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Joint dependence of leave-one-out consensus residuals 2026-11-05

The joint dependence of leave-one-out consensus residuals arises because different laboratories' benchmarks reuse many of the same observations. Excluding each result from its own leave-one-out interlaboratory consensus does not make the complete collection of residuals independent.

Let

a_{i}

be fixed positive coefficients with

\sum_{i} a_{i} = 1

, and let

\overset{ˉ}{X}_{a} = \sum_{i} a_{i} X_{i}

be the full weighted mean. With at least two observations, every

a_{i} < 1

, and the omitted-result residual is

D_{i} = X_{i} - \frac{\sum _{j \neq = i} a _{j} X _{j}}{1 - a _{i}} = \frac{X _{i} - X ˉ _{a}}{1 - a _{i}} .

(1)

Write

P_{a} = I - 1 a^{T}

for the weighted centering matrix, and

M = diag (1 - a_{1}, \dots, 1 - a_{n})

for a diagonal matrix. Then

D = A X, A = M^{- 1} P_{a}, Cov (D) = A Cov (X) A^{T} .

(2)

Each row of

A

sums to zero, so each residual is a contrast: a common additive level cancels.

For independent observations with positive variances

v_{i}

, define

S = \sum_{k} a_{k}^{2} v_{k}

. Expanding linearity of covariance gives

Var (D_{i}) = \frac{v _{i} ( 1 - 2 a _{i} ) + S}{( 1 - a _{i} ) ^{2}}, Cov (D_{i}, D_{j}) = \frac{S - a _{i} v _{i} - a _{j} v _{j}}{( 1 - a _{i} ) ( 1 - a _{j} )} (i \neq = j) .

(3)

Equal coefficients and equal variances reduce these expressions to

Var (D_{i}) = \frac{n}{n - 1} v, Cov (D_{i}, D_{j}) = - \frac{n}{( n - 1 ) ^{2}} v .

(4)

The familiar negative covariance belongs to this model. Arbitrary fixed coefficients can produce a different sign.

An original three-laboratory counterexample sets

v_{1} = v_{2} = v_{3} = 1

and

a = (0.05, 0.05, 0.90)

. Then

S = 0.815

, and

Cov (D_{1}, D_{2}) = \frac{0.815 - 0.05 - 0.05}{0.9 5 ^{2}} = \frac{286}{361} ≃ 0.792 > 0.

(5)

This positive covariance has a simple cause: both benchmarks are dominated by the third result. Increasing

X_{3}

pushes both

D_{1}

and

D_{2}

downward. The coefficients are stipulated to expose the dependence; with equal observation variances, they are not inverse-variance coefficients.

Figure 1.
Leave-one-out residual correlation depends on both coefficients and observation variances
. Original three-laboratory models with independent unit-variance observations. Equal coefficients give negative pairwise correlation; a dominant third coefficient creates positive correlation between the first two residuals. **Details**.

For positive inverse-variance weights

a_{i} = v_{i}^{- 1} / W

, where

W = \sum_{k} v_{k}^{- 1}

, one has

a_{i} v_{i} = 1/ W

and

S = 1/ W

. The off-diagonal expression becomes

Cov (D_{i}, D_{j}) = - \frac{1}{W ( 1 - a _{i} ) ( 1 - a _{j} )} < 0.

(6)

Thus the leave-one-out inverse-variance weighted consensus has a special sign pattern under its independent known-variance assumptions.

The coefficients also impose a deterministic constraint,

\sum_{i} a_{i} (1 - a_{i}) D_{i} = 0.

(7)

Consequently the residual covariance matrix is a singular matrix. Under the independent-observation model with every

v_{i} > 0

, its matrix rank is

n - 1

: the weighted centering matrix removes the common-level direction and the invertible diagonal matrix

M

preserves that rank. For equal coefficients the constraint is simply

\sum_{i} D_{i} = 0

These dependencies matter when evaluating several laboratories together. Products of single-residual probabilities generally do not describe joint events. A specified joint model or a set of appropriately chosen contrasts can represent the available information without treating redundant residuals as independent observations.

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