Functional Propensity Score for Causal Inference

Implements functional propensity score (FPS) weighting for causal inference with functional treatments. The method estimates weights that balance observed confounders by removing their dependence on the functional treatment and uses a dual formulation of the weighting problem for efficient unconstrained optimization. The framework supports scalar, binary, and functional outcomes, as well as functional covariates, and can be used to estimate marginal causal effects in settings with time-varying exposures. The methodology follows Ciardulli, S., Fontana, N., Vantini, S., and Ieva, F. (2026) "Generalized propensity score weighting for functional causal inference framework" .

FPScausal: Functional propensity score weighting for causal inference with functional treatments, covariates, and outcomes

FPScausal implements the Functional Propensity Score (FPS) weighting methodology for causal inference with functional treatments and outcomes. If you use this package, please cite:

Ciardulli S. \& Fontana, N., Vantini S., Ieva, F. (2026). Generalized propensity score weighting for functional causal inference framework. arXiv. https://arxiv.org/abs/2608.03200. 

The package handles:

• Functional treatment
• Scalar and/or functional covariates
• Scalar, binary, or functional outcomes

Installation

Install the released version from CRAN:

install.packages("FPScausal")

Or install the development version from GitHub:

install.packages("devtools") # Install devtools if not already installed
devtools::install_github("NicoleFontana/FPSCausal")

Methodology

1. FPCA decomposition: the functional treatment X(s) is represented via its Karhunen–Loève expansion, retaining the first L FPC scores.
2. Empirical-likelihood balancing: covariate-balancing weights are estimated by maximising the empirical likelihood subject to constraints that balance the FPC scores of the treatment against the observed confounders (and their interactions). The resulting dual problem is a smooth unconstrained minimisation solved via the BFGS quasi-Newton algorithm.
3. Weighted least squares: the causal effect function μ(s) (scalar outcome) or causal effect surface μ(s, t) (functional outcome) is recovered via weighted regression.
4. Bootstrap CIs: residual bootstrap (scalar/binary) or pairs bootstrap (functional outcome).

Quick start

Scalar outcome

library(FPScausal)

# Simulate data — scalar covariates only
dat <- simulate_fps_data(
  n                    = 200,
  setting              = "LL",
  outcome_type         = "scalar",
  include_functional_cov = FALSE,
  seed                 = 42
)

# Step 1: estimate weights (treat_domain inferred from treat_grid)
w <- fps_weighting(
  treatment  = dat$X,
  treat_grid = dat$t_grid,
  covariates = dat$C
)

# Diagnostics
plot(w, type = "balance")
plot(w, type = "fpca_treatment")
plot(w, type = "weights")

# Step 2: estimate causal effect with bootstrap CIs
eff <- fps_effect_estimation(
  outcome    = dat$Y,
  fps_object = w,
  bootstrap  = TRUE,
  B          = 500,
  true_beta  = dat$true_beta,
  seed       = 1
)

plot(eff, type = "effect")       # μ(s) with CI ribbon and legend
plot(eff, type = "comparison")   # weighted vs unweighted
plot(eff, type = "significance") # significant time points

Functional outcome

dat_fn <- simulate_fps_data(
  n                    = 200,
  setting              = "LL",
  outcome_type         = "functional",
  include_functional_cov = FALSE,
  seed                 = 99
)

w_fn <- fps_weighting(
  treatment    = dat_fn$X,
  treat_grid   = dat_fn$t_grid,
  treat_domain = c(0, 1),
  domain_name  = "s",
  covariates   = dat_fn$C
)

plot(w_fn, type = "balance")
plot(w_fn, type = "fpca_treatment")

eff_fn <- fps_effect_estimation(
  outcome             = dat_fn$Y,
  fps_object          = w_fn,
  outcome_t_grid      = dat_fn$t_grid,
  outcome_domain      = c(0, 1),
  outcome_domain_name = "t",
  bootstrap           = TRUE,
  B                   = 500,
  seed                = 2
)

plot(eff_fn, type = "effect")           # μ(s,t) heatmap
plot(eff_fn, type = "bootstrap_slice",  # 1-D slice at t = 0.5
     point = 0.5, which_domain = "outcome")
plot(eff_fn, type = "bootstrap_slice",  # 1-D slice at s = 0.5
     point = 0.5, which_domain = "treatment")
plot(eff_fn, type = "significance")     # 2-D significance map

Functional covariate

dat2 <- simulate_fps_data(
  n                    = 2000,
  setting              = "LL",
  outcome_type         = "scalar",
  include_functional_cov = TRUE,
  seed                 = 7
)

w2 <- fps_weighting(
  treatment   = dat2$X,
  treat_grid  = dat2$t_grid,
  domain_name = "s",
  covariates  = list(scalar = dat2$C, functional = list(dat2$D)),
  cov_grids   = list(dat2$t_grid)
)

plot(w2, type = "balance")
plot(w2, type = "fpca_covariates")

eff2 <- fps_effect_estimation(dat2$Y, w2, true_beta = dat2$true_beta)
plot(eff2, type = "effect")

Package functions

Simulation settings

simulate_fps_data() supports four settings varying whether the treatment–confounder and confounder–outcome relationships are linear (L) or nonlinear (N):

Dependencies

fda, ggplot2, tidyr, MASS, wCorr, patchwork, progress

Reference

Ciardulli, S. and Fontana, N., Vantini S., Ieva F. (2026). Functional propensity score weighting for causal inference with functional treatments, covariates, and outcomes. arXiv:2608.03200. https://arxiv.org/abs/2608.03200

@misc{ciardulli2026generalizedpropensityscoreweighting,
      title={Generalized propensity score weighting for functional causal inference framework}, 
      author={Simone Ciardulli and Nicole Fontana and Simone Vantini and Francesca Ieva},
      year={2026},
      eprint={2608.03200},
      archivePrefix={arXiv},
      primaryClass={stat.ME},
      url={https://arxiv.org/abs/2608.03200}, 
}

License

MIT