Domain sample allocation within primary sampling units in designing domain-level equal probability selection methods 7. Summary and concluding remarks Domain sample allocation within primary sampling units in designing domain-level equal probability selection methods 7. Summary and concluding remarks

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In the design of any survey, there is a need for good representation of analysis domains in the sample. The sample allocation is not that simple because, unlike information about indicators for commonly used strata available in the sampling frame, domain indicators are generally not available or even if available, it may not be practical to stratify by domains due to interviewer travel costs for in-person surveys. What is needed is a method of sample allocation which allows for desired over-(under-) sampling of domains such that the resulting design is self-weighting or $epsem$ for domains. Such designs are desirable for variance efficiency in general. In the case of one phase two stage designs, under certain assumptions, it is possible to allocate equal interviewer workload per selected PSU such that the sample size for all selected PSUs is controlled at the desired level, but domain sizes over all selected PSUs satisfy the desired level only in expectation. On the other hand, in the case of two phase two stage designs, it is possible to allocate domain sample sizes within PSUs such that the domain sizes over all PSUs are controlled at desired levels but the sample size per selected PSU is not controlled as the equal interviewer workload per PSU is not deemed important in this case. Although the $epsem$ design of Kish at the population level is well known, domain level $epsem$ (denoted $depsem$ in this paper) designs are not well known among practitioners.

In this paper, we considered two main scenarios for $depsem$ designs considered by Folsom et al. (1987). First, for two stage designs with known domain level PSU population counts (as well as known frame-level domain identifiers for elementary units) and pre-specified domain sample sizes, the PSU selection probabilities are defined such that the desired PSU sample size (equal per PSU) is allocated to domains within PSUs to obtain a $depsem$ design. Second, for two phase two stage designs with known PSU selection probabilities and pre-specified domain sample sizes, the domain sampling rates are defined such that the desired domain sample size is allocated to PSUs within domains to obtain a $depsem$ design. These two designs were referred to as $depsem-\text{A}1$ and $\text{B}1$ respectively. A simple justification of these two designs was provided. It is based on the key idea for obtaining $depsem$ designs that the sampling rate $\left(n_{id}/N_{id}\right)$ at the PSU by domain level should be made directly proportional to the domain level sampling rate $\left(f_d\right)$ but inversely proportional to the PSU selection probability $\left({\pi }_i\right).$ For $depsem-\text{A}1,$ $f_d$ is known but ${\pi }_i$ is suitably defined (it was termed composite measure of size by Folsom et al. 1987), while for $depsem-\text{B}1,$ ${\pi }_i$ is known but $f_d$ is suitably defined. The corresponding stratified versions (denoted by $\text{A2}/\text{B2})$ can also be easily defined.

As a generalization of $depsem-\text{A}1/\text{B}1$ designs, $depsem-\text{AB}1$ was proposed where domain-level PSU population counts are only approximately known for specifying PSU selection probabilities, but a two phase design is used to allocate desired domain sample sizes to PSUs after obtaining the true domain-level population counts for selected PSUs in the first phase. Also generalizations of $depsem-\text{B}1$ were considered to obtain $depsem-\text{C}1$ when PSU selection probabilities are pre-specified from other considerations. The $depsem-\text{C}{1}^{\prime }$ extends $depsem-\text{C}1$ to cover certain practical realistic situations: 1) subsampling of elementary units within each selected PSU in the first phase to reduce cost, and 2) nonresponse in screening units for domain classification. The $depsem-\text{C}{2}^{\prime }.$ design allows for stratification in addition to practical features of $depsem-\text{C}{1}^{\prime }$ mentioned above. For all $depsem$ designs except for $\text{A}1,$ PSU sample size is not directly controlled, but domain sample size is controlled via stratification of the first phase before the second phase. This is not a limitation in various practical applications where interviews are not conducted face-to-face.

The initial $depsem$ design framework of Folsom et al. (1987) to allocate equal probability samples for multiple domains in two-stage designs in conjunction with one/two phase is a useful technique currently available in the SUDAAN software system (http://www.rti.org/page.cfm/SUDAAN) and employed successfully at RTI International for many years for studies such as the National Survey of Child and Adolescent Well-Being. The generalizations presented here extend the technique to the situation of multiple domains where the domain-level population counts need to be estimated for all selected PSUs, and where PSU selection probabilities are pre-specified from other considerations. These techniques are expected to be useful to sampling statisticians in a variety of situations.

Acknowledgements

The first author is grateful to Ralph Folsom of RTI International for several useful discussions. The authors thank Ned English of NORC at the University of Chicago for his help in creating site areas for the toolbox application. Thanks are also due to the associate editor and two referees for their very useful comments which helped improve clarity and presentation of the paper.

References

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Folsom, R.E., Potter, F.J. and Williams, S.R. (1987). Notes on a composite size measure for self-weighting samples in multiple domains. Proceedings of the Section on Survey Research Methods, American Statistical Association, 792-796.

Harter, R., Eckman, S., English, N. and O’Muircheartaigh, C. (2010). Applied sampling for large-scale multi-stage area probability designs. In Handbook of Survey Research, Second Edition, (Eds., P. Marsden and J. Wright), Emerald: United Kingdom.

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Singh, A.C., and Harter, R.M. (2011). A generalized epsem two-phase design for domain estimation. Proceedings of the Section on Survey Research Methods, American Statistical Association, 3269-3282.

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