Causal Ensembles for Evaluating the Effect of Delayed Switch to Second-Line Antiretroviral Regimens

Abstract

Transitioning from a failing antiretroviral regimen to a new regimen is a critical period in managing treatments to suppress HIV-1 RNA because it can have lasting effects on the durability of disease and likelihood of developing resistant mutations. Evaluating the timing of a switch to the subsequent therapy is difficult because patients are not randomly assigned to switch failing regimens at designed time points. Li et al. (J. Am. Stat. Assoc. 107:542–554, 2012) proposed and applied doubly robust semi-parametric methods to evaluate the effect of early versus late regimen switch in a two-stage design setting. These semi-parametric estimators are consistent if a parametric treatment model is correctly specified and achieve optimal performance if a parametric outcome model is also correctly specified. Here, we propose a new non-parametric estimator of the same causal estimand using an ensemble-type statistical learner. Compared to earlier estimators, the proposed estimator requires fewer model assumptions and can easily accommodate a large number of potential confounders. We illustrate the methods through simulation studies and application to data from the AIDS Clinical Trials Group Study A5095.

Appendix

Boosting is machine learning algorithm from a theory that attempts to construct a strong learner from a series or collection of weak learners and the earliest substantial contribution is widely attributed to [9, 10]. Freund and Schapire developed an early version of an adaptive resampling and combination scheme that became adaptive boosting or AdaBoost. Breiman [3] showed that AdaBoost can be viewed as functional gradient descent in function space while Friedman et al. [12, 13] linked AdaBoost and other boosting algorithms to a statistical framework in function estimation. The work by Breiman [3] and Friedman et al. [12, 13] brought boosting to a wide array of statistical regression and prediction applications beyond classification and our proposed estimator builds on this idea of function estimation.

Bühlmann and Hothorn [4] recently reviewed the literature in boosting and aggregation and their review informed our outline here. Boosting algorithms can be written as functional gradient descent techniques [3, 12, 13] and we adopt this view here. Briefly, the goal of functional gradient descent is to estimate a function by minimizing an expected loss

$$\displaystyle{E[\rho \{Y,f(X)\}],}$$

where ρ(⋅ , ⋅ ) is a (loss) function of data O ≡ { (X ₁, Y ₁), …, (X _n , Y _n )} and convex with respect to the second argument. Friedman [11] provided a generic outline of a descent algorithm through the following steps:

• 1. Initialize $\hat{f}^{0}(\cdot )$ with an offset value. A common choice is

  $$\displaystyle{\hat{f}^{0}(\cdot ) = \mbox{ argmin} \frac{1} {n}\sum \limits _{i=1}^{n}\rho (Y _{ i},c);}$$

  for a constant c or let $\hat{f}^{0}(\cdot ) = 0$. Set m = 0.

• 2. Increase m by 1. Compute the negative gradient (∂∕∂ f)ρ(Y, f) and evaluate it at $\hat{f}^{m-1}(X_{i})$:

  $$\displaystyle{U_{i} = -\frac{\partial \rho (Y _{i},f)} {\partial f} \bigg\vert _{f=\hat{f}^{m-1}(X_{i})},i = 1,\ldots,n.}$$

• 3. Fit the negative gradient vector U ₁, …, U _n to X ₁, …, X _n by the real-valued base procedure

  $$\displaystyle{(X_{i},U_{i})_{i=1}^{n}\stackrel{\mbox{ base procedure}}{\longrightarrow }\hat{g}^{m}(\cdot ).}$$

• 4. Update $\hat{f}^{m}(\cdot ) =\hat{ f}^{m-1}(\cdot ) +\upsilon \hat{ g}^{m}(\cdot )$, where υ is a step-length factor.

• 5. Iterate steps 2–4 until m = m _stop for some stopping iteration m _stop.

We need to determine two user-defined parameters in the algorithm above; namely, m _stop in step 5 and the step-length factor υ in step 4. The stopping iteration m _stop is determined via cross-validation or some information criterion, such as corrected AIC criterion. The choice of the step-length factor υ is chosen to be sufficiently small (e.g., υ = 0. 1). Popular loss functions ρ(y, f) are exp{ − (2y − 1)f} or log₂[1 + exp{ − (2y − 1)f}] for binary outcomes and squared error loss for continuous outcomes.

BlackBoost was developed by Friedman [11] and uses regression trees as the base learner. Bühlmann and Hothorn [4] reviewed both theory and applications and have highlighted the advantage that estimates will be invariant under monotone transformations of variables. In addition, regression trees can handle continuous and categorical covariates in a unified way.