Joint dependence of leave-one-out consensus residuals

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 ${\bar{X}}_a={\sum }_i{a}_i{X}_i$ be the full weighted mean. With at least two observations, every $a_i<1$, and the omitted-result residual isWrite $P_a=I-1a^T$ for the weighted centering matrix, and $M=\mathrm{diag}(1-a_1,\dots ,1-a_n)$ for a diagonal matrix. ThenEach 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 givesEqual coefficients and equal variances reduce these expressions toThe 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 $S=0.815$, andThis 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.

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 becomesThus 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,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.