Optimum allocation for a dual-frame telephone survey 6. SummaryOptimum allocation for a dual-frame telephone survey 6. Summary

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We investigated two designs for a dual-frame telephone survey: a take-all protocol in which every respondent in the cell-phone sample is interviewed and a screening protocol in which respondents in the cell-phone sample are screened for phone status and only CPO respondents are interviewed. For each design, we derived the optimum allocation of the overall survey resources to the two sampling frames.

We studied the allocation problem given the two traditional meanings of the word “optimum”: (1) to minimize variance subject to a constraint on data collection cost, and (2) to minimize data collection cost subject to a constraint on variance. Given fixed variance, we find that the screening approach tends to achieve lower total cost than the take-all approach when the per-unit cost of screening is low relative the unit cost of the survey interview. The take-all approach can achieve the lower total cost when the per-unit cost of screening is relatively high. Similarly, given fixed total cost, the screening protocol tends to be the more efficient approach when the per-unit cost of screening is relatively low, and the take-all protocol can be the more efficient approach as the per-unit cost of screening rises. Both the landline and cell-phone samples have the capacity to produce estimators for the dual-user domain, while only the cell-phone sample can produce estimators for the CPO domain. Thus, when screening is relatively inexpensive on a per-unit basis, then it should be used to produce the largest possible sample from the CPO domain. But when screening is relatively expensive, then it is better to avoid the screening step and invest the survey resources in a larger interview sample. These results were obtained under an assumption of simple random sampling, and they may not carry over exactly to other sampling designs.

The take-all design results in two estimators for the dual-user domain, which are combined using factors of $p$ and $1-p$ for the estimators from the landline and cell-phone samples, respectively. We studied the optimum choice of $p$ and gave expressions for reasonable compromise values of $p.$ When variance (or cost) is considered as a function of $p,$ we found that it is fairly flat in a neighborhood of the optimum. The optimum allocation itself is a function of $p$ and we found that the allocation is relatively insensitive to choices of $p$ within a broad neighborhood of the optimum $p.$

We initiated this work before 2010 at a time when the CPO population in the U.S. was only a fifth to a quarter of the total population of households. At that time it made sense to contemplate a protocol in which the larger landline sample is interviewed in its entirety and the smaller cell-phone sample is screened for CPO status. At this writing, however, the CPO population comprises more than a third of the total population of households and it is still growing. It has become reasonable to consider a new screening protocol in which the landline sample is screened for telephone status and only LLO respondents are interviewed. The foregoing allocations and findings apply to this new protocol by symmetry.

We illustrated the optimum allocations and the two interviewing protocols using the 2011 National Immunization Survey. The survey is designed to minimize cost under a fixed variance constraint. The NIS results are limited to the population of children age $19-35$  months. Similar results may or may not obtain for a general population survey or for a survey with a different structure of per-unit costs.

Acknowledgements

The authors’ kindly acknowledge suggestions for improved readability offered by the Associate Editor and referees. Disclaimer: The findings and conclusions in this paper are those of the author(s), and do not necessarily represent the official position of the Centers for Disease Control and Prevention.

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