# Using HCbeta

HCbeta is the default estimator in `hcinfer()`. This vignette shows how to use it, inspect the stored quantities, and run small sensitivity checks with its method-specific arguments.

## Run HCbeta

Fit an OLS model and request HCbeta explicitly. This is equivalent to the default `hcinfer(fit)` call.

    library(hcinfer)

    schools <- PublicSchools |>
      dplyr::mutate(
        income_scaled = income / 10000,
        income_scaled_sq = income_scaled^2
      )

    fit <- lm(expenditure ~ income_scaled + income_scaled_sq, data = schools)
    result <- hcinfer(fit, type = "hcbeta")

    summary(result)
    #>
    #> ── 🔎 HCbeta robust inference summary ──────────────────────────────────────────
    #>
    #> ── 📐 Model ──
    #>
    #> Formula: `expenditure ~ income_scaled + income_scaled_sq`
    #> Observations: 50 | Parameters: 3 | Residual df: 47
    #>
    #> ── 🥪 Robust covariance ──
    #>
    #> Estimator: HCbeta
    #> Confidence level: 95.0% | Normal critical value: 1.9600
    #> Tests are two-sided normal Wald tests, one coefficient at a time.
    #> Test results use alpha = 0.050.
    #>
    #> ── 🎯 Leverage diagnostics ──
    #>
    #> # A tibble: 6 × 2
    #>   statistic value
    #>   <chr>     <chr>
    #> 1 minimum   0.02669
    #> 2 q1        0.03106
    #> 3 median    0.03912
    #> 4 mean      0.06
    #> 5 q3        0.04962
    #> 6 maximum   0.6508
    #> Maximum leverage: observation 2 (index 2), value 0.6508
    #> Average leverage: 0.0600
    #> Concentration: 10.85 x average leverage
    #>
    #> ── ⚖️ Robust weights ──
    #>
    #> # A tibble: 6 × 2
    #>   statistic value
    #>   <chr>     <chr>
    #> 1 minimum   1.156
    #> 2 q1        1.167
    #> 3 median    1.187
    #> 4 mean      1.276
    #> 5 q3        1.212
    #> 6 maximum   4.581
    #> Maximum weight: observation 2 (index 2), value 4.5807
    #> Median weight: 1.1869
    #> Concentration: 3.86 x median weight
    #>
    #> ── ⚙️ Method parameters ──
    #>
    #> # A tibble: 12 × 3
    #>    parameter value    role
    #>    <chr>     <chr>    <chr>
    #>  1 c1        7        method constant
    #>  2 c2        0.75     method constant
    #>  3 lower     0.01     method constant
    #>  4 upper     0.99     method constant
    #>  5 mu_hat    0.94     estimated quantity
    #>  6 s2_w      0.008504 estimated quantity
    #>  7 phi_hat   5.632    estimated quantity
    #>  8 a_hat     5.294    estimated quantity
    #>  9 b_hat     0.3379   estimated quantity
    #> 10 zeta      0.5      estimated quantity
    #> 11 a_tilde   3.147    estimated quantity
    #> 12 b_tilde   0.669    estimated quantity
    #>
    #> ── 🔎 Coefficient tests ──
    #>
    #> # A tibble: 3 × 9
    #>   term             estimate robust_se z       p_value alpha test_result
    #>   <chr>            <chr>    <chr>     <chr>   <chr>   <chr> <chr>
    #> 1 (Intercept)      832.9    850.7     0.9791  0.328   0.050 ✅ do not reject H0
    #> 2 income_scaled    -1834    2309      -0.7945 0.427   0.050 ✅ do not reject H0
    #> 3 income_scaled_sq 1587     1547      1.026   0.305   0.050 ✅ do not reject H0
    #>   ci             ci_relation
    #>   <chr>          <chr>
    #> 1 [-834.3, 2500] includes null
    #> 2 [-6359, 2691]  includes null
    #> 3 [-1446, 4620]  includes null
    #>
    #> ── 📏 Confidence intervals ──
    #>
    #> # A tibble: 3 × 4
    #>   term             null_value interval       interpretation
    #>   <chr>            <chr>      <chr>          <chr>
    #> 1 (Intercept)      0          [-834.3, 2500] includes null
    #> 2 income_scaled    0          [-6359, 2691]  includes null
    #> 3 income_scaled_sq 0          [-1446, 4620]  includes null
    #> 💡 test_result is based on p_value < alpha. Do not reject H0 does not mean that
    #> H0 is true.

## Extract coefficient test results

Use `tests()` to extract the coefficient-level Wald results as a tibble.

    tests(result)
    #> # A tibble: 3 × 8
    #>   term             estimate null_value std_error z_value p_value alpha reject
    #>   <chr>               <dbl>      <dbl>     <dbl>   <dbl>   <dbl> <dbl> <lgl>
    #> 1 (Intercept)          833.          0      851.   0.979   0.328  0.05 FALSE
    #> 2 income_scaled      -1834.          0     2309.  -0.794   0.427  0.05 FALSE
    #> 3 income_scaled_sq    1587.          0     1547.   1.03    0.305  0.05 FALSE

For example, extract the robust standard error and p-value for the quadratic income term.

    dplyr::select(tests(result, parm = "income_scaled_sq"), term, std_error, p_value)
    #> # A tibble: 1 × 3
    #>   term             std_error p_value
    #>   <chr>                <dbl>   <dbl>
    #> 1 income_scaled_sq     1547.   0.305

## Inspect HCbeta parameters

HCbeta stores both user-facing constants and estimated quantities in `method_params`.

    result$method_params
    #> $c1
    #> [1] 7
    #>
    #> $c2
    #> [1] 0.75
    #>
    #> $lower
    #> [1] 0.01
    #>
    #> $upper
    #> [1] 0.99
    #>
    #> $mu_hat
    #> [1] 0.94
    #>
    #> $s2_w
    #> [1] 0.008503804
    #>
    #> $phi_hat
    #> [1] 5.632326
    #>
    #> $a_hat
    #> [1] 5.294386
    #>
    #> $b_hat
    #> [1] 0.3379395
    #>
    #> $zeta
    #> [1] 0.5
    #>
    #> $a_tilde
    #> [1] 3.147193
    #>
    #> $b_tilde
    #> [1] 0.6689698

The most useful entries for routine inspection are:

Entry | Meaning
--- | ---
`c1`, `c2` | Constants controlling the HCbeta exponent.
`lower`, `upper` | Truncation limits for leverage complements.
`a_tilde`, `b_tilde` | Adjusted Beta shape parameters.
`zeta` | Shrinkage weight used in the Beta parameter adjustment.

These values are diagnostics. In routine use, the defaults should usually be left unchanged.

## Inspect leverage and weights

HCbeta, like the other estimators, stores leverage values and robust weights. This table shows the observations with the largest leverages.

    diagnostics <- tibble::tibble(
      observation = result$observation,
      leverage = result$leverage,
      weight = result$weights,
      residual = result$residuals
    ) |>
      dplyr::mutate(
        state = schools$state[as.integer(observation)],
        .before = leverage
      )

    diagnostics |>
      dplyr::arrange(dplyr::desc(leverage)) |>
      dplyr::slice_head(n = 5)
    #> # A tibble: 5 × 5
    #>   observation state          leverage weight residual
    #>   <chr>       <chr>             <dbl>  <dbl>    <dbl>
    #> 1 2           Alaska           0.651    4.58   110.
    #> 2 48          Washington DC    0.208    1.61  -161.
    #> 3 24          Mississippi      0.200    1.59   -44.0
    #> 4 4           Arkansas         0.0887   1.30   -30.5
    #> 5 40          South Carolina   0.0794   1.28     8.64

You can also sort by robust weight.

    diagnostics |>
      dplyr::arrange(dplyr::desc(weight)) |>
      dplyr::slice_head(n = 5)
    #> # A tibble: 5 × 5
    #>   observation state          leverage weight residual
    #>   <chr>       <chr>             <dbl>  <dbl>    <dbl>
    #> 1 2           Alaska           0.651    4.58   110.
    #> 2 48          Washington DC    0.208    1.61  -161.
    #> 3 24          Mississippi      0.200    1.59   -44.0
    #> 4 4           Arkansas         0.0887   1.30   -30.5
    #> 5 40          South Carolina   0.0794   1.28     8.64

The covariance object can be plotted directly to display adjustment factors against leverages.

    plot(vcov_hc(fit, type = "hcbeta"))

## Use the covariance-only interface

Use `vcov_hc()` when you only need the HCbeta covariance matrix and diagnostics.

    hcbeta_cov <- vcov_hc(fit, type = "hcbeta")
    hcbeta_cov
    #>
    #> ── 🥪 HCbeta robust covariance ─────────────────────────────────────────────────
    #> 📐 Model: `expenditure ~ income_scaled + income_scaled_sq`
    #> Dimension: 3 x 3
    #> Observations: 50
    #> Parameters: 3
    #> 🎯 Maximum leverage: 0.6508
    #> ⚖️ Maximum robust weight: 4.5807
    #> Use `vcov()` to extract the stored covariance matrix.
    vcov(hcbeta_cov)
    #>                  (Intercept) income_scaled income_scaled_sq
    #> (Intercept)         723617.6      -1962262          1312195
    #> income_scaled     -1962262.2       5329884         -3569755
    #> income_scaled_sq   1312195.3      -3569755          2394627

The covariance object stores the same method parameters and diagnostics.

    hcbeta_cov$method_params
    #> $c1
    #> [1] 7
    #>
    #> $c2
    #> [1] 0.75
    #>
    #> $lower
    #> [1] 0.01
    #>
    #> $upper
    #> [1] 0.99
    #>
    #> $mu_hat
    #> [1] 0.94
    #>
    #> $s2_w
    #> [1] 0.008503804
    #>
    #> $phi_hat
    #> [1] 5.632326
    #>
    #> $a_hat
    #> [1] 5.294386
    #>
    #> $b_hat
    #> [1] 0.3379395
    #>
    #> $zeta
    #> [1] 0.5
    #>
    #> $a_tilde
    #> [1] 3.147193
    #>
    #> $b_tilde
    #> [1] 0.6689698
    summary(hcbeta_cov)
    #>
    #> ── 🥪 HCbeta robust covariance summary ─────────────────────────────────────────
    #>
    #> ── 📐 Model ──
    #>
    #> Formula: `expenditure ~ income_scaled + income_scaled_sq`
    #> Observations: 50 | Parameters: 3 | Residual df: 47
    #>
    #> ── 🎯 Leverage diagnostics ──
    #>
    #> # A tibble: 6 × 2
    #>   statistic value
    #>   <chr>     <chr>
    #> 1 minimum   0.02669
    #> 2 q1        0.03106
    #> 3 median    0.03912
    #> 4 mean      0.06
    #> 5 q3        0.04962
    #> 6 maximum   0.6508
    #> Maximum leverage: observation 2 (index 2), value 0.6508
    #> Average leverage: 0.0600
    #> Concentration: 10.85 x average leverage
    #>
    #> ── ⚖️ Robust weights ──
    #>
    #> # A tibble: 6 × 2
    #>   statistic value
    #>   <chr>     <chr>
    #> 1 minimum   1.156
    #> 2 q1        1.167
    #> 3 median    1.187
    #> 4 mean      1.276
    #> 5 q3        1.212
    #> 6 maximum   4.581
    #> Maximum weight: observation 2 (index 2), value 4.5807
    #> Median weight: 1.1869
    #> Concentration: 3.86 x median weight
    #>
    #> ── ⚙️ Method parameters ──
    #>
    #> # A tibble: 12 × 3
    #>    parameter value    role
    #>    <chr>     <chr>    <chr>
    #>  1 c1        7        method constant
    #>  2 c2        0.75     method constant
    #>  3 lower     0.01     method constant
    #>  4 upper     0.99     method constant
    #>  5 mu_hat    0.94     estimated quantity
    #>  6 s2_w      0.008504 estimated quantity
    #>  7 phi_hat   5.632    estimated quantity
    #>  8 a_hat     5.294    estimated quantity
    #>  9 b_hat     0.3379   estimated quantity
    #> 10 zeta      0.5      estimated quantity
    #> 11 a_tilde   3.147    estimated quantity
    #> 12 b_tilde   0.669    estimated quantity

## Run a sensitivity check

The HCbeta constants can be passed through `...`. A sensitivity check compares the default result with a small set of alternative settings.

    sensitivity_results <- list(
      default = hcinfer(fit, type = "hcbeta"),
      c1_5 = hcinfer(fit, type = "hcbeta", c1 = 5),
      tighter_truncation = hcinfer(fit, type = "hcbeta", lower = 0.02, upper = 0.98)
    )

    sensitivity <- purrr::imap(sensitivity_results, \(res, setting) {
      row <- tests(res, parm = "income_scaled_sq")
      ci <- confint(res, parm = "income_scaled_sq")

      tibble::tibble(
        setting = setting,
        std_error = row$std_error,
        p_value = row$p_value,
        conf_low = ci$conf_low,
        conf_high = ci$conf_high,
        max_weight = max(res$weights)
      )
    })
    sensitivity <- dplyr::bind_rows(sensitivity)
    sensitivity
    #> # A tibble: 3 × 6
    #>   setting            std_error p_value conf_low conf_high max_weight
    #>   <chr>                  <dbl>   <dbl>    <dbl>     <dbl>      <dbl>
    #> 1 default                1547.   0.305   -1446.     4620.       4.58
    #> 2 c1_5                   1292.   0.219    -945.     4119.       3.02
    #> 3 tighter_truncation     1547.   0.305   -1446.     4620.       4.58

This table helps identify whether the reported inference is sensitive to small changes in HCbeta tuning constants. The defaults remain the recommended starting point.

## Compare HCbeta with one classical estimator

For a quick comparison, put HCbeta next to a classical estimator such as HC3.

    hc3 <- hcinfer(fit, type = "hc3")

    extract_income <- function(res, method) {
      row <- tests(res, parm = "income_scaled_sq")
      ci <- confint(res, parm = "income_scaled_sq")
      tibble::tibble(
        method = method,
        estimate = row$estimate,
        std_error = row$std_error,
        p_value = row$p_value,
        conf_low = ci$conf_low,
        conf_high = ci$conf_high
      )
    }

    compare_hc3 <- dplyr::bind_rows(
      extract_income(result, "hcbeta"),
      extract_income(hc3, "hc3")
    )
    compare_hc3
    #> # A tibble: 2 × 6
    #>   method estimate std_error p_value conf_low conf_high
    #>   <chr>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>
    #> 1 hcbeta    1587.     1547.   0.305   -1446.     4620.
    #> 2 hc3       1587.     1995.   0.426   -2324.     5498.

## Practical workflow

For routine analyses:

1. Fit an OLS model with `lm()`.
2. Run `hcinfer(fit)` or `hcinfer(fit, type = "hcbeta")`.
3. Use `summary(result)` for readable output.
4. Use `tests(result)`, `confint(result)`, `coef(result)`, and `vcov(result)` for extracted values.
5. Inspect `result$weights`, `result$leverage`, and `result$method_params` when diagnostics are needed.

Use `vignette("hcinfer-comparison")` when you want to compare HCbeta with other estimators implemented in the package.