Balke-Pearl sharp bounds on the average causal effect under an instrument

Sharp (tight) bounds on the average causal effect of D on Y when the assignment Z is a valid instrument and compliance is imperfect. All three variables are binary. Write p_yd.z = P(Y = y, D = d | Z = z), eight numbers with sum_{y,d} p_yd.z = 1 for each z. Balke & Pearl solve the linear program over the sixteen response-type probabilities by enumerating the vertices of the dual constraint polytope, giving a closed form. Their eq. (4), p. 1173, is the lower bound, the maximum of p00.0 + p11.1 - 1, p00.1 + p11.1 - 1, p11.0 + p00.1 - 1, p00.0 + p11.0 - 1, 2 p00.0 + p11.0 + p10.1 + p11.1 - 2, p00.0 + 2 p11.0 + p00.1 + p01.1 - 2, p10.0 + p11.0 + 2 p00.1 + p11.1 - 2, p00.0 + p01.0 + p00.1 + 2 p11.1 - 2; their eq. (5) is the upper bound, the minimum of 1 - p10.0 - p01.1, 1 - p01.0 - p10.1, 1 - p01.0 - p10.0, 1 - p01.1 - p10.1, 2 - 2 p01.0 - p10.0 - p10.1 - p11.1, 2 - p01.0 - 2 p10.0 - p00.1 - p01.1, 2 - p10.0 - p11.0 - 2 p01.1 - p10.1, 2 - p00.0 - p01.0 - p01.1 - 2 p10.1.

Usage

Sfbnds(y, D, Z)

Arguments

y: Binary outcome, one entry per unit.
D: Binary treatment actually received.
Z: Binary instrument (assignment).

Value

List with estimate (interval midpoint), lower, upper, width, lower_manski, upper_manski, width_manski, noncompliance, itt, compliance_gap, late, excludes_zero, the eight p cells, n, n_z0, n_z1, method.

Details

The first four entries of each set are the Robins-Manski bounds; the last four are what makes the Balke-Pearl interval strictly narrower. The width cannot exceed the rate of noncompliance P(d1|z0) + P(d0|z1), which is reported so the property can be checked.

Both equations were read from a rendered image of p. 1173 of the JASA printing; that PDF is a scan with no text layer, so no minus sign passed through a text extractor.

References

Balke, A. & Pearl, J. (1997). Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association 92(439), 1171-1176, eqs. (4)-(5) p. 1173. doi:10.1080/01621459.1997.10474074

Examples

set.seed(1)
Sfbnds(y = rbinom(40, 1, 0.5), D = rbinom(40, 1, 0.5), Z = rbinom(40, 1, 0.5))
#> $estimate
#> [1] 0.01262626
#> 
#> $lower
#> [1] -0.3636364
#> 
#> $upper
#> [1] 0.3888889
#> 
#> $width
#> [1] 0.7525253
#> 
#> $lower_manski
#> [1] -0.3636364
#> 
#> $upper_manski
#> [1] 0.3888889
#> 
#> $width_manski
#> [1] 0.7525253
#> 
#> $noncompliance
#> [1] 1
#> 
#> $itt
#> [1] -0.05555556
#> 
#> $compliance_gap
#> [1] 0
#> 
#> $late
#> [1] NA
#> 
#> $excludes_zero
#> [1] 0
#> 
#> $p00_0
#> [1] 0.3181818
#> 
#> $p01_0
#> [1] 0.1818182
#> 
#> $p10_0
#> [1] 0.1818182
#> 
#> $p11_0
#> [1] 0.3181818
#> 
#> $p00_1
#> [1] 0.2222222
#> 
#> $p01_1
#> [1] 0.3333333
#> 
#> $p10_1
#> [1] 0.2777778
#> 
#> $p11_1
#> [1] 0.1666667
#> 
#> $n
#> [1] 40
#> 
#> $n_z0
#> [1] 22
#> 
#> $n_z1
#> [1] 18
#> 
#> $method
#> [1] "Balke-Pearl sharp bounds on the ACE (Balke & Pearl 1997, eqs. 4-5)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"