Cluster and two-stage sampling

Let the finite element population be partitioned into nonempty finite sets $\mathcal{U}={⨆}_{i=1}^M{U}_i,$ where $M\ge 1$ and $U_i$ is the set of secondary sampling units (SSUs) in the $i$th primary sampling unit (PSU). Thus each element belongs to exactly one PSU. Write $N_i=\mathrm{\mid }{U}_i\mathrm{\mid }$, $N={\sum }_{i=1}^M{N}_i$, and let $y_{ij}\in \mathbb{R}$ be the fixed value for $j\in U_i$. The fixed total of PSU $i$ and the total over all elements are $T_i={\sum }_{j\in U_i}{y}_{ij},T={\sum }_{i=1}^M{T}_i\mathrm{.}$

A one-stage cluster design selects a subset $S_1\subseteq [M]$ of PSUs according to a finite sampling design in the sense of Finite populations and sampling designs. The PSU totals and all SSU values are fixed; design probability is over $S_1$. A selected PSU is completely enumerated, so its observed SSUs are all of $U_i$ for $i\in S_1$.

For a uniform simple random sample without replacement (SRSWOR) of $m$ PSUs, $1\le m\le M$, the first-stage sample space is the set of all $m$-element subsets of $[M]$, each with probability ${\left(\genfrac{}{}{0px}{}{M}{m}\right)}^{-1}$. For $i,r\in [M]$, write $I_i=1\{i\in S_1\}$, ${\pi }_i^{(1)}=P_d(I_i=1),{\pi }_{ir}^{(1)}=P_d(I_i=I_r=1),$ with ${\pi }_{ii}^{(1)}={\pi }_i^{(1)}$.

When $M\ge 2$, the finite-population variance of the PSU totals is $S_T^2=\frac{1}{M-1}{\sum }_{i=1}^M(T_i-\bar{T}{)}^2,\bar{T}=\frac{1}{M}{\sum }_{i=1}^M{T}_i\mathrm{.}$

The finite-population design convention in Finite populations and sampling designs fixes every $y_{ij}$ and averages only over the declared PSU-sampling law. A PSU is the sampling unit in one-stage cluster sampling; its SSUs are all observed after that PSU is selected. This does not treat the SSUs as independently sampled units.

Use the PSU partition and fixed values from Cluster sampling units, cluster totals, and one-stage SRSWOR. A first-stage design has a nonempty finite support ${\mathcal{S}}_1\subseteq 2^{[M]}$ and probability mass function $p_1$, positive on that support and summing to $1$. For each PSU $i$, fix a second-stage sampling design with nonempty finite support ${\mathcal{S}}_{2i}\subseteq 2^{U_i}$ and mass function $p_{2i}$, positive on that support and summing to $1$. Given the selected PSU set $S_1$, the second-stage samples $(S_i{)}_{i\in S_1}$ are independent, with each $S_i$ drawn from $p_{2i}$, independently of the identities of other selected PSUs. The joint law is $P_d(S_1,(S_i{)}_{i\in S_1})=p_1(S_1){\prod }_{i\in S_1}{p}_{2i}(S_i)\mathrm{.}$ It is a probability mass function: summing over the second-stage supports first gives $p_1(S_1)$ for each $S_1$, and summing these values gives $1$. For a function $g$ of the joint outcome, write $E_1[h]={\sum }_{S_1\in {\mathcal{S}}_1}{p}_1(S_1)h(S_1),E_2[g\mid S_1=s]={\sum }_{(S_i{)}_{i\in s}}({\prod }_{i\in s}{p}_{2i}(S_i))g(s,(S_i{)}_{i\in s})\mathrm{.}$ For a first-stage function $h$, define ${\mathrm{Var}}_1(h)=E_1[(h-E_{1}h{)}^2]$. For a function $g$ of the full second-stage outcome and fixed $S_1=s$, define ${\mathrm{Var}}_2(g\mid S_1=s)=E_2[(g-E_2[g\mid S_1=s]{)}^2\mid S_1=s]\mathrm{.}$ The symbol ${\mathrm{Var}}_{2i}$ denotes centered-square variance under $p_{2i}$, and ${\mathrm{Var}}_d$ denotes centered-square variance under the joint law. These use the variance definition in Moments, variance, and covariance on a probability space.

For the first stage, put ${\pi }_i^{(1)}=P_d(i\in S_1),{\pi }_{ir}^{(1)}=P_d(i,r\in S_1)\mathrm{.}$ For the second-stage design in PSU $i$, put ${\pi }_{j\mid i}^{(2)}=P_{2i}(j\in S_i),{\pi }_{j\mathrm{\ell }\mid i}^{(2)}=P_{2i}(j,\mathrm{\ell }\in S_i)\mathrm{.}$ The stage-specific covariance differences are ${\mathrm{\Delta }}_{ir}^{(1)}={\pi }_{ir}^{(1)}-{\pi }_i^{(1)}{\pi }_r^{(1)},{\mathrm{\Delta }}_{j\mathrm{\ell }\mid i}^{(2)}={\pi }_{j\mathrm{\ell }\mid i}^{(2)}-{\pi }_{j\mid i}^{(2)}{\pi }_{\mathrm{\ell }\mid i}^{(2)}\mathrm{.}$

If ${\pi }_i^{(1)}>0$ for every PSU and ${\pi }_{j\mid i}^{(2)}>0$ for every SSU, the within-PSU expansion total and the two-stage expansion total are defined by ${\widehat{T}}_i={\sum }_{j\in S_i}\frac{y_{ij}}{{\pi }_{j\mid i}^{(2)}},\widehat{T}={\sum }_{i\in S_1}\frac{{\widehat{T}}_i}{{\pi }_i^{(1)}}\mathrm{.}$ No positivity assumption on the pair-inclusion probabilities is part of this definition.

The product in the joint law is an explicit conditional-independence assumption across selected PSUs. A design with dependent second-stage samples across PSUs, or a second-stage law depending jointly on the first-stage set, has a different variance formula. The expansion estimator's divisions use only first-order inclusion probabilities.

Let a fixed finite population have $M\ge 2$ PSUs with fixed cluster totals $T_1,\dots ,T_M$. Select $m$ PSUs by uniform SRSWOR, where $1\le m\le M$. For $S_1$ the selected set, define ${\widehat{T}}_C=\frac{M}{m}{\sum }_{i\in S_1}{T}_i,S_T^2=\frac{1}{M-1}{\sum }_{i=1}^M(T_i-\bar{T}{)}^2,\bar{T}=\frac{1}{M}{\sum }_{i=1}^M{T}_i\mathrm{.}$ Then $E_d[{\widehat{T}}_C]=T:={\sum }_{i=1}^M{T}_i$ and ${\mathrm{Var}}_d({\widehat{T}}_C)=\frac{M^2(1-m\mathrm{/}M)}{m}{S}_T^2\mathrm{.}$

The proof is design-based and uses no distributional model for the fixed $T_i$, no normal approximation, and no axiom of choice. At $m=M$ the sample is a census and the displayed variance is zero; $m=1$ is covered by the zero pair-inclusion probability in step 1.1. The statement excludes $M=1$ because $S_T^2$ with denominator $M-1$ is then undefined; that sole-PSU census has zero variance directly.

Let $m_c\ge 2$ and $k\ge 1$. In a no-finite-population-correction superpopulation comparison, suppose the sampled PSU outcome vectors are independent and have a common distribution with $E[Y_{ij}]=\mu ,\mathrm{Var}(Y_{ij})={\sigma }^2>0,\mathrm{Cov}(Y_{ij},Y_{i\mathrm{\ell }})=\rho {\sigma }^2(j\mathrm{\ne }\mathrm{\ell }),$ for each PSU $i$, while distinct PSUs are independent. The cluster estimator is the mean of $k$ complete cluster means, ${\bar{Y}}_C=\frac{1}{k{m}_c}{\sum }_{i=1}^k{\sum }_{j=1}^{m_c}{Y}_{ij},$ and the reference element SRS estimator is the mean of $k{m}_c$ independent draws from the same one-element marginal distribution. Then $\mathrm{DEFF}:=\frac{\mathrm{Var}({\bar{Y}}_C)}{\mathrm{Var}({\bar{Y}}_E)}=1+(m_c-1)\rho \mathrm{.}$ The covariance model's validity entails $-1\mathrm{/}(m_c-1)\le \rho \le 1$. In particular, $\rho =0$ means no ICC inflation: both variances equal ${\sigma }^2\mathrm{/}(k{m}_c)>0$. This formula makes no claim for unequal cluster sizes or weights, nonexchangeable within-cluster covariance, dependent PSUs, or non-negligible finite-population corrections.

Shalabh derives the equal-cluster relation after neglecting finite-population corrections. On PDF p. 7 the source states that $\rho =0$ gives relative efficiency $E=1$ and describes this as “there is no error.” In context this means there is no efficiency loss relative to element SRS due to clustering, not zero sampling variance: both variances remain ${\sigma }^2\mathrm{/}(k{m}_c)>0$ when ${\sigma }^2>0$.

Use the product two-stage design and expansion estimator from Two-stage sampling laws and the expansion estimator. Assume AC and ${\pi }_i^{(1)}>0$ for every PSU $i$ and ${\pi }_{j\mid i}^{(2)}>0$ for every SSU $j\in U_i$. Define ${\mathrm{\Delta }}_{ir}^{(1)}={\pi }_{ir}^{(1)}-{\pi }_i^{(1)}{\pi }_r^{(1)},{\mathrm{\Delta }}_{j\mathrm{\ell }\mid i}^{(2)}={\pi }_{j\mathrm{\ell }\mid i}^{(2)}-{\pi }_{j\mid i}^{(2)}{\pi }_{\mathrm{\ell }\mid i}^{(2)}\mathrm{.}$ Then $E_d[\widehat{T}]=T$ and ${\mathrm{Var}}_d(\widehat{T})={\sum }_{i=1}^M{\sum }_{r=1}^M\frac{{\mathrm{\Delta }}_{ir}^{(1)}}{{\pi }_i^{(1)}{\pi }_r^{(1)}}{T}_i{T}_r+{\sum }_{i=1}^M\frac{1}{{\pi }_i^{(1)}}{\sum }_{j\in U_i}{\sum }_{\mathrm{\ell }\in U_i}\frac{{\mathrm{\Delta }}_{j\mathrm{\ell }\mid i}^{(2)}}{{\pi }_{j\mid i}^{(2)}{\pi }_{\mathrm{\ell }\mid i}^{(2)}}{y}_{ij}{y}_{i\mathrm{\ell }}\mathrm{.}$ The first term is the design variance of the expansion based on the true PSU totals. The second is the expected within-PSU sampling variance. Zero pair inclusion probabilities are permitted because the formula divides only by positive first-order inclusion probabilities.

The AC hypothesis is carried from the published conditional-expectation and conditional-variance suppliers used in step 2.1; it is not used in the one-stage theorem or the direct finite enumeration on the B page. The stage-2 independence assumption is essential: dependent second-stage samples across PSUs add conditional covariance terms.