G-Computation
In some cases, we may want to know the causal effect of a treatment that varies and is confounded over time. For example, a doctor might want to know the effect of a treatment given at multiple times whose status depends on the health of the patient at a given time. One way to get an unbiased estimate of the causal effect is to use G-computation. The basic steps for using G-computation in CausalELM are below.
For a good overview of G-Computation see:
Chatton, Arthur, Florent Le Borgne, Clémence Leyrat, Florence Gillaizeau, Chloé
Rousseau, Laetitia Barbin, David Laplaud, Maxime Léger, Bruno Giraudeau, and Yohann
Foucher. "G-computation, propensity score-based methods, and targeted maximum likelihood
estimator for causal inference with different covariates sets: a comparative simulation
study." Scientific reports 10, no. 1 (2020): 9219.Step 1: Initialize a Model
The GComputation constructor takes at least three arguments: covariates, treatment statuses, outcomes, all of which can be either an AbstractArray or any data structure that implements the Tables.jl interface (e.g. DataFrames). This implementation supports binary treatments and binary, continuous, time to event, and count outcome variables.
Non-binary categorical outcomes are treated as continuous.
You can also specify the causal estimand, which activation function to use, whether the data is of a temporal nature, the number of extreme learning machines to use, the number of features to consider for each extreme learning machine, the number of bootstrapped observations to include in each extreme learning machine, and the number of neurons to use during estimation. These options are specified with the following keyword arguments: quantity_of_interest, activation, temporal, num_machines, num_feats, sample_size, and num_neurons.
# Create some data with a binary treatment
X, T, Y = rand(1000, 5), [rand()<0.4 for i in 1:1000], rand(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))
g_computer = GComputation(X, T, Y)Step 2: Estimate the Causal Effect
To estimate the causal effect, we pass the model above to estimatecausaleffect!.
# Note that we could also estimate the ATT by setting quantity_of_interest="ATT"
estimate_causal_effect!(g_computer)Step 3: 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, standard error, and confidence interval for the treatment effect, you can set the inference keyword to true. However, these values are calculated via randomization inference, which will take a long time. This can be greatly sped up by launching Julia with more threads and lowering the number of iterations using the n keyword (at the expense of accuracy).
summarize(g_computer)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 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 G-computation. 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)