Cluster-resampled wringing allocation

Cluster-resampled wringing allocation uses pilot contacts as the resampling units when assessing how uncertain wringing and readout variability affect a planned gauge block study. Readings from one prepared contact stay grouped. Resampling them as unrelated observations would remove the dependence that motivates nested contact-study allocation.

Write the pilot as vectors $B_i=(y_{i1},\dots ,y_{i{r}_0})$ from $m_0$ independently prepared contacts. A cluster bootstrap draws $m_0$ of these vectors with replacement, copies every reading in each selected vector, recomputes the variance-component estimates and then reevaluates candidate allocations. For an original five-contact illustration, drawn indices $(2,2,5,1,4)$ copy contact2 twice and omit contact3. Each draw is a separate bootstrap occurrence; neither duplication nor omission creates a new physical contact. The empirical-with-replacement construction is the general bootstrap procedure, applied here to contact vectors.^[ref]

For a balanced random-intercept linear mixed model, each resample can yield planning estimatesWith preparation/read costs $c_p,c_r$, a finite candidate $(m,r)$ has cost $m(c_p+c_{r}r)$ and modeled mean variance ${\hat{\tau }}^2/m+{\hat{\sigma }}^2/(mr)$. Comparing candidates within each resample reveals how often the preferred allocation changes, instead of treating one pilot ratio as known exactly.

For an original180-minute budget with $c_p=9$ and $c_r=1$ minute, consider $(m,r)=(15,3),(12,6),(10,9)$. With ${\sigma }^2=16\mathrm{n}{\mathrm{m}}^2$ and ${\tau }^2=4\mathrm{n}{\mathrm{m}}^2$, their mean variances are approximately0.6222,0.5556 and0.5778nm², favoring12 contacts with6 readings. Changing only the planning between-contact variance to1nm² gives0.4222,0.3056 and0.2778nm², favoring10 contacts with9 readings. A resampling distribution of planning inputs can therefore matter to the design decision.

The cost of flattening the data is visible analytically. Under the same model, the true mean variance is $({\tau }^2+{\sigma }^2/r)/m$, while an independence calculation over all $mr$ readings gives $({\tau }^2+{\sigma }^2)/(mr)$. Their ratio isusing intraclass correlation $\rho$. For $r=4$ and $\rho =1/2$, the variance ratio is2.5, a design effect that an independence-based analysis misses.

This nonparametric analysis inherits the limited empirical contact distribution and any boundary estimate $\hat{\tau }=0$. More bootstrap draws do not provide more independent wrings. Drift, shared standards or an unbalanced pilot need an appropriate model or resampling design. NIST documents physical wringing films and their dependence on preparation and elapsed time, which is why contact preparation belongs in the experimental record.^[ref]