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Computation of Measures of Size for the 2007 State Assessment

When designing each school sample, there are five objectives that 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 grade:

  • To meet the target student sample size for each grade;
  • To select an equal-probability sample of students;
  • To limit the number of students that are 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; and
  • 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. In certain jurisdictions, a census of students was taken to meet, as nearly as possible, the target student sample size. Elsewhere, to meet the target student sample and achieve a reasonable compromise among the other four objectives above, the design staff used the following algorithm to assign a measure of size to each school based on its enrollment per grade as indicated on the sampling frame. 

The measures of size differ between grades and between jurisdictions that participated in all three assessments (reading, mathematics, and writing) and jurisdictions that participated in two assessments (reading and mathematics). The former set of jurisdictions are called "reading-mathematics-writing jurisdictions" below (abbreviated in some cases as "rmw"), and the latter set are called "reading-mathematics-only jurisdictions" below (abbreviated in some places as "rmo").

For fourth grade, the preliminary measures of size (MOS) were set as follows:


MOS subscript js equals bracket 4 rows. Row 1 equals X subscript js, if 68 is less than or equal to X subscript js. Row 2 equals 62 if 20 is less than or equal to X subscript js less than 68. Row 3 equals 62 times X subscript js divided by 20, if 10 less than or equal X subscript js less than 20. Row 4 equals 62 divided by 2 if X subscript js less than 10

where xjs is the estimated grade enrollment for jurisdiction j and school s.

For eighth grade reading-mathematics-writing jurisdictions, the preliminary measures of size (MOS) were set as follows:


MOS subscript js equals bracket 4 rows. Row 1 equals X subscript js, if 102 is less than or equal to X subscript js. Row 2 equals 93 if 20 is less than or equal to X subscript js less than 102. Row 3 equals 93 times X subscript js divided by 20, if 10 less than or equal X subscript js less than 20. Row 4 equals 93 divided by 2 if X subscript js less than 10

where xjs is the estimated grade enrollment for jurisdiction j and school s.

For eighth grade reading-mathematics-only jurisdictions, the measures of size were as follows:

MOS subscript js equals bracket 4 rows. Row 1 equals X subscript js, if 68 is less than or equal to X subscript js. Row 2 equals 63 if 20 is less than or equal to X subscript js less than 68. Row 3 equals 63 times X subscript js divided by 20, if 10 less than or equal X subscript js less than 20. Row 4 equals 63 divided by 2 if X subscript js less than 10
 

where xjs is the estimated grade enrollment for jurisdiction j and school s.

For Puerto Rico, the measures of size were set as follows:

MOS subscript js equals bracket 4 rows. Row 1 equals X subscript js, if 34 is less than or equal to X subscript js. Row 2 equals 30 if 20 is less than or equal to X subscript js less than 34. Row 3 equals 30 times X subscript js divided by 20, if 10 less than or equal X subscript js less than 20. Row 4 equals 30 divided by 2 if X subscript js less than 10

where xjs is the estimated grade enrollment for jurisdiction j and school s.

The next task in this development is to describe bj, the constant of proportionality for a specified jurisdiction. It is a sampling parameter that, when multiplied with a school’s preliminary measure of size (MOSjs), 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.

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


E subscript js equals the minimum of b subscript j times M subscript js and u subscript j

The quantity uj (the maximum number of “hits” allowed) in this formula is designed to put an upper bound on the burden for the sampled schools. In most jurisdictions, uj was set to 3. In Alaska, uj was set to 8, and in Puerto Rico, uj was set to 1.

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


E subscript js equals the minimum of b subscript j times MOS subscript js times the inverse of pi subscript djs and u subscript j

used the bj and uj values from the main school sample for the jurisdiction (i.e., the same sampling rates as  for Common Core of Data schools within each jurisdiction). The variable 
Pie superscript minus 1 subscript d j s
is the probability of selection of the district into the new-school district (d) sample.

After applying this procedure, there was one jurisdiction in fourth grade (Department of Defense schools) and two jurisdictions in eighth grade (Hawaii and Rhode Island) where a very high proportion of schools were included, but not all. To achieve a more robust (and readily understood) design, it was determined that in these cases all eighth-grade schools in these jurisdictions were included in the sample with certainty, and the number of hits per school was adjusted accordingly so that the target student sample size would be achieved.

For three jurisdictions in fourth grade (Bureau of Indian Education [BIE], District of Columbia, and Wyoming) and 10 jurisdictions in eighth grade (BIE, District of Columbia, Delaware, Department of Defense schools, North Dakota, Vermont, Wyoming, Atlanta, Boston, and Cleveland), the number of students on the sampling frame was smaller than the target sample. Also, in the eighth-grade sample for Austin, the target sample size accounted for more than 90 percent of all students on the frame. For these jurisdictions, all students in all schools were sampled.


Last updated 12 April 2010 (GF)

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National Center for Education Statistics - http://nces.ed.gov
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