---
title: "Random Forest Survival Analysis with ggRandomForests"
author: "John Ehrlinger"
date: today
format:
  html:
    fig-format: png
    fig-dpi: 96
    toc: true
    toc-depth: 3
    html-math-method: mathjax
    code-fold: false
bibliography: ggRandomForests.bib
vignette: >
  %\VignetteIndexEntry{Random Forest Survival Analysis with ggRandomForests}
  %\VignetteEngine{quarto::html}
  %\VignetteEncoding{UTF-8}
---

::: {.callout-warning}
## Work in progress
This vignette is under active development. Code examples and narrative may
change before the next release.
:::

```{r setup, include=FALSE}
knitr::opts_chunk$set(
  fig.align = "center",
  fig.width = 7,
  fig.height = 5,
  message = FALSE,
  warning = FALSE,
  comment = "#>"
)
options(mc.cores = 1, rf.cores = 1)
```

# Introduction

Random forests [@Breiman:2001] are a non-parametric ensemble method that
requires no distributional assumptions on the relation between covariates and
the response. Random survival forests (RSF) [@Ishwaran:2007a; @Ishwaran:2008]
extend the method to right-censored, time-to-event data by growing trees with a
log-rank splitting rule and aggregating Kaplan--Meier estimates within terminal
nodes.

The forest returns three ensemble quantities for each subject. First, the
survival function $\hat{S}(t)$: the probability of surviving past time $t$,
bounded between 0 and 1. Second, the cumulative hazard function (CHF):
$\hat{H}(t) = -\log \hat{S}(t)$, an unbounded, monotone-increasing summary of
accumulated risk. Third, mortality: the expected number of events a subject
would accumulate if followed indefinitely under their covariate profile,
computed by summing the CHF over the observed time grid. Mortality is not a
probability; it is an unbounded relative-risk score. Read it as a single number
for ranking patients from low to high risk, not as a survival chance.

The **[randomForestSRC](https://www.randomforestsrc.org)** package
[@Ishwaran:RFSRC:2014] provides a unified implementation for survival,
regression, and classification forests.
**ggRandomForests** extracts tidy data objects from `rfsrc` fits and renders
them with **ggplot2** [@Wickham:2009], so you can see how a survival forest is
built, which predictors carry the risk, and how survival shifts across each one.

This vignette demonstrates a complete random survival forest workflow on the
primary biliary cirrhosis (PBC) data set [@fleming:1991]:

1. **Data preparation and exploration** --- cleaning, EDA, Kaplan--Meier curves
2. **Growing the forest** --- fitting an RSF, checking convergence and OOB error
3. **Variable selection** --- VIMP and minimal depth via `max.subtree()`
4. **Dependence plots** --- variable dependence and partial dependence via
   `gg_variable()` and `gg_partial_rfsrc()`
5. **Variable interactions** --- conditioning plots and partial dependence
   surfaces

```{r packages}
library(ggplot2)
library(dplyr)
library(tidyr)
library(randomForestSRC)
library(survival)

if (requireNamespace("ggRandomForests", quietly = TRUE)) {
  library(ggRandomForests)
} else if (requireNamespace("pkgload", quietly = TRUE)) {
  pkgload::load_all(export_all = FALSE, helpers = FALSE, attach_testthat = FALSE)
} else {
  stop("Install ggRandomForests (or pkgload for dev builds) to render this vignette.")
}

theme_set(theme_bw())

# Plotting constants
event_marks  <- c(1, 4)
event_labels <- c("Censored", "Death")
event_colors <- c("steelblue", "firebrick")
```

# Data: Primary Biliary Cirrhosis (PBC)

The PBC study consists of 424 patients referred to Mayo Clinic between 1974 and
1984, of whom 312 were randomized into a trial of D-penicillamine (DPCA) versus
placebo. The data is described in @fleming:1991 Chapter 0.2, with a proportional
hazards model developed in Chapter 4.4. We use the copy bundled with
**randomForestSRC**.

```{r data-load}
data("pbc", package = "randomForestSRC")
```

## Data cleaning

We convert `days` to `years`, recode `treatment` as a factor, and coerce
low-cardinality numeric columns (five or fewer unique values, including binary
0/1 indicators) to factors. We avoid converting binary columns to `logical`
because `randomForestSRC::partial.rfsrc()` does not handle logical predictors
correctly in survival forests.

```{r data-clean}
pbc <- pbc |>
  mutate(
    years     = days / 365.25,
    age       = age / 365.25,
    treatment = factor(
      ifelse(treatment == 1, "DPCA",
             ifelse(treatment == 2, "Placebo", NA)),
      levels = c("DPCA", "Placebo")
    )
  ) |>
  select(-days)

# Low-cardinality numerics (including binary 0/1) to factor.
# NOTE: do NOT convert to logical — partial.rfsrc() fails with logical
# predictors in survival forests (randomForestSRC <= 3.5.1).
# Exclude the response columns (status, years) from conversion.
resp_cols <- c("status", "years")
for (nm in setdiff(names(pbc), resp_cols)) {
  v <- pbc[[nm]]
  if (is.numeric(v) && !is.factor(v) && length(unique(v[!is.na(v)])) <= 5) {
    pbc[[nm]] <- factor(v)
  }
}
```

```{r variable-labels}
# Human-readable labels for plot axes
st_labs <- c(
  status       = "Death Event",
  treatment    = "Treatment",
  age          = "Age (years)",
  sex          = "Female",
  ascites      = "Ascites",
  hepato       = "Hepatomegaly",
  spiders      = "Spiders",
  edema        = "Edema (0, 0.5, 1)",
  bili         = "Serum Bilirubin (mg/dl)",
  chol         = "Serum Cholesterol (mg/dl)",
  albumin      = "Albumin (gm/dl)",
  copper       = "Urine Copper (ug/day)",
  alk.phos     = "Alkaline Phosphatase (U/liter)",
  ast          = "SGOT (U/ml)",
  trig         = "Triglycerides (mg/dl)",
  platelet     = "Platelets (per cubic ml/1000)",
  prothrombin  = "Prothrombin Time (sec)",
  stage        = "Histologic Stage",
  years        = "Follow-up Time (years)"
)
```

## Exploratory data analysis

Good practice before modeling: scan categorical variables as stacked histograms
over follow-up time, and continuous variables as scatter plots colored by event
status.

```{r eda-categorical, fig.height=4, fig.cap="EDA for categorical variables. Bar color indicates factor level; white = missing."}
cls <- sapply(pbc, class)
cnt_idx <- which(cls %in% c("numeric", "integer"))
fct_idx <- setdiff(seq_along(pbc), cnt_idx)
fct_idx <- union(fct_idx, which(names(pbc) == "years"))

dta_cat <- suppressWarnings(
  pbc[, fct_idx] |>
    pivot_longer(-years, names_to = "variable", values_to = "value",
                 values_transform = list(value = as.character))
)

ggplot(dta_cat, aes(x = years, fill = value)) +
  geom_histogram(color = "black", binwidth = 1) +
  labs(y = "", x = st_labs["years"]) +
  scale_fill_brewer(palette = "RdBu", na.value = "white") +
  facet_wrap(~variable, scales = "free_y", nrow = 2) +
  theme(legend.position = "none")
```

```{r eda-continuous, fig.height=4, fig.cap="EDA for continuous variables. Points colored by death event; rug marks indicate missing values."}
cnt_idx <- union(cnt_idx, which(names(pbc) == "status"))
dta_cont <- pbc[, cnt_idx] |>
  pivot_longer(c(-years, -status),
               names_to = "variable", values_to = "value")

ggplot(dta_cont |> filter(!is.na(value)),
       aes(x = years, y = value, color = factor(status), shape = factor(status))) +
  geom_point(alpha = 0.4) +
  geom_rug(data = dta_cont |> filter(is.na(value)), color = "grey50") +
  labs(y = "", x = st_labs["years"], color = "Death", shape = "Death") +
  scale_color_manual(values = event_colors) +
  scale_shape_manual(values = event_marks) +
  facet_wrap(~variable, scales = "free_y", ncol = 4) +
  theme(legend.position = c(0.8, 0.2))
```

Look for patterns of missingness (white bars, rug marks) and extreme values that
fall outside the biological range.

## Kaplan--Meier survival by treatment

The Kaplan--Meier estimate is the marginal survival curve with no covariate
adjustment. It serves as the no-covariate baseline: once we fit a forest,
comparing the KM curves to the forest's OOB predictions lets us judge how much
predictive signal the covariates add beyond the raw event rate.

We restrict to the 312 trial patients and construct KM curves with `gg_survival`.

```{r km-survival, fig.cap="KM survival estimates by treatment group with 95% confidence bands."}
pbc_trial <- pbc |> filter(!is.na(treatment))
pbc_test  <- pbc |> filter(is.na(treatment))

gg_km <- gg_survival(interval = "years", censor = "status",
                     by = "treatment", data = pbc_trial,
                     conf.int = 0.95)

plot(gg_km) +
  labs(y = "Survival Probability", x = "Time (years)",
       color = "Treatment", fill = "Treatment") +
  theme(legend.position = c(0.2, 0.2)) +
  coord_cartesian(ylim = c(0, 1.01))
```

The curves largely overlap, consistent with the original finding that DPCA
offered no clear survival benefit over placebo [@fleming:1991].

```{r km-cumhaz, fig.cap="KM cumulative hazard estimates by treatment group."}
plot(gg_km, type = "cum_haz") +
  labs(y = "Cumulative Hazard", x = "Time (years)",
       color = "Treatment", fill = "Treatment") +
  theme(legend.position = c(0.2, 0.8)) +
  coord_cartesian(ylim = c(-0.02, 1.22))
```

We can also stratify on continuous variables. Here we reproduce the bilirubin
groupings from @fleming:1991 Figure 4.4.2:

```{r km-bili, fig.width=6, fig.cap="KM survival stratified by bilirubin groups."}
pbc_bili <- pbc_trial |>
  mutate(bili_grp = cut(bili, breaks = c(0, 0.8, 1.3, 3.4, 29)))

plot(gg_survival(interval = "years", censor = "status",
                 by = "bili_grp", data = pbc_bili),
     error = "none") +
  labs(y = "Survival Probability", x = "Time (years)", color = "Bilirubin")
```

Higher bilirubin strongly predicts worse survival --- an effect the random forest
will rediscover without any prior specification.

# Growing a Random Survival Forest

Several predictors in the PBC trial data contain missing values (cholesterol,
copper, triglycerides, and others). We handle these with a two-step approach:
first impute using `impute.rfsrc()`, which uses the random forest proximity
structure to fill in missing values, then fit the survival forest on the
complete imputed data. This keeps the fitted forest object free of imputation
state, which is required for `partial.rfsrc()` to work correctly.

```{r rfsrc-fit}
# Step 1: impute missing values via random forest proximity
pbc_imputed <- impute.rfsrc(Surv(years, status) ~ .,
                             data    = pbc_trial,
                             ntree   = 100,
                             nimpute = 2)

# Step 2: grow the survival forest on the complete imputed data
rfsrc_pbc <- rfsrc(Surv(years, status) ~ .,
                   data      = pbc_imputed,
                   ntree     = 150,
                   nsplit    = 10,
                   tree.err  = TRUE,
                   importance = TRUE)
```

The forest grew `r rfsrc_pbc$ntree` trees, splitting on
`r rfsrc_pbc$mtry` randomly selected candidate variables at each node, and
stopping at a minimum terminal node size of `r rfsrc_pbc$nodesize`.

## OOB error convergence

```{r error-plot, fig.height=4, fig.cap="OOB prediction error vs. number of trees."}
plot(gg_error(rfsrc_pbc))
```

The error stabilizes well before 1000 trees, indicating the forest is large
enough for reliable predictions.

## OOB predicted survival

```{r rfsrc-predicted, fig.cap="OOB predicted survival curves. Blue = censored, red = death events."}
gg_rsf <- plot(gg_rfsrc(rfsrc_pbc), alpha = 0.2) +
  scale_color_manual(values = event_colors) +
  theme(legend.position = "none") +
  labs(y = "Survival Probability", x = "Time (years)") +
  coord_cartesian(ylim = c(-0.01, 1.01))
gg_rsf
```

Each curve is one patient's OOB ensemble survival function $\hat{S}(t)$,
extended to the last follow-up time. The forest never saw that patient when
building its prediction, so these are genuine out-of-sample estimates. Red
(death event) curves generally fall faster and reach lower survival
probabilities, confirming the forest separates risk groups. Comparing this
spread to the marginal KM curves from the previous section shows how much the
covariates tighten risk stratification.

```{r rfsrc-by-treatment, fig.cap="Median predicted survival by treatment group with 95% confidence bands."}
plot(gg_rfsrc(rfsrc_pbc, by = "treatment")) +
  theme(legend.position = c(0.2, 0.2)) +
  labs(y = "Survival Probability", x = "Time (years)") +
  coord_cartesian(ylim = c(-0.01, 1.01))
```

## Test set predictions

The non-trial patients (`pbc_test`) have substantial missing data.
`predict.rfsrc()` handles these transparently at prediction time via
`na.action = "na.impute"` — this is distinct from training-time imputation
and does not affect the fitted forest object.

```{r predict-test, fig.cap="Predicted survival for non-trial patients (test set)."}
rfsrc_pbc_test <- predict(rfsrc_pbc, newdata = pbc_test,
                          na.action = "na.impute",
                          importance = TRUE)

plot(gg_rfsrc(rfsrc_pbc_test), alpha = 0.2) +
  scale_color_manual(values = event_colors) +
  theme(legend.position = "none") +
  labs(y = "Survival Probability", x = "Time (years)") +
  coord_cartesian(ylim = c(-0.01, 1.01))
```

Because the training curves use OOB estimates, both plots represent
out-of-sample predictions and are directly comparable.

# Variable Selection

Random forest uses all available predictors. To understand which matter most, we
examine variable importance (VIMP) and minimal depth.

## Variable importance (VIMP)

VIMP measures the increase in prediction error when a variable's values are
randomly permuted across the out-of-bag sample. For survival forests the
prediction error is the Harrell C-statistic complement (1 − C), so permuting
an important variable hurts C and yields a large positive VIMP. A negative VIMP
means the variable is adding noise: the forest would predict better without it.
See the [randomForestSRC](https://www.randomforestsrc.org) documentation for
details on the VIMP calculation for censored outcomes [@Ishwaran:2007a].

```{r vimp-plot, fig.width=6, fig.cap="Variable importance ranking. Blue = positive VIMP, red = negative."}
plot(gg_vimp(rfsrc_pbc), lbls = st_labs) +
  theme(legend.position = c(0.8, 0.2)) +
  labs(fill = "VIMP > 0")
```

Bilirubin ranks highest, followed by copper, prothrombin time, albumin, and age
--- closely matching the variables selected in the @fleming:1991 proportional
hazards model.

## Minimal depth

Minimal depth [@Ishwaran:2010] ranks variables by how close to the root node
they first split, on average. Variables that partition large portions of the
population early are considered most important.

```{r varsel}
md_pbc <- max.subtree(rfsrc_pbc)
```

The `max.subtree()` function computes minimal depth for each variable. The
threshold is `r round(md_pbc$threshold, 2)`, selecting
`r length(md_pbc$topvars)` variables: `r paste(md_pbc$topvars, collapse = ", ")`.

Both selection methods agree on the key predictors: `bili`, `albumin`, `copper`,
`prothrombin`, and `age`. We add `edema` (selected by the @fleming:1991 model)
for the remainder of the analysis.

```{r select-vars}
xvar     <- c("bili", "albumin", "copper", "prothrombin", "age")
xvar_cat <- "edema"
xvar_all <- c(xvar, xvar_cat)
```

# Variable Dependence

## Variable dependence plots

Variable dependence shows each patient's predicted survival probability
$\hat{S}(t_0)$ at a fixed time horizon $t_0$, plotted against a predictor of
interest. Because $\hat{S}(t_0)$ is a probability, the y-axis is always bounded
to [0, 1]. Points are colored by event status; a loess smooth traces the
overall trend while the individual points show how much variance remains after
the forest accounts for all other covariates.

```{r vardep-bili, fig.height=4, fig.cap="Variable dependence of survival at 1 and 3 years on bilirubin."}
gg_v <- gg_variable(rfsrc_pbc, time = c(1, 3),
                    time.labels = c("1 Year", "3 Years"))

plot(gg_v, xvar = "bili", alpha = 0.4) +
  labs(y = "Survival", x = st_labs["bili"]) +
  theme(legend.position = "none") +
  scale_color_manual(values = event_colors, labels = event_labels) +
  scale_shape_manual(values = event_marks, labels = event_labels) +
  coord_cartesian(ylim = c(-0.01, 1.01))
```

Survival drops sharply with increasing bilirubin, and the 3-year curve drops
further than the 1-year curve, suggesting a non-proportional hazards effect.

```{r vardep-panel, fig.height=5, fig.cap="Variable dependence at 1 and 3 years for continuous predictors."}
plot(gg_v, xvar = xvar[-1], panel = TRUE, alpha = 0.4) +
  labs(y = "Survival") +
  theme(legend.position = "none") +
  scale_color_manual(values = event_colors, labels = event_labels) +
  scale_shape_manual(values = event_marks, labels = event_labels) +
  coord_cartesian(ylim = c(-0.05, 1.05))
```

The plots confirm survival increases with albumin and decreases with copper,
prothrombin time, and age. The divergence between time curves for `copper`
further supports a non-proportional hazards mechanism.

```{r vardep-categorical, fig.height=4, fig.cap="Variable dependence on edema (categorical)."}
plot(gg_v, xvar = xvar_cat, alpha = 0.4) +
  labs(y = "Survival") +
  theme(legend.position = "none") +
  scale_color_manual(values = event_colors, labels = event_labels) +
  scale_shape_manual(values = event_marks, labels = event_labels) +
  coord_cartesian(ylim = c(-0.01, 1.02))
```

Patients with edema = 1 (edema despite diuretics) have markedly lower predicted
survival.

## Partial dependence

::: {.callout-warning}
**Known issue (draft):** `randomForestSRC::partial.rfsrc()` currently fails for
survival forests in randomForestSRC ≥ 3.3.  The partial dependence and
surface sections below will show an error until this upstream bug
is resolved.  All other sections of this vignette are fully functional.
:::

Partial dependence integrates out the effects of other covariates, giving a
risk-adjusted view of how each predictor influences the response
[@Friedman:2000]. We use `gg_partial_rfsrc()` which calls
`randomForestSRC::partial.rfsrc()` directly and returns a `gg_partial_rfsrc`
object.  For survival forests, the result includes a `time` column so
`plot(pd)` produces one curve per predictor value over time, faceted by
variable name.

```{r partial-dep, error=TRUE, fig.height=5, fig.cap="Partial dependence of survival at approximately 1 and 3 years on continuous predictors."}
# partial.rfsrc() requires times that match the model's time.interest grid;
# gg_partial_rfsrc() snaps the requested values to the nearest observed times.
ti   <- rfsrc_pbc$time.interest
t1yr <- ti[which.min(abs(ti - 1))]
t3yr <- ti[which.min(abs(ti - 3))]

pd <- gg_partial_rfsrc(rfsrc_pbc, xvar.names = xvar,
                       partial.time = c(t1yr, t3yr))

# Quick S3 plot — survival forests produce time-series curves per predictor value
plot(pd)
```

For a publication-ready layout with custom colour scale, access `pd$continuous`
directly:

```{r partial-dep-custom, error=TRUE, fig.height=5, fig.cap="Partial dependence (custom styling)."}
ggplot(pd$continuous, aes(x = x, y = yhat,
                          color = factor(round(time, 2)),
                          group = factor(time))) +
  geom_line(linewidth = 1) +
  facet_wrap(~name, scales = "free_x") +
  labs(y = "Predicted Survival", x = "", color = "Year") +
  scale_color_manual(values = setNames(c("steelblue", "firebrick"),
                                       as.character(round(c(t1yr, t3yr), 2)))) +
  theme_bw()
```

The partial dependence curves at approximately 1 and 3 years confirm the
variable dependence findings and support the log-transforms used in the
@fleming:1991 model for `bili`, `albumin`, and `prothrombin`. The divergence
between time curves for `bili` and `copper` supports non-proportional hazards.

## Conditional dependence

To investigate interactions, we can condition variable dependence on group
membership in another variable. Here we examine the dependence of survival on
bilirubin, stratified by edema group.

```{r coplot-edema, fig.width=7, fig.height=4, fig.cap="Variable dependence on bilirubin, conditional on edema group."}
gg_v1 <- gg_variable(rfsrc_pbc, time = 1)
gg_v1$edema <- paste("edema =", gg_v1$edema)

plot(gg_v1, xvar = "bili", alpha = 0.5) +
  labs(y = "Survival at 1 Year", x = st_labs["bili"]) +
  theme(legend.position = "none") +
  scale_color_manual(values = event_colors, labels = event_labels) +
  scale_shape_manual(values = event_marks, labels = event_labels) +
  facet_grid(~edema) +
  coord_cartesian(ylim = c(-0.01, 1.01))
```

The decreasing trend with bilirubin holds across all edema groups, but survival
is uniformly lower in the edema = 1 panel, confirming an additive effect.

We can also condition on a continuous variable by binning into quantile groups:

```{r coplot-albumin, fig.width=7, fig.height=5, fig.cap="Variable dependence on bilirubin, conditional on albumin groups."}
albumin_cts <- quantile_pts(gg_v1$albumin, groups = 6, intervals = TRUE)
gg_v1$albumin_grp <- cut(gg_v1$albumin, breaks = albumin_cts)
levels(gg_v1$albumin_grp) <- paste("albumin =",
                                    levels(gg_v1$albumin_grp))

plot(gg_v1, xvar = "bili", alpha = 0.5) +
  labs(y = "Survival at 1 Year", x = st_labs["bili"]) +
  theme(legend.position = "none") +
  scale_color_manual(values = event_colors, labels = event_labels) +
  scale_shape_manual(values = event_marks, labels = event_labels) +
  facet_wrap(~albumin_grp) +
  coord_cartesian(ylim = c(-0.01, 1.01))
```

The effect of bilirubin attenuates at higher albumin levels, suggesting an
interaction between these two liver function markers.

# Partial Dependence Surfaces

For a richer view of the interaction between bilirubin and albumin, we construct
a partial dependence surface. We compute partial dependence on a grid of 8
albumin values, each evaluated at 25 bilirubin points.

```{r surface-data, error=TRUE, cache=FALSE}
# Create grid of albumin values
alb_grid <- quantile_pts(pbc_trial$albumin, groups = 8)

# For each albumin value, compute partial dependence on bili at ~1 year
surface_list <- lapply(alb_grid, function(alb_val) {
  newx <- pbc_trial[, rfsrc_pbc$xvar.names, drop = FALSE]
  newx$albumin <- alb_val
  pd_alb <- gg_partial_rfsrc(rfsrc_pbc, xvar.names = "bili",
                              newx = newx, partial.time = t1yr)
  df <- pd_alb$continuous
  df$albumin <- alb_val
  df
})

surface_df <- bind_rows(surface_list)
```

```{r pd-surface, error=TRUE, fig.height=5, fig.cap="Partial dependence surface: survival at 1 year as a function of bilirubin and albumin. Fill colour is the predicted survival probability."}
if (!exists("surface_df")) {
  message("surface_df not available --- skipping surface (see surface-data chunk error above).")
} else {
  ggplot(surface_df, aes(x = x, y = albumin, fill = yhat)) +
    geom_tile() +
    scale_fill_viridis_c(name = "Survival") +
    labs(x = "Bilirubin", y = "Albumin") +
    theme_bw()
}
```

The surface shows that survival is highest when bilirubin is low and albumin is
high (upper-left corner), and drops steeply as bilirubin increases or albumin
decreases. The curvature of the surface --- particularly the steep
gradient at low albumin and high bilirubin --- confirms the interaction detected
in the conditional plots.

## Brier Score and CRPS

VIMP and minimal depth tell us which predictors matter, but they say nothing
about how well the forest's survival predictions are calibrated. The Brier
score answers that question. At each point on the event-time grid it measures
the mean squared difference between the predicted survival probability
$\hat{S}(t)$ and the binary outcome (survived past $t$ or not). Because
right-censored subjects have unknown true outcomes, the calculation uses
inverse-probability-of-censoring weighting (IPCW): each subject's squared
error is upweighted by the inverse probability of being uncensored at that time,
so the estimate remains unbiased even under heavy censoring [@graf:1999].

Two reference values help interpret the curve. A Brier score of 0 is perfect
prediction. A score around 0.25 is the uninformative ceiling: a model that
assigns every subject a survival probability of 0.5, regardless of their
covariates or the time horizon, scores near 0.25 by construction. A forest
that beats 0.25 is doing better than that floor; one that exceeds it has been
made worse by its covariates, which is a sign of overfitting or a poorly
specified model.

`gg_brier()` wraps `randomForestSRC::get.brier.survival()` and returns a tidy
data frame ready to plot.

```{r brier-overall, fig.width=7, fig.height=3.5, fig.cap="Time-resolved Brier score for the PBC survival forest."}
gg_bs <- gg_brier(rfsrc_pbc)
plot(gg_bs)
```

Read the curve left to right. Early on it sits near zero --- almost everyone
is still alive, so predicting survival is easy. It climbs to a peak around the
median event time, where the outcome is genuinely uncertain, then falls again
as the at-risk pool shrinks and the remaining predictions get easier.

Setting `envelope = TRUE` adds a 15--85% ribbon around that line, showing how
spread out the individual subjects' Brier contributions are at each time. A
narrow ribbon means the forest is assigning similar risks across subjects; a
wide one means the predicted risks are pulling apart.

```{r brier-envelope, fig.width=7, fig.height=3.5, fig.cap="Brier score with 15--85% per-subject envelope."}
plot(gg_bs, envelope = TRUE)
```

The running CRPS (continuous ranked probability score) is the Brier score
integrated over time and divided by elapsed time — the time-average of the
curve above. It collapses the whole trajectory into one running number, which
is handy when you want a single calibration figure rather than a curve to read.
Like the Brier score, a CRPS near 0 is good and a value near 0.25 is the
uninformative ceiling (the same constant-predictor reference as above). The final (integrated) value at the right edge of the
plot is the one most often reported.

```{r brier-crps, fig.width=7, fig.height=3.5, fig.cap="Running CRPS for the PBC survival forest."}
plot(gg_bs, type = "crps")
```

The integrated CRPS --- a scalar summary of overall calibration --- is stored
as an attribute and can be retrieved with:

```{r brier-crps-scalar}
attr(gg_bs, "crps_integrated")
```

# Conclusion

We have walked a full random survival forest analysis with
**randomForestSRC** and **ggRandomForests**, and the results hang together:

- `gg_survival()` drew Kaplan--Meier curves that showed no treatment effect in
  the PBC trial, the same conclusion reached in @fleming:1991.
- `gg_error()` showed the OOB error settling quickly as trees were added.
- VIMP (`gg_vimp()`) and minimal depth (`max.subtree()`) landed on the same
  five predictors (bilirubin, albumin, copper, prothrombin, and age),
  which is also where the proportional hazards model landed.
- `gg_variable()` exposed non-proportional hazards for bilirubin and copper,
  where the gap between time horizons widens.
- Partial dependence from `gg_partial_rfsrc()` gave the risk-adjusted version
  of those curves and backed the log-transforms used in the parametric model.
- Conditioning plots and the partial dependence surface drew out the
  bilirubin--albumin interaction.
- `gg_brier()` measured how accurate the predictions actually were, both
  across time and as a single CRPS summary.

Notice the pattern. Each `gg_*()` function returns a tidy object (often a
data frame, sometimes a small list of data frames); the plotting comes
after. Lean on the package's `plot()` methods when the
default figure works, and drop down to `ggplot2` directly when it does not.

# References
