Cross-validated Brier score for competing risks, RMST and RMTL regression using stratified leave-fold-out fits (binregStrata)

Fits candidate models via binregStrata with cv.fold = TRUE and evaluates each using apparent (in-sample) and k-fold cross-validated IPCW Brier scores relative to a null model. Supports all outcome types available in binreg/binregStrata: cause-specific cumulative incidence (competing risks), restricted mean survival time (RMST), and restricted mean time lost (RMTL).

Usage

brier_binreg(
  formula,
  data,
  time,
  fold = 5,
  rhs = NULL,
  rhs0 = ~+1,
  cens.model = ~+1,
  brier.cens.model = ~+1,
  brier.args = list(),
  cv.split.index = NULL,
  ...
)

Arguments

formula: Two-sided formula of the form Event(time, status) ~ covariates + cluster(id) or Event(time, status, cause) ~ covariates + cluster(id). The cluster(id) term is optional; if absent an internal identifier id__ is added automatically assuming independent observations.
data: A data.frame containing all variables in formula and rhs.
time: Scalar or numeric vector of evaluation time points. When a vector is supplied the function loops over time points reusing the same fold structure and returns a named list of binregCV objects with class "binregCV_list".
fold: Integer. Number of cross-validation folds / strata (default 5).
rhs: Named list of one-sided formulas specifying candidate models, e.g. list(small = ~Z1, full = ~Z1 + Z2). If NULL (default), the right-hand side of formula (minus any cluster() term) is used as a single candidate named "model1".
rhs0: One-sided formula for the null (intercept-only) model (default ~+1).
cens.model: Censoring model passed to binregStrata for the CV fold fits (default ~+1). Since folds are handled internally by binregStrata's strata/cv.fold mechanism, this does not need to include strata(fold).
brier.cens.model: Censoring model passed to resmeanIPCW when computing Brier scores. Default ~+1 gives Kaplan-Meier weights. Supply a stratified model, e.g. ~strata(Z1, Z2), for covariate-adjusted IPCW weights.
brier.args: Named list of additional arguments forwarded to every resmeanIPCW call (beyond formula, data, time, Ydirect, cens.model, and model which are set internally). Typically empty.
cv.split.index: List from folds(n, fold) or a vector of fold indices (1:fold) for each observation, to fix the cross-validation folds. NULL (default) generates folds internally via folds(n, fold).
...: Additional arguments passed to every binreg / binregStrata call, e.g. cause = 2, outcome = "rmst", or offset.

Value

An object of class "binregCV" (single time point) or "binregCV_list" (multiple time points), with the following structure:

null: List with coef, se, and iid for the null model. coef and se are named vectors with entries bbrier (apparent, log scale) and bbrierCV (cross-validated, log scale). iid contains the influence-function matrix for bbrierCV with delta-method correction for the stratified CV prediction step and the log transform.
models: Named list, one element per entry in rhs. Each element contains coef (brier, brierCV, log scale), se, dcoef (differences from null: log Brier ratios), sed, and iid (influence-function matrix for brierCV, log scale, with CV delta-method correction).
iid_mat: Matrix of dimension \(n \times (1 + \code{length(rhs)})\) with columns null and one per entry of rhs, each the log-scale CV Brier influence function for the corresponding model. Used by estimate.binregCV for joint inference.
cv.split.index: List of fold index vectors used.
rhs, rhs0: Processed candidate and null formulas (cluster terms removed).
model: Link function read from the binreg fit: "logit", "exp", or "lin".
call, time, fold, cens.model, brier.cens.model: Call and settings for printing and downstream use.

Details

Cross-validation is delegated directly to binregStrata via its strata and cv.fold = TRUE arguments: each stratum (fold) gets its own coefficient vector \(\beta_s\), estimated on the complement of that fold. Out-of-fold predictions are obtained by routing each observation to the \(\beta_s\) corresponding to its own fold, which was trained on the other folds.

The fold membership is stored in an internal fold column (1-based) in data and passed to binregStrata as a 0-based strata vector. Censoring weights for the underlying competing-risks model are estimated according to cens.model as usual; since strata already define the folds, cens.model need not include strata(fold) explicitly.

Brier scores are computed on the linear scale via resmeanIPCW(..., model = "lin") and then log-transformed using lava::estimate(..., f = log) with the delta method. All stored coefficients, standard errors, and influence functions are therefore on the log scale. Use summary(fit, transform = exp) or estimate(fit, f = exp) to back-transform to the original Brier scale.

The delta-method correction for the cross-validated influence function accounts for the block structure of binregStrata's influence function (\(n_{id} \times (\code{nstrata} \times p)\), one \(p\)-column block per stratum). For each stratum \(s\), only observations belonging to fold \(s\) have a non-zero derivative of their out-of-fold prediction with respect to \(\beta_s\), since \(\beta_s\) is exactly the coefficient used to predict fold \(s\). The full gradient is therefore block-sparse, with one non-zero \(p\)-length block per stratum, and the correction reduces to a single matrix product against iid(outff_fit).

Left-truncation (delayed entry) is not supported and will raise an error. Specifically, Event(entry, time, status) is rejected; use Event(time, status) or Event(time, status, cause).

Outcome types

The function supports all outcome types provided by binreg/ binregStrata:

Competing risks CIF: Default; use Event(time, status, cause) and pass cause via .... The Brier score measures squared error for the cause-specific CIF.
RMST: Pass outcome = "rmst" via .... The Brier score measures squared error for the restricted mean survival time up to time.
RMTL: Pass outcome = "rmtl" via .... The Brier score measures squared error for the restricted mean time lost.

Scale and inference

All output is on the log scale. To obtain Brier scores on the original scale with confidence intervals, use:


  summary(fit, transform = exp)
  estimate(fit, f = exp)

Differences on the log scale correspond to log Brier ratios; exponentiate to obtain Brier ratios. For absolute differences on the original scale, compute contrasts after estimate(fit, f = exp).

Examples

if (FALSE) { # \dontrun{
library(mets)
data(bmt)
bmt$id <- seq_len(nrow(bmt))

## --- competing risks CIF/Survival (default) ---
fit <- brier_binreg(
  Event(time, cause) ~ tcell + platelet + age + cluster(id),
  data = bmt, time = 50,
  rhs = list(small = ~age, full = ~tcell + platelet + age)
)
summary(fit)                        ## log scale
summary(fit, transform = exp)       ## Brier scale

## multiple time points
fitm <- brier_binreg(
  Event(time, cause) ~ tcell + platelet + age + cluster(id),
  data = bmt, time = c(50, 60),
  rhs = list(small = ~age, full = ~tcell + platelet + age)
)
summary(fitm, transform = exp)

## joint inference across models via lava::estimate
e <- estimate(fitm)
e
estimate(e, f = exp)               ## back-transform

## stratified IPCW weights for Brier score
fit_strat <- brier_binreg(
  Event(time, cause) ~ tcell + platelet + age + cluster(id),
  data = bmt, time = 50,
  rhs = list(small = ~age, full = ~tcell + platelet + age),
  brier.cens.model = ~strata(tcell)
)

## --- RMST --- survival case only
fit_rmst <- brier_binreg(
  Event(time, cause) ~ tcell + platelet + age + cluster(id),
  data = bmt, time = 50, cause = 1:2,
  rhs    = list(small = ~age, full = ~tcell + platelet + age),
  outcome = "rmst"
)
summary(fit_rmst, transform = exp)

## --- RMTL --- competing risks or survival (RMTL)
fit_rmtl <- brier_binreg(
  Event(time, cause) ~ tcell + platelet + age + cluster(id),
  data = bmt, time = 50,
  rhs    = list(small = ~age, full = ~tcell + platelet + age),
  outcome = "rmtl"
)
summary(fit_rmtl, transform = exp)
} # }

Cross-validated Brier score for competing risks, RMST and RMTL regression using stratified leave-fold-out fits (`binregStrata`)