Diagnostic visualization for causal estimands

CP plots and local CP plots in R.

CPplot draws estimated treatment effects against estimated propensity scores and returns the diagnostic linear fits used to read bracketing relationships and plug-in ATT, ATO, and ATC estimates.

Run the first example View API reference

Installation

install.packages("remotes")
remotes::install_github("e9tian/CPplot")
library(CPplot)

Observational CP plot

If y is the outcome, z is the binary treatment, and x is a covariate, a CP plot can be drawn with cp_plot(y ~ z + x, data = df). The example below uses three covariates in the same way.

Example

set.seed(1)
n <- 900
df <- data.frame(x1 = rnorm(n), x2 = rnorm(n))
p <- plogis(-0.30 + 1.05 * df$x1 - 0.45 * df$x2)
df$x3 <- p^2
df$z <- rbinom(n, 1, p)
tau <- -0.20 + 1.20 * p + 5.00 * df$x3 + 0.70 * df$x2
df$y <- 1 + 0.5 * df$x1 + 0.4 * df$x2 +
  df$z * tau + rnorm(n, sd = 0.5)

fit <- cp_plot(y ~ z + x1 + x2 + x3, data = df)
fit$plot

Numerical diagnostics

Use fit$slopes to report the fitted-line slopes and fit$intersections to report the three pairwise intersections and their plug-in ATT, ATO, and ATC estimates. Use fit$bracketing to report the corresponding bracketing implications. fit$bracketing_summary gives a compact text summary of the ATO bracketing diagnostic.

fit$slopes
fit$intersections
fit$bracketing
fit$bracketing_summary

# fit$slopes
               fit   n    slope  std_error       p_value
1       Unweighted 900 5.101455 0.09159536 1.501789e-293
2 Treated weighted 900 5.760079 0.09442888 1.304333e-321
3 Control weighted 900 4.346009 0.10703058 1.776072e-205

# fit$intersections
  estimand           line_1           line_2     e_hat effect_hat
1      ATT       Unweighted Treated weighted 0.5764895   2.186164
2      ATO Treated weighted Control weighted 0.4629750   1.532311
3      ATC Control weighted       Unweighted 0.3640090   1.102205

# fit$bracketing
               fit    slope slope_sign       implication
1       Unweighted 5.101455   positive ATT >= ATE >= ATC
2 Treated weighted 5.760079   positive        ATT >= ATO
3 Control weighted 4.346009   positive        ATO >= ATC

# fit$bracketing_summary
[1] "The tilted slopes support ATC <= ATO <= ATT."

Local CP plot for IV studies

If y is the outcome, d is the treatment, z is the binary IV, and x is a covariate, a local CP plot can be drawn with local_cp_plot(y ~ d + x | z + x, data = df). The example below uses three covariates in the same way. In formula mode, CPplot estimates the IV propensity score and both conditional contrasts, then forms the plug-in inputs pi_c_hat = delta_d and tau_c_hat = delta_y / delta_d. By default, points with z = 0 are labeled Unencouraged, and points with z = 1 are labeled Encouraged.

Example

set.seed(2)
n <- 900
df <- data.frame(x1 = rnorm(n), x2 = rnorm(n))
p <- plogis(-0.25 + 0.95 * df$x1 + 0.35 * df$x2)
df$x3 <- p^2
df$z <- rbinom(n, 1, p)
df$d <- rbinom(n, 1, plogis(-0.8 + 1.2 * df$z + 0.4 * df$x1 - 0.2 * df$x2))
tau <- -0.20 + 1.20 * p + 5.00 * df$x3 + 0.70 * df$x2
df$y <- 1 + 0.4 * df$x1 - 0.2 * df$x2 + tau * df$d + rnorm(n, sd = 0.5)

fit <- local_cp_plot(y ~ d + x1 + x2 + x3 | z + x1 + x2 + x3, data = df)
fit$plot

Numerical diagnostics

Use fit$slopes to report the fitted-line slopes and fit$intersections to report the pairwise line intersections and their plug-in local ATT, ATO, and ATC estimates. Use fit$bracketing to report the corresponding local bracketing implications. fit$bracketing_summary gives a compact text summary of the local ATO bracketing diagnostic.

fit$slopes
fit$intersections
fit$bracketing
fit$bracketing_summary

# fit$slopes
                             fit   n    slope  std_error       p_value
1              Complier weighted 900 6.541203 0.1677583 2.216206e-195
2   Encouraged-complier weighted 900 9.086877 0.1688588 3.131736e-283
3 Unencouraged-complier weighted 900 4.304402 0.1737031 1.034667e-103

# fit$intersections
    estimand                         line_1                         line_2
1 tau_ATT^c              Complier weighted   Encouraged-complier weighted
2 tau_ATO^c   Encouraged-complier weighted Unencouraged-complier weighted
3 tau_ATC^c Unencouraged-complier weighted              Complier weighted
      e_hat effect_hat
1 0.5677469   2.462528
2 0.4683526   1.559344
3 0.3552332   1.072433

# fit$bracketing
                             fit    slope slope_sign
1              Complier weighted 6.541203   positive
2   Encouraged-complier weighted 9.086877   positive
3 Unencouraged-complier weighted 4.304402   positive
                          implication
1 tau_ATT^c >= tau_ATE^c >= tau_ATC^c
2              tau_ATT^c >= tau_ATO^c
3              tau_ATO^c >= tau_ATC^c

# fit$bracketing_summary
[1] "The tilted slopes support tau_ATC^c <= tau_ATO^c <= tau_ATT^c."

What the diagnostic lines tell you

Read the slope signs as diagnostics. A positive fitted slope gives the implication listed below; a negative fitted slope reverses the corresponding inequality.

CP plot line	Weight used	Population analog	Positive slope implies
Unweighted linear fit	`1`	`cov{tau(X), e(X)}`	`tau_ATT >= tau_ATE >= tau_ATC`
Treated-weighted linear fit	`e_hat_i`	`cov{tau(X), e(X) \| Z = 1}`	`tau_ATT >= tau_ATO`
Control-weighted linear fit	`1 - e_hat_i`	`cov{tau(X), e(X) \| Z = 0}`	`tau_ATO >= tau_ATC`

Local CP plot line	Weight used	Population analog	Positive slope implies
Complier-weighted linear fit	`pi_c_hat_i`	`cov{tau^c(X), e(X) \| U = c}`	`tau^c_ATT >= tau^c_ATE >= tau^c_ATC`
Encouraged-complier weighted fit	`pi_c_hat_i * e_hat_i`	`cov{tau^c(X), e(X) \| U = c, Z = 1}`	`tau^c_ATT >= tau^c_ATO`
Unencouraged-complier weighted fit	`pi_c_hat_i * (1 - e_hat_i)`	`cov{tau^c(X), e(X) \| U = c, Z = 0}`	`tau^c_ATO >= tau^c_ATC`

What the line intersections tell you

Theorem 3 of the associated paper shows that, when each corresponding pair of population projections is distinct, the vertical coordinate of each pairwise intersection is a weighted causal estimand. The horizontal coordinate is the corresponding propensity-score centroid.

These are three pairwise intersections, not necessarily one common intersection. In the sample, fit$intersections reports their analogs, and each vertical coordinate equals the corresponding plug-in estimate.

Pair of CP lines	Population intersection	Vertical coordinate
Unweighted and treated weighted	`(e_bar_ATT, tau_ATT)`	ATT
Treated weighted and control weighted	`(e_bar_ATO, tau_ATO)`	ATO
Control weighted and unweighted	`(e_bar_ATC, tau_ATC)`	ATC

Local CP plot

The same geometry holds under the complier distribution. The complier-weighted and encouraged-complier weighted lines intersect at the local ATT; the encouraged-complier and unencouraged-complier weighted lines intersect at the local ATO; and the unencouraged-complier and complier-weighted lines intersect at the local ATC.

Pair of local CP lines	Vertical coordinate
Complier weighted and encouraged-complier weighted	`tau_ATT^c`
Encouraged-complier and unencouraged-complier weighted	`tau_ATO^c`
Unencouraged-complier and complier weighted	`tau_ATC^c`

Already have fitted nuisance functions?

Pass fitted columns directly. This is useful after causal forests, BART, SuperLearner, xgboost, or your own estimator.

cp_plot(
  data = df,
  e_hat = "e_hat",
  tau_hat = "tau_hat",
  treatment = "z"
)

local_cp_plot(
  data = df,
  e_hat = "e_hat",
  tau_c_hat = "tau_c_hat",
  pi_c_hat = "pi_c_hat",
  iv = "z"
)

See the API reference for every argument, helper functions, and bundled paper-demo data.

Citation

citation("CPplot")

Tian, P., Yang, F., & Ding, P. (2026). Introducing the CP-plot for Causal Inference with Observational Studies. arXiv preprint arXiv:2606.11715.