Horvitz-Thompson totals and contrasts under a randomised deployment

When the deployment was randomised, every place's exposure probability \(\pi_i(\ell) = P(E_i = \ell)\) is known from the design (or can be computed by re-drawing assignments, see morie_spillover_exposure_probs). The estimator \(\hat T(\ell) = \sum_i 1\{E_i=\ell\}\, y_i / \pi_i(\ell)\) is unbiased for the population total of \(Y(\ell)\) whenever every \(\pi_i(\ell) > 0\), and so is any contrast between levels (Research.P3.Design.ht_unbiased, Research.P3.Design.ht_contrast_unbiased). No ignorability assumption is used: the design supplies it.

Usage

morie_spillover_ht(y, exposure, probs, joint = NULL)

Arguments

y: Observed outcome per place.
exposure: Realised exposure per place (0, 1, 2).
probs: Matrix with one row per place and columns "0", "1", "2" giving \(\pi_i(\ell)\); all entries positive.
joint: Optional n by n by 3 array of joint exposure probabilities \(\pi_{ij}(\ell)\), as returned in the joint element of morie_spillover_exposure_probs(..., joint = TRUE).

Value

A list with totals (\(\hat T(0), \hat T(1), \hat T(2)\)), means (totals divided by the number of places), spillover, direct, total_effect (per-place means), and when joint is given variance and se of the three totals (NA where a joint probability is zero), and theorems.

Details

When the joint exposure probabilities \(\pi_{ij}(\ell) = P(E_i = \ell, E_j = \ell)\) are supplied, the variance of each total is estimated by the Horvitz-Thompson variance estimator \(\hat V(\ell) = \sum_{i,j} 1\{E_i = E_j = \ell\}\,(\pi_{ij} - \pi_i\pi_j)/\pi_{ij}\; y_i y_j/(\pi_i \pi_j)\), which is unbiased for \(\mathrm{Var}\,\hat T(\ell) = \sum_{i,j} (\pi_{ij} - \pi_i\pi_j)\, Y_i(\ell) Y_j(\ell)/(\pi_i\pi_j)\) whenever every \(\pi_{ij}(\ell) > 0\) (Research.P3.Design.ht_variance, Research.P3.Design.ht_variance_estimator_unbiased). A pair of places that can never share a level leaves that variance unidentified, and the function then reports NA for it rather than a number.

Examples

set.seed(2)
edges <- cbind(1:19, 2:20)
pr <- morie_spillover_exposure_probs(n = 20, edges, n_treated = 6, n_draws = 2000, joint = TRUE)
trt <- sample(20, 6); exposure <- morie_spillover_exposure(seq_len(20) %in% trt, edges)$exposure
y <- 5 - 1.5 * (exposure == 2) - 0.5 * (exposure == 1)
morie_spillover_ht(y, exposure, pr$marginal,
                   pr$joint)[c("spillover", "direct", "total_effect", "se")]
#> $spillover
#> [1] 1.841628
#> 
#> $direct
#> [1] -2.226164
#> 
#> $total_effect
#> [1] -0.3845363
#> 
#> $se
#>        T0        T1        T2 
#> 12.328835 36.572164  7.322717 
#>