sensemakr for Julia (Sensemakr) implements a suite of sensitivity analysis tools that extends the traditional omitted variable bias framework and makes it easier to understand the impact of omitted variables in regression models, as discussed in Cinelli, C. and Hazlett, C. (2020) “Making Sense of Sensitivity: Extending Omitted Variable Bias.” Journal of the Royal Statistical Society, Series B (Statistical Methodology).
You will need Julia version 1.7.0 or higher to install this package
The latest version can be downloaded by running:
import Pkg; Pkg.add(url = "https://github.com/d2cml-ai/Sensemakr.jl");The latest stable version can be downloaded by running:
import Pkg; Pkg.add("Sensemakr");# loads Sensemakr
using Sensemakr
# loads DataFrames and GLM
using DataFrames, GLM
# creates dataframe with darfur data
darfur = load_darfur();
# runs OLS
form = @formula(peacefactor ~ directlyharmed + age + farmer_dar + herder_dar + pastvoted + hhsize_darfur + female + village);
fitted_model = lm(form, darfur);
# creates sensemakr object
darfur_sense = sensemakr(fitted_model, "directlyharmed", benchmark_covariates = "female", kd = [1, 2, 3]);
# summary of sensitivity analysis
summary(darfur_sense)Sensitivity Analysis to Unobserved Confounding
Model Formula: peacefactor ~ 1 + directlyharmed + age + farmer_dar + herder_dar + pastvoted + hhsize_darfur + female + village
Null hypothesis: q = 1.0 and reduce = true
-- This means we are considering biases that reduce the absolute value of the current estimate
-- The null hypothesis deemed problematic is H0:tau = 0.0
Unadjusted Estimates of "directlyharmed":
Coef. Estimate: 0.097
Standard Error: 0.023
t-value: 4.184
Sensitivity Statistics:
Partial R2 of treatment with outcome: 0.022
Robustness Value, q = 1.0: 0.139
Robustness Value, q = 1.0 alpha = 0.05: 0.076
Verbal interpretation of sensitivity statistics:
-- Partial R2 of the treatment with the outcome: an extreme confounder (orthogonal to the covariates) that explains 100% of the residual variance of the outcome, would need to explain at least 2.187 % of the residual variance of the treatment to fully account for the observed estimated effect.
-- Robustness Value, q = 1.0: unobserved confounders (orthogonal to the covariates) that of both the treatment and the outcome are strong enough to bring the point estimate to 0.0 (a bias of 100.0% of the original estimate). Conversely, unobserved confounders that do not explain more than 13.878% of the residual variance of both the treatment and the outcome are not strong enough to bring the point estimate to 0.0.
-- Robustness Value,q = 1.0, alpha = 0.05: unobserved confounders (orthogonal to the covariates) that explain more than 7.626% of the residual variance of both the treatment and the outcome are strong enough to bring the estimate to a range where it is no longer 'statistically different' from 0.0 (a bias of 100.0% of the original estimate), at the significance level of alpha = 0.05. Conversely, unobserved confounders that do not explain more than7.626% of the residual varianceof both the treatment and the outcome are not strong enough to bring the estimate to a range where it is no longer 'statistically different' from 0.0, at the significance level of alpha = 0.05.
Bounds on omitted variable bias:
--The table below shows the maximum strength of unobserved confounders with association with the treatment and the outcome bounded by a multiple of the observed explanatory power of the chosen benchmark covariate(s).
3×9 DataFrame
Row │ bound_label r2dz_x r2yz_dx treatment adjusted_estimate adjusted_se adjusted_t adjusted_lower_CI adjusted_upper_CI
│ String Float64 Float64 String Float64 Float64 Float64 Float64 Float64
─────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
1 │ 1.0x female 0.00916429 0.124641 directlyharmed 0.0752203 0.0218733 3.4389 0.032283 0.118158
2 │ 2.0x female 0.0183286 0.249324 directlyharmed 0.0529152 0.0203501 2.60025 0.012968 0.0928623
3 │ 3.0x female 0.0274929 0.37405 directlyharmed 0.030396 0.0186701 1.62806 -0.00625328 0.0670453
# contour plot for estimate
plot(darfur_sense)
# contour plot for t-value
plot(darfur_sense, sensitivity_of = "t-value")
# extreme scenario plot
plot(darfur_sense, plot_type = "extreme")
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