Metalearners
Instead of knowing the average causal effect, we might want to know which units benefit and which units lose by being exposed to a treatment. For example, a cash transfer program might motivate some people to work harder and incentivize others to work less. Thus, we might want to know how the cash transfer program affects individuals instead of it average affect on the population. To do so, we can use metalearners. Depending on the scenario, we may want to use an S-learner, T-learner, X-learner, R-learner, or doubly robust learner. The basic steps to use all five metalearners are below. The difference between the metalearners is how they estimate the CATE and what types of variables they can handle. In the case of S, T, X, and doubly robust learners, they can only handle binary treatments. On the other hand, R-learners can handle binary, categorical, count, or continuous treatments but only supports continuous outcomes.
For a deeper dive on S-learning, T-learning, and X-learning see:
Künzel, Sören R., Jasjeet S. Sekhon, Peter J. Bickel, and Bin Yu. "Metalearners for
estimating heterogeneous treatment effects using machine learning." Proceedings of the
national academy of sciences 116, no. 10 (2019): 4156-4165.To learn more about R-learning see:
Nie, Xinkun, and Stefan Wager. "Quasi-oracle estimation of heterogeneous treatment
effects." Biometrika 108, no. 2 (2021): 299-319.To see the details out doubly robust estimation implemented in CausalELM see:
Kennedy, Edward H. "Towards optimal doubly robust estimation of heterogeneous causal
effects." Electronic Journal of Statistics 17, no. 2 (2023): 3008-3049.Initialize a Metalearner
S-learners, T-learners, X-learners, R-learners, and doubly robust estimators all take at least three arguments—covariates, treatment statuses, and outcomes, all of which can be either an array or any struct that implements the Tables.jl interface (e.g. DataFrames). S, T, X, and doubly robust learners support binary treatment variables and binary, continuous, count, or time to event outcomes. The R-learning estimator supports binary, continuous, or count treatment variables and binary, continuous, count, or time to event outcomes.
Non-binary categorical outcomes are treated as continuous.
You can also specify the the number of folds to use for cross-fitting, the number of extreme learning machines to incorporate in the ensemble, the number of features to consider for each extreme learning machine, the activation function to use, the number of observations to bootstrap in each extreme learning machine, and the number of neurons in each extreme learning machine. These arguments are specified with the folds, nummachines, numfeatures, activation, samplesize, and num\neurons keywords.
# Generate data to use
X, Y, T = rand(1000, 5), rand(1000), [rand()<0.4 for i in 1:1000]
# We could also use DataFrames or any other package that implements the Tables.jl API
# using DataFrames
# X = DataFrame(x1=rand(1000), x2=rand(1000), x3=rand(1000), x4=rand(1000), x5=rand(1000))
# T, Y = DataFrame(t=[rand()<0.4 for i in 1:1000]), DataFrame(y=rand(1000))
s_learner = SLearner(X, Y, T)
t_learner = TLearner(X, Y, T)
x_learner = XLearner(X, Y, T)
r_learner = RLearner(X, Y, T)
dr_learner = DoublyRobustLearner(X, T, Y)Estimate the CATE
We can estimate the CATE for all the models by passing them to estimatecausaleffect!.
estimate_causal_effect!(s_learner)
estimate_causal_effect!(t_learner)
estimate_causal_effect!(x_learner)
estimate_causal_effect!(r_learner)
estimate_causal_effect!(dr_lwarner)Get a Summary
We can get a summary of the model by pasing the model to the summarize method.
!!!note To calculate the p-value and standard error for the treatmetn effect, you can set the inference argument to false. However, p-values and standard errors are calculated via randomization inference, which will take a long time. But can be sped up by launching Julia with a higher number of threads.
summarize(s_learner)
summarize(t_learner)
summarize(x_learner)
summarize(r_learner)
summarize(dr_learner)Step 4: Validate the Model
We can validate the model by examining the plausibility that the main assumptions of causal inference, counterfactual consistency, exchangeability, and positivity, hold. It should be noted that consistency and exchangeability are not directly testable, so instead, these tests do not provide definitive evidence of a violation of these assumptions. To probe the counterfactual consistency assumption, we simulate counterfactual outcomes that are different from the observed outcomes, estimate models with the simulated counterfactual outcomes, and take the averages. If the outcome is continuous, the noise for the simulated counterfactuals is drawn from N(0, dev) for each element in devs and each outcome, multiplied by the original outcome, and added to the original outcome. For discrete variables, each outcome is replaced with a different value in the range of outcomes with probability ϵ for each ϵ in devs, otherwise the default is 0.025, 0.05, 0.075, 0.1. If the average estimate for a given level of violation differs greatly from the effect estimated on the actual data, then the model is very sensitive to violations of the counterfactual consistency assumption for that level of violation. Next, this method tests the model's sensitivity to a violation of the exchangeability assumption by calculating the E-value, which is the minimum strength of association, on the risk ratio scale, that an unobserved confounder would need to have with the treatment and outcome variable to fully explain away the estimated effect. Thus, higher E-values imply the model is more robust to a violation of the exchangeability assumption. Finally, this method tests the positivity assumption by estimating propensity scores. Rows in the matrix are levels of covariates that have a zero or near zero probability of treatment. If the matrix is empty, none of the observations have an estimated zero probability of treatment, which implies the positivity assumption is satisfied.
One can also specify the maxium number of possible treatments to consider for the causal consistency assumption and the minimum and maximum probabilities of treatment for the positivity assumption with the num_treatments, min, and max keyword arguments.
Obtaining correct estimates is dependent on meeting the assumptions for interrupted time series estimation. If the assumptions are not met then any estimates may be biased and lead to incorrect conclusions.
For a thorough review of casual inference assumptions see:
Hernan, Miguel A., and James M. Robins. Causal inference what if. Boca Raton: Taylor and
Francis, 2024.For more information on the E-value test see:
VanderWeele, Tyler J., and Peng Ding. "Sensitivity analysis in observational research:
introducing the E-value." Annals of internal medicine 167, no. 4 (2017): 268-274.validate(s_learner)
validate(t_learner)
validate(x_learner)
validate(r_learner)
validate(dr_learner)