With \(p_{11} = P(A=1|Z=1)\), \(p_{10} = P(A=1|Z=0)\),
\(m_1 = E[Y|A=1,Z=1]\), \(m_0 = E[Y|A=0,Z=0]\) and a bounded outcome,
the average effect lies in an interval whose ends are attained and whose
width is \(P(A=0|Z=1) + P(A=1|Z=0)\)
(Research.P8.NaturalBounds.width).
Usage
morie_iv_natural_bounds(p11, p10, m1, m0)
Arguments
- p11, p10
Treatment probabilities under the two instrument values.
- m1, m0
Observed outcome means among the treated under \(Z=1\) and the
untreated under \(Z=0\), in \([0, 1]\).
Value
A list: lower, upper, width.
References
Pearl J (2009). Causality, 2nd ed., section 8.2. Cambridge.
Examples
morie_iv_natural_bounds(p11 = 0.8, p10 = 0.3, m1 = 0.6, m0 = 0.4)
#> $lower
#> [1] -0.1
#>
#> $upper
#> [1] 0.4
#>
#> $width
#> [1] 0.5
#>
#> attr(,"theorem")
#> [1] "Research.P8.NaturalBounds.width"
