Domain sample allocation within primary sampling units in designing domain-level equal probability selection methods 3. Review of $depsem-\text{B}$  designs with a simple justificationDomain sample allocation within primary sampling units in designing domain-level equal probability selection methods 3. Review of $depsem-\text{B}$  designs with a simple justification

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Now suppose that the schools have already been sampled with usual PSU population counts $N_i$ as size measures, and therefore their probabilities of selection ${\pi }_i$ are known as given in (1.1). Under $depsem-\text{B}1$ involving two phase designs, $f_d’\text{s}$ are not specified, but the desired values of $n_{+d}^{*}$ are pre-specified for all domains. For example, the schools are pre-selected and the desired numbers of sampled male and female teachers are pre-specified, but the sampling rates for male and female teachers are not specified. It is still possible to select $epsem$ samples of male and female teachers using suitable values of $f_d$ in (2.2), but not with equal sample size per school, as shown in Folsom et al. (1987) and described below.

$Depsem-\text{B}1$  Design: Here ${\pi }_i$ is set by the usual size measure as $m{N}_i/N$ for $i=1,\dots ,M.$ Denote it by ${\pi }_i^{B1}.$ As in $depsem-\text{A}1,$ the sampling rate $n_{id}/N_{id}$ in domain $d$ within PSU $i$ should be set proportional to $f_d/{\pi }_i^{B1},$ although here $f_d$ is not known. Under $depsem-\text{B}1,$ each selected PSU is stratified by domains for the selection of elementary units using the first phase domain-screening information while under $depsem-\text{A}1,$ domain memberships of elementary units are assumed to be available in the frame itself. In this case, the condition of equal interviewer workload per PSU is relaxed and the desired PSU sample sizes $n_{i+}^{*}$ over all domains are not pre-specified. Instead, it is the desired domain sample size $n_{+d}^{*}$ that is directly controlled by allocating it to PSUs within each domain $d.$ Thus, $n_{+d}^{*}$ is rendered nonrandom which is clearly preferable for control on resulting precision of domain level estimates. It follows that, analogous to (2.1), the sample allocations ${\left\{n_{id}^{B1}\right\}}_{1\le i\le m}$ of the domain total $n_{+d}^{*}$ for each domain $d$ to selected PSUs are given by

$$n_{id}^{B1}=\left(n_{+d}^{*}\right)\frac{f_d{N}_{id}/{\pi }_i^{B1}}{{{\displaystyle \sum }}_{i^{\prime }=1}^m{f}_d{N}_{i^{\prime }d}/{\pi }_{i^{\prime }}^{B1}}=\left(\frac{n_{+d}^{*}}{{\widehat{N}}_{+d}^{B1}}\right)\frac{N_{id}}{{\pi }_i^{B1}}=\left(\frac{{\widehat{f}}_d^{B1}}{{\pi }_i^{B1}}\right)N_{id},(3.1)$$

where ${\widehat{N}}_{+d}^{B1}={\displaystyle {\sum }_{i^{\prime }=1}^m{N}_{i^{\prime }d}}/{\pi }_{i^{\prime }}^{B1}$ and the unspecified $f_d$ cancels out. However, we can set ${\widehat{f}}_d^{B1},$ the sampling rate for domain $d,$ as $n_{+d}^{*}/{\widehat{N}}_{+d}^{B1}.$ Clearly, ${\left\{n_{id}^{B1}\right\}}_{1\le i\le m}$ satisfies $n_{+d}^{*}$ exactly by construction. However, the allocations do not satisfy $n_{i+}^{*}\left(\text{or }n/m\right),$ even in expectation, because in general

$$E\left(n_{i+}^{B1}\right)= E\left(\frac{{\displaystyle {\sum }_{d=1}^D{\widehat{f}}_d^{B1}{N}_{id}}}{{\pi }_i^{B1}}\right)=\left(\frac{n}{m}\right)\frac{{\displaystyle {\sum }_{d=1}^{D}E\left({\widehat{f}}_d^{B1}\right)N_{id}}}{{\displaystyle {\sum }_{d=1}^{D}f{N}_{id}}}\ne \frac{n}{m},(3.2)$$

unless ${\widehat{f}}_d^{B1}$ is constant and equals $f\left(=n/N\right),$ which is in conflict with the desired disproportionate domain allocations.

Other considerations such as random rounding of $n_{id}^{B1}$ to obtain integer allocations carry over in a manner analogous to $depsem-\text{A}1.$ However, if the requirement of $n_{id}^{B1} \le N_{id}$ for all domains within each $i=1,\dots ,m,$ is not satisfied, one option is to reduce $n_{+d}^{*},$ while the other option is to collapse neighboring PSUs. For example, collapsing $i$ and $i^{\prime },$ and letting $\tilde{i}$ denote the collapsed PSU, we have $N_{\tilde{i}d}=N_{id}+N_{i^{\prime }d}.$ Then ${\pi }_{\tilde{i}}^{B1}$ required for calculating sample allocations in the second phase from (3.2) is now given by ${\pi }_{\tilde{i}}^{B1}={\pi }_i^{B1}+{\pi }_{i^{\prime }}^{B1}-{\pi }_{i{i}^{\prime }}^{B1}$ which, incidentally, also requires knowledge of the second order inclusion probability ${\pi }_{i{i}^{\prime }}^{B1}.$

$Depsem-\text{B2}$  Design: We next consider a generalization of the above case to stratified designs. In our example of the teacher survey, this case corresponds to schools stratified by school districts. This extension carries over in a manner analogous to $depsem-\text{A}2.$ That is, suppose for the first phase sample, $m_h$ PSUs are to be selected from the $h^{\text{th}}$ stratum, $h=1,\dots ,H$ with the usual pre-specified selection probabilities $m_h{N}_{hi}/N_h$ to be denoted by ${\pi }_{hi}^{B2}.$ The sample allocations ${\left\{n_{hid}^{B2}\right\}}_{1\le i\le m_h,1\le h\le H }$ of the domain total $n_{++d}^{*}$ to selected PSUs within each stratum $h,$ analogous to formula (3.1) of $depsem-\text{B}1,$ are given by

$$n_{hid}^{B2}=\left(\frac{{\widehat{f}}_d^{B2}}{{\pi }_{hi}^{B2}}\right)N_{hid},(3.3)$$

where $N_{hid}$ is the domain $d$ population count within PSU $i$ and stratum $h,$

$${\widehat{f}}_d^{B2}=n_{++d}^{*}/{\widehat{N}}_{++d}^{B2},\text{ and }{\widehat{N}}_{++d}^{B2}= {\displaystyle \sum _{h=1}^H{\displaystyle \sum _{i=1}^{m_h}{N}_{hid}}}/{\pi }_{hi}^{B2}.$$

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