Estimate the average causal effect (ACE) of a given treatment variable
*X* on a given outcome *Y* by adjusting for a set *S* of covariates (equivalenly, assuming
*S* blocks all backdoor paths in the causal graph that generates the observations). . Bounds are based on finding conditional instrumental variables
using the faithfulness assumption relaxed to allow for a moderate degree of unfaithfulness.
Candidate models are generated from the method described in `covsearch`

.

1 | ```
bindagCausalEffectBackdoor(problem, prior_table, S)
``` |

`problem` |
a |

`prior_table` |
effective sample size hyperparameter of a Dirichlet prior for testing independence with contingency tables. |

`S` |
an array indicating the indices of the covariates used in the adjustment. |

The algorithm implemented is a naive one. If the dimensionality of *S* is larger than around 20, this may crash the system.
This function is assumed to be used along other methods such as `covsearch`

and `wpp`

, which generate
relativelty small covariate sets.

the estimated ACE.

Pearl, J. (2000) *Causality: Models, Reasoning and Inference*. Cambridge University Press.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
## Generate a synthetic problem
problem <- simulateWitnessModel(p = 4, q = 4, par_max = 3, M = 1000)
## Idealized case: suppose we know the true distribution,
## get "exact" ACE estimands for different adjustment sets
sol_pop <- covsearch(problem, pop_solve = TRUE)
effect_pop <- synthetizeCausalEffect(problem)
cat(sprintf(
"ACE (true) = %1.2f\nACE (adjusting for all) = %1.2f\nACE (adjusting for nothing) = %1.2f\n",
effect_pop$effect_real, effect_pop$effect_naive, effect_pop$effect_naive2))
## Perform inference and report results
covariate_hat <- covsearch(problem, cred_calc = TRUE, M = 1000)
if (length(covariate_hat$witness) > 1) {
sol_ACE <- bindagCausalEffectBackdoor(problem, prior_table = 10, S = covariate_hat$Z[[1]])
cat(sprintf("Estimated ACE = %1.2f\n", sol_ACE))
}
``` |

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