Robustness values for ecological inference

The robustness value is the minimum bound for both c_outcome and c_predictor in ei_sens() such that the bias bound is a certain value. For example, if the provided bias bound is 0.5, meaning a bias of magnitude 0.5 would be considered problematic, then the robustness value is the minimum amount of confounding of outcome and predictor (more specifically, the Riesz representer) that would lead to bias of magnitude 0.5.

Usage

ei_sens_rv(est, bias_bound, confounding = 1)

Arguments

est: A set of estimates from ei_est() using both regression and Riesz representer.
bias_bound: <data-masking> A bias bound: an amount of bias which is considered substantial. Evaluated in the context of est, so that one can to refer to std.error and estimate as needed.
confounding: The confounding parameter (\(\rho\)), which must be between 0 and 1 (the adversarial worst-case).

Value

A data frame of the same format as est, but with a new rv column containing the robustness values.

References

Chernozhukov, V., Cinelli, C., Newey, W., Sharma, A., & Syrgkanis, V. (2024). Long story short: Omitted variable bias in causal machine learning (No. w30302). National Bureau of Economic Research.

Examples

data(elec_1968)

spec = ei_spec(elec_1968, vap_white:vap_other, pres_ind_wal,
               total = pres_total, covariates = c(state, pop_urban, farm))
m = ei_ridge(spec)
rr = ei_riesz(spec, penalty = m$penalty)
est = ei_est(m, rr, spec)

ei_sens_rv(est, 0.1) # how much confounding for bias of 0.1
#> # A tibble: 3 × 5
#>   predictor outcome      estimate std.error      rv
#>   <chr>     <chr>           <dbl>     <dbl>   <dbl>
#> 1 vap_white pres_ind_wal    0.387    0.0246 0.331  
#> 2 vap_black pres_ind_wal    0.490    0.0505 0.108  
#> 3 vap_other pres_ind_wal   -1.19     0.528  0.00491
ei_sens_rv(est, 2 * std.error) # how much confounding for bias of 2 SE
#> # A tibble: 3 × 5
#>   predictor outcome      estimate std.error     rv
#>   <chr>     <chr>           <dbl>     <dbl>  <dbl>
#> 1 vap_white pres_ind_wal    0.387    0.0246 0.180 
#> 2 vap_black pres_ind_wal    0.490    0.0505 0.109 
#> 3 vap_other pres_ind_wal   -1.19     0.528  0.0506

# How much confounding to equalize all estimates (no polarization)
y_avg = weighted.mean(elec_1968$pres_ind_wal, elec_1968$pres_total)
ei_sens_rv(est, estimate - y_avg)
#> # A tibble: 3 × 5
#>   predictor outcome      estimate std.error     rv
#>   <chr>     <chr>           <dbl>     <dbl>  <dbl>
#> 1 vap_white pres_ind_wal    0.387    0.0246 0.164 
#> 2 vap_black pres_ind_wal    0.490    0.0505 0.154 
#> 3 vap_other pres_ind_wal   -1.19     0.528  0.0725

# Extract as matrix
as.matrix(ei_sens_rv(est, 0.2), "rv")
#>            outcome
#> predictor   pres_ind_wal
#>   vap_white  0.545120562
#>   vap_black  0.203654212
#>   vap_other  0.009796171