Relates confounding of an omitted variable with predictor or outcome to bias in ecological estimates, using the nonparametric sensitivity analysis of Chernozhukov et al. (2024).
Arguments
- est
A set of estimates from
ei_est()using both regression and Riesz representer.- c_outcome
The (nonparametric) partial \(R^2\) of the omitted variables with the outcome variables. Must be between 0 and 1. Can be a vector, in which case all combinations of values with
c_predictorare used.- c_predictor
How much variation latent variables create in the Riesz representer, i.e. \(1-R^2\) of the true Riesz representer on the estimated one without the omitted variable. Must be between 0 and 1. Can be a vector, in which case all combinations of values with
c_outcomeare used.- bias_bound
If provided, overrides
c_predictorand finds values ofc_predictorthat correspond to (the absolute value of) the provided amount of bias.- confounding
The confounding parameter (\(\rho\)), which must be between 0 and 1 (the adversarial worst-case).
- expand_ci
If
TRUEand confidence intervals are present inest, expand the width of the intervals in each direction by the calculated bias bound.
Value
A data frame of the same format as est, but with additional
columns: c_outcome and c_predictor, matching all combinations of those
arguments, and bias_bound, containing the bound on the amount of bias.
The data frame has additional class ei_sens, which supports a
plot.ei_sens() method.
Details
The parameter c_predictor equals \(1 - R^2_{\alpha\sim\alpha_s}\), where
\(\alpha\) is the true Riesz representer and \(\alpha_s\) is the Riesz
representer with the observed covariates. The RR can be equivalently
expressed as $$
\alpha = \partial_{\bar x_j} \log f(\bar x_j\mid z, u),
$$ where \(U\) is the unobserved confounder and \(f\) is the conditional
density. The corresponding c_predictor is then $$
1 - R^2_{\alpha\sim\alpha_s} = 1 - \
\frac{\mathbb{E}[(\partial_{\bar x_j} \log f(\bar x_j\mid z))^2]}{
\mathbb{E}[(\partial_{\bar x_j} \log f(\bar x_j\mid z, u))^2]}.
$$
The bounds here are plug-in estimates and do not incorporate sampling uncertainty. As such, they may fail to cover the true value in finite samples, even under large enough sensitivity parameters; see Section 5 of Chernozhukov et al. (2024).
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(est, c_outcome=0.2)
#> # A tibble: 303 × 7
#> predictor outcome estimate std.error c_outcome c_predictor bias_bound
#> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 vap_white pres_ind_wal 0.387 0.0246 0.2 0 0
#> 2 vap_black pres_ind_wal 0.490 0.0505 0.2 0 0
#> 3 vap_other pres_ind_wal -1.19 0.528 0.2 0 0
#> 4 vap_white pres_ind_wal 0.387 0.0246 0.2 0.0001 0.00111
#> 5 vap_black pres_ind_wal 0.490 0.0505 0.2 0.0001 0.00392
#> 6 vap_other pres_ind_wal -1.19 0.528 0.2 0.0001 0.0909
#> 7 vap_white pres_ind_wal 0.387 0.0246 0.2 0.0004 0.00221
#> 8 vap_black pres_ind_wal 0.490 0.0505 0.2 0.0004 0.00784
#> 9 vap_other pres_ind_wal -1.19 0.528 0.2 0.0004 0.182
#> 10 vap_white pres_ind_wal 0.387 0.0246 0.2 0.0009 0.00332
#> # ℹ 293 more rows
# How much variation would the regression residual need to explain of
# Riesz representer to cause bias of 0.4?
ei_sens(est, c_outcome=1, bias_bound=0.4)
#> # A tibble: 3 × 7
#> predictor outcome estimate std.error c_outcome c_predictor bias_bound
#> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 vap_white pres_ind_wal 0.387 0.0246 1 0.723 0.4
#> 2 vap_black pres_ind_wal 0.490 0.0505 1 0.172 0.4
#> 3 vap_other pres_ind_wal -1.19 0.528 1 0.000388 0.4
# Update confidence intervals and extract as matrix
est = ei_est(m, rr, spec, conf_level=0.95)
sens = ei_sens(est, c_outcome=0.5, c_predictor=0.2)
as.matrix(sens, "conf.high")
#> outcome
#> predictor pres_ind_wal
#> vap_white 0.5228791
#> vap_black 0.8984785
#> vap_other 7.0309701