Date: 2025-11-04

# SelectBoost for Beta regression

#### Frédéric Bertrand

#### 2025-11-04

## Overview

The new `sb_beta()` helper glues the beta-regression selectors provided by this package to a SelectBoost-style correlated-resampling loop implemented directly in `SelectBoost.beta`. It takes care of squeezing the response inside the open unit interval (unless `squeeze = FALSE`) and tagging the output with the selector that was used.

This vignette walks through two complementary perspectives:

1. Reconstructing the SelectBoost workflow step by step with `betareg_step_aic()` to highlight where correlated resampling happens.
2. Calling `sb_beta()` to obtain the same result with a single function call.

Throughout the examples we rely on the built-in simulator to generate correlated design matrices with a handful of truly associated predictors.

    sim <- simulation_DATA.beta(
      n = 150, p = 6, s = 3, beta_size = c(1, -0.8, 0.6),
      corr = "ar1", rho = 0.25,
      mechanism = "jitter"
    )
    str(sim$X)
    #>  num [1:150, 1:6] -0.123 1.635 1.428 -0.508 -0.243 ...
    #>  - attr(*, "dimnames")=List of 2
    #>   ..$ : NULL
    #>   ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
    summary(sim$Y)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #> 0.04934 0.25725 0.49003 0.48969 0.70233 0.99998

## Manual SelectBoost workflow with beta selectors

The classic SelectBoost algorithm first normalises the design matrix, computes pairwise correlations, groups variables above a chosen threshold and finally resamples the predictors before applying the selector. All of those stages are available directly in `SelectBoost.beta`.

    # Normalise the predictors (centre + L2 scale)
    X_norm <- sb_normalize(sim$X)

    # Compute correlations
    corr_mat <- sb_compute_corr(X_norm)

    # Group variables whose absolute correlation exceeds 0.6
    raw_groups <- sb_group_variables(corr_mat, c0 = 0.6)

    # Draw eight correlated replicas for the grouped variables
    X_draws <- sb_resample_groups(X_norm, raw_groups, B = 8, seed = 11)
    #> Warning: All groups are singletons; correlated resampling degenerates to
    #> repeated `X_norm`.

    dim(X_draws[[1]])
    #> [1] 150   6

Each element of `X_draws` stores a correlated copy of the normalised design. Feeding these matrices to `sb_apply_selector_manual()` together with a beta-regression selector yields coefficient estimates for every resampled data set.

    coef_path <- sb_apply_selector_manual(
      X_norm, X_draws, sim$Y, selector = betareg_step_aic
    )

    dim(coef_path)
    #> [1] 8 9
    coef_path[, 1:3]
    #>                        sim0        sim1        sim2
    #> (Intercept)     -0.03588528 -0.03588528 -0.03588528
    #> x1              11.34931343 11.34931343 11.34931343
    #> x2              -8.95724666 -8.95724666 -8.95724666
    #> x3               7.17554325  7.17554325  7.17554325
    #> x4               0.00000000  0.00000000  0.00000000
    #> x5               0.87055660  0.87055660  0.87055660
    #> x6               0.00000000  0.00000000  0.00000000
    #> phi|(Intercept)  2.95165950  2.95165950  2.95165950

The leading column `sim0` records the coefficients fitted on the original normalised design, providing a convenient baseline against which the resampled paths can be compared.

Finally, the `sb_selection_frequency()` helper counts how often each variable appears with a non-zero coefficient across the replicates. Because `betareg_step_aic()` returns a `glmnet`-style coefficient vector (intercept plus predictors), we set `version = "glmnet"` when computing the selection frequencies.

    sel_freq <- sb_selection_frequency(coef_path, version = "glmnet")
    sel_freq
    #>              x1              x2              x3              x4              x5
    #>               1               1               1               0               1
    #>              x6 phi|(Intercept)
    #>               0               1

This manual exercise confirms that the correlated resampling loop from the original SelectBoost package plugs seamlessly into the beta selectors shipped in `SelectBoost.beta`.

## Running the entire loop with `sb_beta()`

The `sb_beta()` wrapper performs the same steps internally while exposing the arguments most relevant to beta regression. By default it uses `betareg_step_aic()` as the base selector, but any of the exported functions (`"betareg_step_bic"`, `betareg_glmnet`, etc.) can be passed either by name or as a function.

    sb <- sb_beta(
      sim$X, sim$Y,
      B = 60,
      step.num = 0.5,
      steps.seq = c(0.9, 0.7, 0.5)
    )

    class(sb)
    #> [1] "sb_beta" "matrix"  "array"
    attr(sb, "selector")
    #> [1] "betareg_step_aic"
    rownames(sb)
    #> [1] "c0 = 1.000" "c0 = 0.900" "c0 = 0.700" "c0 = 0.500" "c0 = 0.000"
    round(sb, 3)
    #> SelectBoost beta selection frequencies
    #> Selector: betareg_step_aic
    #> Resamples per threshold: 60
    #> Interval mode: none
    #> c0 grid: 1.0, 0.9, 0.7, 0.5, 0.0
    #> Inner thresholds: 0.9, 0.7, 0.5
    #>             x1    x2  x3   x4  x5  x6 phi|(Intercept)
    #> c0 = 1.000 1.0 1.000 1.0 0.00 1.0 0.0               1
    #> c0 = 0.900 1.0 1.000 1.0 0.00 1.0 0.0               1
    #> c0 = 0.700 1.0 1.000 1.0 0.00 1.0 0.0               1
    #> c0 = 0.500 1.0 1.000 1.0 0.00 1.0 0.0               1
    #> c0 = 0.000 0.2 0.167 0.2 0.25 0.2 0.2               1
    #> attr(,"c0.seq")
    #> [1] 1.0 0.9 0.7 0.5 0.0
    #> attr(,"steps.seq")
    #> [1] 0.9 0.7 0.5
    #> attr(,"B")
    #> [1] 60
    #> attr(,"selector")
    #> [1] "betareg_step_aic"
    #> attr(,"resample_diagnostics")
    #> attr(,"resample_diagnostics")$`c0 = 1.000`
    #> [1] group                   size                    regenerated
    #> [4] cached                  mean_abs_corr_orig      mean_abs_corr_surrogate
    #> [7] mean_abs_corr_cross
    #> <0 rows> (or 0-length row.names)
    #>
    #> attr(,"resample_diagnostics")$`c0 = 0.900`
    #> [1] group                   size                    regenerated
    #> [4] cached                  mean_abs_corr_orig      mean_abs_corr_surrogate
    #> [7] mean_abs_corr_cross
    #> <0 rows> (or 0-length row.names)
    #>
    #> attr(,"resample_diagnostics")$`c0 = 0.700`
    #> [1] group                   size                    regenerated
    #> [4] cached                  mean_abs_corr_orig      mean_abs_corr_surrogate
    #> [7] mean_abs_corr_cross
    #> <0 rows> (or 0-length row.names)
    #>
    #> attr(,"resample_diagnostics")$`c0 = 0.500`
    #> [1] group                   size                    regenerated
    #> [4] cached                  mean_abs_corr_orig      mean_abs_corr_surrogate
    #> [7] mean_abs_corr_cross
    #> <0 rows> (or 0-length row.names)
    #>
    #> attr(,"resample_diagnostics")$`c0 = 0.000`
    #>               group size regenerated cached mean_abs_corr_orig
    #> 1 x1,x2,x3,x4,x5,x6    6          60  FALSE          0.1189262
    #>   mean_abs_corr_surrogate mean_abs_corr_cross
    #> 1               0.1375874          0.06768812
    #>
    #> attr(,"interval")
    #> [1] "none"

The resulting matrix comes with several attributes that document how the frequencies were generated. `attr(sb, "c0.seq")` returns the correlation threshold grid, `attr(sb, "B")` stores the number of correlated resamples per threshold, `attr(sb, "interval")` highlights whether interval sampling was activated, and `attr(sb, "resample_diagnostics")` keeps summary statistics on the cached surrogate draws. These metadata mirror the legacy SelectBoost beta implementation and are now documented in `?sb_beta()`.

Changing the selector is simply a matter of passing a different routine. The call below uses the GAMLSS-based elastic-net variant and asks `sb_beta()` to pass `choose = "bic"` to the underlying `betareg_glmnet()` implementation.

    sb_enet <- sb_beta(
      sim$X, sim$Y,
      selector = betareg_glmnet,
      B = 60,
      step.num = 0.5,
      version = "glmnet",
      choose = "bic",
      prestandardize = TRUE
    )

    attr(sb_enet, "selector")
    #> [1] "betareg_glmnet"
    colMeans(sb_enet)
    #>         x1         x2         x3         x4         x5         x6
    #> 0.35000000 0.33888889 0.33333333 0.34444444 0.34444444 0.01666667

Because the wrapper always builds on the same correlated resamples, results are directly comparable across selectors as long as they adopt the `glmnet`-style coefficient convention. This makes it straightforward to run stability analyses for interval responses by pairing `sb_beta()` with the convenience wrapper `sb_beta_interval()` (or the lower-level `fastboost_interval()`) or to compare several beta selectors under the exact same resampled design matrices.

## Conference communications

The SelectBoost4Beta workflow and its correlated resampling foundations were presented by Frédéric Bertrand and Myriam Maumy in 2023 at two conferences:

- Joint Statistical Meetings 2023 (Toronto, Canada) — “Improving variable selection in Beta regression models using correlated resampling”.
- BioC2023 (Boston, USA) — “SelectBoost4Beta: Improving variable selection in Beta regression models”.

Both communications emphasised how leveraging correlation-aware resampling improves the recall and precision of variable selection in high-dimensional Beta regression settings.