Extreme Value Bounds with Double Sampling with Sensitivity

Interpolates between worst-case bounds and ignorability. delta is the fraction of the follow-up nonrespondents whose outcomes are left unmodeled; the remaining 1 - delta are assumed to be drawn from a distribution with the mean and variance observed among the follow-up respondents. At delta = 1 the estimator reproduces estimator_ds, and at delta = 0 it returns a point estimate.

Usage

estimator_ds_sens(
  Y,
  Z,
  R1,
  Attempt,
  R2,
  minY,
  maxY,
  delta,
  strata = NULL,
  alpha = 0.05,
  data
)

Arguments

Y

The (unquoted) outcome variable, or a formula outcome ~ treatment for use with declare_estimator(.method = estimator_ds_sens). Must be numeric.

Z

The (unquoted) assignment indicator variable. Must be numeric and take values 0 or 1. Ignored when Y is a formula.

R1

The initial sample response indicator: unquoted column name, or a quoted string column name when using the formula interface. Must be numeric and take values 0 or 1.

Attempt

The follow-up attempt indicator: unquoted column name, or quoted string. Must be numeric and take values 0 or 1.

R2

The follow-up response indicator: unquoted column name, or quoted string. Must be numeric and take values 0 or 1.

minY

The minimum possible value of the outcome (Y) variable.

maxY

The maximum possible value of the outcome (Y) variable.

delta

Sensitivity parameter in [0, 1]. At delta = 1 worst-case bounds apply; at delta = 0 ignorability holds for all follow-up non-responders.

strata

Stratification variable: unquoted column name or a quoted string column name.

alpha

The desired significance level. 0.05 by default.

data

A dataframe. Must be given by name: data is the last argument, so passing it positionally assigns it to another argument.

Value

A named numeric vector with elements estimate_lower and estimate_upper, the two ends of the identification region; std.error_lower and std.error_upper, their standard errors; and conf.low and conf.high, the joint Imbens-Manski confidence interval. The names are those of the bounds row of tidy(), which returns the same quantities as a data frame.

References

Coppock, Alexander, Alan S. Gerber, Donald P. Green, and Holger L. Kern (2017). Combining Double Sampling and Bounds to Address Nonignorable Missing Outcomes in Randomized Experiments. Political Analysis 25(2):188-206. doi:10.1017/pan.2016.6

Imbens, Guido W., and Charles F. Manski (2004). Confidence Intervals for Partially Identified Parameters. Econometrica 72(6):1845-1857. doi:10.1111/j.1468-0262.2004.00555.x

Examples

set.seed(343)
N <- 1000
Y_0 <- sample(1:5, N, replace = TRUE, prob = c(0.1, 0.3, 0.3, 0.2, 0.1))
Y_1 <- sample(1:5, N, replace = TRUE, prob = c(0.1, 0.1, 0.4, 0.3, 0.1))
Z <- rbinom(N, 1, 0.5)
Y_star <- Z * Y_1 + (1 - Z) * Y_0
R1 <- rbinom(N, 1, prob = 0.7 + 0.1 * Z)
Y <- Y_star
Y[R1 == 0] <- NA

# Follow up intensively with a random half of the initial non-responders
Attempt <- rep(0, N)
Attempt[R1 == 0] <- rbinom(sum(R1 == 0), 1, 0.5)
R2 <- rep(0, N)
R2[Attempt == 1] <- rbinom(sum(Attempt == 1), 1, 0.9)
Y[Attempt == 1 & R2 == 1] <- Y_star[Attempt == 1 & R2 == 1]
df <- data.frame(Y, Z, R1, Attempt, R2)

# delta = 1 reproduces the worst-case double-sampling bounds
estimator_ds_sens(Y, Z, R1, Attempt, R2, minY = 1, maxY = 5, delta = 1, data = df)
#>  estimate_lower  estimate_upper std.error_lower std.error_upper        conf.low 
#>    0.1356982015    0.4100265693    0.0826967632    0.0811528054   -0.0003261488 
#>       conf.high 
#>    0.5435113299 

# delta = 0 assumes ignorability among follow-up non-responders
estimator_ds_sens(Y, Z, R1, Attempt, R2, minY = 1, maxY = 5, delta = 0, data = df)
#>  estimate_lower  estimate_upper std.error_lower std.error_upper        conf.low 
#>      0.28072912      0.28072912      0.07778958      0.07778958      0.12826435 
#>       conf.high 
#>      0.43319388