NAEP Assessment Weighting Procedures → Weighting Procedures for the 2011 Assessment → Computation of Replicate Weights for Variance Estimation for the 2011 Assessment → Computing School-Level Replicate Factors for the 2011 Assessment

NAEP Technical DocumentationComputing School-Level Replicate Factors for the 2011 Assessment 

The calculation of school-level replicate factors for an assessment depended on the variance replication procedure, with or without finite population corrections for the first stage of sampling.

Mathematics, Reading, and Science Assessments

The replicate variance estimation approach for the mathematics, reading, and science assessments involved finite population corrections at the school level. The calculation of school-level replicate factors for these assessments depended upon whether or not a school was selected with certainty. For certainty schools, the school-level replicate factors for all replicates are set to unity – this is true regardless of whether or not the variance replication method uses finite population corrections – since certainty schools are not subject to sampling variability. Alternatively, one can view the finite population correction factor for such schools as being equal to zero. Thus, for each certainty school in a given assessment, the school-level replicate factor for each of the 62 replicates (r = 1, ..., 62) was assigned as follows:

 

where SCH_REPFAC_js(r) is the school-level replicate factor for school s in primary stratum j for the r-th replicate.

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

where

•  min(π_j1, π_j2) is the smallest school probability between the two schools comprising R_jr,

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

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

For noncertainty schools in variance strata comprising three schools, the school-level replicate factors, SCH_REPFAC_js(r), r = 1,..., 62, were calculated as follows:

For school s from primary stratum j, variance stratum r,

while for r' = r + 31 (mod 62):

and for all other r* other than r and r':

where

•  min(π_j1, π_j2,π_j3) is the smallest school probability among the three schools comprising R_jr,

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

• U_js is the variance unit (1, 2, or 3) for school s in primary stratum j.

Computer-Based Writing Assessments

The replicate variance estimation approach for the computer-based writing assessments did not involve school-level finite population corrections. As described in the section Defining Variance Strata and Forming Replicates, variance strata were defined by grouping first-stage units (schools or geographic PSUs) into pairs or triplets. The calculation of the school-level replicate factors for each of the 62 replicates was based on this grouping.

For schools in variance strata comprising pairs of first-stage units, the school-level replicate factors, SCH_REPFAC_js(r), r = 1,..., 62, were calculated as follows:

where

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

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

For schools in variance strata comprising three first-stage units, school-level replicate factors for school were calculated as follows:

For school s in primary stratum j, variance stratum r,

while for r' = r + 31 (mod 62):

and for all other r* other than r and r':

where

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

• U_jsis the variance unit (1, 2, or 3) for school s in primary stratum j.

In primary strata with fewer than 62 variance strata, the replicate weights for the “unused” variance strata (the remaining ones up to 62) for these schools were set equal to the school base weight (so that those replicates contribute nothing to the variance estimate).

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