Decision making and uncertainty quantification for individualized treatments using Bayesian Additive Regression Trees

Abstract

Individualized treatment rules can improve health outcomes by recognizing that patients may respond differently to treatment and assigning therapy with the most desirable predicted outcome for each individual. Flexible and efficient prediction models are desired as a basis for such individualized treatment rules to handle potentially complex interactions between patient factors and treatment. Modern Bayesian semiparametric and nonparametric regression models provide an attractive avenue in this regard as these allow natural posterior uncertainty quantification of patient specific treatment decisions as well as the population wide value of the prediction-based individualized treatment rule. In addition, via the use of such models, inference is also available for the value of the optimal individualized treatment rules. We propose such an approach and implement it using Bayesian Additive Regression Trees as this model has been shown to perform well in fitting nonparametric regression functions to continuous and binary responses, even with many covariates. It is also computationally efficient for use in practice. With Bayesian Additive Regression Trees, we investigate a treatment strategy which utilizes individualized predictions of patient outcomes from Bayesian Additive Regression Trees models. Posterior distributions of patient outcomes under each treatment are used to assign the treatment that maximizes the expected posterior utility. We also describe how to approximate such a treatment policy with a clinically interpretable individualized treatment rule, and quantify its expected outcome. The proposed method performs very well in extensive simulation studies in comparison with several existing methods. We illustrate the usage of the proposed method to identify an individualized choice of conditioning regimen for patients undergoing hematopoietic cell transplantation and quantify the value of this method of choice in relation to the optimal individualized treatment rule as well as non-individualized treatment strategies.

Keywords: BART; Individualized treatment rules; boosting; optimal ITR; outcome weighted learning; prediction models; random forests; subgroup analysis; value function estimation.

Figure 1

An example of a single tree with branch decision rules and terminal nodes

Figure 2

Value function relative to the optimal value function for simulation settings in (a) scenarios as in [14], (b) same as (a) but with treatment effects cut by 25%, and (c) scenarios as in [4]

Figure 3

BART posterior means vs. true probabilities for complex interaction model, with n = 500 (left side) and n = 5000 (right side). First two rows show predictions for individual treatment outcomes, while row 3 shows predictions for treatment differences, all using a single training dataset. Row 4 shows the posterior means for the treatment difference averaged over 400 repeated data simulations of the training set.

Figure 4

BART posterior means vs. true probabilities for no interaction model, with n = 500 (left side) and n = 5000 (right side). First row shows predictions for treatment differences using a single training dataset, while row 2 shows the posterior means for the treatment difference averaged over 400 repeated data simulations of the training set.

Figure 5

Results of BART ITR for HCT example. (a) Waterfall plot of 1 yr survival differences (FluMel-FluBu) by patient (posterior mean differences, along with inter-quartile ranges and 95% posterior intervals), (b) Waterfall plot of posterior probabilities that survival is higher for Flu/Mel, (c) Density plot of value functions for three treatment strategies (FluMel, FluBu, BART ITR) as well as Optimal ITR, and (d) Density plot of difference in value functions for treatment strategies compared to BART ITR. The posterior mean of the value function distributions for each treatment strategy are: FluBu: 0.651, FluMel: 0.667, BART1I9TR: 0.677, Optimal ITR: 0.682.

Figure 6

Tree fit to the posterior mean treatment differences. Values in each node represent the posterior mean and 95% credible intervals for the average treatment effect of the subgroup of patients represented in that node.