Uniform inference for ranked subgroups

A scoring rule ranks the evaluation population before outcomes are used to choose a subgroup size. URATE() estimates the effect among the highest-ranked fraction and constructs a uniform confidence band over all subgroup sizes. The rule may be fitted on independent training data. Inference conditions on that realized rule and does not require it to consistently estimate CATE. The evaluation data must satisfy the manuscript’s random-sampling and complete- randomization assumptions. Cross-validated scores require a different argument and are not supported by this uniform-band procedure.

library(evalHTE)
set.seed(91)
n <- 120
score <- rnorm(n)  # a prespecified rule in this simulated example
D <- sample(rep(0:1, each=n/2))
Y <- rnorm(n) + D*(1+score)
band <- URATE(D, score, Y, alpha=.1, L=1000, seed=801)
k <- band$selected
data.frame(n_selected=k,proportion=band$p[k],effect=band$rate[k],
           lower=band$lower[k])
#>   n_selected proportion   effect    lower
#> 1         30       0.25 2.261076 0.951516

The selected size maximizes the uniform lower bound, with ties resolved in favor of the smallest subgroup. The uniform guarantee permits this selection. A pointwise interval or p-value at that size would not provide the same post-selection guarantee. Coverage applies to one rule and outcome; choosing among multiple models or covariates is not covered by that guarantee.

plot_CI(band)

The plot uses the stored calibration. Supplying alpha to plot_CI() or to summary() for an evaluated HTE object recalibrates the same reference with the same seed. Two-sided bands require side="two.sided" at evaluation and are calibrated using absolute Gaussian paths. An upper band uses side="upper".

For a baseline covariate, use D=NULL. The following curve compares each subgroup’s covariate mean with the evaluation population mean. The endpoint contrast is zero by construction. Selection based on this curve minimizes the upper bound; evaluating a subgroup selected by another curve instead uses its rank index directly.

X <- score + rnorm(n)
composition <- URATE(NULL, score, X, adjustment="difference", side="upper",
                     alpha=.1, L=1000, seed=2801)
c(estimate=composition$rate[k],upper=composition$upper[k])
#> estimate    upper 
#> 1.398961 1.873475

To use fitted models, retain an independent evaluation sample through the existing sample-splitting workflow:

fit <- estimate_hte(treatment="D",form=Y ~ D + X,data=training_and_evaluation,
                    algorithms="causal_forest",split_ratio=.4,ngates=5)
result <- evaluate_hte(fit,alpha=.1,L=50000,seed=801)
summary(result)
plot_CI(result)

URATE() uses uncentered outcomes and the observed evaluation treatment fraction. It estimates the marginal effect in consecutive rank blocks with a finite-block residual correction. Gaussian draws retain the dependence from the estimated ranking threshold and fixed treatment counts. Each endpoint contrast transforms the entire draw before calibration.

The default objective minimizes cumulative boundary area. objective="gates" minimizes the area of the GATES band above p_min; this integration limit does not restrict the coverage or selection domain. The finite slope grid and Monte Carlo quantiles approximate the asymptotic procedure. A warning at the largest slope calls for a wider grid, and final analyses should increase L.

The revised prostate application uses alpha=.1, L=50000, the cumulative objective, and model-specific seeds 801 through 804. Exact replication also requires the paper’s saved split, preprocessing, fitted scores, and model settings. The generic training defaults do not reproduce those fitted models. The paper adds a fixed Gaussian perturbation to its scores before evaluation; pass those saved scores without adding another perturbation. Package tie handling randomizes exact ties only. Neither evaluation nor plotting changes the caller’s random-number state.