Regression to the mean at selected hot spots

Places selected because their first-period count exceeds threshold show, under the null of exchangeable periods, a non-positive mean change in the second period (Research.P19.selected_change_nonpos); places selected for being low show a non-negative one (low_selected_change_nonneg). The function reports the observed change on the selected places, the mirror change (selection on the second period, read backwards), and the symmetrised change, which is the same statistic on the exchangeable population made of the data and its period-swapped copy, and is therefore non-positive by the theorem. The difference between the observed and the symmetrised change is the part of the drop that selection symmetry does not deliver; a treatment effect is identified only against a control group selected the same way.

Usage

morie_regression_to_mean(x1, x2, threshold, weights = NULL)

Arguments

x1, x2

Counts per place in the first and second period.

threshold

Selection cut: places with x1 > threshold are the hot spots.

weights

Optional non-negative weights.

Value

A list with n_selected, selected_change (mean of \(x_2 - x_1\) on the selected places), mirror_change, symmetrised_change (always \(\le 0\)), excess_over_symmetry, low_selected_change (places with x1 < threshold, always \(\ge 0\) after symmetrisation: low_symmetrised_change) and theorems.

Examples

set.seed(5)
mu <- rgamma(300, 2, 0.5)                   # stable place means
x1 <- rpois(300, mu); x2 <- rpois(300, mu)  # no treatment, no trend
morie_regression_to_mean(x1, x2, threshold = quantile(x1, 0.9))[c("selected_change", "symmetrised_change")]
#> $selected_change
#> [1] -2.555556
#> 
#> $symmetrised_change
#> [1] -2.211538
#>