Feasible computation of propensity score and outcome regression models with the Highly Adaptive Lasso

Background: Estimation of causal effects in healthcare database studies often requires modeling nuisance functions (propensity score or outcome regression models) to adjust for baseline confounders. Recent literature suggests that nonparametric regression models are advantageous to parametric ones since they require less assumptions to converge to the true propensity score or outcome function. However, the flexibility of nonparametric methods comes at the cost of slower convergence rates, which creates challenges for valid inference. To overcome these challenges, nonparametric regression methods are needed with faster convergence rates (with rates n^{-1/4}). The recently proposed Highly Adaptive Lasso (HAL) of Benkeser and van der Laan (2016) has such statistical properties. However, its favorable statistical properties come at an exponential runtime, thus making its computation infeasible in practice.

Objectives: To investigate strategies for reducing the computational effort and to develop an improved heuristic of the HAL estimate which allows one to compute both propensity score and outcome regression models at a considerably reduced computational cost while only incurring a negligible trade-off in accuracy.

Methods: HAL minimizes prediction error by optimizing a global distance measure (the variational norm) between the estimating function and the data. However, not all the input data is usually necessary for this optimization task. To reduce computation time we first utilize data-driven heuristics to remove data points which do not contribute considerably to the predictive performance. Second, bounding the variational norm allows one to quantify in advance the incurred trade-off in accuracy.

Results: We consider the same data example as in the original paper of Benkeser and van der Laan (2016), consisting of a sinusoidal regression model which is to be reconstructed from 1000 data points. We aim to demonstrate that the HAL estimate can be computed at a comparable precision using only a fraction of the data and thus at a fraction of the computation time. Indeed, we show that the global measure (the variational norm) of the fitted models decreases from 8.13 (100 data points) to 7.05 (300 data points) to 6.69 (600 data points). This is only a 4% increase in comparison to the fully fitted model (variational norm of 6.43 for the full 1000 data points) despite using only 60% of the data. A visual inspection confirms the good accordance of the heuristic models with the fully fitted model.

Conclusions: Results indicate that despite its worst-case (exponential) computational complexity, the HAL estimate can be computed on a reduced (logarithmic size) input, thus allowing one to reduce the computational complexity.