= Participant influence on a consensus assigned value
\b[Participant influence on a consensus assigned value] is the dependence created when a result being assessed also contributes to its benchmark. The deviation from that benchmark is then generally smaller than the deviation from a corresponding consensus formed solely from the other participants. This is a property of the estimator, not evidence that the laboratory improved.
For an original equal-weight calculation with $n>1$ results, write
$$
\bar x=\frac{x_i+(n-1)\bar x_{-i}}n,
\qquad x_i-\bar x=\frac{n-1}{n}(x_i-\bar x_{-i}),
$$
where $\bar x_{-i}$ is the <leave-one-out interlaboratory consensus> computed as the <arithmetic mean> of the other results. With results $(8,10,10,12)$, the overall mean is 10. The participant reporting 12 differs from it by 2, but differs from the other three laboratories' mean $28/3$ by $8/3$. The factor $3/4$ accounts for the difference exactly.
A fixed-weight version makes the effect more general. If normalized weights $a_j$ sum to one, $0\le a_i<1$, and
$$
x_c=a_ix_i+(1-a_i)x_{c,-i},
$$
then $x_i-x_c=(1-a_i)(x_i-x_{c,-i})$. A participant with weight 0.8 retains only one fifth of its difference from the remaining weighted consensus. These are fixed-weight identities; <robust statistics> with data-dependent weights requires analysis of the actual estimator, for example through an <influence function>.
Dependence also enters uncertainty. For independent results $X_i$ having common mean $\theta$ and variance $\sigma^2$,
$$
\operatorname{cov}(X_i,\bar X)=\frac{\sigma^2}{n},\qquad
\operatorname{Var}(X_i-\bar X)=\sigma^2\left(1-\frac1n\right).
$$
Treating $X_i$ and $\bar X$ as independent would instead add their variances and give the wrong result. This is a stated sampling model, not a universal uncertainty formula for every <interlaboratory comparison>.
The primary guide distinguishes assigned values based on participant consensus, expert laboratories, formulation and <certified reference materials>. It also requires the assigned value's uncertainty to be evaluated and reported.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§7.2.2 and Annex D.2, assignment methods; §7.2.6, uncertainty]{ref} Using an independent reference or a documented leave-one-out method changes the dependence, but neither is automatically appropriate for every scheme.
Finally, reducing self-influence does not resolve <shared bias in an interlaboratory consensus>. All remaining laboratories can still share the same calibration error. Benchmark independence, robustness to an extreme result and absolute accuracy are separate questions.
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