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Bootstrap ratio bias with unequal cluster sizes arises because the pooled mean in a cluster bootstrap is a ratio with a random denominator. Let the empirical distribution over the $K$ observed groups assign mass $1/K$ to each pair $(n_j,T_j)$ of group size and response sum. One bootstrap replicate draws $K$ such pairs and computes ${\overline{T}}^{\ast }/{\overline{n}}^{\ast }$. Its conditional expectation need not equal $r=\overline{T}/\overline{n}$, even though the numerator and denominator averages are individually unbiased for their empirical means.

A second-order Taylor expansion of $f(t,n)=t/n$ explains the leading effect. Write $v_n$ and $c_{Tn}$ for the empirical variance of one group size and empirical covariance of one group size with its response sum, using denominator $K$. The conditional bootstrap bias is approximatelyThe second derivatives are $f_{nn}=2\overline{T}/{\overline{n}}^3$ and $f_{tn}=-1/{\overline{n}}^2$; the variances/covariance of the averages bring in $1/K$. This delta method approximation requires a denominator sufficiently concentrated away from zero and control of the remainder. It is not an exact identity for a few highly unequal groups.

For an original three-group example, take $(n,T)=(1,0),(2,2),(4,16)$. Then $r=18/7\approx 2.57143$, $\overline{n}=7/3$, $v_n=14/9$, and $c_{Tn}=26/3$. The second-order bias estimate is $-2/7\approx -0.28571$. Enumerating all $3^3=27$ ordered bootstrap draws instead gives mean about $2.24797$, hence exact conditional bias about $-0.32346$. The approximation identifies the direction and mechanism but misses a visible remainder at this small group count.

Equal group sizes remove the random denominator and this particular ratio bias. If every group has the same response mean, $T_j=r{n}_j$, every replicate ratio is exactly $r$ regardless of size. Neither case justifies replacing a row-weighted estimand with an equally weighted group mean in general; see unequal-cluster weighting in a cluster bootstrap. The primary Stata specification samples entire clusters and allows replicate row counts to vary.^[ref]

A cluster bootstrap resamples entire observed groups, retaining all observations and their dependence within each selected group. The sampling unit is the group: draw $K$ group labels with replacement from the $K$ observed groups, concatenate their observations, and recompute the desired statistic. This is a form of bootstrapping suited to independent groups whose internal observations need not be independent. The Stata manual implements it with cluster(); idcluster() distinguishes two resampled copies of the same original group when the fitted procedure needs group identifiers.^[ref]

For an original example, three sensors each supply two perfectly shared readings: $A=(0,0)$, $B=(2,2)$, and $C=(4,4)$. The overall sample mean is $2$. A cluster draw $(A,A,C)$ produces readings $(0,0,0,0,4,4)$ and sample mean $4/3$; the two copies of $A$ remain distinct resampling slots. Conditional on these data, the empirical variance of a group mean drawn uniformly from $0,2,4$ is $8/3$. Averaging three independent draws therefore gives bootstrap varianceResampling six individual rows instead givesThe row procedure halves the conditional variance by pretending that two identical within-sensor readings are independent information. These fractions use the empirical drawing law directly, not a Bessel-corrected estimate of an unknown population variance.

With unequal group sizes, drawing the same number of groups can produce a different number of rows in each replicate. A pooled row mean and a mean of group means are different statistics: unequal-cluster weighting in a cluster bootstrap must follow the intended estimand. Many rows from a few sensors do not replace having many independent sensors; arbitrary dependence between sensors, a dominant group, or too few groups can defeat the intended inference.

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]

Effective repetition count across independent calibration blocks accounts for dependence confined to each acquisition block rather than shared by every stored trace. Suppose block $j$ contains $n_j$ empty-interval indicator random variables, all with common probability $0<p<1$ and within-block correlation coefficient $\rho \ge 0$. Assume different blocks are independent. With $N={\sum }_j{n}_j$, summing within-block covariances givesMatching this to the variance of an average of independent indicators yieldsFor $K$ equal blocks of size $m$, this reduces to $N_{\mathrm{eff}}=Km/[1+(m-1)\rho ]$. The denominator involves the block size, not the total $Km$; the single-block expression in effective repetition count for detector intervals with shared drift must not be applied as though the same drift state affected every block.

For an original comparison, fix $N=120$ and $\rho =0.05$. One block gives $N_{\mathrm{eff}}=120/6.95\approx 17.27$. Six independent blocks of twenty give $N_{\mathrm{eff}}=120/1.95\approx 61.54$. Six very uneven blocks of sizes $(100,4,4,4,4,4)$ instead giveAll three designs store the same number of traces. The difference comes from how often the same fluctuating condition is reused. For fixed $N$, $K$, and positive $\rho$, minimizing ${\sum }_j{n}_j^2$ favors balanced blocks; this motivates optimal allocation of detector repetitions across calibration blocks when block-preparation costs also matter.

Independence must come from the physical acquisition model. Starting another file does not erase a shared clock, reference calibration, or slowly varying environmental influence. NIST's uncertainty law retains shared-input covariance explicitly.^[ref] If drift persists between nominal blocks, those cross-block terms must be added and the displayed formula changes. A cluster bootstrap likewise requires blocks that are suitable independent resampling units.

The effective repetition count for detector intervals with shared drift is the number of independent repetitions that would give the same variance for an average indicator as the actual dependent repetitions. In overdispersion of detector-interval counts under shared drift, let $X_i$ indicate an empty interval, with common probability $\overline{p}$ satisfying $0<\overline{p}<1$. If the covariance between distinct indicators is $v=\mathrm{Var}(p(H))$, their correlation coefficient isThe parent's law of total variance calculation therefore givesThis is an observable-specific variance equivalence, not a literal count of independent traces or a guarantee of confidence interval coverage. A different statistic or a different dependence pattern may require a different equivalence.

For an original feasible mixture, take $\overline{p}=0.1$ and $\rho =0.1$. These moments can arise when the block probability is $0$ with probability $9/19$ and $0.19$ with probability $10/19$: its variance is $0.009$. With $N=50$, the variance-inflation multiplier is $5.9$ andDoubling the repetitions to $100$ only raises $N_{\mathrm{eff}}$ to about $9.17$. For a fixed positive $\rho$, its large-$N$ limit is $1/\rho$, here $10$.

The experiment must therefore record which traces share a reference or drift state. Acquiring genuinely independent calibration blocks addresses a different limitation from oversampling one block; effective repetition count across independent calibration blocks depends on that grouping. A cluster bootstrap can preserve the observed within-block dependence when blocks are suitable independent resampling units. NIST's uncertainty law explicitly retains shared-input covariance, supporting the need to model reference reuse rather than discard it.^[ref]

Unequal-cluster weighting in a cluster bootstrap must preserve the statistic's intended population target while resampling the independent groups. For group $j$, let $n_j$ be its observed size, $T_j$ its observation sum, and ${\overline{X}}_j=T_j/n_j$. Two distinct estimates areThe first weights groups by their numbers of observations; the second gives each group equal weight. If group size is related to group response, the corresponding population targets need not agree. For identically distributed independent groups with suitable finite moments, they correspond to $E(T)/E(n)$ and $E(T/n)$, respectively.

For each cluster bootstrap replicate, draw $K$ group slots with replacement and recompute the selected statistic, including its denominator. The Stata manual explicitly notes that unequal group sizes cause replicate row counts to vary; keeping a fixed original row denominator would therefore compute a different statistic.^[ref]

For an original two-group example, take $A=(0)$ and $B=(10,10,10)$. The original pooled sample mean is $7.5$, while the equally weighted group-mean estimate is $5$. The four ordered two-group draws $AA,AB,BA,BB$ have equal probability. Their pooled sample means are $0,7.5,7.5,10$, whereas their equally weighted group-mean estimates are $0,5,5,10$.

Consequently the conditional bootstrap expectation of the pooled estimate is $6.25$, not $7.5$. This bootstrap ratio bias with unequal cluster sizes is visible with only two observed groups: averaging a random ratio does not generally give the ratio of the averages. It is not a reason to substitute the equally weighted estimate, which changes the estimand, or a claim that every large-group bootstrap is invalid.

Resampling groups in proportion to $n_j$ and then retaining all their rows introduces an additional size preference; that design needs its own justification. Replicated group copies also need distinct identifiers when the fitted model uses those identifiers. The correct resampling unit, the target mean, and the statistic recomputed in each draw must all be specified.