Visualization Tools for Sensitivity Analysis of Unmeasured Confounding.

confoundvis

Overview

confoundvis is an R package for visualizing sensitivity analysis to unmeasured confounding in observational studies. It implements visualization tools for three major sensitivity frameworks — the Impact Threshold (Frank, 2000), Partial R-squared / Robustness Value (Cinelli & Hazlett, 2020), and E-value style metrics (VanderWeele & Ding, 2017) — helping researchers assess how strong omitted confounding would need to be to attenuate, invalidate, or reverse an estimated causal effect, and communicate those findings through clear, publication-ready graphics.

Core functions:

Installation

Install the development version from GitHub:

# install.packages("pak")
pak::pak("causalfragility-lab/confoundvis")

Quick Start

library(confoundvis)

# Construct a sensitivity analysis object
obj <- new_confoundsens(
  estimate       = 0.5,
  se             = 0.1,
  df             = 200,
  r2dz_x         = seq(0.01, 0.4, by = 0.01),
  r2yz_dx        = seq(0.01, 0.4, by = 0.01),
  benchmark_r2dz = 0.1,
  benchmark_r2yz = 0.15
)

# Contour plot of robustness values
plot_sensitivity_contour(obj)

# Robustness curve across effect sizes
plot_robustness_curve(obj)

# Love-plot style sensitivity display
plot_sensitivity_love(obj)

# Local Taylor approximation
plot_local_taylor(obj)

Plot Methods

# Contour plot
plot_sensitivity_contour(obj)

# Robustness curve
plot_robustness_curve(obj)

# Love plot
plot_sensitivity_love(obj)

# Taylor approximation
plot_local_taylor(obj)

Sensitivity Frameworks

The package operationalises three sensitivity frameworks:

Simulation Results

Taylor Approximation Bias Across Window Widths

A simulation with theta0 = 0.5, slope = -0.7, kappa3 = 0.6, and window widths w ∈ {0.05, 0.1, 0.2, 0.3, 0.4, 0.5} confirms that local quadratic approximation consistently outperforms first-order (tangent) approximation as the local window widens.

library(confoundvis)
library(ggplot2)
library(scales)

set.seed(2025)

theta0     <- 0.5
slope      <- -0.7
kappa3     <- 0.6
windows    <- c(0.05, 0.1, 0.2, 0.3, 0.4, 0.5)
kappa_vals <- c(-0.4, 0, 0.4)
delta      <- seq(0, 0.5, length.out = 1000)

results <- list()
for (kap in kappa_vals) {
  theta_true <- theta0 + slope * delta + kap * delta^2 + kappa3 * delta^3
  simdat <- data.frame(delta = delta, theta = theta_true)
  for (w in windows) {
    local_dat <- subset(simdat, delta <= w)
    pred_t <- theta0 + slope * local_dat$delta
    mae_t  <- mean(abs(local_dat$theta - pred_t))
    fit_q  <- lm(theta ~ delta + I(delta^2), data = local_dat)
    mae_q  <- mean(abs(local_dat$theta - predict(fit_q, newdata = local_dat)))
    results[[length(results) + 1]] <- data.frame(
      kappa = kap, window = w, mae_tangent = mae_t, mae_quadratic = mae_q
    )
  }
}

df <- do.call(rbind, results)
df_long <- rbind(
  data.frame(kappa = df$kappa, window = df$window,
             type = "Tangent",   mae = df$mae_tangent),
  data.frame(kappa = df$kappa, window = df$window,
             type = "Quadratic", mae = df$mae_quadratic)
)
df_long$kappa_lab <- factor(
  df_long$kappa,
  levels = c(-0.4, 0, 0.4),
  labels = c("Concave (K < 0)", "Linear (K = 0)", "Convex (K > 0)")
)

ggplot(df_long, aes(x = window, y = mae, color = type, linetype = type)) +
  geom_line(linewidth = 0.9) +
  geom_point(size = 1.8) +
  scale_y_log10(labels = label_scientific()) +
  facet_wrap(~ kappa_lab, nrow = 1) +
  labs(
    title    = "Taylor Approximation MAE vs Local Window Width",
    subtitle = "True path includes cubic term (kappa3 = 0.6)",
    x        = "Local window width (w)",
    y        = "Mean Absolute Error (log scale)",
    color    = "Approximation",
    linetype = "Approximation"
  ) +
  theme_bw(base_size = 11) +
  theme(
    legend.position  = "bottom",
    panel.grid.minor = element_blank(),
    strip.background = element_rect(fill = "grey92", colour = NA),
    plot.title       = element_text(face = "bold")
  )

Key finding: Quadratic approximation yields substantially lower error than the tangent at every window width. Approximation error grows with window width, and curvature accelerates the deterioration of first-order linear approximation.

Zero-Crossing Error Under Linearity Mis-specification

When the true sensitivity path is nonlinear, using a linear approximation to locate the zero-crossing introduces systematic bias.

theta0_vals <- c(0.2, 0.4, 0.6)
slope_vals  <- c(-0.3, -0.6, -0.9)
kappa_vals2 <- seq(-0.6, 0.6, by = 0.2)

find_true <- function(t0, s, k) {
  if (abs(k) < 1e-10) return(-t0 / s)
  disc  <- s^2 - 4 * k * t0
  if (disc < 0) return(NA_real_)
  roots <- c((-s + sqrt(disc)) / (2 * k), (-s - sqrt(disc)) / (2 * k))
  roots <- roots[roots > 0]
  if (length(roots) == 0) return(NA_real_)
  min(roots)
}

res2 <- list()
for (t0 in theta0_vals) {
  for (s in slope_vals) {
    for (k in kappa_vals2) {
      l_lin  <- -t0 / s
      l_true <- find_true(t0, s, k)
      if (is.na(l_true)) next
      res2[[length(res2) + 1]] <- data.frame(
        theta0 = t0, slope = s, kappa = k,
        lambda_linear = l_lin, lambda_true = l_true,
        error = l_lin - l_true
      )
    }
  }
}

df2 <- do.call(rbind, res2)
df2$theta0_lab <- factor(
  df2$theta0,
  levels = c(0.2, 0.4, 0.6),
  labels = c("theta0 = 0.2", "theta0 = 0.4", "theta0 = 0.6")
)

ggplot(df2, aes(x = kappa, y = error,
                color = factor(slope), group = factor(slope))) +
  geom_hline(yintercept = 0, linetype = "dashed", colour = "grey50") +
  geom_line(linewidth = 0.9) +
  geom_point(size = 1.8) +
  facet_wrap(~ theta0_lab, nrow = 1) +
  labs(
    title    = "Signed Error in lambda* Under Linearity Mis-specification",
    subtitle = "Positive = over-estimation by linear approximation",
    x        = "True curvature (kappa)",
    y        = "Estimated lambda* (linear) - true lambda*",
    color    = "Slope"
  ) +
  theme_bw(base_size = 11) +
  theme(
    legend.position  = "bottom",
    panel.grid.minor = element_blank(),
    strip.background = element_rect(fill = "grey92", colour = NA),
    plot.title       = element_text(face = "bold")
  )

Key finding: Under concavity, linear approximation over-estimates the fragility threshold; under convexity, it under-estimates. The bias grows with both curvature magnitude and initial effect size.

References

• Frank, K. A. (2000). Impact of a confounding variable on a regression coefficient. Sociological Methods & Research, 29(2), 147–194. https://doi.org/10.1177/0049124100029002001

• Cinelli, C., & Hazlett, C. (2020). Making sense of sensitivity: Extending omitted variable bias. Journal of the Royal Statistical Society: Series B, 82(1), 39–67. https://doi.org/10.1111/rssb.12348

• VanderWeele, T. J., & Ding, P. (2017). Sensitivity analysis in observational research: Introducing the E-value. Annals of Internal Medicine, 167(4), 268–274. https://doi.org/10.7326/M16-2607

Citation

citation("confoundvis")

Hait, S. (2025). confoundvis: Visualization Tools for Sensitivity Analysis to Unmeasured Confounding. R package version 0.1.0. https://github.com/causalfragility-lab/confoundvis

Author

Subir Hait
Michigan State University
[email protected]
ORCID: 0009-0004-9871-9677