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

For the mathematics, reading, and writing assessment samples, which involved school-level finite population corrections, the student-level replication factors were calculated the same way regardless of whether or not the student was in a certainty school.

For students in student-level variance strata comprising pairs of students, the student-level replicate factors, STU_REPFACjsk(r), r = 1,..., 62, were calculated as follows: 

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

 

where

  • πs is the probability of selection for school s,

  • Rjsr is the set of students within the r-th variance stratum for school s in primary stratum j, and

  • Ujsk is the variance unit (1 or 2) for student k in school s in stratum j.

For students in variance strata comprising three students, the student-level replicate factors STU_REPFACjsk(r), r = 1,..., 62, were calculated as follows:

 

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

 

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

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

 

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


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

where

  • πs is the probability of selection for school s,

  • Rjsr is the set of students within the r-th replicate stratum for school s in stratum j, and

  • Ujsk is the variance unit (1, 2, or 3) for student k in school s in stratum j.

Note, for students in certainty schools, where πs = 1, the student replicate factors are 2 and 0 in the case of pairs, and 1.5, 1.5, and 0 in the case of triples.

Last updated 28 July 2022 (ML)