## Computation of School Weights

The school weight wi (school i) for the 2001 assessment had five basic components:

• a component for PSU selection;

• a component for school selection given PSU selection;

• a component for selection of schools to a field test session only;

• a component to account for school nonresponse (that is, a refusal of the school to participate in the study); and

• a component to trim school weights that contribute too much to overall variance.

For 2001 assessment schools, the overall school weight can be given as

where

• The quantity πi is the overall probability of selection of the school and is the product of the probability of selection of the PSU (see Sampling of Primary Sampling Units) containing the school and the conditional probability of selection of school i given that the PSU was selected on the final PSU-subsetted school frame (see Sampling of Schools). [There were two special cases: one in the eighth-grade sample and one in the twelfth-grade sample. In the eighth-grade sample, a sampled school split into two schools. One part with half probability was selected. A weight factor of 2 is attached. In the twelfth-grade sample, a sampled school was ineligible, but another unsampled school was mistakenly fielded (which was present on the frame but not sampled). The assessments from this school were retained but the conditional probability was reset to 1 to make this school self-representing at this level of sampling.]

• The quantity SUBADJi equals 1 for all originally sampled schools and equals the ratio of estimated grade enrollment of the original to the substitute school for participating substitute schools.

• The quantity FTADJi equals 1 for all schools with two or more sessions and equals the reciprocal of the probability of being selected for the main assessment (rather than the field test assessment) for schools with one session.

• SNk is a weighted response factor in a selected set of nonresponse cells k accounting for school refusals (see School Trimming).

• The trimming factor TRIMi equals 1 for most schools and was set to a value <1 for schools that were contributing too much to overall variance (see School Nonresponse Adjustments).

This table shows the distribution of school base weights πi–1 over the full school sample for each of the three grades. The summations are estimates of the total number of schools in the United States that include the grade.

Distribution of school base weights, by school base weights and grade, national main assessment: 2001
School base weights Grade 4 Grade 8 Grade 12
Number 441 465 522
Minimum 15.48 8.61 1.001
5th percentile 46.67 18.79 8.22
10th percentile 59.14 22.29 9.91
25th percentile 83.83 36.48 14.73
50th percentile 143.91 93.39 41.13
75th percentile 169.31 93.39 60.85
90th percentile 402.95 261.5 174.94
95th percentile 411.17 290.56 184.15
Maximum 1611.79 1129.27 699.76
Sum 80,676 50,881 32,461
Mean 182.94 109.42 62.19
Standard deviation 190.13 145.9 80.38
1 This school was a self-representing school because of a sampling error.
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2001.

SUBADJi is added as a weighting factor to allow the student sample from the participating substitute school to fully represent the nonparticipating original school. It equals the ratio of the estimated grade enrollment for the originally sampled school to the estimated grade enrollment for the substitute school. The student sample from the substitute school then "represents" the grade-eligible group from the originally sampled school. As an extreme example, suppose the originally sampled school has one grade-eligible student, with the substitute school having a full sample of 60 students. Then SUBADJi will be 1/60 in this case: the 60 students' assessment are "representing" the one single student from the originally sampled school. The substitute selection process was designed to minimize the differences between the original sampled school enrollment and the substitute's enrollment (see School Substitution for Adjusting School Nonresponse).

This table shows the distribution of SUBADJi values across the substitute schools for each grade.

Distribution of substitute school adjustment factor SUBADJi by substitute school adjustment factor and grade, national main assessment: 2001
Substitute school adjustment factor Grade 4 Grade 8 Grade 12
Number of schools 24 26 29
Minimum 0.613 0.125 0.177
25th percentile 0.934 0.885 0.954
50th percentile 1.028 1.017 1.106
75th percentile 1.258 1.147 1.270
Maximum 5.000 4.714 2.750
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2001.

Last updated 26 August 2008 (FW)