For a finite instrument \(Z\), treatment \(X\) and outcome \(Y\) with a
latent confounder independent of \(Z\), the observed law obeys
\(\sum_y \max_z P(y, x \mid z) \le 1\) for every \(x\)
(Research.P8.IVInequality.instrumental_inequality). A table that violates
it refutes the instrument.
Usage
morie_instrumental_inequality(p, tol = 1e-08)
Arguments
- p
A three-way array of \(P(y, x \mid z)\) with dimensions
(y, x, z); every z-slice sums to one.
- tol
Tolerance for the slice sums and the test.
Value
A list: per_x (the sums \(\sum_y \max_z P(y,x|z)\)),
max and passes.
References
Pearl J (2009). Causality, 2nd ed., section 8.4. Cambridge.
Examples
p <- array(c(0.4, 0.1, 0.3, 0.2, 0.2, 0.3, 0.1, 0.4), dim = c(2, 2, 2))
morie_instrumental_inequality(p)
#> $per_x
#> [1] 0.7 0.7
#>
#> $max
#> [1] 0.7
#>
#> $passes
#> [1] TRUE
#>
#> attr(,"theorem")
#> [1] "Research.P8.IVInequality.instrumental_inequality"
