Small-sample behaviour of phylogenetic covariance estimates

This page documents measured small-sample behaviour of the among-axis phylogenetic covariance Σ_a in bivariate location–scale fits. It exists so that a number you get from a small tree is not mistaken for a number you can act on.

Nothing here is a bug report. These are properties of first-order Laplace/REML estimation of variance components, they shrink as sample size grows, and they are shared with comparable implementations. They are documented rather than patched because removing them requires a higher-order correction, not a guard.

What was measured

1000 replicates, N = 128, true Σ_a diagonal in 0.16–0.25, reported as log-Cholesky entries Lij.

entry	axis	Cholesky depth	bias
L11	μ₁×μ₁	0	−0.093
L22	μ₂×μ₂	1	−0.282
L33	σ₁×σ₁	2	−0.899
L44	σ₂×σ₂	3	−1.852

All four are biased downward — variance components are under-estimated on small trees.

It is small-sample, not structural

Bias × N is near-constant across an N-ladder (233 / 247 / 193 for L44 at N = 128 / 256 / 512), i.e. bias ≈ O(1/N). It shrinks with more tips; it does not run away.

The pattern follows Cholesky depth, not "mean vs scale"

It is tempting to read the table as "mean axes are fine, scale axes collapse". That is not what the numbers say. L22 is a mean axis and already jumps 3×, comparable to the mean→scale step. Each diagonal entry is Var(axis k | axes 1..k−1), so it absorbs the compounding error of every axis conditioned before it. Depth explains more of the escalation than the mean/scale distinction.

Three contributors, measured separately

• Cholesky-depth compounding — affects every axis, mean and scale alike. Dominant.

• The exp() link on scale axes — real and correctly signed. The likelihood is exactly quadratic in a mean-axis latent (Laplace is exact there, to machine precision) but not in a scale-axis latent. On its own this accounts for only ~0.02 in log-SD at N = 128 — an order of magnitude too small to explain the observed −1.85.

• Phylogenetic tip correlation — an amplifier, roughly 2× for scale axes at ntip = 16. On a star topology the scale-axis bias roughly halves. Notably the L44/L33 ratio (≈1.85) is unchanged by tree correlation, confirming that part is structural to the fixed axis ordering.

REML helps substantially and is the right default here

Same-seed comparison at ntip = 16:

entry	ML	REML	reduction
L33	−3.33	−1.42	−57 %
L44	−6.02	−2.48	−59 %

The bordered Schur correction does include the mean↔scale cross-blocks and removes most of ML's downward bias. What remains originates in the Laplace approximation of the random-effect integral itself, which REML does not target.

If you are estimating Σ_a on a small tree, prefer method = :REML.

Standard errors on the off-diagonals are not informative at small N

This matters more in practice than the bias itself, and it is easy to miss.

Correlation between |error| and the reported SE, over the same replicates:

entry	corr(|error|, SE)	reading
L11 (diagonal, mean)	+0.702	SE tracks error — usable
L44 (diagonal, scale)	+0.786	SE tracks error — usable
L43 (off-diagonal)	+0.032	SE carries almost no information about error

For L44 specifically, the mean reported SE (3.498) is far larger than the empirical SD of the estimates (1.243). Coverage is therefore ~100 % — which sounds reassuring and is not: an interval wide enough to always contain the truth conceals a bias of −1.85 (about 1.5 sampling SDs) rather than revealing it.

Do not read a wide interval on a small tree as evidence the estimate is fine.

Practical guidance

• Diagonal entries (variances) at small ntip are under-estimated; treat them as lower bounds.

• Off-diagonal entries (covariances between axes) at small ntip: treat the point estimate as weak and the SE as roughly uninformative. Do not build an inference on a single small-tree fit.

• Prefer method = :REML for variance components; it removes ~55–60 % of ML's bias here.

• Scale axes need more tips than mean axes for the same reliability, both because of the exp() link and because they sit deeper in the Cholesky ordering.

• Reordering the axes does not fix this — it moves which entry absorbs the worst compounding.

Provenance

Measured 2026-08-25 on a prespecified N-ladder and coverage study. The reproducibility record retains the full calibration and bias investigation.

The three-factor decomposition is not fully separated — the isolated exp()-link mechanism is 10–50× too small alone, so the explanation is a compound. The star-vs-balanced comparison is 15 replicates and is suggestive rather than production-grade.