---
title: "Validating the Founder-Genome-Equivalent Standard Error"
---

```{r}
#| label: setup
#| include: false
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

## Why this article exists {#sec-why}

Founder genome equivalents (`FG`, [Lacy 1989]) summarize how much of a colony's
founding genetic diversity survives in the living population. Like genome
uniqueness, `FG` is estimated by a **gene-drop simulation**, so it carries Monte
Carlo sampling noise that shrinks as the number of iterations grows.
`calcFGSE()` reports that noise as a standard error, so `FG` can be shown as
`FG ± SE`.

Unlike genome uniqueness -- a simple average, whose standard error is a column
variance -- `FG` is a **non-linear function** of the random per-founder retention
vector `r`:

$$ FG = \left( \sum_f p_f^2 / r_f \right)^{-1}, $$

so its standard error is obtained by the **delta method** (first-order
linearization), folding in the *within-iteration covariance* among founders. A
standard error derived this way is only trustworthy if it is **calibrated** --
the number it reports must match the real run-to-run spread of `FG`. This article
is the recorded evidence that it is, on a real deep pedigree, **before** the SE is
surfaced to users. It is the "validate before expose" gate in the
compute → validate → surface sequence.

## What "calibrated" means here {#sec-checks}

The estimator was put through seven checks at `K = 1000` iterations. Each runs
`B = 300` independent gene drops (one per seed), so the *measured* spread of the
`FG` estimates can be compared against the *reported* standard error.

| # | Check | What it verifies | Pass band |
|---|-------|------------------|-----------|
| 1 | **Agreement** | `mean(SE)` over the 300 seeds equals `sd(FG)`, the Monte Carlo truth | ratio in [0.92, 1.08] |
| 2 | **Coverage** | `FG ± 1.96·SE` contains a high-iteration reference `FG` about 95% of the time | [0.93, 0.97] |
| 3 | **Scaling** | quadrupling iterations (`K → 4K`) halves both the empirical spread and the reported SE | ratio in [1.8, 2.2] |
| 4 | **Degeneracy** | no run hits the silent-collapse case (a contributing founder retained in zero drops) | fraction = 0 |
| 5 | **Bootstrap** | an independent column bootstrap of the SE agrees with the delta-method SE | `boot/delta` in [0.85, 1.15] |
| 6 | **Off-diagonal** | the full (covariance-aware) SE versus the diagonal-only approximation | reported (see below) |
| 7 | **Reproducibility** | a fixed seed list makes the whole study byte-identical run to run | exact |

The pass bands were specified in advance for this validation study. The agreement
and coverage bands are calibrated for `B = 300`: the agreement ratio's own
sampling fluctuation is about `1/sqrt(2(B-1)) ≈ 0.04`, so a calibrated estimator
sits near 1 with the band roughly two standard errors wide.

## The two pedigrees {#sec-pedigrees}

A standard error can look perfect on an easy pedigree and still be wrong on a hard
one, so the study runs on two deliberately different pedigrees.

- **`lacy1989Ped`** -- the 7-animal, 3-founder pedigree from Lacy's original
  paper (`FG = 2.18`). All three founders are well retained (`r` ≈ 0.75) and
  barely correlated, so this is a fast, fully deterministic **anchor**. It cannot,
  on its own, exercise founder covariance or thin-retention skew.
- **`examplePedigree`** (assembled exactly as `reportGV()` analyses it: 704
  animals, **202 founders**, 327 probands) -- a real **deep, bottlenecked**
  pedigree. Several founders are retained in well under 10% of drops
  (minimum `r` ≈ 0.013, i.e. about 13 of 1000 iterations), and the founders'
  retentions are strongly correlated. This is the pedigree that exercises the
  covariance term, the skew stratum, and the degeneracy audit -- the checks that
  matter.

Validating on `lacy1989` alone would *falsely bless* the estimator (check 6 makes
this concrete).

## Results {#sec-results}

```{r}
#| label: reproduce
#| eval: false
# Reproduce (build-ignored; not run on render). ~7 min, fixed seeds.
#   Rscript data-raw/fgSEValidation.R
# Drives the validation harness (tests/testthat/helper-fgSEValidation.R) over
# B = 300 seeds on each pedigree at K = 1000 and 4000, and records the numbers
# below in data-raw/fgSEValidation-results.rds.
```

**`lacy1989Ped`** (`B = 300`, `K = 1000`; reference `FG = 2.1866` at `K = 20000`):

| Check | Value | Band | Verdict |
|-------|-------|------|---------|
| Agreement `mean(SE)/sd(FG)` | 1.0076 | [0.92, 1.08] | **PASS** |
| Coverage | 0.9333 | [0.93, 0.97] | **PASS** |
| Scaling (empirical / delta) | 1.846 / 2.004 | [1.8, 2.2] | **PASS** |
| Degeneracy fraction | 0.0000 | = 0 | **PASS** |
| Bootstrap `boot/delta` (dropped 0.000) | 0.9910 | [0.85, 1.15] | **PASS** |
| Off-diagonal `full/diag` | 1.001 | reported | -- |

**`examplePedigree`** (`B = 300`, `K = 1000`; reference `FG = 47.836` at `K = 20000`):

| Check | Value | Band | Verdict |
|-------|-------|------|---------|
| Agreement `mean(SE)/sd(FG)` | 1.0169 | [0.92, 1.08] | **PASS** |
| Coverage | 0.9500 | [0.93, 0.97] | **PASS** |
| Scaling (empirical / delta) | 1.981 / 2.008 | [1.8, 2.2] | **PASS** |
| Degeneracy fraction | 0.0000 | = 0 | **PASS** |
| Bootstrap `boot/delta` (dropped 0.000) | 1.0043 | [0.85, 1.15] | **PASS** |
| Off-diagonal `full/diag` | 0.692 | reported | -- |

Every gating check passes on both pedigrees. On the deep pedigree the agreement
ratio (1.017) and coverage (0.950) land essentially on their nominal targets, and
the SE halves cleanly when iterations quadruple. The `lacy1989` empirical-scaling
(1.846) and coverage (0.9333) sit nearer their band edges -- expected, because with
only three founders and `B = 300` those two statistics are the noisiest; the
deep-pedigree figures are the load-bearing evidence.

### The off-diagonal term is not optional {#sec-offdiag}

Check 6 is the reason the full delta method is used rather than a cheaper
diagonal (independence) approximation. The diagonal form drops the covariance
between founders; founders compete for a finite pool of descendant gene copies,
so that covariance is typically negative and the diagonal SE is an **overestimate**.

| Pedigree | full SE | diagonal SE | full / diag |
|----------|---------|-------------|-------------|
| `lacy1989Ped` | 0.013961 | 0.013941 | 1.001 |
| `examplePedigree` | 0.06542 | 0.094498 | 0.692 |

On `lacy1989` the gap is 0.1% -- invisible. On the real pedigree the diagonal SE is
**45% too large** (`0.0945 / 0.0654`). An estimator validated only on `lacy1989`
would happily ship the diagonal approximation and overstate the uncertainty by
nearly half on a real colony. This is exactly why the gate requires a deep
pedigree, and why `calcFGSE()` uses the covariance-aware influence form.

## Caveats carried forward {#sec-caveats}

- **Finite-iteration bias.** `FG` computed from `K` iterations is a slightly
  biased estimate of the true `FG` by `O(1/K)` (a Jensen effect of the non-linear
  `1/r`), which the standard error does not capture. At the default `K = 1000` it
  is small; it shrinks with more iterations.
- **Thin-retention founders.** When a contributing founder is retained in only a
  handful of drops (effective count `K·r` below roughly 5--10), `FG`'s sampling
  law is right-skewed and the first-order delta SE can under-cover; the column
  bootstrap (check 5) is the backstop, and the remedy is more iterations. The
  degeneracy audit (check 4) catches the limiting case where such a founder is
  retained in *zero* drops -- `calcFGSE()` returns `NA` with a warning there
  rather than a confident-but-meaningless number.
- **Independent seeds.** The math assumes the gene-drop iterations are
  independent. The study uses a distinct seed per replicate; `geneDrop()` draws
  each iteration column independently.

## Verdict {#sec-verdict}

**PASS.** The founder-genome-equivalent sampling standard error reported by
`calcFGSE()` is calibrated on both a clean anchor and a real deep, bottlenecked
pedigree: it matches the Monte Carlo spread of `FG`, covers a high-iteration
reference at the nominal rate, shrinks as `1/sqrt(K)`, never reports a finite SE
for a collapsed `FG`, and agrees with an independent bootstrap. The standard
error is cleared to be surfaced beside `FG` in the reports and the Shiny app.

## References {#sec-references}

Lacy, R.C. (1989) "Analysis of Founder Representation in Pedigrees: Founder
Equivalents and Founder Genome Equivalents." *Zoo Biology* 8:111--123.

See also `calcFGSE()`, `calcFG()`, `calcFEFG()`, `calcRetention()`, and the
companion gene-drop convergence diagnostic `gvaConvergence()`.