DEEPCAUSAL Procedure
Estimating Unknown Functions in Propensity Score and Outcome Models
There are two main models in causal inference: the propensity score model and the outcome model. Two approximate DNNs are constructed to estimate these two models. The DEEPCAUSAL procedure supports one binary treatment and one continuous outcome. The unconfoundedness is assumed; it is also called ignorability, selection on the observables, or no omitted variable bias.
Propensity Score Model
The propensity score model estimates the treatment assigning probability conditional on some covariates:
where 
denotes the propensity score, T denotes the binary treatment, and 
is a vector of the observed covariates. The superscript 
in 
denotes the covariates in Set 1, which are the covariates used in the propensity score model. The 
is one unknown function to be estimated by a DNN.
The DNN for the propensity score model, a binary classification predictor, predicts the binary treatment by using the covariates in Set 1. It trains the observations in the input data set. The DNN is a fully connected multilayer perceptron (MLP). The input layer of that DNN corresponds to , and the output layer corresponds to the observed treatment variable. The activation functions for the hidden layers are all rectified linear unit (ReLU) functions, and the activation function for the output layer is a sigmoid function. The loss function is the negative log-likelihood function.
If no covariates are used to predict the binary treatment (
), then no DNN needs to be trained for the propensity score model. A random assignment of the treatment and many other reasons might result in the empty ; that is, all covariates are independent of the treatment. The propensity score is estimated by
where 
denotes the treatment value of the ith observation and n denotes the number of observations. The indicator function 
if A is true; otherwise, it is 0.
Outcome Model
The outcome model describes how the outcome depends on the treatment and some covariates in Set 2, which are the covariates used in the propensity score model. The covariates in Set 1 and Set 2 might be different. The DEEPCAUSAL procedure supports only one continuous outcome variable, and the outcome model is as follows:
where Y is the outcome, t is the treatment, and 
is a known function. For example, 
might be the identity function; that is, 
. The superscript 
in 
denotes the covariates in Set 2. In PROC DEEPCAUSAL, is the identity function. The 
and 
are unknown functions to be estimated by a DNN.
The DNN for the outcome model, a regression predictor, predicts the outcome by using the covariates in Set 2 and the treatment variable. It trains the observations in the input data set. The input layer of that DNN corresponds to . After the hidden layers, there is a parameter layer with two neurons that correspond to 
and 
. The activation functions for this parameter layer are linear functions. Subsequently, t is included, and 
is calculated as the result of the final output layer. is set to the identity function in PROC DEEPCAUSAL, so 
is output as the prediction.
If no covariates are used to predict the outcome (
), then no DNN needs to be trained for the outcome model. The outcome model is predicted by
Moreover, when , and are constants and are computed by
where 
denotes the outcome value of the ith observation, denotes the treatment value of the ith observation, and n denotes the number of observations. The indicator function if A is true; otherwise, it is 0.