Extreme Value Bounds with Double Sampling

Bounds the average treatment effect when some outcomes are missing and a random sample of the initial nonrespondents was pursued a second time. Because that sample was drawn at random, the outcomes it recovers estimate the mean outcome among all nonrespondents, so only the subjects who refused twice need worst-case treatment and the identification region narrows accordingly. Reports the region with a joint Imbens-Manski confidence interval.

Usage

estimator_ds(
  Y,
  Z,
  R1,
  Attempt,
  R2,
  minY,
  maxY,
  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). 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.

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

Manski, Charles F. (1990). Nonparametric Bounds on Treatment Effects. American Economic Review Papers and Proceedings 80(2):319-323.

Neyman, Jerzy (1938). Contribution to the Theory of Sampling Human Populations. Journal of the American Statistical Association 33(201):101-116. doi:10.1080/01621459.1938.10503378

Hansen, Morris H., and William N. Hurwitz (1946). The Problem of Non-Response in Sample Surveys. Journal of the American Statistical Association 41(236):517-529. doi:10.1080/01621459.1946.10501894

Miratrix, Luke W., Jasjeet S. Sekhon, and Bin Yu (2013). Adjusting Treatment Effect Estimates by Post-Stratification in Randomized Experiments. Journal of the Royal Statistical Society, Series B 75(2):369-396. doi:10.1111/j.1467-9868.2012.01048.x

Examples

set.seed(343) # For reproducibility
N <- 1000

# Potential Outcomes
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))

R1_0 <- rbinom(N, 1, prob = 0.7)
R1_1 <- rbinom(N, 1, prob = 0.8)

R2_0 <- rbinom(N, 1, prob = 0.9)
R2_1 <- rbinom(N, 1, prob = 0.95)

# Covariate
strata <- as.numeric(Y_0 > 2)

# Random Assignment
Z <- rbinom(N, 1, .5)

# Reveal Initial Sample Outcomes
R1 <- Z*R1_1 + (1-Z)*R1_0 # Initial sample response
Y_star <- Z*Y_1 + (1-Z)*Y_0 # True outcomes
Y <- Y_star
Y[R1==0] <- NA # Mask outcome of non-responders

# Conduct Double Sampling
Attempt <- rep(0, N)
Attempt[is.na(Y)] <- rbinom(sum(is.na(Y)), 1, .5)

R2 <- rep(0, N)
R2[Attempt==1] <-  (Z*R2_1 + (1-Z)*R2_0)[Attempt==1]

Y[R2==1 & Attempt==1] <- Y_star[R2==1 & Attempt==1]

df <- data.frame(Y, Z, R1, Attempt, R2, strata)

# Without post-stratification
estimator_ds(Y, Z, R1, Attempt, R2, minY=1, maxY=5, data=df)
#>  estimate_lower  estimate_upper std.error_lower std.error_upper        conf.low 
#>      0.18156835      0.30409948      0.08044860      0.08056503      0.04865040 
#>       conf.high 
#>      0.43720979 

# With post-stratification
estimator_ds(Y, Z, R1, Attempt, R2, minY=1, maxY=5, strata=strata, data=df)
#>  estimate_lower  estimate_upper std.error_lower std.error_upper        conf.low 
#>      0.24698753      0.37580136      0.06847708      0.06507285      0.13421353 
#>       conf.high 
#>      0.48296897