Proportional Allocation to Strata

Proportional allocation is a procedure for dividing a sample among the strata in a stratified sample survey. A sample survey collects data from a population in order to estimate population characteristics. A stratified sample selects separate samples from subgroups of the population, which are called “strata” and can often increase the accuracy of survey results. In order to implement stratified sampling, it is necessary to be able to divide the population at least implicitly into strata before sampling. Given a budget that allows gathering data on n subjects or a budget amount SB, there is a need to decide how to allocate the resources for data gathering to the strata. Three factors typically affect the distribution of resources to the strata: (1) the population size, (2) the variability of values, and (3) the data collection per unit cost in the strata. One also can have special interest in characteristics of some particular strata that could affect allocations.

Assuming the goal of the survey is to estimate a total or average for the entire population, the variability of values are not known to differ substantially by strata, and data collection costs are believed to be roughly equal by strata, one could consider allocating sample size to strata proportional to strata population sizes. That is, if there are H strata with population size Nh in stratum h, h = 1,2,…,H, and one can afford to collect data on n units, then the proportional allocation sample size for stratum h is nu = n(Nh/N), where is the total population size. As a result, strata with large numbers of units in their populations receive more sample, whereas small strata receive less sample. With roughly equal per-unit data collection costs, a budget amount $B corresponds to a total sample size n. If the nh's are not integer, then one must round the numbers to integers for sample selection. Rounding does not necessarily move all values to the closest integer for all strata, because the total sample size n needs to be allocated.

Suppose you want to collect data on students at a large public university. Questions of interest could be hours worked per week, amount expended per semester on textbooks, amount of time spent eating at restaurants in a week, number of trips to the airport in a semester, and whether or not friends smoke cigarettes. The students selected for the survey could be contacted via their university email addresses and asked to complete an online Web survey. A survey can be preferable to contacting every student, because better efforts can often be made for a sample to encourage response and check data quality. Administrative records contain college year designations (1st, 2nd, 3rd, 4th) for each student in the target population; college years can be used as strata. Suppose the total sample size is allowed to be 1,600 students. Equal allocation to strata would sample 400 students from each year. Table 1 presents proportional allocations of students to the four strata based on total enrollments by college year; these numbers are similar to 2006 enrollment at Iowa State University. As can be seen in the table, the stratum of fourth-year students receives the largest sample (n4 = 503), where as the stratum of second-year students receives the smallest (n2 = 324).

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