Bounded two-arm confidence rectangle

Construct a two-arm variance confidence rectangle for bounded outcomes using Maurer–Pontil or Martinez-Taboada–Ramdas one-arm bounds.

Usage

rectangle_bounded_two_arm(
  y,
  d,
  alpha = 0.05,
  method = c("bounded", "maurer_pontil", "mp", "martinez_taboada_ramdas", "mtr"),
  beta = NULL,
  correction = c("bonferroni", "sidak_arms"),
  normalize = FALSE,
  lower = NULL,
  upper = NULL,
  na.rm = TRUE
)

rectangle_bounded_binary(
  y,
  d,
  alpha = 0.05,
  method = c("bounded", "maurer_pontil", "mp", "martinez_taboada_ramdas", "mtr"),
  beta = NULL,
  correction = c("bonferroni", "sidak_arms"),
  normalize = FALSE,
  lower = NULL,
  upper = NULL,
  na.rm = TRUE
)

Arguments

y

Pilot outcomes.

d

Pilot treatment indicator; treatment is 1 and control is 0.

alpha

Target joint error level.

method

Bounded-outcome method. "bounded", "maurer_pontil", and "mp" are synonyms; "martinez_taboada_ramdas" and "mtr" use MTR bounds.

beta

Optional endpoint error allocation. If NULL, error is split according to correction.

correction

Endpoint error correction, either "bonferroni" or "sidak_arms".

normalize

If TRUE, normalize outcomes to [0, 1] before computing variances. For guarantee-bearing bounded CMR on a non-unit scale, provide known lower and upper bounds.

lower, upper

Optional lower and upper outcome bounds used when normalize = TRUE. If either is omitted, the pilot minimum and/or maximum is used with a warning; that data-dependent normalization is exploratory and does not carry the finite-sample bounded-outcome CMR guarantee.

na.rm

If TRUE, drop rows with missing y or d.

Value

A cmr_binary_rectangle list with rectangle, one-arm bound objects for treatment and control, endpoint error allocation, sample sizes, pilot variance estimates, normalization details, and method metadata. For Maurer–Pontil bounds, the top-level vhat entries are raw Bessel sample variances and can exceed 0.25; the projected unit-interval values are in treatment$statistic$projected_vhat and control$statistic$projected_vhat.

See also

Other rectangle helpers: binary_rectangle_corners(), cmr_multiarm_from_rectangle(), cmr_stratified_from_rectangle(), cmr_unbounded_from_rectangle(), folded_binomial_pmf(), multiarm_variance_objective(), rectangle_bernoulli_binary(), rectangle_multiarm(), rectangle_multiple_outcomes(), rectangle_proxy(), rectangle_stratified(), rectangle_unbounded(), stratified_variance_objective(), variance_bounds_bernoulli_exact(), variance_bounds_maurer_pontil(), variance_bounds_unbounded_mom()

Examples

d <- rep(c(1, 0), each = 5)
y <- c(0.20, 0.40, 0.30, 0.10, 0.60, 0.50, 0.30, 0.20, 0.40, 0.10)
rectangle_bounded_binary(y, d, method = "bounded")
#> $rectangle
#> v_l1 v_u1 v_l0 v_u0 
#> 0.00 0.25 0.00 0.25 
#> 
#> $treatment
#> $treatment$L
#> [1] 0
#> 
#> $treatment$U
#> [1] 0.25
#> 
#> $treatment$vhat
#> [1] 0.037
#> 
#> $treatment$method
#> [1] "bounded"
#> 
#> $treatment$n
#> [1] 5
#> 
#> $treatment$statistic
#> $treatment$statistic$vhat
#> [1] 0.037
#> 
#> $treatment$statistic$projected_vhat
#> [1] 0.037
#> 
#> $treatment$statistic$sdhat
#> [1] 0.1923538
#> 
#> $treatment$statistic$beta_l
#> [1] 0.0125
#> 
#> $treatment$statistic$beta_u
#> [1] 0.0125
#> 
#> 
#> 
#> $control
#> $control$L
#> [1] 0
#> 
#> $control$U
#> [1] 0.25
#> 
#> $control$vhat
#> [1] 0.025
#> 
#> $control$method
#> [1] "bounded"
#> 
#> $control$n
#> [1] 5
#> 
#> $control$statistic
#> $control$statistic$vhat
#> [1] 0.025
#> 
#> $control$statistic$projected_vhat
#> [1] 0.025
#> 
#> $control$statistic$sdhat
#> [1] 0.1581139
#> 
#> $control$statistic$beta_l
#> [1] 0.0125
#> 
#> $control$statistic$beta_u
#> [1] 0.0125
#> 
#> 
#> 
#> $alpha
#> [1] 0.05
#> 
#> $beta
#> beta_l1 beta_u1 beta_l0 beta_u0 
#>  0.0125  0.0125  0.0125  0.0125 
#> 
#> $correction
#> [1] "bonferroni"
#> 
#> $joint_error_bound
#> [1] 0.05
#> 
#> $method
#> [1] "bounded"
#> 
#> $n
#> n1 n0 
#>  5  5 
#> 
#> $vhat
#> vhat1 vhat0 
#> 0.037 0.025 
#> 
#> $normalization
#> NULL
#> 
#> attr(,"class")
#> [1] "cmr_binary_rectangle" "list"