Estimate Counterfactual Sensitivity

Estimates the sensitivity (true positive rate) of a binary classifier at one or more thresholds under a hypothetical intervention where treatment is set to a specific level.

Usage

cf_sensitivity(
  predictions,
  outcomes,
  treatment,
  covariates,
  threshold = 0.5,
  treatment_level = 0,
  estimator = c("dr", "om", "ipw", "naive"),
  propensity_model = NULL,
  outcome_model = NULL,
  se_method = c("none", "bootstrap", "influence"),
  n_boot = 200,
  conf_level = 0.95,
  cross_fit = FALSE,
  n_folds = 5,
  parallel = FALSE,
  ncores = NULL,
  ps_trim = NULL,
  ...
)

cf_tpr(
  predictions,
  outcomes,
  treatment,
  covariates,
  threshold = 0.5,
  treatment_level = 0,
  estimator = c("dr", "om", "ipw", "naive"),
  propensity_model = NULL,
  outcome_model = NULL,
  se_method = c("none", "bootstrap", "influence"),
  n_boot = 200,
  conf_level = 0.95,
  cross_fit = FALSE,
  n_folds = 5,
  parallel = FALSE,
  ncores = NULL,
  ps_trim = NULL,
  ...
)

Arguments

predictions

Numeric vector of model predictions.

outcomes

Numeric vector of observed outcomes.

treatment

Numeric vector of treatment indicators (0/1).

covariates

A matrix or data frame of baseline covariates (confounders).

threshold

Numeric vector of classification thresholds. Predictions above this value are classified as positive. Can be a single value or a vector for computing sensitivity at multiple thresholds simultaneously. Default is 0.5.

treatment_level

The counterfactual treatment level (default: 0).

estimator

Character string specifying the estimator:

"naive": Naive estimator (biased)
"cl": Conditional loss estimator
"ipw": Inverse probability weighting estimator
"dr": Doubly robust estimator (default)

propensity_model

Optional fitted propensity score model. If NULL, a logistic regression model is fit using the covariates.

outcome_model

Optional fitted outcome model. If NULL, a regression model is fit using the covariates among treated/untreated. For binary outcomes, this should be a model for E[Y|X,A] (binomial family). For continuous outcomes, this should be a model for E[L|X,A] (gaussian family).

se_method

Method for standard error estimation:

"bootstrap": Bootstrap standard errors (default)
"influence": Influence function-based standard errors
"none": No standard error estimation

n_boot

Number of bootstrap replications (default: 500).

conf_level

Confidence level for intervals (default: 0.95).

cross_fit

Logical indicating whether to use cross-fitting for nuisance model estimation (default: FALSE). Cross-fitting enables valid inference when using flexible machine learning estimators.

n_folds

Number of folds for cross-fitting (default: 5).

parallel

Logical indicating whether to use parallel processing for bootstrap (default: FALSE).

ncores

Number of cores for parallel processing (default: NULL, which uses all available cores minus one).

ps_trim

Propensity score trimming specification. Controls how extreme propensity scores are handled. Can be:

NULL (default): Uses absolute bounds c(0.01, 0.99)
"none": No trimming applied
"quantile": Quantile-based trimming with default c(0.01, 0.99)
"absolute": Explicit absolute bounds with default c(0.01, 0.99)
A numeric vector of length 2: c(lower, upper) absolute bounds
A single numeric: Symmetric bounds c(x, 1-x)
A list with method ("absolute"/"quantile"/"none") and bounds

...

Additional arguments passed to internal functions.

Value

An object of class c("cf_sensitivity", "cf_performance") containing:

estimate: Point estimate(s) of counterfactual sensitivity
se: Standard error(s) (if computed)
ci_lower: Lower confidence interval bound(s)
ci_upper: Upper confidence interval bound(s)
threshold: Threshold value(s) used
estimator: Estimator used
naive_estimate: Naive sensitivity for comparison
n_obs: Number of observations
treatment_level: Counterfactual treatment level

Details

Sensitivity (also known as true positive rate or recall) is defined as: $$Sensitivity(c) = P(\hat{Y} > c | Y^{(a)} = 1)$$

where $Y^{(a)}$ is the potential outcome under treatment level $a$.

The function implements three estimators following Coston et al. (2020):

Outcome Model (OM) Estimator: Weights by the predicted probability of being a case under the counterfactual: $$\hat{\psi}_{sens,om} = \frac{\sum_i I(\hat{h}(X_i) > c) \hat{m}(X_i)}{\sum_i \hat{m}(X_i)}$$ where $\hat{m}(X) = P(Y=1|X, A=a)$.

IPW Estimator: Weights by the inverse probability of treatment: $$\hat{\psi}_{sens,ipw} = \frac{\sum_i I(\hat{h}(X_i) > c, Y_i=1, A_i=a) / \hat{e}(X_i)}{\sum_i I(Y_i=1, A_i=a) / \hat{e}(X_i)}$$ where $\hat{e}(X) = P(A=a|X)$.

Doubly Robust (DR) Estimator: Combines OM and IPW for protection against misspecification of either model.

References

Coston, A., Mishler, A., Kennedy, E. H., & Chouldechova, A. (2020). "Counterfactual risk assessments, evaluation, and fairness." Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency, 582-593.

Boyer, C. B., Dahabreh, I. J., & Steingrimsson, J. A. (2025). "Estimating and evaluating counterfactual prediction models." Statistics in Medicine, 44(23-24), e70287. doi:10.1002/sim.70287

Examples

# Generate example data
set.seed(123)
n <- 1000
x <- rnorm(n)
a <- rbinom(n, 1, plogis(-0.5 + 0.5 * x))
y <- rbinom(n, 1, plogis(-1 + x - 0.5 * a))
pred <- plogis(-1 + 0.8 * x)

# Estimate counterfactual sensitivity at default threshold (0.5)
result <- cf_sensitivity(
  predictions = pred,
  outcomes = y,
  treatment = a,
  covariates = data.frame(x = x),
  treatment_level = 0,
  estimator = "dr",
  se_method = "none"
)
print(result)
#> 
#> Counterfactual Sensitivity Estimate
#> ====================================
#> 
#> Estimator: DR 
#> Treatment level: 0 
#> N: 1000 
#> 
#> Threshold: 0.5 
#> Estimate: 0.21 
#> Naive estimate: 0.2291 
#> 

# Estimate at multiple thresholds
result_multi <- cf_sensitivity(
  predictions = pred,
  outcomes = y,
  treatment = a,
  covariates = data.frame(x = x),
  threshold = c(0.3, 0.5, 0.7),
  se_method = "none"
)
print(result_multi)
#> 
#> Counterfactual Sensitivity Estimate
#> ====================================
#> 
#> Estimator: DR 
#> Treatment level: 0 
#> N: 1000 
#> 
#> Results by threshold:
#>  Threshold Estimate  Naive
#>        0.3   0.6649 0.6545
#>        0.5   0.2100 0.2291
#>        0.7   0.0416 0.0400
#>