\( \def\RR{\bf R} \def\real{\mathbb{R}} \def\bold#1{\bf #1} \def\d{\mbox{Cord}} \def\hd{\widehat \mbox{Cord}} \DeclareMathOperator{\cov}{cov} \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cor}{cor} \newcommand{\ac}[1]{\left\{#1\right\}} \)
Pathway Lasso: Estimate Brain Information Flow Pathways
Xi (Rossi) Luo
Brown University
Department of Biostatistics
Center for Statistical Sciences
Computation in Brain and Mind
Brown Institute for Brain Science
NESS at Yale University
April 22, 2016
Funding: NSF/DMS (BD2K) 1557467; NIH P20GM103645, P01AA019072, P30AI042853; AHA
Coauthor
Yi Zhao
(3rd Yr PhD Student)
Brown University
Task fMRI
 Task fMRI: performs tasks under brain scanning
 Story vs Math task:
listen to story (treatment stimulus) or math questions (control), eye closed
 Not restingstate: "rest" in scanner
Goal: how brain processes story/math differently?
fMRI data: bloodoxygenlevel dependent (BOLD) signals from each
cube/voxel (~millimeters),
$10^5$ ~ $10^6$ voxels in total.
fMRI Studies
Time 1
2
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~200
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This talk: one subject, two sessions (to test replicability)
Network Model with Stimulus
Question: quantify
red,
blue, and other pathways
from stimulus to orange outcome circle/region Heim et al, 09
Pathway=Activation+Connectivity
 Activation: stimulus $\rightarrow$ brain region activity
 Connectivity: one brain region $\rightarrow$ another region
 Whether not two or more brain regions "correlate"
 Pathway: stimulus $\rightarrow$ brain region A $\rightarrow$ region B
 Strong path: strong activation
and strong conn
 Zero path: zero activation
or zero conn, including
 Zero activation + strong conn = zero
 Strong activation + zero conn = zero
Mediation Analysis and SEM
$$\begin{align*}M &= Z a + {U + \epsilon_1}\\ R &= Z c + M b + {U g + \epsilon_2}\end{align*}$$
 Pathway effect: $a \times b$indirect; residual: $c$direct
 Mediation analysis
 Baron&Kenny, 86; Sobel, 82; Holland 88; Preacher&Hayes 08; Imai et al, 10; VanderWeele, 15;...
Mediation Analysis in fMRI
 Parametric Wager et al, 09 and functional Lindquist, 12 mediation, under (approx.) independent errors
 Stimulus $\rightarrow$ brain $\rightarrow$ user reported ratings, one mediator
 Usual assumption: $U=0$ and $\epsilon_1 \bot \epsilon_2$
 Parametric and multilevel mediation Yi and Luo, 15, with correlated errors for two brain regions
 Stimulus $\rightarrow$ brain region A $\rightarrow$ brain region B, one mediator
 Correlations between $\epsilon_1$ and $\epsilon_2$
 This talk: multiple mediator and multiple pathways
 High dimensional: more mediators than sample size
 Alt: dimension reduction by arXiv1511.09354 Chen, Crainiceanu, Ogburn, Caffo, Wager, Lindquist, 15
Full Pathway Model
 Stimulus $Z$, $K$ mediating brain regions $M_1, \dotsc, M_K$, Outcome region $R$
 Strength of activation ($a_k$) and connectivity ($b_k$, $d_{ij}$)
 Too complex, even for small $K = 2$ Daniel et al, 14
Reduced Pathway Model
$$\begin{align}M_k & = Z A_k + E_{1k},\, k=1,\dotsc, K\\ R & = Z C + \sum_{k=1}^{K} M_k B_k + E_2 \end{align}$$
 $A_k$: total inflow to mediator $M_k$; $B_k$: total conn
 Pathway effect: $A_k \times B_k$; Residual: $C$
Relation to Full Model
 Proposition: Our "total" parameters has explicit forms of "individual" flow parameters in the full model
 Proposition: Our $E_k$'s are correlated, but won't affect estimation (will affect variance)
 Reduced model: a step to select spatial mediators
 Strong overall inflow and strong conn flow
 Favor reduced: challenging to determine the temporal order because of low temporal resolution
Regularized Regression
 Minimize the penalized least squares criterion
$$\sum_{k=1}^K \ M_k  Z A_k \_2^2 + \ R  Z C  \sum_k M_k B_k \_2^2 + \mbox{Pen}(A, B)$$
The choice of penalty $\mbox{Pen}(\cdot)$ to be discussed
 All data are normalized (mean=0, sd=1)
 Want to select sparse
pathways for highdim $K$
 Alternative approach: twostage LASSO Tibshirani, 96 to select sparse
inflow and
connection separately: $$ \sum_{k=1}^K \ M_k  Z A_k \_2^2 + \lambda \sum_k  A_k  \\ \ R  Z C  \sum_k M_k B_k \_2^2 + \lambda \sum_k B_k $$
Penalty: Pathway LASSO
 Select strong pathways effects: $A_k \times B_k$
 TSLASSO: shrink to zero when $A$&$B$ moderate but $A\times B$ large
 Penalty (prototype) $$ \lambda \sum_{k=1}^K A_k B_k $$
 Nonconvex in $A_k$ and $B_k$
 Computationally heavy and nonunique solutions
 Hard to prove theory
 We propose the following general class of penalties$$ \lambda \sum_{k=1}^K ( A_k B_k + \phi A_k^2 + \phi B_k^2) $$
Theorem $$v(a,b) = a b + \phi (a^2 + b^2)$$ is convex
if and only if $\phi\ge 1/2$. Strictly convex if $\phi > 1/2$.
Contour Plot of Different Penalties

Nondifferentiable at points when $a\times b = 0$
 Shrink $a\times b$ to zero
 Special cases: $\ell_1$ or $\ell_2$
 TSLASSO: different $ab$ effects though $a+b$ same
Algorithm: ADMM + AL
 SEM/regression loss: $u$; Nondiffernetiable penalty: $v$
 ADMM to address differentiability $$ \begin{aligned} \text{minimize} \quad & u(\Theta,D)+v(\alpha,\beta) \\ \text{subject to} \quad & \Theta=\alpha, \\ & D=\beta, \\ & \Theta e_{1}=1, \end{aligned}$$
 Augmented Lagrangian for multiple constraints
 Iteratively update the parameters
 We derive theorem on explicit (not simple) updates
Complications
 Mixed norm penalty $$\mbox{PathLasso} + \omega \sum_k (A_k + B_k)$$
 Tuning parameter selection by cross validation
 Reduce false positives via thresholding Johnston and Lu, 09
 Inference/CI: bootstrap after refitting
 Remove false positives with CIs covering zero Bunea et al, 10
Simulations
 Our PathLasso compares with TSLasso
 Simulate with varying error correlations
 Tuningfree comparison: performance vs tuning parameter (estimated effect size)
 PathLasso outperforms under CV
Pathway Recovery
ROC
F1 Score
MSE
Our PathLasso (red) outperforms twostage Lasso (blue)
Other curves: variants of PathLasso and correlation settings
Data: Human Connectome Project
 Two sessions (LR/RL), story/math task Binder et al, 11
 gICA reduces voxel dimensions to 76 brain maps
 ROIs/clusters after thresholding
 Apply to two sess separately, compare replicability
 Jaccard: whether selected pathways in two runs overlap
 $\ell_2$ diff: difference between estimated path effects
 Tuningfree comparisons
Jaccard
$\ell_2$ Diff
Regardless of tuning, our PathLasso (red) has smaller crosssess diff (selection and estimation) than TSLasso (blue)
StimM25R and StimM65R significant shown largest weight areas
 M65 responsible for language processing, larger flow under story
 M25 responsible for uncertainty, larger flow under math
Summary
 High dimensional pathway model
 Penalized SEM for pathway selection and estimation
 Convex optimization for nonconvex products
 Sufficient and necessary condition
 Algorithmic development for complex optimization
 Improved estimation and selection accuracy
 Higher replicability in HCP data
 Manuscript: Pathway Lasso (arXiv 1603.07749)
Thank you!
Slides at: bit.ly/xlNESS16
More info: BigComplexData.com