Sensitivity Analysis: How Robust Is Your Causal Inference?
sensitivity-analysis.Rmd🎯 Sensitivity Analysis: Why It Matters
Even when using careful matching or regression, unobserved confounding may bias our treatment effect estimates. Sensitivity analysis helps us quantify how much unobserved bias would be required to nullify our results.
1. 🧪 Simulate Confounded Data
set.seed(123)
n <- 1000
U <- rnorm(n) # unobserved confounder
X <- rbinom(n, 1, prob = plogis(0.3 * U)) # treatment depends on U
Y <- 2 * X + 1.5 * U + rnorm(n) # outcome depends on X and U
data <- data.frame(X, Y, U)2. ⚠️ Naive Regression (No U Adjustment)
##
## Call:
## lm(formula = Y ~ X, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5542 -1.2648 0.0258 1.1609 5.8993
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.09461 0.07598 -1.245 0.213
## X 2.24198 0.10967 20.444 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.733 on 998 degrees of freedom
## Multiple R-squared: 0.2952, Adjusted R-squared: 0.2945
## F-statistic: 418 on 1 and 998 DF, p-value: < 2.2e-163. 📏 True Model with U (for comparison)
##
## Call:
## lm(formula = Y ~ X + U, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0533 -0.6312 -0.0234 0.6837 3.2464
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.005461 0.043212 0.126 0.899
## X 1.984994 0.062544 31.737 <2e-16 ***
## U 1.443323 0.031525 45.784 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9841 on 997 degrees of freedom
## Multiple R-squared: 0.7728, Adjusted R-squared: 0.7724
## F-statistic: 1696 on 2 and 997 DF, p-value: < 2.2e-164. 📉 Sensitivity with Manual Bias Factor
We’ll manually simulate how adding different levels of confounding changes the treatment effect:
bias_sim <- data.frame(
rho = seq(0, 1, by = 0.05)
) %>%
mutate(
bias = rho * sd(U),
adjusted_beta = coef(naive_model)[["X"]] - bias
)
ggplot(bias_sim, aes(x = rho, y = adjusted_beta))
geom_line(color = "steelblue", size = 1)## geom_line: na.rm = FALSE, orientation = NA
## stat_identity: na.rm = FALSE
## position_identity
geom_hline(yintercept = coef(true_model)[["X"]], linetype = "dashed", color = "red")## mapping: yintercept = ~yintercept
## geom_hline: na.rm = FALSE
## stat_identity: na.rm = FALSE
## position_identity
labs(title = "Effect of Unobserved Confounding on Estimated Treatment Effect",
x = "Assumed Correlation with U (rho)", y = "Adjusted Beta for X")## $x
## [1] "Assumed Correlation with U (rho)"
##
## $y
## [1] "Adjusted Beta for X"
##
## $title
## [1] "Effect of Unobserved Confounding on Estimated Treatment Effect"
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