Multiplicity Sampling

Multiplicity sampling is a probability sampling technique that is used to enhance an existing sampling frame by adding elements through a form of network sampling. It is especially useful when surveying for rare attributes (e.g. rare hereditary diseases).

In sample surveys, population elements are linked to units in the sampling frame, or frame units. For example, persons can be linked to a specific household by familial relationship among persons residing in the same housing unit. A counting rule identifies the linkage between population elements and frame units. In most surveys, population elements are linked to one and only one frame unit, and thus there is a one-to-one correspondence of element to unit. For multiplicity sampling, a counting rule is established that defines the linkage between population elements and frame units in which one or more population elements are linked to one or more frame units. The counting rule defines a one-to-many or a many-to-many correspondence between population elements and frame units. The count of frame units linked by the counting rule to each population element is called the multiplicity of a frame unit.

For an unbiased estimate of the number of population elements, the design-based sampling weight for each selected frame unit is adjusted for the number of frame units linked to the population element by dividing the sampling weights by the multiplicity. The multiplicity is needed for only those units selected in the sample. Multiplicity sampling uses the linkage of the same population element to two or more frame units to allow the sample of frame units to identify more population elements.

For example, in a household survey to estimate the frequency of a target condition in a population, the standard household survey would enumerate only persons with the target condition in sampled households. With multiplicity sampling, a counting rule based on adult biological siblings residing in households would identify a person with a specific attribute (the population element) linked to their own household and to the households of his or her adult biological siblings. Each sampled household member would be asked, (a) if you or an adult biological sibling have the specific condition, (b) the number of adult siblings with the condition, and (c) the number of households containing adult biological siblings. The person with the attribute would be [Page 492]identified with all of these households, not only their own household. Each frame unit in the sample would be assigned the count of adult siblings with the condition, and the multiplicity would be the number of households containing adult biological siblings of the person with the condition. The multiplicity-adjusted sampling weight is the design-based sampling weight for the household member divided by the multiplicity.

The sampling variance would be computed using the standard variance estimator appropriate for the sampling design. Because the multiplicity for each sampled frame unit will vary, the multiplicity-adjusted sampling weights often exhibit more variation than the design-based sampling weights before the multiplicity adjustment and can be expected to increase the sampling variance relative to the sampling.

Multiplicity sampling is an option when population elements with the target condition are rare and the costs of the large sample to identify an adequate number of population elements are beyond the survey resources. Multiplicity sampling requires a clear workable counting rule that can achieve an accurate count of the multiplicity for each sampling unit.

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