Causal Mediation Analysis for Functional Data

Xi (Rossi) LUO

The University of Texas
Health Science Center
School of Public Health
Dept of Biostatistics
and Data Science
ABCD Research Group 

The 2nd JCSDSC, Kunming, China
July 12, 2024

Funding: NIH R01MH126970, R01EB022911

Collaborators

Yi Zhao
Indiana Univ

Michael Sobel
Columbia Univ

Martin Lindquist
Johns Hopkins Univ

Brian Caffo
Johns Hopkins Univ

Slides viewable on web:
bit.ly/fmjcsds

fMRI Experiments

Goal: infer dynamic brain activation and connectivity

fMRI data: blood-oxygen-level dependent (BOLD) signals from each cube. Each subject: observe matrix regions $\times$ time

Functional Data

Time (seconds)

Hemodynamic Response Function (HRF)

Stimulus sequence convolutions are functional data

Functional Mediation: Conceptual

All are functional data
Arrows represent quantifiable causal effects (curves)

Literature Review

• Mediation is an active area
  • Classical mediation deals with scalar variables Baron and Kenny, 1986
  • Recent work on sparse longitudinal data Zheng and van der Laan, 2017; VanderWeele and Tchetgen Tchetgen, 2017
  • Multilevel data Zhao and Luo, 2023
  • High dimensional settings Zhao and Luo, 2022
• Intersection with functional/time series models
  • Parametric time series models Zhao and Luo, 2019
  • Sparse and irregular longitudinal data Zeng and others, 2021
  • Functional mediation with scalar treatments and outcomes Lindquist, 2012
  • Spatial-temporal mediator with scalar variables Jiang and Colditz, 2023; Xu and Kang, 2023
  • This work: functional treatments, functional mediators, and functional outcomes

Causal Estimands and Assumptions

Notations

• Let $Z_{t}\in\mathcal{D}_{t}^{z}$, $M_{t}\in\mathcal{D}_{t}^{m}$, and $Y_{t}\in\mathcal{D}_{t}^{y}$ observed at time $t \in [0, T].$
• Let $\bar{Z}_{[s,t]}=\{Z_{u}:s\leq u\leq t\}$, similarly for $\bar{M}_{[s,t]}$, and $\bar{Y}_{[s,t]}$.
• When $s=0$, write $\bar{Z}_{[0,t]}\equiv\bar{Z}_{t}$, similarly for $\bar{M}_{[0,t]}$ and $\bar{Y}_{[0,t]}$.
• Let $M_{t}(\bar{z}_{T})$ and $Y_{t}(\bar{z}_{T})\equiv Y_{t}(\bar{z}_{T},\bar{M}_{T}(\bar{z}_{T}))$ denote, respectively, the potential values of the mediator and outcome at time $t$ when the treatment path is set to $\bar{z}_{T}$ and let $Y_{t}(\bar{z}_{T},\bar{m}_{T})$ denote the potential value of the outcome at time $t$ when the treatment path is set to $\bar{z}_{T}$ and the mediator path to $\bar{m}_{T}$.

Total Effect, Indirect/Direct Effects

• Total effect (TE) of treatment path $\bar{z}_{t}$ versus path $\bar{z}_{t}^{*}$ on $Y_{t}$
• These are functional versions where funtional potential outcomes depend on functional treatment paths

Assumptions

• A1: $\Pr(\bar{Z}_{T} \in \tilde{\mathcal{D}}_{T}^{z}) > 0$; for any Borel measurable subset $\tilde{\mathcal{D}}^{zm}_{T}$ of $\bar{\mathcal{D}}^{zm}_{T}$ with positive measure, $\Pr((\bar{Z}_{T}, \bar{M}_{T}) \in \tilde{\mathcal{D}}_{T}^{zm}) > 0$.
• A2: For all $\bar{z}_{T}\in \bar{\mathcal{D}}^{z}_{T}$, $$ \begin{equation} \label{eq:causal_assump1a} \bar{Y}_{t}(\bar{z}_{T},\bar{M}_{T}(\bar{z}_{T}))~\bot~\bar{Z}_{T}, \end{equation} $$ and for all $(\bar{z}_{T}, \bar{m}_{T}) \in \bar{\mathcal{D}}_{T}^{zm}$, $$ \begin{equation}\label{eq:causal_assump1b} \bar{Y}_{T}(\bar{z}_{T},\bar{m}_{T})~\bot~\bar{Z}_{T}, \end{equation} $$
• A3: For all $t\in[0,T]$ and ($\bar{z}_{t}, \bar{m}_{t}) \in \bar{\mathcal D}_{t}^{zm}$, $$ \begin{equation}\label{eq:causal_assump2} Y_{t}(\bar{z}_{t},\bar{m}_{t})~\bot~\bar{M}_{t}(\bar{z}_{t}) \mid \bar{Z}_{t}. \end{equation} $$
• A4: For all $\bar{z}_{T}\in\bar{\mathcal{D}}_{T}^{z}$, $$ \begin{equation}\label{eq:causal_assump3} \bar{M}_{T}(\bar{z}_{T})~\bot~\bar{Z}_{T}. \end{equation} $$
• A5: For all $t\in[0,T]$ and triples $(\bar{z}_{t}, \bar{z}^{*}_{t}, \bar{m}_{t}) \in \bar{\mathcal{D}}_{t}^{z} \times \bar{\mathcal{D}}_{t}^{z} \times \bar{\mathcal{D}}_{t}^{m}$ with $\bar{z}_{t} \neq \bar{z}_{t}^{*}$, $$ \begin{equation}\label{eq:causal_assump4} \bar{Y}_{t}(\bar{z}_{t},\bar{m}_{t})~\bot~\bar{M}_{t}(\bar{z}_{t}^{*}). \end{equation} $$
• These are extensions of scalar ones to functional data, and weaker versions are also possible in our paper.

Causal Identifiability

Model

Causal SEMs: Concurrent Model

These are defined on potential outcomes with functional curves $\alpha(t)$, $\beta(t)$, $\gamma(t)$

Causal SEMs: Historical Model

Historical Model: Direct and Indirect Effects

Direct Effect

Indirect Effect

Causal effects are 1D/2D integration of the shaded

Method: Basis

• We estimate two SEM equations separately. The concurrent model is easy.
• Using the matrix form, $X = (M, Z)$, the historical model $$ Y(t)=\int_{\Omega_{t}} X(s)\theta(s,t)~\mathrm{d}s+\epsilon(t), $$
• Bivariate function $\theta_{j}(s,t)$ expanded as $$ \theta_{j}(s,t)=\sum_{k=1}^{K_{1j}}\sum_{l=1}^{K_{2j}}g_{klj}\phi_{kj}(s)\eta_{lj}(t) $$

• Solve the weighted regularized fitting criterion Ramsay, 2006, Wang et al, 2016 $$\scriptsize \mathrm{LMSSE}(\theta)=\int_{0}^{T}\| Y(t)-\mathbf{D}(t) \mathbf{G} \|_{2}^{2}~\mathrm{d}t+\lambda_{s}\mathrm{PEN}_{s}(\theta)+\lambda_{t}\mathrm{PEN}_{t}(\theta), $$ $$\scriptsize \mathbf{G}=\begin{pmatrix} \mathrm{vec}(\mathbf{G}_{1}) \\ \vdots \\ \mathrm{vec}(\mathbf{G}_{q}) \end{pmatrix}, ~ \mathbf{D}(t)=\begin{pmatrix} \boldsymbol{\eta}_{1}^\top(t)\otimes X_{1}^{*}(t) & \cdots & \boldsymbol{\eta}_{q}^\top(t)\otimes X_{q}^{*}(t) \end{pmatrix} $$
• The smoothness penalty $$\scriptsize \begin{align} \mathrm{PEN}_{s}(\theta) =& \int_{0}^{T}\int_{0}^{T}\{\mathcal{L}_{s}\theta(s,t)\}\{\mathcal{L}_{s}\theta^\top(s,t)\}~\mathrm{d}s\mathrm{d}t \end{align} $$
• Select $\lambda$ by cross validation
• Solve $ \mathbf{G}$ explicitly

Simulations

Simulations

• Compare with two other popular scalar methods:
  • KKB: extension of Baron and Kenny for multilevel data
  • Beta-KKB: summarize functions by feature vectors and then apply standard mediation
• These methods produce constant causal effect estimates
• Performance metrics include integrated MSE, MAE, and Bias
• Simulate from a functional mediation model with different sample sizes

Comparison

Challenging for scalar mediation methods to capture functional causal effects

Curve Estimation

Estimated curves (gray) closely track the true curves (red)

Real Data

• N=121 subjects responding to the instructions of pressing and not pressing buttons while their brain activities were scanned by fMRI
• Two causal paths
  • IE: stimulus → preSMA brain region → M1 brain region
  • DE: stimulus → M1 brain region
• We choose the tuning parameters, smoothness and historical window size, by cross-validation
• Inference is through bootstrapping, point-wise confidence bands

Brain Mediation Model

Goal: quantify functional direct and indirect effects

• Red vertical bars: stimulus instructions of withholding from pressing buttons
• Elevated IE/DE (solid lines) after the stimuli, especially when multiple stimuli presented in a short period prior
• Shaded areas: 95% point-wise confidence bands

Summary

• Functional model for mediation analysis
• Causal assumptions for identifiable effects
• Estimate dynamic causal effects as functional curves
• Enhance understanding of how the brain processes stimulus infomation
• R package available: cfma

Thank you!

Comments? Questions?