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8
Multivariate Mediation
Jirapat Samranvedhya (Pat)
Department of Statistics
North Carolina State University
Statistical Learning Group
November 3, 2017
Motivation
Pneumonia outcome
Flu vaccine treatment
Other lung/throat pathogens
Introduction
Pneumonia
Pneumonia
3rd leading cause of hospitalization
6th leading cause of death
Influenza
Influenza
2-10% of influenza leads to pneumonia
10-20% of all-cause pneumonia is related to flu
Sanofi makes the vaccine
Vaccine study
Vaccine study
Double blind, randomized trial
32,000 adults, 65+ years
Flu-like symptom -> NP swab
Flu vaccine might not affect pneumonia outcome only through flu
Data
Data
Sample selection
- Consent for future study
- All with pneumonia, NP swab within certain timeframe of symptom onset
- Random samples of those without pneumonia
Pending lab test (end of 2017 or early 2018)
- Vaccine treatment (categorical)
- Pneumonia outcome (categorical)
- Other pathogens (continuous)
Objectives
Objectives
1. Do pathogens mediate the effect?
2. Which one?
Model
Multivariate mediation with
multinomial outcome
Model
Pretty diagram
Related work
Related work
Continuous outcome
Assume known variance
PCA-like transformation (w):
- Maximize likelihood instead of variance
- Answers both objectives!
Solve with Lagrange multiplier
Optimization problem for first direction
Assume normality and known variances
Formulation
Formulation
Logistic model for categorical outcome
Optimized over alpha, beta and w
Variance considered hyperparameter
Optimization problem for first direction
Assume normality
Optimization
Preliminary
Preliminary
Holding other parameters constant,
- alpha: convex, least squares analytic solution -> profiled likelihood
- beta: convex, solved iteratively (logistic regression)
- w: nonconvex from norm constraint
Logistic + norm constraint is hard to solve
Convex relaxation
Convex relaxation
Convexity is nice
- Local minimum is global minimum
- First-order conditions give optimality
So the plan is:
- Profile out alpha
- Relax constraint to inequality
- Rescale beta and w post-optimization
Variable splitting
Solve smaller, easier problems
Variable splitting
Motivation
Motivation
Dr. Chi's Advanced computing, lecture 21
ADMM: review
Augmented Lagrangian
Augmented Lagrangian
Augmented Lagrangian Method
Augmented Lagrangian
Descent
ADMM
ADMM
What if updating (x, z) together is hard?
Alternating Direction Method of Multipliers
Scaled form
Scaled form
Combine linear and quadratic term Ax + Bz - c, completing the squares so to speak,
with change of variable u = y/rho
ADMM: applied
ADMM: applied
f(w1, b) = -loglik
g(w2) = Lagrangian norm constraint
such that w1 - w2 = 0
This splits logistic and norm constraint, both of which we know how to do!
x-update (beta, w1): BFGS
z-update (w2): Lagrangian, KKT conditions
u-update: straightforward
Simulation
Still waiting for data, so this is all we got
Simulation
Setup
Setup
Gaussian copula to describe joint distribution of M and X, basically specify X and M|X
Specify true w and beta
Compute multinomial probability to draw Y
n = 500 in each of 1000 iterations
ADMM parameter rho = 2
Variance: 0.1, 1, 10, 100, 100 with same starting seed
Initialize: beta as vector of ones,
w as vector of ones (normalized to unit norm),
u as vector of 0.001
Result
Intel i5-4300U and 4GB memory (Windows 7 Enterprise)
Result
0.1 and 1
0.1 and 1
10 and 100
10 and 100
Discussion
Discussion
Estimates centered around the truth
Hyperparameter:
- W estimates are robust
- Beta ones are more sensitive, especially in the extremes (min/max)
- Speed-precision tradeoff
Takeaway
Convexity is nice
Use ADMM to solve smaller problem
Prezi looks fancy
but there's no LaTeX/math support
Get flu shot
Take advanced computing
References
Boyd, et al. Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. 2010.
http://stanford.edu/~boyd/papers/pdf/admm_distr_stats.pdf
Chen, et al. High-dimensional Multivariate Mediation with Application to Neuroimaging Data. 2016. https://arxiv.org/pdf/1511.09354.pdf