Bayesian additive regression trees for causal inference with multiple treatments

The setting: 3 prostate cancer treatments

The setting: 3 prostate cancer treatments

Issue 1: Selection bias

Issue 2: Non-overlapping distributions

Issue 3: Large weights

Issue 3: Large weights

Notation

Consider causal effect of \(A \in \{1, \ldots, Z\}\) on binary outcome \(Y \in \{0,1\}\)

• \(i = 1, \ldots, n\) for \(n\) total subjects
  • \(n = n_1 + \ldots + n_Z\)
• \(\{Y_i(1), \ldots, Y_i(Z)\}\) as potential outcomes for \(i\)
• Covariates \(X_i\)

Notation

Interest: average treatment effect among treated

• \(ATT_{1|1, a}\): effect of \(T = 1\) versus \(T = a\) among those with \(T = 1\)
  • \(ATT_{1|1, a}\) = \(\frac{1}{n_1} \sum_{i:A_i = 1} (Y_i(1) - Y_i(a))\) for \(a = \{2, \ldots, Z\}\)

• Ex: \(Z\) = 3, \(ATT_{1| 1, 2}\), \(ATT_{1| 1, 3}\), \(ATT_{1| 2, 3}\)
  • \(ATT_{1|1, 2}\) = \(\frac{1}{n_1} \sum_{i:A_i = 1} (Y_i(1) - Y_i(2))\)
  • \(ATT_{1|1, 3}\) = \(\frac{1}{n_1} \sum_{i:A_i = 1} (Y_i(1) - Y_i(3))\)
  • \(ATT_{1|2, 3}\) = \(\frac{1}{n_1} \sum_{i:A_i = 1} (Y_i(2) - Y_i(3))\)

Causal inference with multiple treatments

Why not binary approaches?

• May not fully account for differences in patient characteristics
• Comparisons of disctinct cohorts with dissimilar characteristics
• Challenging to identify optimal treatment
• Main issue: matching on scalar alone insufficient
• See Lopez & Gutman, 2017 for more

Causal inference with multiple treatments

• Inverse probability of treatment weighting (Feng et al, 2012)
• Generalized boosted models (McCaffrey et al, 2013)
• Matching (Yang et al, 2016; Lopez & Gutman, 2017)

Bayesian Additive Regression Trees

BART model:

• \(P(Y=1|A=a, X=x) = \Phi(f(a, x))\)
• \(f(a,x)\) appromated using sum of trees
• \(ATT\)’s estimated using counterfactuals
  • \(ATT_{1|1, a} = \frac{1}{n_1} \sum_{i:A_i = 1}^{n} \Phi(f(1, x_i)) - \Phi(f(a, x_i))\)
  • \(ATT_{1|2, a} = \frac{1}{n_1} \sum_{i:A_i = 1}^{n} \Phi(f(2, x_i)) - \Phi(f(a, x_i))\)

Bayesian Additive Regression Trees

Why BART for causal inference? see Hill, 2012

• Flexibly models response surface ✔️
• Large number of continuous and categorical predictors ✔️
• No ambiguity with respect to balance assessment ✔️
• Accessibility ✔️
• Accuracy ✔️

Bayesian Additive Regression Trees

Why BART for multiple treatments?

• Coherent posterior intervals ❓
• Heterogenous treatment effects ❓
• Accessibility ❓
• Accuracy ❓

Simulation study

6 factorial design using dbarts package in R

• Ratio of \(n_1\) : \(n_2\) : \(n_3\)
• \(n\)
• No. predictors
• \(P(A|X)\)
• Predictor strength alignment
• Response surfaces (parallel?)

Simulation results

3 prostate cancer treatments

ATT’s: generalizable to population receiving RT 1

Comments

• Attentuation of effect comparing RT 1 v. Prostatectomy
  • distinct cohorts
• Less attentuation for RT 1 v RT 2
  • similar cohorts

• BART shows promise for causal inference with multiple treatments
  • expanded simulations required
  • formal paper ready Spring 2018

Comments

Github site: https://github.com/statsbylopez/ci-bart

Twitter: @statsbylopez