​​​​​​​NAEP Technical DocumentationRaking Adjustment Factor Calculation

For assessed, excluded, and full-time remote students in a given subject, the raking adjustment factor \(STU\_RAKE_{k}\) was computed as below.

First, the weight for student \(k\) was initialized as

\begin{equation} STUSAWT_{k}^{adj(0)} = STU\_BWT_{k} \times SCH\_TRIM_{k} \times SCH\_NRAF_{k} \times STU\_NRAF_{k} \times SUBJFAC_{k} , \end{equation}

where

• ​ \(STU\_BWT_{k}\) is the student base weight for a given student \(k\);
• \(SCH\_TRIM_{k}\) is the school-level weight trimming factor for the school associated with student \(k\);
• \(SCH\_NRAF_{k}\) is the school-level nonresponse adjustment factor for the school associated with student \(k\);
• \(STU\_NRAF_{k}\) is the student-level nonresponse adjustment factor for student \(k\); and
• \(SUBJFAC_{k}\) is the subject factor for student \(k\).

Then, the sequence of weights for the first iteration was calculated as follows for student \(k\) in category \(c\) of dimension \(d\):

for dimension 1: \begin{equation} STUSAWT_{k}^{adj(1)} = \dfrac {TOTAL_{c(1)}} { \sum_{ R_{c(1)} \smile E_{c(1)}} {STUSAWT_{k}^{adj(0)} } } \times STUSAWT_{k}^{adj(0)} , \end{equation}

for dimension 2: \begin{equation} STUSAWT_{k}^{adj(2)} = \dfrac {TOTAL_{c(2​)}} { \sum_{ R_{c(2)} \smile E_{c(2)}} {STUSAWT_{k}^{adj(1)} } } \times STUSAWT_{k}^{adj(1)} , \end{equation}

for dimension 3: \begin{equation} STUSAWT_{k}^{adj(3)} = \dfrac {TOTAL_{c(3)}} { \sum_{ R_{c(3)} \smile E_{c(3)}} {STUSAWT_{k}^{adj(2)} } } \times STUSAWT_{k}^{adj(2)} , \end{equation}

where

• \(R_{c(d)}\) is the set of all assessed students in category \(c\) of dimension \(d\);
• \(E_{c(d)}\) is the set of all excluded or full-time remote students in category \(c\) of dimension \(d\); and
• \(TOTAL_{c(d)}\) is the control total for category \(c\) of dimension \(d\).

The process is said to converge if the maximum difference between the sum of adjusted weights and the control totals is 1.0 for each category in each dimension. If after the sequence of adjustments the maximum difference was greater than 1.0, the process continues to the next iteration, cycling back to the first dimension with the initial weight for student \(k\) equaling \(STUSAWT_{k}^{adj(3)}\) from the previous iteration. The process continued until convergence was reached.

Once the process converged, the adjustment factor was computed as

\begin{equation} STU\_RAKE_{k} = \dfrac {STUSAWT_{k}} { STU\_BWT_{k} \times SCH\_TRIM_{k} \times SCH\_NRAF_{k} \times STU\_NRAF_{k} \times SUBJFAC_{k} } , \end{equation}

where

• \(STUSAWT_{k}\) is the weight for student \(k\) after convergence.

The process was done independently for the full sample and for each replicate.

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