Wringing-film covariance model

A wringing-film covariance model specifies how uncertain effective contact-layer contributions vary together in a gauge block wringing measurement. Individual contact uncertainties do not determine the uncertainty of a stack unless their dependence is also specified.

Let the vector of contact differences be $\mathbf{\delta }=({\delta }_1,\dots ,{\delta }_n{)}^T$ and its covariance matrix be $\Sigma$. For a correction $\Delta ={\mathbf{c}}^T\mathbf{\delta }$, propagation givesThe coefficients encode the actual geometry and calibration subtraction. They need not all be positive. This is the general correlated-input law applied to wringing-film uncertainty.^[ref]

A simple original model writes ${\delta }_j=b+{ϵ}_j$, where one common contribution $b$ has variance $u_b^2$ and independent contact contributions have variance $u_{ϵ}^2$. ThenFor three equal-sign contacts with $u_b=3\mathrm{nm}$ and $u_{ϵ}=2\mathrm{nm}$, the variance is $93\mathrm{n}{\mathrm{m}}^2$. For a difference of two contacts with coefficients $(1,-1)$, the common contribution cancels and the variance is only $8\mathrm{n}{\mathrm{m}}^2$. These are chosen covariance structures, not universal properties of wrung surfaces.

Calibration and use can share only part of their contact history. If $n$ calibration contacts share effect $b_c$ and $n$ use contacts share effect $b_u$, their common-effect correction is $n(b_u-b_c)$. Its variance isPerfectly shared equal contributions cancel; independent preparation effects do not. Assuming either relation requires evidence about preparation, reference surfaces and measurement history.

Repeated readouts of one unchanged wring mostly probe readout scatter. Rewringing samples a different level of variability; changes of surface preparation or elapsed time may add another. The NIST handbook documents that contact layers depend on conditions and can evolve after thermal equilibration.^[ref] A nested wringing-repeatability study can help separate those levels, but repeatability data alone do not identify every shared calibration contribution.

Retain the covariance assumptions with the result and test whether the proposed matrix is a valid positive semidefinite matrix. A small combined uncertainty obtained by assigning cancelling correlations is meaningful only when those correlations describe the physical comparison.

A nested wringing-repeatability study separates variation between independently prepared gauge block wringings from readout scatter within one unchanged wring. Repeating a detector reading many times does not sample the contact variability introduced by separating and rejoining the surfaces.

For $m$ wrings with $r$ readings each, an original balanced random effect model iswhere independent wring effects have variance ${\tau }^2$ and independent readout errors have variance ${\sigma }^2$. Assume stable conditions and independence between those levels. One-way analysis of variance givesThus $M{S}_W$ estimates readout variance and $(M{S}_B-M{S}_W)/r$ estimates the between-wring variance in this model. A negative variance estimate reflects sampling variability or model limitations, not a physically negative contact variance.

For a small invented data set, duplicate readings in nm relative to an arbitrary reference are $(-3,-1)$, $(1,3)$ and $(5,7)$. Wring means are $-2,2,6$; $M{S}_W=2$ and $M{S}_B=32$, so the method-of-moments between-wring variance is $15\mathrm{n}{\mathrm{m}}^2$. These are arithmetic examples, not a recommended sample size or a NIST calibration result.

For the overall mean of the balanced study,With stipulated $\tau =6\mathrm{nm}$, $\sigma =2\mathrm{nm}$, five wrings and four readings per wring, its standard deviation is $\sqrt{7.4}\approx 2.72\mathrm{nm}$. Increasing to forty readings on the same five wrings lowers it only to about 2.69 nm. Fifty wrings with four readings each would instead give about 0.86 nm, if the same model and stable preparation remain valid.

A shared reference or persistent preparation contribution does not disappear under either averaging rule. An independent common uncertainty $u_0$ adds $u_0^2$ to the displayed variance. This connects the nested study with the wringing-film covariance model and the gauge-block reference uncertainty floor.

NIST's handbook documents contact-layer dependence on preparation and time, including evolution after thermal equilibration.^[ref] Record rewringing events, timestamps and surface preparation; use appropriate controls for drift and damage. A nested contact-study allocation should balance new wrings against repeat readings, rather than treating every reading as an independent contact experiment.

Nested contact-study allocation chooses how many independent contacts to prepare and how many readings to take within each contact. In a nested wringing-repeatability study, new contacts sample between-wring variability, whereas extra readings of one unchanged contact mainly reduce readout scatter.

For $m$ wrings and $r$ readings per wring, suppose independent between-wring variance is ${\tau }^2$ and independent readout variance is ${\sigma }^2$. The overall mean hasAn original planning model assigns cost $c_w$ to preparing a wring and cost $c_r$ to each reading. With fixed budget $C=m(c_w+r{c}_r)$, a continuous relaxation givesFor positive costs and nonzero $\tau ,\sigma$, differentiating with respect to $r$ yieldsThis balances the cost of new contacts against the benefit of repeated readings. It is not a universal required replication count; integer counts, minimum study sizes and other constraints still apply.

For stipulated costs 9 minutes per wring and 1 minute per reading, with $\tau =2\mathrm{nm}$, $\sigma =6\mathrm{nm}$ and budget 180 minutes, the continuous optimum is nine readings per wring. Ten wrings then fit exactly, giving standard deviation $\sqrt{0.8}\approx 0.894\mathrm{nm}$ for the mean. Eighteen wrings with one reading each give about 1.491 nm; four wrings with thirty-six readings each give about 1.118 nm.

The variance components need pilot information. Wringing-allocation pilot uncertainty can change the preferred allocation, especially if a small pilot wrongly suggests negligible between-wring variation. Include enough independent preparations to assess that level rather than optimizing solely for an assumed mean variance.

The physical contact can also change with time, handling or surface damage. NIST documents dependence of wringing films on conditions and elapsed time.^[ref] Drift, preparation trends and shared reference uncertainty require additional terms; they do not necessarily decrease by adding readings according to this independent-effects formula. State the objective and the constraints with the final design.

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 modelwith all random effects independent. The within-wring mean square $M{S}_W$ has expected value ${\sigma }^2$; the between-wring mean square $M{S}_B$ has expected value ${\sigma }^2+r_0{\tau }^2$. Thus method-of-moments planning estimates areTruncating 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 $r_0=2$, $M{S}_W=36\mathrm{n}{\mathrm{m}}^2$ and $M{S}_B=38\mathrm{n}{\mathrm{m}}^2$. The estimates are $\hat{\sigma }=6$ nm and $\hat{\tau }=1$ nm. If another short pilot gives $M{S}_B=34\mathrm{n}{\mathrm{m}}^2$ with the same $M{S}_W$, truncation instead gives $\hat{\tau }=0$. 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, $M{S}_W$ 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 optimumranges 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 $(0.867,0.583,0.500)\mathrm{n}{\mathrm{m}}^2$ when $\tau =1$ nm, but $(1.867,1.833,2.000)\mathrm{n}{\mathrm{m}}^2$ 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.^[ref] Drift and common reference errors require their own model terms rather than being silently absorbed into independent readout noise.

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]