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Computing School-Level Replicate Base Weights for the 2005 Assessment

For the 2005 NAEP, the calculation of school-level replicate base weights varied by whether a school was selected with certainty or not since replicates were formed using first-stage sample units. In certainty schools, where replicates were formed at the student-level, the school base weight for each of the 62 replicates (r = 1, ..., 62) was set equal to the full-sample school base weight. That is,

SCH_BWT(r) = SCH_BWT

For noncertainty schools, where replicate strata were formed at the school-level by grouping schools into pairs or triplets, school base weights were calculated for each of the 62 replicates based on this grouping. For schools in replicate strata comprising pairs of schools, the school-level replicate base weights SCH_BWTjs(r), r = 1, ..., 62, were calculated as follows:

SCH underscore BWT subscript js, into bracket r subscript 2, bracket closes, equals left brace 3 rows, 3 column matrix,row 1 column 1 is: 2 times SCH underscore BWT subscript js, row 1 column 2 is: for js is an element of R subscript jr, row 1 column 3: U subscript js equals 1.

where

  • SCH_BWTjs is the school base weight for school s in primary stratum j,

  • Rjr is the set of schools within the r-th replicate stratum for primary stratum j, and

  • Ujs is the variance unit (1 or 2) for school s in primary stratum j.

For schools in replicate strata comprising three schools, two sets of school-level replicate base weights were computed (see replicate variance estimation for details): one for the first replicate r1 and another for the second replicate r2. The two sets of school-level replicate base weights SCH_BWTjs(r1), r1 = 1,..., 62, and SCH_BWTjs(r2), r2 = 1,..., 62, were calculated as described below.

  SCH underscore BWT subscript js open paren r subscript one close paren equals open paren one point five times SCH underscore BWT subscript js for js is an element of cap R subscript jr subsubscript one and cap U subscript js equals one close paren or equals open paren one point five times SCH underscore BWT subscript js for js is an element of R subscript jr subsubscript one and cap U subscript js equals two close paren or equals open paren zero for js is an element of R subscript jr subsubscript.

SCH underscore BWT subscript js, into brackets r subscript 2 bracket closes, equals left brace 4 row 3 column matrix, row 1 column 1 is: 1.5 times SCH underscore BWT subscript js, row 1 column 2 is: for js is an element of R subscript jr subscript 2, row 1 column 3 is: U subscript js equals 1, row 2 column 1 is: 0, row 2 column 2 is: for js is an element of R subscript jr subscript2, row 2 column 3 is: U subscript js equals 2, row 3 column 1 is: 1.5 times SCH underscore BWT subscript js, row 3 c.

To illustrate, consider a primary stratum with 111 noncertainty schools in which the first 54 final replicate strata contain two schools each (doublets), and replicate stratum 55 contains 3 schools (triplet). Replicate stratum 55 is paired with replicate stratum 23 (i.e., r2 = 23).

In the first replicate stratum, for the first school:

  • the variance unit = 1,
  • the replicate weight for the first replicate stratum = 2 times the school base weight, and
  • the replicate weight for all other replicate strata = the school base weight.

In the first replicate stratum, for the second school:

  • the variance unit = 2,
  • the replicate weight for the first replicate stratum = 0, and
  • the replicate weight for all other replicate strata = the school base weight.

In the 55th replicate stratum, for the first school:

  • the variance unit = 1,
  • the replicate weight for the 55th replicate stratum = 1.5 times the school base weight,
  • the replicate weight for the 23rd replicate stratum = 1.5 times the school base weight, and
  • the replicate weight for all other replicate strata = the school base weight.

In the 55th replicate stratum, for the second school:

  • the variance unit = 2,
  • the replicate weight for the 55th replicate stratum = 1.5 times the school base weight,
  • the replicate weight for the 23rd replicate stratum = 0, and
  • the replicate weight for all other replicate strata = the school base weight.

In the 55th replicate stratum, for the third school:

  • the variance unit = 3,
  • the replicate weight for the 55th replicate stratum = 0,
  • the replicate weight for the 23rd replicate stratum 1.5 times the school base weight, and
  • the replicate weight for all other replicate strata = the school base weight.

In primary strata with fewer than 62 replicate strata, the replicate weights for the “unused” replicate strata (the remaining ones up to 62) for these schools were set equal to the school base weight (so that those replicates contribute zero to any replicate sum of squares).


Last updated 05 November 2009 (GF)

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