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Computation of Replicate School Base Weights

          

Replicate Variance Estimation

A total of 62 replicate weights r = 1,...,62 were computed for the purpose of variance estimation for the NAEP 2002 studies. A
general review of the theory that underlies the jackknife variance estimators used in NAEP studies can be found in Replicate Variance Estimation. For the State NAEP 2002 school samples, the starting point for generating the replicate weights was the school base weights ws, equal to the reciprocals of the school probabilities of selection pie subcript s superscript minus 1 .

The definition of the replicate strata and replicate units is given in Defining of Replicate Strata and Forming of Replicates. Within each doublet preliminary stratum, the first school was assigned as the first variance unit and the second school as the second variance unit. Within each triplet preliminary stratum, the three schools were assigned variance units 1 through 3. Given ws as the full sample base weight (the inverse of the school's probability of selection) school s, Rr as the set of schools within the r-th replicate stratum and Us as the variance unit (1 or 2), the replicate weights w(r), r = 1,...,R, were computed as follows:

w subscript s left parenthesis r right parenthesis equals open left bracket 2 times w subscript s if s is a member of set R subscript r and U subscript s equals 1, 0 if s is a member of set R subscript r and U subscript s equals 2, or w subscript s if s is not a member of set R subscript r

For triplets, each triplet replicate stratum Rr was assigned a paired replicate stratum r2 . The replicate weights for schools in Rr follow:

w subscript s left parenthesis r right parenthesis equals left open bracket 1.5 times w subscript s if s is a member of set R subscript r and U subscript s equals 1 or 1.5 times w subscript s if s is a member of set R subscript r and U subscript s equals 2 or 0 if s is a member of set R subscript r and U subscript s equals 3 or w subscript s if s is not a member of set R subscript

w subscript s left parenthesis r subscript 2 right parenthesis equals left open bracket 1.5 times w subscript s if s is a member of set R subscript r and U subscript s equals 1 or 0 if s is a member of set R subscript r and U subscript s equals or 1.5 times w subscript s if s is a member of set R subscript r and U subscript s equals 3 or w subscript s if s is not a member of set R subscript r

For example, consider a state with 111 noncertainty schools in which the first 54 final replicate strata contain two schools each (doublets), and replicate stratum 55 contains three 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.

Last updated 06 November 2008 (RF)

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