# Additional Considerations

In the context of using stratified random sampling with proportional allocation where strata are defined and samples are drawn using someone's self-reported race or ethnicity, several additional considerations are worth noting.^{[1]} One is what to do when not all of the calculated strata sample sizes * rig* are integers, as will likely be the case. For example, suppose in the example in Table E.1., the strata sizes in the sampling frame were instead 5,860 for stratum A, 1,095 for stratum B, and 3,045 for stratum C, with corresponding strata sample sizes of 175.80, 32.85, and 91.35. This poses a dilemma, because it is impossible to select 175.80 people. A reasonable approach is to round down the strata sample sizes to the next lowest integer, with the remaining selections attributed randomly to the strata, proportional to the fraction lost by the rounding. In this example, rounding down yields integer sample sizes of 175, 32, and 91, for a total of 298, with two selections left to allocate. Each of the remaining two selections would be randomly allocated to strata A, B, and C with probabilities 0.400, 0.425, and 0.175, respectively. The probabilities are determined by dividing the remaining fraction after rounding by the number left to allocate.

^{[2]}Such an allocation procedure would preserve the characteristic that everyone in the sampling frame has an equal probability of selection (feature 3 above). Feature 2 would be retained approximately, with a small sampling error.

^{[3]}

Another consideration for implementing stratified random sampling is the necessity for properly defined mutually exclusive groupings determining the strata. In the context of the applicant pool, this requirement may not be met, for several reasons. First, when using race/ethnicity as strata, some applicants may belong to multiple racial ethnic groups. For example, an applicant may be both black and Hispanic. If a category of black Hispanic is not offered when racial/ethnic data are collected, such applicants would be placed into whichever of these two strata to which they self-identify for the purpose of sampling; however, sampling such an applicant would increase the numbers of both blacks and Hispanics sampled.

Second, applicants may decline to provide racial/ethnic data, leaving them without a stratum. One solution for this case would be to create an additional “declined” stratum to be treated as any other in the stratified sampling process. Feature 2 would still technically hold given the definition of this new stratum; in practical terms, the racial/ ethnic distribution among non-decliners in the applicant pool would still match the racial/ethnic distribution among non-decliners in the resulting sample, while the racial/ethnic distribution among decliners in the applicant pool and in the sample would naturally differ due to sampling variability.^{[4]}

Third, the requirement to have properly defined racial/ethnic strata may not be met because self-reporting of this information allows for potential misreporting. Such misclassification would * not* alter the probability of selection for any individual in the applicant pool, including those misclassified. However, in the presence of misclassification, the reason for conducting stratified random sampling instead of taking a simple random sample (i.e., the goal of ensuring that distribution of race/ethnicity in the sample does not deviate from the distribution in the sampling frame because of chance alone, as would be the case for simple random sampling explained in Appendix D) would not be fully achieved. Although the reported distributions of race/ethnicity in the applicant pool and the sample would still match, the strata into which applicants are misclassified would actually contain applicants of multiple races/ethnicities due to the misclassification, introducing sampling variability into the number of applicants actually chosen from each race/ethnicity in each of those strata. The practical significance of the introduction of variability in the number of applicants actually sampled from each race/ethnicity depends on the amount of the misclassification. As misclassification grows, the chances for larger deviations between the desired number sampled from each race/ethnicity and the actual number sampled increase, although such deviations are equally likely to be positive or negative (as seen in Appendix D).

A final example consideration is what size sample is needed overall, which in turn sets the desired proportion that should be sampled from each of the groups. In the illustration offered above, we set the proportion at 0.03, or 3 percent. In actuality, that proportion might be driven by any number of considerations, such as the desired minimum number of people needed in the smallest stratum, the maximum number needed in the largest stratum, or the total desired in the overall sample.

- [1] These are some examples of those potential concerns, but they are not an exhaustive list.
- [2] For example, stratum B had a calculated sample size of 32.85; rounding down to an alio- cation of 32 left a remaining fraction of 0.85. Diving 0.85 by 2 remaining allocations gives 0.425 to use as the allocation probability.
- [3] An alternative strategy here would be to randomly assign the first of the two remaining allocations among the three strata using these probabilities, and then assign the final allocation to one of the other two not chosen, rescaling their respective probabilities to sum to one. Feature 2 (approximately) and feature 3 would again be retained.
- [4] Luck of the draw could technically produce a match, but this is highly unlikely in all but the most trivial of cases.