Unequal-cluster weighting in a cluster bootstrap

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.

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]