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Correct a "combined table" for multiplicity **and** recompute *E*-values

Source: R/margot_correct_combined_table.R
margot_correct_combined_table.Rd

`margot_correct_combined_table()` takes the **combined_table** produced by the various *margot* models (or by your own code) and

  1. applies the chosen confidence-interval adjustment, **and**

  2. recalculates *E*-values (and their lower bounds) so they match the new interval.

By default it implements the single–step **Bonferroni** correction at \(\alpha = 0.05\) as advocated by VanderWeele & Mathur (2019).

Usage

margot_correct_combined_table(
  combined_table,
  adjust = c("bonferroni", "holm", "BH", "none"),
  alpha = 0.05,
  scale = c("RD", "RR"),
  delta = 1,
  sd = 1,
  m = NULL
)

Arguments

combined_table

A data frame with *at least* the columns

  • `E[Y(1)]-E[Y(0)]` **or** `E[Y(1)]/E[Y(0)]`

  • `2.5

Extra columns (e.g. the original *E*-values) are carried through.

adjust

Multiplicity method: `"bonferroni"` (default), `"holm"`, `"BH"`, or `"none"`. Bonferroni and Holm provide strong FWER control; BH provides FDR control. `"none"` retains the supplied confidence limits.

alpha

Family-wise error-rate (for bonferroni/holm) or false discovery rate (for BH) to control. Default `0.05`.

scale

Scale used to recompute the *E*-value. `"RD"` is the legacy label for the standardised-continuous-outcome approximation from an outcome-mean difference or ATE; `"RR"` treats the estimate as a risk ratio.

delta

Exposure contrast represented by an outcome-mean difference, used only when `scale = "RD"`.

sd

Outcome standard deviation used to standardise an outcome-mean difference, used only when `scale = "RD"`.

m

Positive whole number giving the total number of tests in the Bonferroni family. It must be at least the number of table rows. When `NULL`, Margot uses the number of rows. Holm and BH continue to use the rows supplied in `combined_table` as their adjustment family.

Value

A data frame with the same rows (and order) as `combined_table`, but with

  • updated `2.5

  • freshly computed `E_Value` and `E_Val_bound`.

Numeric columns retain their computational precision. Apply display rounding only when formatting the returned table for presentation.

E-value calculation

For a risk ratio \(r\), let \(r^* = r\) when \(r \ge 1\) and \(r^* = 1/r\) otherwise. Margot computes the null E-value as $$r^* + \sqrt{r^*(r^*-1)}.$$ The confidence-bound E-value uses the confidence limit closest to 1 and equals 1 when the interval includes 1.

For an outcome-mean difference \(b\), exposure contrast \(\delta\), and outcome standard deviation \(s\), Margot first forms the standardised difference \(d = b|\delta|/s\). It approximates the risk ratio as \(\exp(0.91d)\) and the risk-ratio confidence limits as \(\exp(0.91d \pm 1.78\,\mathrm{SE}(d))\), then applies the same null E-value equation. This calculation treats \(s\) as known.

How the correction is applied

For Bonferroni, let \(m\) be the total number of tests in the multiplicity family.

  • **Bonferroni** uses $$ z^* = \Phi^{-1}\!\bigl(1-\alpha/(2m)\bigr) $$ and rescales the original half-width.

  • **Holm** first step-down adjusts the (two-sided) *p*-value for each test, then back-calculates a *symmetric* CI whose coverage matches the adjusted *p*. Point estimates **never** change.

  • **BH** (Benjamini-Hochberg) applies FDR correction to *p*-values, then back-calculates symmetric CIs. Controls false discovery rate rather than family-wise error rate.

References

VanderWeele TJ, Mathur MB (2019). *Some desirable properties of the Bonferroni correction: Is the Bonferroni correction really so bad?* **Am J Epidemiol** 188(3): 617–618.

VanderWeele TJ, Ding P (2017). Sensitivity analysis in observational research: introducing the E-value. *Annals of Internal Medicine* 167(4): 268–274. doi:10.7326/M16-2607 .

Chinn S (2000). A simple method for converting an odds ratio to effect size for use in meta-analysis. *Statistics in Medicine* 19(22): 3127–3131.

VanderWeele TJ (2017). On a square-root transformation of the odds ratio for a common outcome. *Epidemiology* 28(6): e58.