> 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 or functional assumptions on how covariates relate to the response. The method builds a large collection of de-correlated decision trees via bootstrap aggregation (bagging) and random feature selection, then averages their predictions to produce a stable, low-variance estimator. 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 forest regression workflow on the Boston Housing data set (Harrison and Rubinfeld 1978; Belsley et al. 1980):

1. Data exploration — EDA scatter panels, variable descriptions
2. Growing the forest — fitting an RF, checking OOB error convergence
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)

    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())

# Data: Boston Housing Values

The Boston Housing data (Harrison and Rubinfeld 1978; Belsley et al. 1980) is a standard benchmark for regression. It contains data for 506 census tracts of Boston from the 1970 census. The objective is to predict the median value of owner-occupied homes (`medv`, in $1000s) from 13 predictors covering crime, zoning, industry, environmental quality, and housing characteristics. We use the copy from the MASS package (Venables and Ripley 2002).

    data(Boston, package = "MASS")
    Boston$chas <- as.logical(Boston$chas) # nolint: object_name_linter

    st_labs <- c(
      crim    = "Crime rate by town",
      zn      = "Residential land zoned > 25k sq ft (%)",
      indus   = "Non-retail business acres (%)",
      chas    = "Borders Charles River",
      nox     = "Nitrogen oxides (10 ppm)",
      rm      = "Rooms per dwelling",
      age     = "Units built before 1940 (%)",
      dis     = "Distance to employment centers",
      rad     = "Highway accessibility index",
      tax     = "Property tax rate per $10,000",
      ptratio = "Pupil-teacher ratio",
      black   = "Proportion of Black residents",
      lstat   = "Lower status population (%)",
      medv    = "Median home value ($1000s)"
    )

## Exploratory data analysis

We plot each predictor against the response (`medv`), coloring by the sole categorical variable (`chas`, whether the tract borders the Charles River). A loess smooth highlights the marginal trend.

    dta <- Boston |>
      pivot_longer(c(-medv, -chas), names_to = "variable", values_to = "value")

    ggplot(dta, aes(x = medv, y = value, color = chas)) +
      geom_point(alpha = 0.4) +
      geom_smooth(aes(x = medv, y = value), color = "grey30",
                  inherit.aes = FALSE, se = FALSE) +
      labs(y = "", x = st_labs["medv"]) +
      scale_color_brewer(palette = "Set2") +
      facet_wrap(~variable, scales = "free_y", ncol = 3)

EDA: each predictor vs. median home value, colored by Charles River indicator.

Even from this simple view, the strong relationship between `medv` and both `lstat` (lower status %) and `rm` (rooms per dwelling) is apparent. We expect the random forest to confirm these as the most important predictors.

# Growing a Random Forest

We grow a regression forest using all 13 predictors. The `rfsrc()` function detects the regression family from the continuous response.

    rfsrc_Boston <- rfsrc(medv ~ ., data = Boston, # nolint: object_name_linter
                          importance = TRUE, err.block = 5)
    rfsrc_Boston

    #>                          Sample size: 506
    #>                      Number of trees: 500
    #>            Forest terminal node size: 5
    #>        Average no. of terminal nodes: 66.638
    #> No. of variables tried at each split: 5
    #>               Total no. of variables: 13
    #>        Resampling used to grow trees: swor
    #>     Resample size used to grow trees: 320
    #>                             Analysis: RF-R
    #>                               Family: regr
    #>                       Splitting rule: mse *random*
    #>        Number of random split points: 10
    #>                      (OOB) R squared: 0.86630631
    #>    (OOB) Requested performance error: 11.3087108

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

## OOB error convergence

    gg_e <- gg_error(rfsrc_Boston)
    gg_e <- gg_e |> filter(!is.na(error))
    class(gg_e) <- c("gg_error", class(gg_e))
    plot(gg_e)

OOB mean squared error vs. number of trees.

The error stabilizes well before 500 trees, indicating the forest is large enough for reliable predictions.

## OOB predictions

    plot(gg_rfsrc(rfsrc_Boston), alpha = 0.5) +
      coord_cartesian(ylim = c(5, 49))

OOB predicted median home values. Points are jittered; boxplot shows the distribution.

Each point is a single tract’s OOB prediction. The distribution is a sanity check — we are more interested in the why behind these predictions.

# Variable Selection

## Variable importance (VIMP)

VIMP measures the increase in prediction error when a variable is randomly permuted (Breiman 2001). Large positive values mean the variable is essential; negative values suggest noise is more informative.

    plot(gg_vimp(rfsrc_Boston), lbls = st_labs)

VIMP ranking. Longer blue bars indicate more important variables.

`lstat` and `rm` dominate, with a clear gap to the remaining predictors. All VIMP values are positive, indicating every predictor contributes at least marginally.

## 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_Boston <- max.subtree(rfsrc_Boston) # nolint: object_name_linter

The threshold is 3.01, selecting 6 variables: crim, nox, rm, dis, ptratio, lstat.

Both VIMP and minimal depth agree on the dominance of `lstat` and `rm`. We use the minimal depth top variables for the remainder of the analysis.

    xvar <- md_Boston$topvars

# Variable Dependence

## Variable dependence plots

Variable dependence shows each tract’s OOB predicted `medv` plotted against a predictor, with a loess smooth indicating the trend.

    gg_v <- gg_variable(rfsrc_Boston)

    plot(gg_v, xvar = xvar, panel = TRUE, alpha = 0.5) +
      labs(y = st_labs["medv"], x = "")

Variable dependence for top predictors (minimal depth rank order).

The panels confirm what EDA suggested: `medv` decreases sharply with `lstat` and increases with `rm`, both in strongly non-linear ways. The remaining variables show weaker but still discernible trends.

    plot(gg_v, xvar = "chas", alpha = 0.4) +
      labs(y = st_labs["medv"])

Variable dependence for Charles River (categorical).

Most tracts do not border the Charles River, and the predicted value distributions largely overlap — consistent with `chas` ranking last in both VIMP and minimal depth.

## Partial dependence

Partial dependence integrates out the effects of all other covariates, giving a risk-adjusted view of each predictor’s marginal effect (Friedman 2001):

\[ \tilde{f}(x) = \frac{1}{n} \sum_{i=1}^n \hat{f}(x, x_{i,o}) \]

We use `gg_partial_rfsrc()`, which calls `randomForestSRC::partial.rfsrc()` directly and returns a `gg_partial_rfsrc` object. The quickest path to a plot is `plot(pd)`, which handles continuous and categorical variables automatically. For a custom layout we can also access `pd$continuous` directly.

    pd <- gg_partial_rfsrc(rfsrc_Boston, xvar.names = xvar)

    # Quick S3 plot — works out of the box for the standard regression case
    plot(pd)

Partial dependence for top predictors.

For a publication-ready layout with custom axis labels, access the underlying data frame directly:

    ggplot(pd$continuous, aes(x = x, y = yhat)) +
      geom_line(color = "steelblue", linewidth = 1) +
      facet_wrap(~name, scales = "free_x") +
      labs(y = st_labs["medv"], x = "") +
      theme_bw()

Partial dependence (custom styling).

`lstat` shows a strongly concave relationship, while `rm` has a flat region below ~6 rooms before increasing sharply. These non-linear shapes are difficult to capture with simple parametric transforms but are naturally handled by the random forest.

# Variable Interactions and Conditioning Plots

## Conditioning on a categorical variable

The simplest coplot conditions on a categorical variable. Here we examine `medv` vs. `lstat`, split by Charles River status:

    gg_v$chas_label <- ifelse(gg_v$chas, "Borders Charles River",
                              "Does not border")

    plot(gg_v, xvar = "lstat", alpha = 0.5) +
      labs(y = st_labs["medv"], x = st_labs["lstat"]) +
      theme(legend.position = "none") +
      facet_wrap(~chas_label)

Variable dependence on lstat, conditional on Charles River.

The decreasing trend holds in both groups, with slightly higher values along the Charles River at every `lstat` level.

## Conditioning on a continuous variable

To investigate the `lstat`–`rm` interaction, we bin `rm` into six quantile groups using `quantile_pts()` and facet:

    rm_pts <- quantile_pts(rfsrc_Boston$xvar$rm, groups = 6, intervals = TRUE)
    gg_v$rm_grp <- cut(rfsrc_Boston$xvar$rm, breaks = rm_pts)
    levels(gg_v$rm_grp) <- paste("rm in", levels(gg_v$rm_grp))

    plot(gg_v, xvar = "lstat", alpha = 0.5) +
      labs(y = st_labs["medv"], x = st_labs["lstat"]) +
      theme(legend.position = "none") +
      scale_color_brewer(palette = "Set3") +
      facet_wrap(~rm_grp)

Predicted medv vs. lstat, conditional on rooms-per-dwelling groups.

Median values decrease with `lstat` within every `rm` group, but the intercept shifts upward with more rooms. Smaller homes in low-`lstat` (high-status) neighborhoods still command high prices.

The complement view — `medv` vs. `rm`, conditional on `lstat` groups — completes the picture:

    lstat_pts <- quantile_pts(rfsrc_Boston$xvar$lstat, groups = 6,
                              intervals = TRUE)
    gg_v$lstat_grp <- cut(rfsrc_Boston$xvar$lstat, breaks = lstat_pts)
    levels(gg_v$lstat_grp) <- paste("lstat in", levels(gg_v$lstat_grp))

    plot(gg_v, xvar = "rm", alpha = 0.5) +
      labs(y = st_labs["medv"], x = st_labs["rm"]) +
      theme(legend.position = "none") +
      scale_color_brewer(palette = "Set3") +
      facet_wrap(~lstat_grp)

Predicted medv vs. rooms, conditional on lower-status groups.

The `rm` effect is strongest in low-`lstat` tracts (bottom-left panels) and nearly flat in high-`lstat` tracts, confirming a meaningful interaction.

# Interactive Partial Dependence Surface

To visualize the joint partial dependence of `medv` on `lstat` and `rm`, we compute partial dependence on a grid: 25 values of `rm`, each evaluated at 25 points along `lstat`.

    rm_grid <- quantile_pts(rfsrc_Boston$xvar$rm, groups = 25)

    surface_list <- lapply(rm_grid, function(rm_val) {
      newx <- rfsrc_Boston$xvar
      newx$rm <- rm_val
      pd_rm <- gg_partial_rfsrc(rfsrc_Boston, xvar.names = "lstat", newx = newx)
      df <- pd_rm$continuous
      df$rm <- rm_val
      df
    })

    surface_df <- bind_rows(surface_list)

    if (requireNamespace("plotly", quietly = TRUE)) {
      library(plotly)

      surface_wide <- surface_df |>
        select(lstat = x, rm, medv = yhat) |>
        arrange(rm, lstat)

      lstat_vals <- sort(unique(surface_wide$lstat))
      rm_vals    <- sort(unique(surface_wide$rm))
      z_matrix   <- matrix(surface_wide$medv,
                            nrow = length(rm_vals),
                            ncol = length(lstat_vals),
                            byrow = TRUE)

      plot_ly(x = lstat_vals, y = rm_vals, z = z_matrix) |>
        add_surface(colorscale = "Viridis", showscale = TRUE) |>
        layout(
          scene = list(
            xaxis = list(title = "Lower Status (%)"),
            yaxis = list(title = "Rooms per Dwelling"),
            zaxis = list(title = "Median Value ($1000s)")
          )
        )
    } else {
      message("Install the plotly package for interactive 3D surfaces.")
      ggplot(surface_df, aes(x = x, y = rm, fill = yhat)) +
        geom_tile() +
        scale_fill_viridis_c(name = "Median Value") +
        labs(x = "Lower Status (%)", y = "Rooms per Dwelling") +
        theme_bw()
    }

Interactive partial dependence surface: median home value as a function of lstat and rm.

The surface confirms the strong interaction: home values are highest when `lstat` is low and `rm` is high (upper-left corner), dropping steeply along both axes. The non-planar curvature — particularly the sharp step near `rm` = 7 — demonstrates the kind of complex, non-linear structure that random forests capture naturally.

# Conclusion

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

- OOB error via `gg_error()` showed the forest stabilized quickly.
- VIMP via `gg_vimp()` and minimal depth via `max.subtree()` both identified `lstat` and `rm` as the dominant predictors, with a clear gap to the remaining variables.
- Variable dependence via `gg_variable()` revealed strongly non-linear predictor–response relationships that match the raw data EDA.
- Partial dependence via `gg_partial_rfsrc()` provided risk-adjusted confirmation, showing concave `lstat` and threshold-like `rm` effects.
- Conditioning plots and interactive surfaces exposed the `lstat`–`rm` interaction, where the room-size effect is strongest in high-status neighborhoods.

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

Belsley, David A., Edwin Kuh, and Roy E. Welsch. 1980. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. John Wiley & Sons. https://doi.org/10.1002/0471725153.
Breiman, Leo. 2001. “Random Forests.” Machine Learning 45 (1): 5–32. https://doi.org/10.1023/A:1010933404324.
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.
Harrison, David, and Daniel L. Rubinfeld. 1978. “Hedonic Housing Prices and the Demand for a Clean Environment.” Journal of Environmental Economics and Management 5 (1): 81–102. https://doi.org/10.1016/0095-0696(78)90006-2.
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, 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.
Venables, William N., and Brian D. Ripley. 2002. Modern Applied Statistics with S. 4th ed. Springer. https://doi.org/10.1007/978-0-387-21706-2.
Wickham, Hadley. 2016. ggplot2: Elegant Graphics for Data Analysis. 2nd ed. Springer. https://doi.org/10.1007/978-3-319-24277-4.