IPW sensitivity analysis for a non-ignorable selection mechanism

The Robins-Rotnitzky-Scharfstein selection-bias function q(y) indexes departures from missing at random. With the exponential tilt q(y) = lambda y the observed-data weights become C exp(-lambda Y) / pi(X), and the resulting Hajek mean traces out a sensitivity curve. lambda = 0 is exactly the MAR inverse-probability-weighted estimator; positive lambda encodes "large Y are less likely to be observed".

Usage

IpwSn(Y, X, C, lam_grid)

Arguments

Y

Outcome; entries with C = 0 are unused.

X

Covariate block for the selection model; intercept added.

C

Selection indicator, 0 or 1.

lam_grid

Values of the selection-bias parameter.

Value

List with estimate (mu at lambda 0), mu, lambda, mu_min, mu_max, range, propensity, gamma, n_observed, n, method.

Details

Formula: mu(lam) = sum C exp(-lam Y) Y / pi / sum C exp(-lam Y) / pi.

References

Robins, Rotnitzky and Scharfstein (2000), Sensitivity analysis for selection bias and unmeasured confounding, IMA Volumes in Mathematics and its Applications 116, Springer, pp. 1-94, doi:10.1007/978-1-4612-1284-3_1 ; Scharfstein, Rotnitzky and Robins (1999), JASA 94(448):1096-1120. doi:10.1080/01621459.1999.10473862

Examples

set.seed(1)
IpwSn(Y = rnorm(20), X = matrix(rnorm(40), 20, 2), C = rbinom(20, 1, 0.5),
      lam_grid = c(0, 0.5, 1))
#> $estimate
#> [1] 0.238061
#> 
#> $mu
#> [1]  0.23806103 -0.03173159 -0.26154496
#> 
#> $lambda
#> [1] 0.0 0.5 1.0
#> 
#> $mu_min
#> [1] -0.261545
#> 
#> $mu_max
#> [1] 0.238061
#> 
#> $range
#> [1] 0.499606
#> 
#> $propensity
#>  [1] 0.6391507 0.4935464 0.5712183 0.4734203 0.4968432 0.5888540 0.4841266
#>  [8] 0.4769124 0.6294605 0.5759148 0.6240853 0.5090481 0.6077562 0.5689145
#> [15] 0.4463575 0.4873697 0.4721528 0.6456007 0.5277706 0.6814972
#> 
#> $gamma
#> [1] 0.1149724 0.2826471 0.1825664
#> 
#> $n_observed
#> [1] 11
#> 
#> $n
#> [1] 20
#> 
#> $method
#> [1] "mu(lam) = sum C exp(-lam Y) Y / pi / sum C exp(-lam Y) / pi, Robins, Rotnitzky & Scharfstein (2000)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"