# ACE_bounds: Bounds for the Average Causal Effect (ACE). In noncompliance: Causal Inference in the Presence of Treatment Noncompliance Under the Binary Instrumental Variable Model

## Description

The empirical bounds for the Average Causal Effect (ACE), under the assumptions of the Instrumental Variable (IV) model.

## Usage

 ```1 2``` ```ACE_bounds(n_y0x0z0, n_y1x0z0 = NA, n_y0x1z0 = NA, n_y1x1z0 = NA, n_y0x0z1 = NA, n_y1x0z1 = NA, n_y0x1z1 = NA, n_y1x1z1 = NA) ```

## Arguments

 `n_y0x0z0` Number of individuals with Y=0, X=0, Z=0. Alternatively, a vector with elements (either counts, p(y, x , z) or p(y, x | z)) in the order of the arguments. `n_y1x0z0` Number of individuals with Y=1, X=0, Z=0. `n_y0x1z0` Number of individuals with Y=0, X=1, Z=0. `n_y1x1z0` Number of individuals with Y=1, X=1, Z=0. `n_y0x0z1` Number of individuals with Y=0, X=0, Z=1. `n_y1x0z1` Number of individuals with Y=1, X=0, Z=1. `n_y0x1z1` Number of individuals with Y=0, X=1, Z=1. `n_y1x1z1` Number of individuals with Y=1, X=1, Z=1.

## Value

The empirical bounds for the ACE.

## References

Richardson, T. S.; Robins, J. M. (2014). ACE Bounds; SEMs with Equilibrium Conditions. Statist. Sci. 29, no. 3, 363-366..

## Examples

 ```1 2 3 4``` ```ACE_bounds(158, 14, 0, 0, 52, 12, 23, 78) ACE_bounds(c(158, 14, 0, 0, 52, 12, 23, 78)) ACE_bounds(99, 1027, 30, 233, 84, 935, 31, 422) ACE_bounds(c(99, 1027, 30, 233, 84, 935, 31, 422)) ```

### Example output

```[1] 0.3913319 0.7792107
[1] 0.3913319 0.7792107
[1] -0.6420409  0.2390209
[1] -0.6420409  0.2390209
```

noncompliance documentation built on May 2, 2019, 2:38 a.m.