NAEP Technical DocumentationSampling of Private Schools for the 2012 Long-Term Trend (LTT) Assessment

In the design of each school sample, five objectives underlie the process of determining the probability of selection for each school and how many students are to be sampled from each selected school containing the respective age-eligible students. They are as follows:

• to meet the target student sample size

• to select an equal-probability sample of students

• to limit the number of students who are selected from a 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

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

Therefore, to meet the target student sample size objective and achieve a reasonable compromise among the other four objectives, the following algorithm was used to assign a measure of size to each school based on its age-specific enrollment as indicated on the sampling frame. 

The measures of size vary by enrollment size. In general, very small schools were assigned a measure of size that is one quarter of the target sample size of 64 students. The preliminary measures of size (MOS_js) were set as follows:

where x_js is the estimated age-eligible enrollment for age j in school s, PSCHWT_S = the PSS area frame weight for school s, computed by the U.S. Census Bureau, and PSU_WT_S = the PSU weight for school s.

The next task in this development is to describe b_j, the constant of proportionality for each age group. It is a sampling parameter that, when multiplied with a school’s preliminary measure of size (MOS_js), yields the school’s final measure of size. It is computed in such a way that, when used with the systematic sampling procedure, the target student sample size is achieved. For private schools, this parameter varied by age group and private school affiliation (Catholic, non-Catholic, and unknown affiliation).

The final measure of size, E_js, is defined as:

The quantity u_j (the maximum number of “hits” allowed) in this formula is designed to put an upper bound on the burden for the sampled schools. For private schools, u_j is 1.

Schools were ordered using the serpentine sort described under the stratification of private schools. A systematic sample was then drawn using this serpentine sorted list and the measures of size. The numbers of private schools selected were 137, 130, and 93 for ages 9, 13, and 17, respectively.

The estimated number of age-eligible students was computed using a derived linear function of estimated grade enrollments.  For 9-year-olds, the frame contained schools with a second, third, fourth, or fifth grade. M_js(2), M_js(3), M_js(4), and M_js(5) denote the estimated enrollments from the frame for the four grades respectively within each school s for age sample j. F_j(2), F_j(3), F_j(4), and F_j(5) denote the global proportion of students who were 9-years-old (birth months January 2002 through December 2002) within each grade, Census region, and within private schools. These proportions were derived from the NAEP 2009 reading, mathematics, and science assessments. These values are shown in the table below for private schools. Age Distribution Fractions for Ages 9 and 13 describes how these proportions are computed.

The estimated age-eligible enrollment for the 9-year-olds component of the long-term trend (LTT) study for each school was computed as follows:

For 13-year-olds, the frame contained schools with a sixth, seventh, eighth, or ninth grade. M_js(6), M_js(7), M_js(8), and M_js(9) denote the estimated enrollments from the frame for the four grades respectively within each school s for age sample j. G_j(6), G_j(7), G_j(8), and G_j(9) denote the global proportion of students who were 13-years-old (birth months January 1998 through December 1998) within each grade, Census region, and within private schools. These proportions were derived from the NAEP 2009 reading, mathematics, and science assessments. These values are shown in the table below for private schools.  Age Distribution Fractions for Ages 9 and 13 describes how these proportions are computed.

The estimated age-eligible enrollment for the 13-year-olds component of the LTT study for each school was computed as follows:

For the age 17 component, the frame contained schools with a ninth, tenth, eleventh, or twelfth grade. M_js(9), M_js(10), M_js(11), and M_js(12) denote the estimated enrollments from the frame for the four grades respectively within each school s for age sample j . J_j(9), J_j(10), J_j(11), and J_j(12) denote the global proportion of students born between October 1994 and September 1995 within each grade and Census region. The Current Population Survey (CPS) School Supplement (October 2009) was utilized to provide the proportions within each of ninth, tenth, eleventh, and twelfth grades for the age 17 component. The same table was used for both public and private schools since overall age distribution fractions were derived without differentiation of school enrollment by school type. The derivation of the Age Distribution Fractions for Age 17 describes how these proportions are computed.

The estimated age-eligible enrollment for the age 17 component of the LTT study for each school was computed as follows:

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