> Work in progress

> This vignette is under active development. Code examples and narrative may change before the next release.

# 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 et al. 2008; Ishwaran and Kogalur 2007) 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 randomForestSRC package (Ishwaran and Kogalur 2024) provides a unified implementation for survival, regression, and classification forests. ggRandomForests extracts tidy data objects from `rfsrc` fits and renders them with ggplot2 (Wickham 2016), making it straightforward to explore how a forest is constructed, which variables matter, and how the response depends on individual predictors.

This vignette demonstrates a complete random survival forest workflow on the primary biliary cirrhosis (PBC) data set (Fleming and Harrington 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 interactive 3-D partial dependence surfaces with plotly

    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 and Harrington (1991) Chapter 0.2, with a proportional hazards model developed in Chapter 4.4. We use the copy bundled with randomForestSRC.

    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.

    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)
      }
    }

    # 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.

    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")

EDA for categorical variables. Bar color indicates factor level; white = missing.

    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))

EDA for continuous variables. Points colored by death event; rug marks indicate missing values.

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

## Kaplan–Meier survival by treatment

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

    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))

KM survival estimates by treatment group with 95% confidence bands.

The curves largely overlap, consistent with the original finding that DPCA offered no clear survival benefit over placebo (Fleming and Harrington 1991).

    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))

KM cumulative hazard estimates by treatment group.

We can also stratify on continuous variables. Here we reproduce the bilirubin groupings from Fleming and Harrington (1991) Figure 4.4.2:

    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")

KM survival stratified by bilirubin groups.

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.

    # Step 1: impute missing values via random forest proximity
    pbc_imputed <- impute.rfsrc(Surv(years, status) ~ .,
                                 data    = pbc_trial,
                                 ntree   = 500,
                                 nimpute = 2)

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

The forest grew 500 trees, splitting on 5 randomly selected candidate variables at each node, and stopping at a minimum terminal node size of 15.

## OOB error convergence

    plot(gg_error(rfsrc_pbc))

OOB prediction error vs. number of trees.

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

## OOB predicted survival

    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

OOB predicted survival curves. Blue = censored, red = death events.

Each curve represents one patient’s OOB prediction, extended to the last follow-up time. Red (death event) curves generally fall faster, confirming the forest discriminates between risk groups.

    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))

Median predicted survival by treatment group with 95% confidence bands.

## 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.

    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))

Predicted survival for non-trial patients (test set).

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 is randomly permuted. Large positive values mean the variable is essential for accuracy; negative values suggest noise is more informative than the variable.

    plot(gg_vimp(rfsrc_pbc), lbls = st_labs) +
      theme(legend.position = c(0.8, 0.2)) +
      labs(fill = "VIMP > 0")

Variable importance ranking. Blue = positive VIMP, red = negative.

Bilirubin ranks highest, followed by copper, prothrombin time, albumin, and age — closely matching the variables selected in the Fleming and Harrington (1991) proportional hazards model.

## Minimal depth

Minimal depth (Ishwaran et al. 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.

    md_pbc <- max.subtree(rfsrc_pbc)

The `max.subtree()` function computes minimal depth for each variable. The threshold is 5.88, selecting 8 variables: age, ascites, edema, bili, chol, albumin, copper, prothrombin.

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

    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 at a given time horizon plotted against a predictor of interest. Points are colored by event status; a loess smooth indicates the trend.

    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))

Variable dependence of survival at 1 and 3 years on bilirubin.

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

    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))

Variable dependence at 1 and 3 years for continuous predictors.

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.

    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))

Variable dependence on edema (categorical).

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

## Partial dependence

> Warning

> Known issue (draft): `randomForestSRC::partial.rfsrc()` currently fails for survival forests in randomForestSRC ≥ 3.3. The partial dependence and interactive 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 2001). 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.

    # 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)

Partial dependence of survival at approximately 1 and 3 years on continuous predictors.

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

    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()

Partial dependence (custom styling).

The partial dependence curves at approximately 1 and 3 years confirm the variable dependence findings and support the log-transforms used in the Fleming and Harrington (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.

    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))

Variable dependence on bilirubin, conditional on edema group.

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:

    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))

Variable dependence on bilirubin, conditional on albumin groups.

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

# Interactive 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 25 albumin values, each evaluated at 25 bilirubin points.

    # Create grid of albumin values
    alb_grid <- quantile_pts(pbc_trial$albumin, groups = 25)

    # 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)

    if (!exists("surface_df")) {
      message("surface_df not available — skipping plotly surface (see surface-data chunk error above).")
    } else if (requireNamespace("plotly", quietly = TRUE)) {
      # Reshape for surface
      library(plotly)

      surface_wide <- surface_df |>
        select(bili = x, albumin, survival = yhat) |>
        arrange(albumin, bili)

      # Create matrix form
      bili_vals <- sort(unique(surface_wide$bili))
      alb_vals  <- sort(unique(surface_wide$albumin))
      z_matrix  <- matrix(surface_wide$survival,
                           nrow = length(alb_vals),
                           ncol = length(bili_vals),
                           byrow = TRUE)

      plot_ly(x = bili_vals, y = alb_vals, z = z_matrix) |>
        add_surface(colorscale = "Viridis", showscale = TRUE) |>
        layout(
          scene = list(
            xaxis = list(title = "Bilirubin"),
            yaxis = list(title = "Albumin"),
            zaxis = list(title = "Survival")
          )
        )
    } else {
      message("Install the plotly package for interactive 3D surfaces.")
      # Fallback: contour plot with ggplot2
      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()
    }

Interactive partial dependence surface: survival as a function of bilirubin and albumin.

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 non-planar shape 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

The Brier score measures prediction accuracy at each point on the event-time grid by comparing predicted survival probabilities to observed outcomes, with inverse-probability-of-censoring weighting (IPCW) to account for right-censoring (Graf et al. 1999). `gg_brier()` wraps `randomForestSRC::get.brier.survival()` and returns a tidy data frame ready for plotting.

    gg_bs <- gg_brier(rfsrc_pbc)
    plot(gg_bs)

Time-resolved Brier score for the PBC survival forest.

The score starts near zero (all subjects alive, predictions trivially correct), peaks around the median event time where uncertainty is greatest, then decreases as the remaining at-risk pool shrinks.

The 15–85% per-subject envelope shows how spread the individual Brier contributions are at each time. A narrow ribbon indicates the forest assigns similar predicted risks across subjects; a wide ribbon points to heterogeneous risk predictions.

    plot(gg_bs, envelope = TRUE)

Brier score with 15–85% per-subject envelope.

The running CRPS (continuous ranked probability score) integrates the Brier score over time, normalised by elapsed time. It provides a single time-averaged accuracy measure that penalises poorly calibrated predictions throughout the follow-up period.

    plot(gg_bs, type = "crps")

Running CRPS for the PBC survival forest.

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

    attr(gg_bs, "crps_integrated")

    #> [1] 1.401364

# Conclusion

This vignette demonstrated a complete random survival forest analysis using randomForestSRC and ggRandomForests:

- Kaplan–Meier curves via `gg_survival()` confirmed no treatment effect in the PBC trial, consistent with Fleming and Harrington (1991).
- OOB error via `gg_error()` showed the forest stabilized quickly.
- VIMP via `gg_vimp()` and minimal depth via `max.subtree()` both identified bilirubin, albumin, copper, prothrombin, and age as key predictors — matching the proportional hazards model.
- Variable dependence via `gg_variable()` revealed non-proportional hazards effects for bilirubin and copper.
- Partial dependence via `gg_partial_rfsrc()` provided risk-adjusted confirmation and supported the log-transforms used in parametric models.
- Conditional plots and interactive surfaces exposed the bilirubin–albumin interaction.
- Brier score and CRPS via `gg_brier()` quantified time-resolved prediction accuracy and overall calibration of the survival forest.

The ggRandomForests design separates data extraction from plotting: every `gg_*()` function returns a tidy data frame that can be plotted with the package’s `plot()` methods or fed directly into custom `ggplot2` workflows.

# References

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Friedman, Jerome H. 2001. “Greedy Function Approximation: A Gradient Boosting Machine.” The Annals of Statistics 29 (5): 1189–232. https://doi.org/10.1214/aos/1013203451.
Graf, Erika, Claudia Schmoor, Willi Sauerbrei, and Martin Schumacher. 1999. “Assessment and Comparison of Prognostic Classification Schemes for Survival Data.” Statistics in Medicine 18 (17–18): 2529–45. https://doi.org/10.1002/(SICI)1097-0258(19990915/30)18:17/18<2529::AID-SIM274>3.0.CO;2-5.
Ishwaran, Hemant, and Udaya B. Kogalur. 2007. “Random Survival Forests for R.” R News 7 (2): 25–31.
Ishwaran, Hemant, and Udaya B. Kogalur. 2024. randomForestSRC: Fast Unified Random Forests for Survival, Regression, and Classification (RF-SRC). https://cran.r-project.org/package=randomForestSRC.
Ishwaran, Hemant, Udaya B. Kogalur, Eugene H. Blackstone, and Michael S. Lauer. 2008. “Random Survival Forests.” The Annals of Applied Statistics 2 (3): 841–60. https://doi.org/10.1214/08-AOAS169.
Ishwaran, Hemant, Udaya B. Kogalur, Eiran Z. Gorodeski, Andy J. Minn, and Michael S. Lauer. 2010. “High-Dimensional Variable Selection for Survival Data.” Journal of the American Statistical Association 105 (489): 205–17. https://doi.org/10.1198/jasa.2009.tm08622.
Wickham, Hadley. 2016. ggplot2: Elegant Graphics for Data Analysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-24277-4.