Multiple-Frame Sampling

Most survey samples are selected from a single sampling frame that presumably covers all of the units in the target population. Multiple-frame sampling refers to surveys in which two or more frames are used and independent samples are respectively taken from each of the frames. Inferences about the target population are based on the combined sample data. The method is referred to as dual-frame sampling when the survey uses two frames.

Sampling designs are often dictated by several key factors, including the target population and parameters of interest, the population frame or frames for sampling selection of units, the mode of data collection, inference tools available for analyzing data under the chosen design, and the total cost. There are two major motivations behind the use of multiple-frame sampling method: (1) to achieve a desired level of precision with reduced cost and (2) to have a better coverage of the target population and hence to reduce possible biases due to coverage errors. Even if a complete frame, such as a household address list, is available, it is often more cost-effective to take a sample of reduced size from the complete frame and supplement the sample by additional data taken from other frames, such as telephone directories or institutional lists that might be incomplete but less expensive to sample from. For surveys of human populations in which the goal is to study special characteristics of individuals, such as persons with certain rare diseases, a sample taken from the frame for general population health surveys is usually not very informative. Other frames, such as lists of general hospitals and/or special treatment centers, often provide more informed data as well as extended coverage of the target population.

There are, however, unique features, issues, and problems with inferences under multiple-frame sampling, which require unique treatments and special techniques. Let

be the population total of a study variable y, where N is the overall population size. Suppose there are three frames: A, B, and C. Each of them may be incomplete, but together they cover the entire target population. Let sA, sB, and sC be the three independent samples taken respectively from frames A, B, and C. The basic question is how to estimate Y using all three samples. It turns out that none of the samples can directly be used if the frames are incomplete. The most general picture is that the three frames divide the target population into seven disjoint domains: A, AB, ABC, AC, B, C, and BC, where A contains population units from frame A but not covered by B or C, AB includes all units from both A and B but not C, ABC represents the set of units covered by all three frames, and so on. If, for instance, frames B and C are nonoverlapping, then the domain BC vanishes. We can rewrite the overall population total as Y = YA + YB + Yc + YAB + YAC + YBC + YABC> where, for instance, YA is the population total for domain A. Each of the three samples can also be partitioned according to the involved population domains: sA = sa∪sab∪sac∪sabc, sB = sb∪sba∪sbc∪sbac and sC=sc∪sca ∪scb∪scab where, for instance, units in both sab and sba are selected from the domain AB, sab is from frame A, whereas sba is from frame B, indicated by the first letter in the subscript. Estimation of Y is typically carried out through the estimation of domain totals using relevant sample data.

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