Bounds the average treatment effect when some outcomes are missing and nothing is assumed about why. Filling every missing outcome in the treatment group with the lowest value the outcome can take and every missing outcome in the control group with the highest gives the smallest average effect the data can support; reversing the fills gives the largest. Reports the resulting identification region with a joint Imbens-Manski confidence interval.
Arguments
- Y
The (unquoted) outcome variable, or a formula
outcome ~ treatmentfor use withdeclare_estimator(.method = estimator_ev). Must be numeric.- Z
The (unquoted) assignment indicator variable. Must be numeric and take values 0 or 1. Ignored when
Yis a formula.- R
The response indicator variable: unquoted column name, or a quoted string column name when using the formula interface. 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:
datais 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
Manski, Charles F. (1990). Nonparametric Bounds on Treatment Effects. American Economic Review Papers and Proceedings 80(2):319-323.
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
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
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
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
# Treated units respond at a higher rate, so the missingness is nonignorable
R <- rbinom(N, 1, prob = 0.7 + 0.1 * Z)
Y <- Y_star
Y[R == 0] <- NA
df <- data.frame(Y, Z, R)
estimator_ev(Y, Z, R, minY = 1, maxY = 5, data = df)
#> estimate_lower estimate_upper std.error_lower std.error_upper conf.low
#> -0.78261705 1.24723513 0.08283104 0.07795614 -0.91886198
#> conf.high
#> 1.37546156
# Equivalently, via the formula interface
estimator_ev(Y ~ Z, R = "R", minY = 1, maxY = 5, data = df)
#> estimate_lower estimate_upper std.error_lower std.error_upper conf.low
#> -0.78261705 1.24723513 0.08283104 0.07795614 -0.91886198
#> conf.high
#> 1.37546156