# Studies of Welfare Populations: Data Collection and Research Issues. Base Weights

The base weight for a sample unit (e.g., a sampled low-income family) is defined as the reciprocal of the probability of including the unit in the sample. The base weight for the i-th unit in the sample is given by

where is the known probability of including unit i in the sample. If the sample units are selected with equal probability, the probability of selection is

for all sample units, where n is the sample size and N is the number of units in the sampling frame. The base weight, therefore, is for all sampled units. In this case, In the family income survey (FIS) example given earlier, assume that a sample of n = 820 families was selected with equal probabilities of selection from a population of N = 41,000 families. Then the probability of selection for each unit in the sample is equal to n/N = 820/41,000, and the base weight would be equal to N/n = 50 for each sampled family. Thus, each family selected into the sample represents 50 families in the administrative file used for sampling.

State surveys may be designed to provide an equal probability sample (similar to the previous example) or a disproportionate sample of low-income persons with respect to a selected set of characteristics (e.g., demographic characteristics). In an equal-probability sample, the distribution of the sample is expected to be similar to the administrative frame. For example, if the administrative frame in state S includes 10 percent Hispanics and 90 percent non-Hispanics, an equal probability sample is expected to include about 10 percent Hispanics and 90 percent non-Hispanics. However, if state S is interested in analyzing the well-being of the low-income Hispanic population, the survey is likely to include an oversampling of low-income Hispanic persons. The oversampling can be accomplished by stratifying the frame into two strata, Hispanics and non-Hispanics, and applying a larger sampling rate to Hispanics. In this case, the sample will contain a disproportionate representation of Hispanics. When disproportionate sampling is applied in stratified sampling, different weights (referred to as base weights) are used to compensate for the unequal representation in the sample. Otherwise, estimates will be biased. Returning to the FIS example, assume that Hispanic families are sampled at a rate of 1 in 30 and that non-Hispanics are sampled at a rate of 1 in 60. Then the base weight for the Hispanics is equal to 30, and the base weight for non-Hispanics is equal to 60. Thus, each sampled Hispanic family represents 30 Hispanic families in the population, and each non-Hispanic family in the sample represents 60 non-Hispanic families in the population. For more information on disproportionate sampling, refer to Kish (1992).

Although the base weights are theoretically unbiased weights that ''inflate" the sample observations to population levels, in practice, most survey practitioners find it useful to modify the base weights. Nonresponse in the survey, for example, results in losses in the sample data that can be partially compensated for by adjusting the weights of the respondents. If the sampling frame is deficient because it is outdated or its coverage of certain population subgroups is inadequate, further adjustment of the weights may be desirable to compensate for these deficiencies. The following section provides brief descriptions of various weight adjustment procedures commonly used in large-scale surveys.

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