Performance Assessment for Continuous and Survival Outcomes

Evaluates prediction accuracy for continuous outcomes and the predictive performance of survival models in a unified framework. Implements the R-squared, L-squared, pseudo R-squared and related explained variation measures of Li and Wang (2019) , the concordance indices of Harrell et al. (1982) and Uno et al. (2011) , the time-dependent area under the receiver operating characteristic curve (AUC) of Heagerty, Lumley and Pepe (2000) , and the Brier score of Graf et al. (1999) . Supports right-censored data; competing risks, following Wolbers et al. (2014) , Wu and Li (2018) and Zhuang et al. (2025) ; and two-phase sampling designs such as case-cohort and nested case-control studies, following Lee, Zeleniuch-Jacquotte and Liu (2021) . Revives and expands the original 'PAmeasures' package with a unified interface for survival prediction assessment.

PAmeasures

PAmeasures evaluates prediction accuracy for continuous outcomes and predictive performance across right-censored survival data, competing risks, and two-phase sampling designs (case-cohort and nested case-control).

This repository contains PAmeasures 2.0.0, extending the original package with survival evaluation methods adapted from TimeMetric. Continuous-outcome accuracy is available through pa_nlm() and the supported historical pam.nlm() entry point. All twenty primary interfaces use the pa_ prefix. The historical pam.censor(), pam.coxph() and pam.survreg() interfaces are not exported; use the corresponding survival capabilities through the pa_ interfaces. See the API transition guide for compatibility details and the migration record for source provenance and historical validation.

It separates prediction from evaluation: the evaluation functions take predicted survival probabilities or cumulative incidence functions from any source, so a model fitted outside the package can be assessed with the same metrics as one fitted by it.

Installation

Install the version in this repository from GitHub:

# install.packages("remotes")
remotes::install_github("Larryzza/PAmeasures")

Quick start

For observed and predicted continuous outcomes from any regression model:

library(PAmeasures)
y <- c(1, 2, 4, 5)
prediction <- c(1, 2, 3, 4)
pa_nlm(y, prediction)   # unrounded numeric R.squared and L.squared
pam.nlm(y, prediction)  # historical four-decimal character format

These functions use affine correction of the supplied predictions, retaining the original PAmeasures R-squared and L-squared definitions. Undefined denominators produce a warning and an NA for the affected measure; see ?pa_nlm for constant predictions, perfect predictions and input requirements.

For right-censored survival predictions:

library(PAmeasures)

# simulate right-censored data and fit a model
d   <- pa_sim_cox_weibull(n = 200, pi_c = 0.3, v = 2,
                          beta = c(0.5, -0.5), seed = 1001)
d   <- d[, c("time", "status", "x1", "x2")]
fit <- survival::coxph(survival::Surv(time, status) ~ x1 + x2,
                       data = d, x = TRUE, y = TRUE)

# predicted survival probabilities over the observed time grid
pred <- pa_predict_coxph(model = fit, covs = c("x1", "x2"), new_data = d)

# evaluate
pa_summarize(list(cox = pred))
#>            Metric  cox
#> 1       pseudo_r2 0.24
#> 2 pseudo_r2_point 0.16
#> 3       harrell_c 0.66
#> 4           uno_c 0.66
#> 5     brier_score 0.21
#> 6          td_auc 0.74
#> 7             ici 0.03

Metrics

pa_metric_names() returns the canonical identifiers used by the evaluation interfaces and emitted in the Metric column; each setting supports a subset:

Not every metric applies to every setting: competing risks use c_index rather than the concordance indices defined for right-censored data. Calibration is available for ordinary right-censored and competing-risks cohorts, and for right-censored two-phase sampling designs. Default evaluation and comparison tables include ici; request metrics = c("ici", "e50", "e90") for all three, or metrics = "all" for these alongside the other supported metrics.

Calibration at a specified time

The calibration functions accept predicted event probabilities from any source, without the original fitted model. t_star is a single evaluation horizon in the same units as follow-up; repeat the call for multiple horizons. ICI averages calibration error across subjects, not across follow-up time.

# Ordinary right-censoring: p = 1 - S(t_star) for each subject.
t_star <- 1
p <- 1 - as.numeric(pec::predictSurvProb(fit, newdata = d, times = t_star))
cal <- pa_calibration(p, d$time, d$status, t_star)
cal$metrics                   # ici, e50, e90, without display rounding
head(cal$curve)               # predicted and calibrated, in subject order

# Add all three to a model comparison table.
pa_summarize(list(cox = pred), t_star = t_star,
             metrics = c("ici", "e50", "e90"), digits = 4)

For competing risks, supply the target-cause CIF and retain all other event types in status (0 denotes censoring):

# A runnable example with known exponential cause-specific hazards.
set.seed(29)
rate1 <- exp(.5 * d$x1 - .5 * d$x2)
target <- stats::rexp(nrow(d), rate1)
competing <- stats::rexp(nrow(d), .4)
censor <- stats::rexp(nrow(d), .3)
cr_time <- pmin(target, competing, censor)
cr_status <- ifelse(censor < pmin(target, competing), 0L,
                    ifelse(target <= competing, 1L, 2L))
t_star <- 1
p_cif <- rate1 / (rate1 + .4) * (1 - exp(-(rate1 + .4) * t_star))
cal_cr <- pa_calibration_cr(p_cif, cr_time, cr_status,
                            t_star = t_star, event_type = 1)
cal_cr$metrics

Right-censored curves use pmcalibration::pmcalibration() with restricted cubic splines (default five knots, controlled by nk). Competing-risks curves use prodlim::prodlim() and its nearest-neighborhood Aalen--Johansen estimator, controlled by bandwidth. Both summarize errors with pmcalibration::cal_metrics(), exposing only ICI (Eavg in the backend), E50, and E90. This competing-risks curve estimator differs from Fine-Gray spline calibration. Constant right-censored predictions use marginal Kaplan--Meier risk; small or few-valued samples use simpler calibration as documented in ?pa_calibration.

The matrix evaluators interpret supplied probability grids as right-continuous steps and use the last grid value at or before t_star for calibration. pa_calibration() and pa_calibration_cr() accept probabilities already evaluated exactly at the requested time. Exact 0/1 probabilities are retained; the right-censored backend moves them slightly inward for fitting only and warns. A failed curve fit warns and produces missing calibration values in the evaluation table without dropping the other metrics.

Use held-out or out-of-fold predictions where possible. Calibration assumes the respective estimator's censoring conditions; the competing-risks censoring/censor_covariates arguments for generalized R-squared do not change the calibration estimator.

For a right-censored case-cohort or NCC validation sample, provide the positive inverse-inclusion weights for the sampled subjects in prediction order:

cal_w <- pa_calibration(p, event_time, status, t_star,
                        case_weights = sampling_weights)
cal_w$metrics
# Calibration-only evaluation needs no separate censoring fit or mean times.
pa_sample_design(list(external = prediction), case_weights = sampling_weights,
                 metrics = "ici", t_star = t_star)

The weighted implementation fits its own Cox calibration model, using the same complementary log-log risk transformation and spline-knot rule as the ordinary backend. Both coefficients and baseline cumulative hazard use sampling weights; ICI then averages absolute errors with those weights. Constant risks use weighted Kaplan-Meier. E50/E90 use normalized weighted order-statistic interpolation. calibration_args = list(nk = 3) controls the two-phase curve. Weights must represent inclusion under the actual sampling mechanism, rather than combined sampling/censoring weights or risk-set weights from a different estimator. This assumes an appropriate calibration Cox model and conditionally independent censoring given its predictor. No sampling-adjusted competing-risks calibration or design-based confidence intervals are implemented.

Functions

Continuous outcomes: pa_nlm(y, y.predict) returns numeric R-squared and L-squared. pam.nlm() is the sole supported historical function-name exception.

Evaluation — take predictions, return a Metric/Value table:

Prediction — produce the input the evaluators expect:

Summaries and weights:

Plotting and simulation: pa_plot_pred(), pa_plot_summary(), pa_sim_cox_weibull(), pa_simulate_fine_gray().

Two-phase designs

Case-cohort and nested case-control evaluation weights each subject by the inverse of its sampling probability:

set.seed(2001)
in_subcohort <- rbinom(nrow(d), 1, 0.4) == 1
w <- pa_case_cohort_weights(d$time, d$status, subcohort = in_subcohort)
keep <- which(in_subcohort | d$status == 1)
keep <- keep[order(d$time[keep])]
cc_pred <- pa_predict_coxph(fit, covs = c("x1", "x2"), new_data = d[keep, ])
km <- survival::survfit(survival::Surv(d$time, 1 - d$status) ~ 1)
pa_evaluate_two_phase(pred_results = cc_pred, km_cens_fit = km,
                       case_weights = w[keep])

pa_nested_case_control_weights(time, status, m = 2) computes inverse-selection weights for two controls per case. Supply matching strata via strata when needed, subset to the subjects actually sampled, and align weights with the prediction order.

Optional backends

pa_predict_cif() accepts alternative model types through suggested packages:

• cmprsk for a Fine-Gray fit, via the fg_model argument
• randomForestSRC for a survival forest, via the cr_model argument

Both are optional; the function reports which package to install if one is needed and absent.

Function naming

All 20 primary functions use pa_. The former 19 tm_ entry points map directly to the corresponding pa_ names, with the same arguments and survival calculations. For example, use pa_survival_eval() and pa_predict_coxph(). The earlier 15 TimeMetric script aliases have been removed. pam.nlm() is retained as the sole historical public alias, with its four-decimal character format. See the API transition guide.

Paper analyses and figures

To rerun every simulation and all three real-data analyses with one command, run from this checkout:

# Check dependencies and historical PBC source without fitting anything.
Rscript --vanilla figures/run_all.R

# Refit real-data models, then run all four full simulation parts (100 repeats).
Rscript --vanilla figures/run_all.R --mode=full --output=results/reruns/pa_full

The unified runner loads this checkout's PAmeasures code. Outputs are grouped under the selected directory: real_data/ for fits and diagnostics, paper/ for real-data manuscript figures/tables, and simulations/ for simulation checkpoints, metrics and figures. run_all_status.csv records both exit codes; a failed real-data step does not prevent the simulation step from running. Existing global manuscript exports and manuscript files are not replaced. Use a new output directory for a fresh run or after computation sources or package versions change. Reusing an unchanged run resumes simulation checkpoints; the real-data analyses are refitted each time. Simulation 3 can return a nonzero status for unestimable ICI repetitions even when Figure 6 is produced; inspect its diagnostics and finite-count file as described in the simulation notes.

In RStudio, from this checkout:

source("figures/run_all.R")
run_all() # check only
run_all(mode = "full", output = "results/reruns/pa_full")

The runner propagates .libPaths() to its child R sessions. For a dependency in a separate library, add that directory to .libPaths() in RStudio, or set R_LIBS=/path/to/library before Rscript in the terminal.

Run these commands from this checkout. They load its local package code rather than an older installed PAmeasures version. The analyses also need pkgload, cmprsk, randomForestSRC, and riskCommunicator (the Framingham data source).

# Recompute all real-data analyses and export their manuscript figures/tables.
Rscript --vanilla figures/run_real_data.R

# Add ICI to saved PBC/FHS results, preserving existing fits and other metrics.
# The GBSG sampling-design result is carried forward unchanged.
Rscript --vanilla figures/reevaluate_real_calibration.R --output=results/real_data/runs/cohort_ici

# Publication is deferred until GBSG is ready. Pass the cohort output explicitly.
# Use new output directories; existing runs are never overwritten.
Rscript --vanilla figures/reevaluate_sample_calibration.R --input=results/real_data/runs/cohort_ici --output=results/real_data/runs/all_ici

# Run or resume the full simulation including ICI.
Rscript --vanilla figures/run_simulations.R --mode=full --output=results/simulations/ici_full

# Redraw completed results only, without refitting or rerunning simulations.
Rscript --vanilla figures/redraw_original.R
# Optional: --section=real or --section=simulations
# For the new ICI run, add --sim-output=results/simulations/ici_full

The presentation entry points in figures/ preserve the selected manuscript .updated.pdf style, including R/L component dots in Simulation 2. The lower-level drivers in analysis/ produce diagnostic plots; use the redraw entry afterward if those drivers were run directly. paper.code.Rmd displays the completed results through the same figure/table outputs; it does not rerun models or simulations. With rmarkdown, knitr and Pandoc available, render it with:

rmarkdown::render("paper.code.Rmd", output_dir = "results/report")
# Optional: params = list(redraw_figures = TRUE)
# For ICI simulations: params = list(simulation_output = "results/simulations/ici_full")

Rendering requires all four full simulation parts and all three real-data exports; smoke or incomplete results are rejected.

The migrated manuscript figure-refresh driver is manuscript/update_all_figures.R. It accepts an explicit manuscript directory and saved-result paths; see manuscript/README.md. Historical results and clinical inputs are not distributed in this repository. Existing TimeMetric checkpoint signatures differ after migration, so use a new output directory for new runs. paper.md contains the software-paper draft.

Local outputs, excluded from Git:

The recorded current real-data run is selected through results/real_data/paper/source_manifest.csv; redraw and FHS horizon commands read that record rather than a historical directory. FHS uses examination 3 as time zero and a 12-year horizon (4380 days). To re-evaluate its existing training fit at a selected horizon (changed source/settings create a separate output):

Rscript --vanilla figures/run_fhs_horizon.R --years=12

Keep full simulation models and checkpoints for resumption. Statistical inputs in R/, analysis/ and DESCRIPTION are fingerprinted; changes require a new simulation output directory rather than silently mixing versions. Smoke runs are optional, write to a separate directory, and are not manuscript results. The existing simulation designs are all right-censored. They now include ICI with a five-knot Cox spline calibration curve at the existing t_star for each job. E50/E90 are not included. ICI is retained without rounding for aggregation. Per-model horizons, prediction-grid times, calibration methods and errors are written to tables/calibration_diagnostics.csv. In Simulation 3, ICI follows the existing large-sample uncensored reference and error-summary calculation. Old checkpoints without ICI are not treated as complete, and the updated report requires ICI rows rather than silently showing historical results.

Only display the plan, without fitting or evaluating anything:

Rscript --vanilla figures/run_simulations.R --mode=plan

To select a part or a repetition range, append --part=sim2_linear,sim2_nonlinear or --repeats=1:10 to the full-run command. Repeat the same command to resume an unchanged run. Before launching, ensure pmcalibration, Hmisc and the other package dependencies are available in the R library used by Rscript. See simulation launch instructions for terminal and RStudio commands.

To run package checks without creating paper results:

Rscript --vanilla -e 'testthat::test_local(".", reporter="summary")'

Contributors

Developed by Li's Lab (UCLA): Tong Zhu, Zian Zhuang, Wen Su, Xiaowu Dai and Gang Li.

The original PAmeasures 0.1.0 was authored by Xiaoyan Wang and Gang Li; their contributions are retained in the package authorship and provenance records.

License

PAmeasures is distributed under GPL-3. Historical TimeMetric MIT notices are preserved in LICENSE.md and inst/licenses/TimeMetric-MIT.txt; see inst/COPYRIGHTS for the source lineage.

References

Brier, G. W. (1950). Monthly Weather Review 78(1), 1–3. Graf, E., Schmoor, C., Sauerbrei, W. & Schumacher, M. (1999). Statistics in Medicine 18(17–18), 2529–2545. Harrell, F. E., Califf, R. M., Pryor, D. B., Lee, K. L. & Rosati, R. A. (1982). JAMA 247(18), 2543–2546. Heagerty, P. J., Lumley, T. & Pepe, M. S. (2000). Biometrics 56(2), 337–344. Li, G. & Wang, X. (2019). JASA 114(528), 1815–1825. Uno, H., Cai, T., Pencina, M. J., D'Agostino, R. B. & Wei, L. J. (2011). Statistics in Medicine 30(10), 1105–1117. Zhuang, Z., Su, W., Kawaguchi, E. & Li, G. (2025). arXiv:2507.15040.