Double Machine Learning
Double machine learning, also called debiased or orthogonalized machine learning, enables estimating causal effects when the dimensionality of the covariates is too high for linear regression or the treatment or outcomes cannot be easily modeled parametrically. Double machine learning estimates models of the treatment assignment and outcome and then combines them in a final model. This is a semiparametric model in the sense that the first stage models can take on any functional form but the final stage model is a linear combination of the residuals from the first stage models.
For more information see:
Chernozhukov, Victor, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. "Double/debiased machine learning for treatment and structural parameters." (2018): C1-C68.
Step 1: Initialize a Model
The DoubleMachineLearning constructor takes at least three arguments—covariates, a treatment statuses, and outcomes, all of which may be either an array or any struct that implements the Tables.jl interface (e.g. DataFrames). This estimator supports binary, count, or continuous treatments and binary, count, continuous, 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.
# Create some data with a binary treatment
X, T, Y, W = rand(100, 5), [rand()<0.4 for i in 1:100], rand(100), rand(100, 4)
# We could also use DataFrames or any other package implementing the Tables.jl API
# using DataFrames
# X = DataFrame(x1=rand(100), x2=rand(100), x3=rand(100), x4=rand(100), x5=rand(100))
# T, Y = DataFrame(t=[rand()<0.4 for i in 1:100]), DataFrame(y=rand(100))
dml = DoubleMachineLearning(X, T, Y)Step 2: Estimate the Causal Effect
To estimate the causal effect, we call estimatecausaleffect! on the model above.
# we could also estimate the ATT by passing quantity_of_interest="ATT"
estimate_causal_effect!(dml)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.
# Can also use the British spelling
# summarise(dml)
summarize(dml)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 double machine learning. 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(g_computer)