# Estimating GPS

The propensity score (PS) is the conditional probability of assignment to a particular treatment given a vector of observed covariates (Rosenbaum and Rubin 1983). Hirano and Imbens (2004) extended the idea to studies with continuous treatment (or exposure) and labeled it as the generalized propensity score (GPS), which is a probability density function. In this package, we use either a parametric model (a standard linear regression model) or a non-parametric model (a flexible machine learning model) to train the GPS model as a density estimation procedure (Kennedy et al. 2017). After the model training, we can estimate GPS values based on the model prediction. The machine learning models are developed using the SuperLearner Package (Van der Laan, Polley, and Hubbard 2007). For more details on the problem framework and assumptions, please see Wu et al. (2020).

Whether the prediction models’ performance should be considered the primary parameter in the training of the prediction model is an open research question. In this package, the users have complete control over the hyperparameters, which can fine-tune the prediction models to achieve different performance levels.

## Available SuperLearner Libraries

The users can use any library in the SuperLearner package. However, in order to have control on internal libraries we generate customized wrappers. The following table represents the available customized wrappers as well as hyperparameters.

Package name sl_lib name prefix available hyperparameters
XGBoost m_xgboost xgb_ nrounds, eta, max_depth, min_child_weight, verbose
ranger m_ranger rgr_ num.trees, write.forest, replace, verbose, family

## Implementation

Both XGBoost and ranger libraries are developed for efficient processing on multiple cores. The only requirement is making sure that OpenMP is installed on the system. User needs to pass the number of threads (nthread) in running the estimate_gps function.

In the following section, we conduct several analyses to test the scalability and performance. These analyses can be used to have a rough estimate of what to expect in different data sizes and computational resources.

## References

Hirano, Keisuke, and Guido W Imbens. 2004. “The Propensity Score with Continuous Treatments.” Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives 226164: 73–84. https://doi.org/10.1002/0470090456.ch7.
Kennedy, Edward H, Zongming Ma, Matthew D McHugh, and Dylan S Small. 2017. “Nonparametric Methods for Doubly Robust Estimation of Continuous Treatment Effects.” Journal of the Royal Statistical Society. Series B, Statistical Methodology 79 (4): 1229. https://doi.org/10.1111/rssb.12212.
Rosenbaum, Paul R, and Donald B Rubin. 1983. “The Central Role of the Propensity Score in Observational Studies for Causal Effects.” Biometrika 70 (1): 41–55.
Van der Laan, Mark J, Eric C Polley, and Alan E Hubbard. 2007. “Super Learner.” Statistical Applications in Genetics and Molecular Biology 6 (1).
Wu, Xiao, Fabrizia Mealli, Marianthi-Anna Kioumourtzoglou, Francesca Dominici, and Danielle Braun. 2020. “Matching on Generalized Propensity Scores with Continuous Exposures.” https://arxiv.org/abs/1812.06575.