Validating the Founder-Genome-Equivalent Standard Error

Why this article exists

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

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

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

# 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

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

• 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

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

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().