Dual-Frame Sampling

Dual-frame sampling designs are a subset of multiple-frame designs in which units within the population of interest are selected via independent probability samples taken from each of two frames. These two frames make up the population of interest, and they typically overlap. The dual-frame sampling approach is often useful when the amount of undercoverage from using a single frame is substantially improved by the introduction of two (or more) frames. The degree of overlap in the two frames is usually not known a priori to the sampling, but should this information be available, estimates of the amount of undercoverage to be expected from the dual-frame approach can be assessed more accurately. The resulting estimates from each of the two frames in the dual-frame sampling design are combined to form a single composite dual-frame estimate of the population parameter(s) of interest. A generic figure illustrating the basic structure of a two-frame design is provided in Figure 1.

Considering this figure, we can see that there are three possible “overlap” situations that may occur when using two frames in the sampling design, including the following:

• 1.

  Illustrated in Figure 1 is the circumstance in which neither of the two frames is completely included in the other, implying that Frame A and Frame B have some degree of overlap (i.e. like cell phone and landline phone ownership). This approach serves to improve the overall coverage of the target population, thus reducing undercoverage; this situation is very common for dual-frame designs in practice. Another spin on this approach comes when estimates from a rare population are desired. For example, using random-digit dialing (RDD) to survey the state to estimate the quality of life of breast cancer survivors one year beyond their cancer diagnosis is possible through the use of a health eligibility screener—however, within a given state, the proportion of adult citizens who are one-year breast cancer survivors may be small, making the screener approach alone prohibitively expensive. The State Cancer Registry, however, provides a list of those diagnosed with cancer and is considered “complete” somewhere around 2 years post-diagnosis. So using this frame at the one-year point would certainly be accompanied by a degree of undercoverage and may contain errors in diagnosis, in general but it would include more individuals from the target population of interest. Using a dual-frame approach with an RDD frame with a health screener along with the cancer registry frame may be a more viable and precise approach for estimating the quality of life parameter of interest.

  Figure 1 Illustration of two frames for a dual-frame sampling design

  
• 2.
  Not illustrated in Figure 1 is the circumstance in which Frame A is a complete subset of Frame B (i.e. a rare segment of the population, like homeless, institutionalized, or members of a health maintenance organization who were prescribed a particular type of drug). In this case, Frame B may provide complete coverage of the population frame (i.e. complete household address list for customers within a business district of a large retail corporation), while Frame A may consist of a subset of population units from [Page 213]Frame B (i.e. an email register of frequent shoppers). If the company wanted to select a random sample of customers, it may be more expensive to sample solely from Frame B based on costs associated with in-person or mailed surveys; to reduce expected costs, a sample from Frame B could be augmented with a sample from Frame A, since emailed versions of the survey would be less expensive to administer than mailed versions.
• 3.
  Also not illustrated in Figure 1 is the circumstance in which Frame A and Frame B have no overlap (i.e. list frame of hospital addresses in the northern region and a telephone directory of hospitals in the southern region of the country). In this case, the dual-frame sampling design would simplify to a stratified sampling design in which two strata (northern and southern regions) use different mechanisms for sampling (using addresses versus phone numbers, for example).

A very common estimator of a population total based on a dual-frame sampling design is the composite estimator first proposed by H. O. Hartley. This estimator combines estimates of regions (a) and (b) of Figure 1 with a linear combination of two estimates of region 2 derived from the probability samples taken from frames A and B, respectively. Specifically, the estimator is given

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