Domain sample allocation within primary sampling units in designing domain-level equal probability selection methods 1. IntroductionDomain sample allocation within primary sampling units in designing domain-level equal probability selection methods 1. Introduction

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For multi-stage design of surveys, an equal probability selection method (or $epsem,$  Kish 1965, page 21) is typically desired toward the goal of variance reduction or variance efficiency. In practice, for two or more stage designs, selection probabilities for primary (or first stage) sampling units (PSUs) are often driven by considerations of over- (under-) sampling to obtain adequate domain sample sizes, and operational efficiency such as equal interviewer workload per PSU. The simplest type of an $epsem$  design is a single stage simple random sampling (SRS) design without replacement of elementary units with selection probabilities $n/N$  where $n,N$  denote respectively the sample and population sizes. Another example is single stage stratified SRS with proportional allocation; i.e., $n_h/N_h\propto 1,$  or $n_h=f{N}_h$  where $f=n/N,$  and $n_h, N_h$  are sample and population sizes, respectively, for the $h^{\text{th}}$  stratum. These and other $epsem$  designs are described in fundamental sampling texts such as those by Cochran (1977) and Lohr (2010).

Yet another example of an $epsem$ design is single stage SRS of whole clusters. For area sampling in field surveys, clusters are useful for operational efficiency due to reduced travel cost in interviewing neighboring housing units although there are some drawbacks. Cluster sizes could vary considerably making the logistics difficult for equalizing interviewer workloads. Moreover, a complete enumeration of each cluster may not be desirable due to cost, and inefficient estimation due to reduced effective sample size as a result of intra-cluster correlations. In general, the probability proportional to size (pps) sampling of clusters followed by equal sample allocation of elementary units per cluster to equalize interviewer assignments is a reasonable and practical compromise for area cluster sampling.

Above considerations lead to two stage designs with first stage selection probabilities to be denoted by ${\pi }_i$ for the $i^{\text{th}}$ PSU, and second stage conditional selection probabilities to be denoted by ${\pi }_{j|i}$ for the $j^{\text{th}}$ elementary unit within the $i^{\text{th}}$ selected PSU. For example, in a survey of teachers, PSUs could be schools, while ultimate sampling or elementary units could be teachers within schools. For SRS of size $n_i^{*}$ within each PSU $i$ with population count $N_i,$ the probabilities ${\pi }_{j|i}$ and ${\pi }_i$ can be defined as follows to obtain an $epsem$ design; see Kish (1965, page 222). Here $n_i^{*}$ are common and equal to $n/m$ where $m$ is the desired number of selected PSUs out of a total of $M$ PSUs in the population. We have

$${\pi }_i=m\frac{N_i}{N}, {\pi }_{j|i}=\frac{n_i^{*}}{N_i}=\frac{n}{m{N}_i}.(1.1)$$

It is easily seen, as expected, that the sum of ${\pi }_i’\text{s}$ over all $M$ PSUs $i$ is the fixed sample size $m$ at the first stage, and the sum of ${\pi }_{j|i}’\text{s}$ over all $N_i$ elementary units $j$ within the $i^{\text{th}}$ PSU is the fixed sample size $n_i^{*}$ at the second stage. Moreover, the unconditional (same as joint because of nesting of units within PSUs) selection probability for the $j^{\text{th}}$ unit in the $i^{\text{th}}$ PSU is the product ${\pi }_i{\pi }_{j|i};$ i.e., $n/N$ or $f,$ which is equal for all units, as desired. For generalizations of self-weighting estimation considered in this paper, it is useful to express the implied sample allocation $n_i$ to the $i^{\text{th}}$ PSU from (1.1) as

$$n_i=\left(\frac{f}{{\pi }_i}\right)N_i,(1.2)$$

based on the observation ${\pi }_i\times n_i/N_i$ equals $f$ where $f$ is the desired sampling fraction $n/N.$ Here, the value of $n_i$ is obtained as $n_i^{*}\left(=n/m\right).$ Note that if all PSUs are selected with certainty; i.e., ${\pi }_i=1,$ the above $\text{PSU}-$ level allocation reduces to proportional allocation in stratified designs with the number of strata being the total number $M$ of PSUs.

The basic idea for making any design $epsem$ is to work backwards; that is, before specifying selection probabilities ${\pi }_i$ for PSUs, it is ensured that the sampling rate within any given PSU $i$ is inversely proportional to ${\pi }_i$ so that ${\pi }_i$ cancels out in the unconditional selection probability ${\pi }_i{\pi }_{j|i}$ within the PSU. In this way the unconditional selection probabilities for elementary units can be made common for all sampled units from different PSUs. We will use this strategy throughout the paper.

From (1.1), observe that in order for $n_i\le N_i,$ we must have $f\le {\pi }_i$ for all $i=1,\dots ,M.$ This condition can be satisfied at the design stage by collapsing neighboring PSUs in order to increase $N_i$ (and hence ${\pi }_i)$ or by reducing $f$ if necessary. In other words, the sample allocated to the $i^{\text{th}}$ PSU must be a fraction of the PSU population size $N_i$ where the fraction is given by the ratio of the desired sampling rate $f$ and the PSU selection probability ${\pi }_i.$

So far we considered $epsem$ designs for a single domain; i.e., estimation at the population level only. However, often survey designs are intended to support analytical goals for multiple domains within the target population. For example, in the case of a teacher survey, domains could be male and female teachers. For domain-level $epsem$ designs (to be termed $depsem$ in this paper), Folsom, Potter and Williams (1987) presented a method for allocating a sample of units to PSUs under two separate designs - $depsem-\text{A}1$ and $depsem-\text{B}1$ defined as follows; the numeric extension in the notation is used to differentiate them from other variations presented later.

The $depsem-\text{A}1$ Design is defined as a one phase two stage design where domain-level PSU population counts $(N_{id}$ for the $i^{\text{th}}$ PSU and $d^{\text{th}}$ domain), desired domain sample size $\left(n_{+d}^{*}\right)$ where ‘+’ denotes sum over $m$ selected PSUs, and equal PSU sample allocation $n_{i+}^{*}\left(=n/m\right)$ over all domains (i.e., equal interviewer load) are specified. Thus, the desired sampling rate $\left(f_d\right)$ for each domain is pre-specified but the PSU selection probabilities $\left({\pi }_i\right)$ are not pre-specified and are suitably defined to obtain the $depsem$ property. Here it is also assumed that frame-level domain identifiers for elementary units are available. Such a design is applicable to situations where in-person interviews with a list frame are desirable.

The $depsem-\text{B}1$  Design is defined as a two phase two stage design where PSU population counts $\left(N_{i+}\right)$ and desired domain sample size $\left(n_{+d}^{*}\right)$ over $m$ selected PSUs are specified. Domain-level population counts $\left(N_{+d}\right)$ are not specified (which of course implies that domain-level PSU population counts $N_{id}$ are not specified), and $\text{PSU}-$ level sample allocations $(n_{i+}$ over all domains) are also not pre-specified. In addition, the desired sampling rates for each domain $\left(f_d\right)$ are not pre-specified. However, PSU selection probabilities are specified by using PSU population counts as size measures, and for selected PSUs in the first stage, domain-level population counts $N_{id}$ become available after the first phase census. Here the domain sampling rates $f_d$ are suitably defined to obtain the $depsem$ property. The two phase aspect of the design is used to obtain domain membership of selected units in the first phase through screening. Such a design may be applicable more generally than the previous one.

The school/teacher example can be used to make these two $depsem$ designs concrete. In $depsem-\text{A}1,$ we know in advance how many male and female teachers are in each school from the list frame, and also we know which teachers are male and which are female. The desired sampling rates of male and female teachers, and the equal number of teachers to be selected per school are known. Then school or PSU selection probabilities are obtained to satisfy the $depsem$ property. In $depsem-\text{B}1,$ we know the probability of selecting each school based on the total number of teachers per school. We do not know how many male and female teachers are in each school, but the desired numbers of male and female teachers in the sample over all selected schools are specified. Then, after screening all teachers in the selected schools for male/female classification, the sampling rates for male and female domains for each pre-selected school are obtained to satisfy the $depsem$ property.

For $depsem-\text{A}1,$ Folsom et al. (1987) provide a composite measure of size for selecting PSUs such that its inverse appears in the specification of domain sample allocations within each PSU. The sample allocation to domains within PSUs satisfies the desired PSU sample size or interviewer workload exactly. However, the desired domain sample size is achieved only in expectation because the sample size of elementary units within domains is not directly controlled, but the PSU sample size is controlled instead to obtain equal interviewer workload.

For $depsem-\text{B}1,$ the same basic method is inverted to produce $depsem$ samples. Here, in the first phase, a census of selected PSUs at the first stage is conducted so that all elementary units within selected PSUs are stratified into domains to obtain domain-level PSU counts and are subsampled such that the desired domain sample size over all PSUs is satisfied. However, any constraint on the PSU sample size is relaxed in the interest of obtaining a $depsem$ sample. $Depsem-\text{B}1$ may be particularly useful for non-face-to-face interview modes such as telephone surveys in the second phase, where the first phase sample of elementary units is used to obtain contact information and domain classification. The first phase results may be based on a self-administered screening questionnaire sent by mail or dropped off after an in-person contact effort to all or a large sample of units in each selected PSU. If the main interview is conducted by phone in the second phase, having equal interviewer workload per PSU is of no practical consequence. Folsom et al. (1987) also considered natural generalizations of both $depsem$ designs to the case of stratified population of PSUs in the first phase.

In this paper, we introduce a systematic general framework for defining $depsem$ designs which provides a simple justification for the $depsem$ property of the above two designs. We then propose generalizations of the two designs under the above framework to obtain new useful variations of $depsem$ designs encountered in practice; see Singh and Harter (2011) for an earlier development. See also Fahimi and Judkins (1991) for an interesting simulation study comparing traditional and nontraditional measures of size with respect to between PSU variance contributions. The organization of this paper is as follows. Section 2 reviews the original composite measure of size method for selecting PSUs as proposed by Folsom et al. (1987) for the $depsem-\text{A}1$ design including its stratified version. Section 3 presents the inverted method of Folsom et al. (1987) for $depsem-\text{B}1$ to obtain domain-level sampling rates over all pre-selected PSUs. Section 4 presents a generalization to a hybrid $depsem-\text{AB}$ design where the domain-level PSU counts for all PSUs are assumed to be only approximately known, and are used first to specify PSU selection probabilities obtained as composite measures of size as in $depsem-\text{A}1,$ and then sampling rates from selected PSUs are specified as in $depsem-\text{B}1$ by obtaining true domain-level PSU counts for selected PSUs through first phase screening. Another generalization considered in Section 4 is when PSUs in the first phase are selected with arbitrarily pre-specified selection probabilities. Section 5 further generalizes $depsem-\text{B}1$ to designs where the second phase sample within each selected PSU is not a census (i.e., there is subsampling within PSUs) or when it is a census but is subject to nonresponse, or both. Generalizations to stratified designs are also considered in Section 5. Section 6 presents a hypothetical but realistic example based on a study for which the proposed $depsem$ designs were developed under a two-phase two stage design to establish nationally representative norms for an English and Spanish instrument toolbox for assessing behavioral and cognitive functions. We conclude with remarks in Section 7.

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