= Wringing-allocation pilot uncertainty
\b[Wringing-allocation pilot uncertainty] is uncertainty in the noise components used to choose how many <gauge block> contacts to prepare and how many readings to take per contact. The optimum in <nested contact-study allocation> depends on estimated between-contact and within-contact variability; a short pilot does not determine their ratio exactly.
For a balanced pilot with $m_0$ independent wrings and $r_0$ readings per wring, use the <random-intercept linear mixed model>
$$
y_{ij}=\mu+b_i+e_{ij},\qquad
\operatorname{Var}(b_i)=\tau^2,\quad \operatorname{Var}(e_{ij})=\sigma^2,
$$
with all random effects independent. The within-wring mean square $MS_W$ has <expected value> $\sigma^2$; the between-wring mean square $MS_B$ has <expected value> $\sigma^2+r_0\tau^2$. Thus method-of-moments planning estimates are
$$
\widehat\sigma^2=MS_W,\qquad
\widehat\tau^2=\max\left(0,\frac{MS_B-MS_W}{r_0}\right).
$$
Truncating a negative estimate is a boundary convention, not proof that true between-wring <variance> vanishes. The expectations follow by decomposing the independent within-wring residuals and wring means in this stated model.
An original pilot has $r_0=2$, $MS_W=36\,\mathrm{nm^2}$ and $MS_B=38\,\mathrm{nm^2}$. The estimates are $\widehat\sigma=6$ nm and $\widehat\tau=1$ nm. If another short pilot gives $MS_B=34\,\mathrm{nm^2}$ with the same $MS_W$, truncation instead gives $\widehat\tau=0$. Substituting that boundary value into the continuous planning rule drives the preferred readings per wring toward infinity, revealing a fragile design decision rather than an experimentally established absence of contact variability.
With independent normal residuals, $MS_W$ has relative standard deviation $\sqrt{2/\nu_W}$, where $\nu_W=m_0(r_0-1)$. Five wrings with two readings each give $\nu_W=5$ and approximately 63% relative scatter in this variance estimate. The between-component estimate subtracts two noisy mean squares and can be still less stable. More readings of the same contact do not substitute for observing more independent contacts.
A useful planning sensitivity analysis holds readout standard deviation at a stipulated 6 nm while testing plausible $\tau$ from 1 to 4 nm. With preparation cost 9 minutes and reading cost 1 minute, the continuous optimum
$$
r_*=(\sigma/\tau)\sqrt{9/1}
$$
ranges from 18 to 4.5 readings per wring. Under a 180-minute budget, feasible allocations $(m,r)=(15,3),(12,6),(10,9)$ give mean variances approximately $(0.867,0.583,0.500)\,\mathrm{nm^2}$ when $\tau=1$ nm, but $(1.867,1.833,2.000)\,\mathrm{nm^2}$ when $\tau=4$ nm. The best choice among these candidates changes with the pilot assumption; no one set of counts is universally optimal.
<Cluster-resampled wringing allocation> can retain the pilot's contact grouping when assessing planning sensitivity; treating all readings as independent destroys the nesting. Any resampling still inherits the pilot's limited contact population. Physically, record fresh preparation, waiting time and handling: NIST documents changes in wringing films with conditions and elapsed time.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[Appendix B, Physics of Wringing Films]{ref} Drift and common reference errors require their own model terms rather than being silently absorbed into independent readout noise.
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