​​​​​​​​​​​​​​NAEP Technical DocumentationComputing Student-Level Replicate Factors

For the ​mathematics, reading, and science 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, \(STU\_REPFAC_{jsk}(r), r=1,...,62\), were calculated as

\begin{equation} STU\_REPFAC_{jsk}(r) = \left\{\begin{array}{llll} 1 + \sqrt {\pi_{s}}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 1 \\ 1 - \sqrt {\pi_{s}}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 2 \\ 1, & \text{for } jsk \notin R_{jsr} \end{array}\right. , \end{equation}​​​

where

  • \(\pi_{s}​\) ​is the probability of selection for school \(s\);
  • \(R_{jsr}\) is the set of students within the \(r\)-th variance stratum for school \(s\) in primary stratum \(j\); and
  • \(U_{jsk}\) ​is the variance unit (1 or 2) for student \(k\) in school ​\(s\) in stratum \(j\)​​.

For triples (i.e., variance strata comprising three students), the replicate factors are perturbed to something other than 1.0 for two different variance strata, rather than just for one stratum as in the case of pairs (i.e., variance strata comprising 2 students). The replicate factors are perturbed in variance stratum \(r\) and variance stratum \(r'\), where \(r'\) is furthest away from variance stratum \(r\) in either direction (i.e., before or after stratum \(r\)). Because there are 62 replicates, the stratum furthest away from stratum \(r\) would be the stratum whose number is the number of stratum \(r\) plus or minus half of 62, depending on whether \(r\) is greater or less than 31. In other words, \(r'=r+31\text{ }mod\text{ }62\). For example, if variance stratum 1 has three students, replicate factors are perturbed in variance stratum 1 (\(r\)) and variance stratum 32 (\(r'\)). The student-level replicate factors \(STU\_REPFAC_{jsk}(r), r=1,...,62\), were calculated as follows:

\begin{equation} STU\_REPFAC_{jsk}(r) = \left\{\begin{array}{llll} 1 + \dfrac {\sqrt {\pi_{s}}} {2}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 1 \\ 1 + \dfrac {\sqrt {\pi_{s}}} {2}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 2 \\ 1 - \sqrt​ {\pi_{s}}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 3 \end{array}\right. , \end{equation}
​

while for variance stratum \(r'=r+31\text{ }mod\text{ }62​\), 

\begin{equation} STU\_REPFAC_{jsk}(r') = \left\{\begin{array}{llll} 1 + \dfrac {\sqrt {\pi_{s}}} {2}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 1 \\ 1 - \sqrt {\pi_{s}}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 2 \\ 1 + \dfrac {\sqrt {\pi_{s}}} {2}, & \text{for } jsk \in R_{jsr}, U_{jsk} = 3 \end{array}\right. ,  \end{equation}


and for all other variance strata, further called \(r\) with an asterisk (\(r^*\)), (that is, variance strata other than strata \(r\) and \(r'\)),

\begin{equation} STU\_REPFAC_{jsk}(r^*) = 1 , \end{equation}
​

where

  • \(\pi_{s}\) ​is the probability of selection for school \(s\);
  • \(R_{jsr}\) is the set of students within the \(r\)-th replicate stratum for school \(s\) in stratum \(j\); and
  • \(U_{jsk}\)​ is the variance unit (1, 2, or 3) for student \(k\) in school \(s\) in stratum \(j\).​

Note, for students in certainty schools, where \(\pi_{s}=1\), 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. 

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Last updated 12 February 2026 (LW)
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