Proportional Sampling

Proportional sampling is a method of sampling in which the investigator divides a finite population [Page 1120]into subpopulations and then applies random sampling techniques to each subpopulation. Proportional sampling is similar to proportional allocation in finite population sampling, but in a different context, it also refers to other survey sampling situations. For a finite population with population size N, the population is divided into H strata (subpopulations) according to certain attributes. The size of the hth stratum is denoted as Nhand

and

Following that, a simple random sample with sample size nhwould be selected within each stratum. This is essentially the same as the stratified random sampling design with proportional allocation, and the inference of the population quantity of interest can be established accordingly. nhusually would not be an integer, so the closest integer can be used. For example, in an opinion survey, an investigator would usually prefer a sample with the sample structure as close as possible to the population structure with respect to different population characteristics, such as age, gender, race, and so forth. Suppose the population structure of gender by age for people more than 20 years of age is as summarized in Table 1, and the total sample size n is determined to be 1,068. The sample sizes of each stratum can be determined as shown in Table 2.

For example, the sample size of females between the ages of 30 and 39 can be determined by

where 245,321 is the total population size.

For other sampling survey situationsfor instance, the selection of sites to monitor the air population level in a region or time points to observe a production process for quality control purposes during a period of operationthe study region can be viewed as a compact region and the number of possible sampling units is infinite. Suppose that the study region Dis divided into Hdisjoint domains D p, h = 1, …, H,according to the practical or natural demands; that is,

For example, for ecological research on Florida alligators, the study region can be a collection of several lakes, and the domains are the lakes. To study the ecosystem in a certain lake, the investigator might divide the study region, a three-dimensional compact space of the lake, by water depth. The investigator would like to allocate the sample sizes proportional to the area Dp,to ensure a fair devotion of sampling effort to each domain. Proportional sampling can also be used such that

and

For example, an investigator would like to take samples of water from several lakes for a water pollution study. Suppose that there are five lakes, [Page 1121]denoted from A to E, and the areas of these lakes are summarized in Table 3. If the sample size is determined to be 150, then the sample sizes in each lake are as summarized in Table 4.

For example, the sample size in Lake C can be determined as

where

is the total area of the study region.

The principle of proportional sampling is widely used in practical sampling survey cases, either to determine the sample sizes of each domain or to evenly spread sampling units over the study region. Numerous researchers recommend proportional sampling because it is relatively more efficient than the uniform sampling or simple random sampling on the whole study region. To name a few, Jose Schoereder and colleagues described the application of proportional sampling in an ecological survey. Saul Blumenthal discussed proportional sampling on a time interval, in which the study region is a one-dimensional compact space. T. Y. Chen and colleagues studied the performance of proportional sampling on software testing and found it is a better choice than the random testing.

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