Cluster sampling units, cluster totals, and one-stage SRSWOR

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.