NAEP Technical DocumentationComputation of Measures of Size for the 2013 Twelfth-Grade Public School National Assessment

In designing the twelfth-grade public school sample, six objectives underlie the process of determining the probability of selection for each school and the number of students to be sampled from each selected school:

• to meet the target student sample size for each explicit sampling stratum;
• to select an equal-probability sample of students from each explicit sampling stratum;
• to limit the number of students selected from any one school;
• to ensure that the sample within a school does not include a very high percentage of the students in the school, unless all students are included;
• to reduce the rate of sampling of small schools, in recognition of the greater cost and burden per student of conducting assessments in such schools; and
• to increase the number of Black and Hispanic students in the sample.

The goal in determining the school's measure of size is to optimize across the middle four objectives in terms of maintaining the accuracy of estimates and the cost effectiveness of the sample design. 

To increase the number of Black and Hispanic students in the sample, the measure of size for schools with relatively high proportions of Black and Hispanic students (15 percent or more) were doubled. This oversampling was limited to only the remainder stratum, (that is, the stratum comprising schools that are not in states with state assessments). The target student sample sizes for the state-based strata are large (4,600 assessed) and should yield a sufficient number of Black and Hispanic students.

For schools with high proportions of Black and Hispanic students in the remainder stratum, the preliminary measures of size (MOS) were calculated as follows: 

where x_js is the estimated grade 12 student enrollment for school s in stratum j.

For all other schools (those in the state-based strata or with a low proportion of Black and Hispanic students in the remainder stratum), the preliminary measures of size (MOS) were calculated set as follows:.

where x_js is the estimated grade 12 student enrollment for school s in stratum j.

The preliminary school measure of size was rescaled to create an expected number of hits by applying a multiplicative constant b_j, which varies by stratum j. The design for the twelfth grade school sample allowed multiple hits. For example, a school with two hits will have twice as many students sampled as a single-hit school. To limit respondent burden, constraints were placed on the number of hits allowed per school. For schools in the state-based sampling strata, the limit was three hits. For schools in the remainder stratum, it was one hit.  

It follows that the final measure of size, E_js, was defined as:

where u_j is the maximum number of hits allowed.

In addition, new and newly-eligible schools were sampled from the new-school frame. The assigned measures of size for these schools,

,

used the b_j and u_j values from the CCD-based school frame for stratum j (i.e., the same sampling rate as  for the CCD-based school sample within each stratum). The variable π_djs is the probability of selection of the district into the new-school district (d) sample.

In addition, an adjustment was made to the initial measures of size in an attempt to reduce school burden by minimizing the number of schools selected for both the High School Longitudinal Study (HSLS) and the grade 12 public school NAEP assessments. The NAEP sampling procedures used an adaptation of the Keyfitz process to compute conditional measures of size that, by design, minimized the overlap of schools selected for both the NAEP and HSLS assessments.

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