Network attributable fraction with spillover

Under interference an individual's outcome depends on the exposure of the people they are connected to, so the ordinary attributable fraction – which assumes each unit answers only to its own exposure – understates what removing the exposure would do. Following the direct/indirect decomposition of Halloran & Hudgens, the neighbourhood exposure of unit i is the fraction of its neighbours exposed, nu_i = sum_j A_ij e_j / sum_j A_ij (zero if isolated), and the outcome is regressed additively on both channels, E[Y_i] = b0 + b1 e_i + b2 nu_i.

Usage

Netparf(y, exposure, network)

Arguments

y

Outcome per unit, length n.

exposure

Own exposure per unit, typically 0/1.

network

Non-negative n by n adjacency; the diagonal is ignored, so a unit is not its own neighbour.

Value

List with estimate, paf, paf_direct, paf_spillover, b0, b1, b2, mean_y, mean_exposure, mean_nu, n.

Details

Formula: PAF = (mean(Y) - b0) / mean(Y), which splits exactly into b1 mean(e) / mean(Y) and b2 mean(nu) / mean(Y). With b2 = 0 it collapses to the ordinary attributable fraction.

References

Halloran, M. E. & Hudgens, M. G. (2016). Dependent happenings: a recent methodological review. Current Epidemiology Reports, 3(4), 297-305. doi:10.1007/s40471-016-0086-4

Examples

set.seed(19)
n <- 100
net <- matrix(rbinom(n * n, 1, 0.05), n); net <- (net + t(net) > 0) * 1
e <- rbinom(n, 1, 0.3)
y <- 1 + 2 * e + 0.5 * as.numeric(net %*% e) / pmax(rowSums(net), 1) + rnorm(n)
Netparf(y, e, net)[c("paf_direct", "paf_spillover")]
#> $paf_direct
#> [1] 0.4139323
#> 
#> $paf_spillover
#> [1] -0.1701741
#>