Half-Open Interval

The half-open interval is a linking procedure that is used in some surveys to address issues of noncoverage. Sampling frames or lists are not perfect, and survey researchers often use frames with problems such as missing elementary units, blanks for relevant information, clusters of elementary units, and duplicate listings. Of these problems, sample frames that are missing elementary units—known as noncoverage—frequently present important practical problems. For example, housing lists incorporating addresses are often used in household surveys. The housing list is often out of date, and when an interviewer visits the housing unit selected from the list, there can be newly constructed housing units that were not on the original list used for sampling. When there is noncoverage of the target population due to an imperfect sample frame, specific remedies are required to improve the frame coverage.

To account for the units missing from a frame, researchers may use a linking procedure. This is a useful device in many situations where the missing units are scattered individually or in small clusters. The linking procedure is often called the half-open interval, which indicates the interval by the boundary between the selected unit and the next unit in the list.

For example, suppose that 100 Kish Street has been selected from the housing list. From an area frame perspective, the address of 100 Kish Street has a boundary denned by its property lines up to the property of the next address on the housing list, 102 Kish Street. This boundary denotes the half-open interval, which does not include the next address on the list. If there is a new or missed housing unit within the interval (e.g. at 100 ½ Kish Street), an interviewer treats it as a sampled unit and conducts an interview with all the units in the interval, including the pre-specified unit (i.e. 100 Kish Street). Thus, the missed units have the same probability of selection as the pre-specified unit.

Occasionally, ordinary linking procedures cannot deal with the missed units adequately. For example, in the case of an address where a single household is expected, finding a newly built apartment house with 20 dwellings presents a real dilemma, since the interviewer technically would need to conduct 21 interviews instead of 1 interview. In such cases, the additional dwellings may be subsampled to reduce the interviewer workload, and weighting must be implemented to compensate for any unequal probabilities of selection. Instead of linking procedures, a large number of newly constructed units can be put into a supplementary stratum from which they are selected with varying probabilities, although it is better if a check of the frame is available to avoid such unpleasant occurrences.

These linking procedures can also be applied to the instances where ordered lists serve as frames. For example, on a payroll listing, a new employee in a department can be missed if the frame is slightly out of date. If a unique employment position on the list, such as the employee listed last in each department, is selected as a sample unit, then the new employee is also added to the frame and selected with the same probability as the one listed last by the linking rule. Another example is the list a public school has of its pupils' households. When the household of a selected child is visited, recently born or missed children can be discovered. The linking procedure must fit the missed children into the half-open interval, thereby reducing the noncoverage of the original list.

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