Ring mis-specification bound for the spillover estimand

The Horvitz-Thompson spillover estimator is design-unbiased for the estimand written with the d-ring exposure map. If the outcome also moves by \(\gamma_i\) when unit \(i\)'s \((d+1)\)-ring is treated, the d-ring estimand misses \((1/N)\sum_i \gamma_i \pi_i\) in expectation, with \(\pi_i\) the outer-ring exposure share of unit \(i\) under the design (Research.P3.Ring.expected_gap_eq), and the miss is at most \(G \cdot \bar\pi\) when every \(|\gamma_i| \le G\) (Research.P3.Ring.expected_gap_le).

Usage

morie_spillover_ring_bound(outer_share, gamma_max, gamma = NULL)

Arguments

outer_share

Per-unit probability that the \((d+1)\)-ring is exposed under the design, for instance the exposed-class column of morie_spillover_exposure_probs run with the larger ring.

gamma_max

Bound \(G\) on the outer-ring effects.

gamma

Optional per-unit outer-ring effects; with them the expected gap itself is returned beside the bound.

Value

A list with mean_share, bound (\(G \bar\pi\)), expected_gap (NA without gamma) and theorems.

Examples

morie_spillover_ring_bound(c(0.1, 0.3, 0.2), gamma_max = 2)
#> $mean_share
#> [1] 0.2
#> 
#> $bound
#> [1] 0.4
#> 
#> $expected_gap
#> [1] NA
#> 
#> $theorems
#> [1] "Research.P3.Ring.expected_gap_eq" "Research.P3.Ring.expected_gap_le"
#>