These schools also were considered to be part of the state sample for these states as well.
The procedure for defining school measures of size based on the sample sizes for the TUDA districts and on the estimated grade enrollments for the schools within these districts was identical to that defined for the main samples. The maximum number of hits was 1 for all districts for the fourth-grade sample, and for Chicago and New York City in the eighth-grade sample. Houston and Los Angeles had a maximum number of two hits for eighth grade, and Atlanta had a maximum number of five hits for eighth grade.
Preliminary probabilities of selection πjs were set for each school equal to
The subscript s indicates school and the subscript j indicates jurisdiction. The quantity bj is a sampling parameter that was computed separately for each TUDA district (see Sampling Parameter Values for TUDA Districts), and MOSjs is the measure of size as it was computed for the school in State NAEP (see Computation of School Measures of Size). These target probabilities of selection reflected the desired probabilities for each school, with the added constraints of maximizing overlap with the State NAEP jurisdiction samples for the states (or “alpha” school samples), and minimizing overlap with the field and link test samples. It was necessary to maximize overlap with the State NAEP samples as the schools selected were utilized for both the State NAEP and the TUDA district samples and given the same operational assessments. Therefore maximizing overlap minimized the combined school sample size. Overlap was minimized with the beta school samples, as it was necessary to cancel the field and link assessments for any overlap schools (taking these schools then as nonrespondents for the beta sample), to avoid undue burden on the schools.
The technique for controlling overlap was called Keyfitzing (see Keyfitz 1951 and Rust and Johnson 1992). In this approach, the relevant sample design was viewed as encompassing all possible alpha, beta, and district samples. Under this universal sample design, the unconditional probabilities of selection for the schools were equal to the desired πjs values given above. Conditional probabilities of selection for inclusion were assigned in the district sample based on whether or not the school was sampled in either the alpha sample or in the beta sample. The conditional probability was higher than πjs if the school was sampled in alpha and not sampled in beta (the case in which it was desirable to maximize the school’s chance for inclusion in the district sample). The conditional probability was lower than πjs if the school was not sampled in alpha but was sampled in beta. If the school was not sampled in either alpha or beta, the conditional probability may have been higher or lower than πjs. The conditional probabilities were computed so that the unconditional probabilities over the universal sample design were equal to the desired πjs values.