NAEP Technical DocumentationComputing School-Level Replicate Factors for the 2017 Assessment

The replicate variance estimation approach for the mathematics, reading, and writing 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: 

SCH underscore REPFAC subscript j s open paren r close paren equals one 
 

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

For noncertainty schools, where preliminary 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_REPFACjs(r), r = 1,..., 62, were calculated as follows:

 

SCH underscore REPFAC subscript j s open paren r close paren equals open paren one plus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two close paren close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals one close paren or equals open paren one minus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two close paren close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals two close paren or equals open paren one if j s is not an element of cap r underscore j r close paren 

 where

  •  min(πj1, πj2) is the smallest school probability between the two schools comprising Rjr,

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

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

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

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

 

SCH underscore REPFAC subscript j s open paren r close paren equals open paren one plus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two comma pi subscript j three close paren divided by two close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals one close paren or equals open paren one plus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two comma pi subscript j three close paren divided by two close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals two close paren or equals open paren one minus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two comma pi subscript j three close paren close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals three close paren 
 

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

 

SCH underscore REPFAC subscript j s open paren r prime close paren equals open paren one plus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two comma pi subscript j three close paren divided by two close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals one close paren or equals open paren one minus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two comma pi subscript j three close paren close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals two close paren or equals open paren one plus square root of open paren one minus minimum of open paren pi subscript j one comma pi subscript j two comma pi subscript j three close paren divided by two close paren for j s is an element of cap r subscript j r comma cap u subscript j s equals three close paren 
 

 

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

SCH underscore REPFAC subscript j s open paren r superscript asterisk close paren equals one 
 

 where

  •  min(πj1, πj2, πj3) is the smallest school probability among the three schools comprising Rjr,

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

  • Ujs is 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).

Last updated 28 July 2022 (PG)