IPW average treatment effect on the Crump-trimmed overlap sample

Discards units with estimated propensity score outside the interval from alpha to 1 - alpha (Crump, Hotz, Imbens and Mitnik 2009; rule-of-thumb cutoff 0.1) and estimates the ATE on the retained sample by the ratio (Hajek) IPW form \(\tau = \frac{\sum T Y / e}{\sum T / e} - \frac{\sum (1-T) Y / (1-e)}{\sum (1-T)/(1-e)}\). With alpha = NULL the optimal cutoff of their Corollary 5.1 (homoskedastic case) is computed from the scores: alpha = 0 when \(\sup 1/(e(1-e)) \le 2 E[1/(e(1-e))]\), otherwise \(1/(\alpha(1-\alpha)) = 2 E[1/(e(1-e)) \mid 1/(e(1-e)) \le 1/(\alpha(1-\alpha))]\). No analytic standard error is reported; inference on the trimmed estimand is left to the caller.

Usage

Causipw(treat, y, ps, alpha = 0.1)

Arguments

treat

Binary treatment indicator, 0/1.

y

Outcome.

ps

Estimated propensity scores strictly inside (0, 1).

alpha

Trimming cutoff in 0 to 0.5 (default 0.1), or NULL for the Corollary 5.1 optimal cutoff.

Value

List with estimate, alpha, n, n_kept, n_treat_kept, n_control_kept, mean_treated, mean_control, method.

References

Crump, R. K., Hotz, V. J., Imbens, G. W. and Mitnik, O. A. (2009), Dealing with limited overlap in estimation of average treatment effects, Biometrika 96(1), 187-199, doi:10.1093/biomet/asn055; Corollary 5.1 and Section 5 in the NBER TWP 330 (2006) version.

Examples

set.seed(1)
Causipw(treat = rbinom(50, 1, 0.5), y = rnorm(50), ps = runif(50, 0.2, 0.8))
#> $estimate
#> [1] -0.05196861
#> 
#> $alpha
#> [1] 0.1
#> 
#> $n
#> [1] 50
#> 
#> $n_kept
#> [1] 50
#> 
#> $n_treat_kept
#> [1] 27
#> 
#> $n_control_kept
#> [1] 23
#> 
#> $mean_treated
#> [1] 0.1501948
#> 
#> $mean_control
#> [1] 0.2021634
#> 
#> $method
#> [1] "Crump et al. (2009) overlap trimming + Hajek IPW"
#>