sensitivity-analyst

name	sensitivity-analyst
description	Sensitivity analysis frameworks and assumption-testing methods

Sensitivity Analyst

Comprehensive sensitivity analysis frameworks for causal inference and mediation studies

Use this skill when working on: unmeasured confounding assessment, E-values, tipping point analysis, measurement error sensitivity, model misspecification checks, Rosenbaum bounds, or any robustness evaluation for causal claims.

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Unmeasured Confounding

The Fundamental Problem

In causal inference, we can never directly test the assumption of no unmeasured confounding. Sensitivity analysis quantifies: "How strong would unmeasured confounding need to be to explain away our findings?"

Bias Factor Framework

For an unmeasured confounder $U$ affecting both treatment $A$ and outcome $Y$:

$$\text{Bias} = \frac{E[Y \mid A=1, U=1] - E[Y \mid A=1, U=0]}{E[Y \mid A=0, U=1] - E[Y \mid A=0, U=0]} \times \frac{P(U=1 \mid A=1) - P(U=1 \mid A=0)}{1}$$

The observed effect $\hat{\theta}$ relates to the true effect $\theta$ via:

$$\hat{\theta} = \theta \times \text{Bias Factor}$$

Sensitivity Parameters

Parameter	Definition	Range
$\gamma$	Odds ratio for U-A association	$[1, \infty)$
$\delta$	Odds ratio for U-Y association	$[1, \infty)$
$\rho$	Correlation of U with unmeasured factors	$[-1, 1]$

R Implementation

#' Sensitivity Analysis for Unmeasured Confounding
#'
#' @param estimate Point estimate of causal effect
#' @param se Standard error of estimate
#' @param gamma_range Range of treatment-confounder association
#' @param delta_range Range of outcome-confounder association
#' @return Sensitivity analysis results
sensitivity_unmeasured <- function(estimate, se,
                                    gamma_range = seq(1, 3, 0.25),
                                    delta_range = seq(1, 3, 0.25)) {

  # Create grid of sensitivity parameters
  grid <- expand.grid(gamma = gamma_range, delta = delta_range)

  # Compute bias factor for each combination
  # Using Ding & VanderWeele (2016) bias formula
  grid$bias_factor <- with(grid, (gamma * delta) / (gamma + delta - 1))

  # Adjusted estimates
  grid$adjusted_estimate <- estimate / grid$bias_factor

  # Find tipping point where CI includes 0
  z_crit <- qnorm(0.975)
  ci_lower <- estimate - z_crit * se

  # Tipping point: where adjusted CI lower bound = 0
  tipping_points <- grid[grid$adjusted_estimate - z_crit * se / grid$bias_factor <= 0, ]

  list(
    original = list(estimate = estimate, ci = c(ci_lower, estimate + z_crit * se)),
    sensitivity_grid = grid,
    tipping_points = tipping_points,
    minimum_bias_to_nullify = min(grid$bias_factor[grid$adjusted_estimate <= 0])
  )
}

---

E-Values

E-Value Definition

The E-value (VanderWeele & Ding, 2017) is the minimum strength of association that an unmeasured confounder would need to have with both treatment and outcome to fully explain away the observed effect.

For a risk ratio $RR$:

$$E\text{-value} = RR + \sqrt{RR \times (RR - 1)}$$

For the confidence interval limit:

$$E\text{-value}{CI} = RR{lower} + \sqrt{RR_{lower} \times (RR_{lower} - 1)}$$

Interpretation Guidelines

E-value	Interpretation
< 1.5	Very weak; easily explained by modest confounding
1.5 - 2.0	Weak; moderate confounding could explain
2.0 - 3.0	Moderate; substantial confounding needed
3.0 - 4.0	Strong; would require strong confounding
> 4.0	Very strong; unlikely to be fully explained

E-Value for Different Effect Measures

Measure	Formula
Risk Ratio (RR)	$E = RR + \sqrt{RR(RR-1)}$
Odds Ratio (OR)	Convert to RR approximation first
Hazard Ratio (HR)	$E = HR + \sqrt{HR(HR-1)}$ (when rare outcome)
Risk Difference	Convert to RR using baseline risk
Standardized Mean Difference	$E = \exp(0.91 \times d) + \sqrt{\exp(0.91 \times d)(\exp(0.91 \times d)-1)}$

R Implementation

#' Compute E-value for Causal Effect Estimate
#'
#' @param estimate Effect estimate (RR, OR, HR, or SMD)
#' @param lo Lower confidence limit
#' @param hi Upper confidence limit
#' @param type Type of effect measure
#' @param rare For OR/HR, is outcome rare (<15%)?
#' @return E-value and interpretation
compute_evalue <- function(estimate, lo = NULL, hi = NULL,
                           type = c("RR", "OR", "HR", "SMD"),
                           rare = TRUE) {
  type <- match.arg(type)

  # Convert to RR scale
  rr <- switch(type,
    "RR"  estimate
       rare estimate  estimate  
       rare estimate  estimate
        estimate
  

  
   rr   rr  1rr

  
  evalue  rr  rr  rr  

  
  evalue_ci  
   lo 
    rr_lo  type
        lo
         rare lo  lo
         rare lo  lo
          lo
    
     rr_lo   rr_lo  1rr_lo
     rr_lo   
      evalue_ci  rr_lo  rr_lo  rr_lo  
      
      evalue_ci  1
    
  

  
  interpretation  case_when
    evalue    
    evalue    
    evalue    
    evalue    
      
  

  
    evalue_point  evalue
    evalue_ci  evalue_ci
    interpretation  interpretation
    message  sprintf
      
      evalue
    
  

---

Tipping Point Analysis

Concept

Tipping point analysis identifies the specific parameter values at which conclusions would change (e.g., confidence interval includes null, effect reverses sign).

Tipping Point Types

Type	Definition	Use Case
Null tipping point	Where CI includes 0	Statistical significance
Clinical tipping point	Where effect < MCID	Clinical significance
Direction tipping point	Where effect changes sign	Effect direction

Visualization

#' Create Tipping Point Contour Plot
#'
#' @param estimate Point estimate
#' @param se Standard error
#' @param gamma_range Confounder-treatment association range
#' @param delta_range Confounder-outcome association range
#' @return ggplot2 contour plot
plot_tipping_point <- function(estimate, se,
                                gamma_range = seq(1, 5, 0.1),
                                delta_range = seq(1, 5, 0.1)) {
  library(ggplot2)

  grid <- expand.grid(gamma = gamma_range, delta = delta_range)
  grid$bias_factor <- with(grid, (  delta    delta  
  gridadjusted  estimate  gridbias_factor
  gridsignificant  gridadjusted    se  gridbias_factor  

  ggplotgrid aesx   y  delta z  adjusted 
    geom_contour_filledbreaks    estimate estimate  
    geom_contouraesz  significant
                 breaks   color   linewidth   
    labs
      x  
      y  
      title  
      subtitle  
     
    theme_minimal

---

Measurement Error Sensitivity

Types of Measurement Error

Type	Description	Effect on Estimates
Non-differential	Error unrelated to other variables	Usually biases toward null
Differential	Error depends on treatment/outcome	Can bias in either direction
Systematic	Consistent over/under-reporting	Shifts estimates
Random	Noise around true value	Attenuates relationships

Measurement Error in Mediation

For mediation with misclassified mediator:

$$\hat{a}\hat{b} = ab \times \text{Attenuation Factor}$$

where:

$$\text{Attenuation} = \frac{\text{Sensitivity} + \text{Specificity} - 1}{1}$$

R Implementation for Misclassification

#' Sensitivity Analysis for Mediator Misclassification
#'
#' @param indirect_effect Observed indirect effect
#' @param se_indirect Standard error
#' @param sens_range Sensitivity range to explore
#' @param spec_range Specificity range to explore
#' @return Corrected estimates grid
sensitivity_misclassification <- function(indirect_effect, se_indirect,
                                           sens_range = seq(0.7, 1, 0.05),
                                           spec_range = seq(0.7, 1, 0.05)) {

  grid <- expand.grid(sensitivity = sens_range, specificity = spec_range)

  # Attenuation factor
  grid$attenuation <- grid$sensitivity + grid$specificity  


  
  gridcorrected_effect  indirect_effect  gridattenuation

  
  gridcorrected_se  se_indirect  gridattenuation

  
  gridci_lower  gridcorrected_effect    gridcorrected_se
  gridci_upper  gridcorrected_effect    gridcorrected_se

  
  original_significant  indirect_effect    se_indirect  
  gridconclusion_change  gridci_lower    original_significant

  
    original  
      effect  indirect_effect
      se  se_indirect
      significant  original_significant
    
    sensitivity_grid  grid
    robust_region  gridgridconclusion_change 
    vulnerable_region  gridgridconclusion_change 
  

---

Model Misspecification

Types of Misspecification

1. Functional form: Linear when true relationship is nonlinear
2. Omitted variables: Missing important confounders
3. Incorrect distribution: Wrong error distribution assumed
4. Heterogeneity: Ignoring effect modification

Specification Curve Analysis

Test how results vary across defensible model specifications:

#' Specification Curve Analysis
#'
#' @param data Dataset
#' @param outcome Outcome variable name
#' @param treatment Treatment variable name
#' @param mediator Mediator variable name
#' @param covariate_sets List of covariate sets to try
#' @param model_types Vector of model types to try
#' @return Specification curve results
specification_curve <- function(data, outcome, treatment, mediator,
                                 covariate_sets,
                                 model_types = c("linear", "logistic")) {

  results <- list()
  spec_id <- 1

  for (covs in covariate_sets) {
    for (model_type in model_types) 

      
      m_formula  as.formulapastemediator  treatment 
                                     pastecovs collapse  

      
      y_formula  as.formulapasteoutcome  treatment  mediator 
                                     pastecovs collapse  

       model_type   
        m_model  lmm_formula data  data
        y_model  lmy_formula data  data
        
        m_model  glmm_formula data  data family  binomial
        y_model  glmy_formula data  data family  binomial
      

      
      a  coefm_modeltreatment
      b  coefy_modelmediator
      indirect  a  b

      resultsspec_id  data.frame
        spec_id  spec_id
        covariates  pastecovs collapse  
        model_type  model_type
        a_path  a
        b_path  b
        indirect_effect  indirect
        n_covariates  covs
      

      spec_id  spec_id  
    
  

  results_df  do.callrbind results

  
  
    specifications  results_df
    summary  data.frame
      median_effect  medianresults_dfindirect_effect
      mean_effect  meanresults_dfindirect_effect
      sd_effect  sdresults_dfindirect_effect
      min_effect  results_dfindirect_effect
      max_effect  results_dfindirect_effect
      prop_positive  meanresults_dfindirect_effect  
      prop_significant  
    
    n_specifications  nrowresults_df
  

---

Rosenbaum Bounds

Framework

Rosenbaum bounds quantify how sensitive causal conclusions are to hidden bias in observational studies using matched designs.

Sensitivity Parameter Γ

$\Gamma$ represents the maximum ratio of odds of treatment for two matched subjects:

$$\frac{1}{\Gamma} \leq \frac{P(A_i = 1 \mid X_i)}{P(A_j = 1 \mid X_j)} \times \frac{1 - P(A_j = 1 \mid X_j)}{1 - P(A_i = 1 \mid X_i)} \leq \Gamma$$

Interpretation

Γ Value	Interpretation
1.0	No hidden bias (randomization)
1.1 - 1.3	Sensitive to small hidden bias
1.3 - 2.0	Moderately robust
2.0 - 3.0	Robust to substantial bias
> 3.0	Very robust

R Implementation

#' Rosenbaum Bounds for Matched Study
#'
#' @param outcome_diff Vector of within-pair outcome differences
#' @param gamma_range Range of sensitivity parameter
#' @return Bounds on p-values
rosenbaum_bounds <- function(outcome_diff, gamma_range = seq(1, 3, 0.1)) {

  n <- length(outcome_diff)

  # Signed rank statistic
  ranks <- rank(abs(outcome_diff))
  W_obs <- sum(ranks[outcome_diff > 0])

  results <- data.frame(gamma = gamma_range)
  results$p_upper <- NA
  results$p_lower <- NA

  for i  gamma_range 
      gamma_rangei

    
    
    p_treat      

    
    E_W  ranks  p_treat
    V_W  ranks  p_treat    p_treat

    
    z_upper  W_obs  E_W  V_W
    resultsp_upperi    pnormz_upper

    
    p_treat_low  1    
    E_W_low  ranks  p_treat_low
    V_W_low  ranks  p_treat_low    p_treat_low
    z_lower  W_obs  E_W_low  V_W_low
    resultsp_loweri    pnormz_lower
  

  
  resultssignificant_upper  resultsp_upper  
  critical_gamma  resultsresultssignificant_upper na.rm  

  
    bounds  results
    critical_gamma  critical_gamma
    interpretation  sprintf
      
      critical_gamma
    
  

---

Sensitivity Analysis for Mediation

Sequential Ignorability Sensitivity

For natural indirect effects, sensitivity to violation of sequential ignorability:

#' Mediation Sensitivity Analysis
#'
#' Following Imai, Keele, & Yamamoto (2010)
#'
#' @param mediation_result Result from mediation analysis
#' @param rho_range Correlation between M and Y errors
#' @return Sensitivity results
sensitivity_mediation <- function(a_coef, b_coef, indirect_effect,
                                   se_indirect,
                                   rho_range = seq(-0.5, 0.5, 0.05)) {

  results <- data.frame(rho = rho_range)

  # Bias from correlation
  # Approximate bias formula from Imai et al.
  results$adjusted_indirect <- indirect_effect * (1 - rho_range^2)

  # Adjusted inference
  results$ci_lower <- results$adjusted_indirect - 1.96  se_indirect
  resultsci_upper  resultsadjusted_indirect    se_indirect
  resultssignificant  resultsci_lower    resultsci_upper  

  
  original_sign  indirect_effect
  critical_rho  rho_rangeresultsadjusted_indirect  original_sign

  
    sensitivity_grid  results
    critical_rho  critical_rho
    interpretation  sprintf
      
      critical_rho
    
    robust_range  resultsresultssignificant 
  

---

Sensitivity Analysis Checklist

Pre-Analysis Planning

• Identify key assumptions being made
• Determine which assumptions are most uncertain
• Select appropriate sensitivity analysis methods
• Define tipping points of interest
• Plan visualization strategy

Analysis Execution

• Compute E-values for main effects
• Conduct unmeasured confounding analysis
• Test sensitivity to measurement error (if applicable)
• Run specification curve analysis
• Compute Rosenbaum bounds (if matched design)

Reporting Template

## Sensitivity Analysis

### Unmeasured Confounding

The E-value for our main finding (RR = X.XX) is E = X.XX, indicating that
an unmeasured confounder would need to be associated with both treatment
and outcome by a risk ratio of at least X.XX each to fully explain away
the observed effect. This represents [interpretation] confounding.

### Tipping Point Analysis

Figure X shows the sensitivity of our conclusions to unmeasured confounding.
The observed effect would be nullified if [specific conditions].

### Measurement Error

Under the assumption of [non-differential/differential] measurement error,
our results remain significant for mediator sensitivity values above X%
and specificity above X%.

### Robustness to Model Specification

Across [N] alternative model specifications varying [covariates/functional
forms/etc.], the median indirect effect was X.XX (range: X.XX to X.XX).
[XX%] of specifications yielded statistically significant positive effects.

---

References

Foundational Papers

• VanderWeele, T. J., & Ding, P. (2017). Sensitivity analysis in observational research: Introducing the E-value. Annals of Internal Medicine
• Rosenbaum, P. R. (2002). Observational Studies
• Ding, P., & VanderWeele, T. J. (2016). Sensitivity analysis without assumptions. Epidemiology

Mediation-Specific

• Imai, K., Keele, L., & Yamamoto, T. (2010). Identification, inference and sensitivity analysis for causal mediation effects. Statistical Science
• VanderWeele, T. J. (2010). Bias formulas for sensitivity analysis for direct and indirect effects. Epidemiology

Measurement Error

• Carroll, R. J., et al. (2006). Measurement Error in Nonlinear Models
• Buonaccorsi, J. P. (2010). Measurement Error: Models, Methods, and Applications

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Version: 1.0.0 Created: 2025-12-08 Domain: Sensitivity analysis for causal inference Applications: Mediation analysis, observational studies, matched designs