Worst-case identification bounds for a binary outcome under two sentences

Each person receives one sentence and reveals only the outcome under it. With no assumption on how sentences were assigned, the probability of the outcome under sentence t is identified only up to \([P(y=1, z=t),\; P(y=1, z=t) + P(z \ne t)]\), both ends attainable (Research.P11.Pop.outcome_bounds); the contrast between the two sentences lies in an interval of width exactly one that always contains zero (ate_width_one, ate_contains_zero). The interval is the honest report of an observational sentencing comparison before any selection assumption is added; assumptions narrow it, data alone cannot.

Usage

morie_sentence_effect_bounds(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, joint and pz per sentence, outcome_bounds (a 2 x 2 matrix, one row per sentence), ate_bounds, ate_width (always 1), naive_difference (the difference of observed rates, which the interval always contains) and theorems.

Examples

set.seed(1)
z <- sample(c("community", "custody"), 500, replace = TRUE, prob = c(0.6, 0.4))
y <- rbinom(500, 1, ifelse(z == "custody", 0.55, 0.35))
morie_sentence_effect_bounds(y, z)
#> $levels
#>  comparison   treatment 
#> "community"   "custody" 
#> 
#> $joint
#> community   custody 
#>     0.208     0.192 
#> 
#> $pz
#> community   custody 
#>     0.624     0.376 
#> 
#> $outcome_bounds
#>           lower upper
#> community 0.208 0.584
#> custody   0.192 0.816
#> 
#> $ate_bounds
#>  lower  upper 
#> -0.392  0.608 
#> 
#> $ate_width
#> [1] 1
#> 
#> $naive_difference
#> [1] 0.177305
#> 
#> $theorems
#> [1] "Research.P11.Pop.outcome_bounds"    "Research.P11.Pop.lower_attained"   
#> [3] "Research.P11.Pop.upper_attained"    "Research.P11.Pop.ate_width_one"    
#> [5] "Research.P11.Pop.ate_contains_zero"
#>