Monotone-treatment-selection bounds for a sentencing contrast

Monotone treatment selection (Manski & Pepper 2000) says that the people who received the harsher sentence are, under every sentence, at least as likely to show the outcome as the people who received the lighter one: the judge selected on risk in the direction of severity. Then the observed mean of the harsher group is an upper bound on \(E[y(b)]\), the observed mean of the lighter group a lower bound on \(E[y(a)]\) (Research.P11.Pop.mts_mean_b_le, mts_mean_a_ge), and the naive difference of observed means overstates the effect of the harsher sentence (mts_ate_le_naive). MTS alone gives no sign. Adding monotone treatment response (morie_sentence_effect_mtr) gives the interval \([0, \text{naive difference}]\) (mtr_mts_bounds): the assumption-free interval of width one, cut down to the observed gap by two named monotonicity assumptions, each stated where it enters.

Usage

morie_sentence_effect_mts(y, z, weights = NULL, contrast = NULL)

Arguments

y

Binary outcome per person (0/1, e.g. reconviction within two years).

z

Sentence per person, a vector with exactly two distinct values.

weights

Optional non-negative weights.

contrast

Which of the two sentence values is the treatment (default the second in sorted order); the other is the comparison.

Value

A list with levels, observed_means, pz, mean_b_bounds, mean_a_bounds (under MTS alone), ate_bounds_mts (MTS alone: lower end from the worst case, upper end the naive difference), ate_bounds_mtr_mts (both assumptions), naive_difference and theorems.

Examples

set.seed(1)
z <- sample(c("community", "custody"), 500, replace = TRUE)
y <- rbinom(500, 1, ifelse(z == "custody", 0.55, 0.35))
morie_sentence_effect_mts(y, z)[c("ate_bounds_mts", "ate_bounds_mtr_mts")]
#> $ate_bounds_mts
#>      lower      upper 
#> -0.4000000  0.1985784 
#> 
#> $ate_bounds_mtr_mts
#>     lower     upper 
#> 0.0000000 0.1985784 
#>