never be zero, because of sampling variation The χ 2 tells us if the variation is significantl different from zero Force the value to be zero Input parameters = 1 Parameters estimated = 0 The program gives a χ 2 statistic The significance of difference between the data and the model Distributed with df = known parameters - input parameters χ 2 = 9.337, df = 1-0 = 1, p = So what? A correlation of 0.3 is significant? Hardl a Revelation No. We have tested a correlation for significance. Something which is much more easil done in other was But We have introduced a ver fleible technique Can be used in a range of other was Testing Other Than Zero Estimated parameters usuall tested against zero Reasonable? Model testing allows us to test against other values a χ 2 = 2.3, n.s b Eample 2: Comparing correlations variables mothers' sensitivit mothers' parental bonding fathers' sensitivit fathers' parental bonding Does the correlation differ between mothers and fathers? 0.5 M PB M S F S 0.2 F PB 0.3

4 dave M PB M S F S F PB dave χ 2 = 1.82, df = 1 p = dave = 0.41 (s.e. 0.08) Latent Variables The true power of SEM comes from latent variable modelling Variables in pscholog are rarel (never?) measured directl the effects of the variable are measured Intelligence, self-esteem, depression Reaction time, diagnostic skill Measuring a Latent Variable Latent variables are drawn as ellipses hpothesised causal relationship with measured variables Measured variable has two causes latent variable other stuff random error Latent Measured Measured = t + e True Score Reliabilit is: the square root of proportion of variance in that is accounted the correlation between and e Error The Multivariate Case Much more comple and unpredictable 1 1 d a c e Value of a Car Causes tpe, size, age, rustiness no reason the should, or should not, be correlated Effects assessment of value b people who know 2 b 2

5 Α B Level of Depression Questionnaire items causes or effects? been feeling unhapp and depressed? been having restless and disturbed nights? found everthing getting 'on top' of ou? MIMIC SEM & fmri Theor to Analsis Functional and Effective Connectivit Eamine the influences between brain areas Interregional correlation ((Horwitz,, et et al, 1984) Structural equation modeling (McIntosh & Gonzalez-Lima, 1991, Buchel & Friston, 1997) Multiple regression and etensions (e.g., Kalman filters, Buchel & Friston, 1998) Baes networks (Dnamic Causal Modeling, Friston, Penn, et al, 2003) Identification of interacting regions Partial Least Squares (McIntosh, Bookstein,, et et al, 1996) Canonical Variates Analsis (Strother( et et al, 1995) Independent Components Analsis - 32 flavours (McKeown et et al, 1998, Calhoun et et al, 2001, Beckmann, Smith, et et al., 2002) Structural Equation Modeling Structural Equation Modeling Multivariate multiple regression Combines interregional covariances with anatomical framework Provides means to assess whether effective connections are modified b task-demands or differ between groups Is not meant to be a model test in the coventional sense Goodness of fit not as relevant w = A = = C A B C D B A 1.00 z = B C D D Structural Equations A = B + C + ψ B = wa + zd + ψ

6 Relation of effective connectivit changes to behavior Buchel, Coull, Friston, Science, 1999 Effective vs. functional connectivit Model: Model: A = = fmri fmri time-series time-series B = = ** A + + e1 e1 C = = ** A + + e2 e2 A Correct model C B χ 2 =0.5, ns. Correlations: A B C Dnamic changes in effective connectivit Attentional modulation of V5 responses to visual motion Stimuli 250 radiall moving dots at 4.7 degrees/s Pre-Scanning 5 30s trials with 5 speed changes (reducing to 1%) Task - detect change in radial velocit Scanning (no speed changes) 6 normal subjects, scan sessions; each session comprising 10 scans of 4 different condition e.g. F A F N F A F N S... F - fiation point onl A - motion stimuli with attention (detect changes) N - motion stimuli without attention S - no motion

7 Structural equation modelling (SEM) Attention - No attention Minimise the difference between the observed (S)( ) and implied (Σ)( covariances b adjusting the path coefficients (a, b, c) The implied covariance structure: =.B + z = z.(i - B) -1 : matri of time-series of regions U, V and W B: matri of unidirectional path coefficients (a,b,c) Variance-covariance covariance structure: T. = Σ = (I-B) - T. C.(I-B) -1 where C = z T z U u c a W w b V v T. is the implied variance covariance structure Σ C contains the residual variances (u,v,w) and covariances The free parameters are estimated b minimising a [maimum likelihood] ihood] function of S and Σ No attention Attention The use of moderator or interaction variables Hierarchical architectures 0.14 χ 2 =11, p<0.01 PP = PP Modulator influence of parietal corte on to V5 V5 V V5 PP χ 2 =13.6, p<0.01 χ 2 =5.9, p<0.01 LGN PFC Changes in effective connectivit over time: Learning SEM: Encoding Earl vs. Late Paired associates learning Pairing Object (Snodgrass) with Location fmri, 48 aial slices, TR 4.1s, 8 scans/cond cond 8 ccles (E)ncoding( (C)ontrol (R)etrieval 3 sessions (each with new objects & locations) PP LP DE ITp ITp ITa E R E R C C C 0.57 DE ITp PP Earl LP ITa χ 2 =6.3 p<0.05 diff. = DE Single subjects: +0.27*, +0.21, +0.37*, +0.24*, +0.19, +0.31* * p < ITp PP Late LP ITa

8 Changes in effective connectivit predict learning learning rate k 1 k =.35 k =.60 k =.63 k= r = 0.64 % correct k =.71 k =.44 learning block Length of EARLY (in learning blocks) that maimised the EARLY vs. LATE difference in connectivit (PP -> ITP) Dnamic Baes Networks

9 Goals Machine learning techniques applied to neuroimaging problems Analze Buckner et al. s fmri dementia data 1 Dartmouth fmri Data Center How do neural networks change with dementia? How to model networks of relationships? Create classifiers to discriminate between health and demented patients Prepping the Data Too man voels Use Talairach Database Lancaster et al., U of Teas 2 Average activit across regions Table of regional time series How to model time series? 1) Buckner, R. L., Snder, A., Sanders, A., Marcus, R., Morris, J. (2000). Journal of Cognitive Neuroscience 2) Lancaster JL, Woldorff MG, Parsons LM, Liotti M, Freitas CS, Raine L, Kochunov PV, Nickerson D, Mikiten SA, Fo PT (2000). Human Brain Mapping Relationship Modeling Discrete Multivariate Model Tpical analsis methods Generalized Linear Model (GLM) Structural Equation Modeling (SEM) Assumptions GLM and SEM frequentl make Fied temporal model Linearit Gaussian distribution Correlation accuratel modeled with covariance Use multivariate temporal models Each ROI is discrete temporal random variable Requires decimation of data Continuous ROI amplitude mapped to small set of discrete states 12 bit continuous 2 bit discrete Multinomials define ROI s behavior Not limited to pair-wise relationships Amplitude Ver High 0.1 High 0.4 Low 0.4 Ver Low 0.1 Dnamic Baesian Network Dnamic Baesian Network Framework for multinomial analsis Eplicitl models time Links indicate correlation Eample is a DBN famil Parents: and dentate Child: Parents predict values of child +1 One famil for ever ROI Families ma share parents Which links should be added? Correlations var in strength DBN is a set of families? +1

10 Dnamic Baesian Network Dnamic Baesian Network One famil for ever ROI Families ma share parents +1 One famil for ever ROI Families ma share parents +1 Which links should be added? Correlations var in strength? +1 Which links should be added? Correlations var in strength +1 DBN is a set of families DBN is a set of families +1? Dnamic Baesian Network Approach Demented Health Basic search idea Suggest man structures Score each one Score measures correlation strength Choose structure with > score Issues Too man structures Finding best is hard Separate data into: Health subjects Diseased patients 2. Learn DBN structure for each group 3. Look for change in DBN structure 4. Classif using likelihood tests Structure Results Prevalence in Dementia Cuneus Parietal Lobe Parietal Lobe Inf. Parietal Lobule BA 31 Post- Central grus Cingulate Grus BA 13 Gra Matter Uvula of Vermis BA 40 Uvula of Vermis BA 40 Declive BA 7 Insula Right Cerebrum Lat. Dorsal Nucleus Post- Central grus BA 12 Lateral Dorsal Nucleus Lateral Dorsal Nucleus Inf. Semi Lunar Lobule Inf. Semi Lunar Lobule Uvula Of Vermis Best health famil Best demented famil 7 of top 10 families 49% of all families

11 Activit in Health BA 33 BA 38 Rectal Grus Validation 1. Quantif confidence in each famil Measure likelihood correlations due to chance Top 10 families all have p values << Compare classification efficac with Support vector machines 65% accurac Gaussian naïve Baesian networks 73% accurac DBN 73% accurac Onl in 1 famil