In addition to the full-sample weight, a set of 62 replicate weights was provided for each student. These replicate weights are used in calculating the sampling variance of estimates obtained from the data, using the jackknife repeated replication method. The method of deriving these weights was aimed at reflecting the features of the sample design appropriately for each sample, so that when the jackknife variance estimation procedure is implemented, approximately unbiased estimates of sampling variance are obtained. This section gives the specifics for generating the replicate weights for the 2006 assessment samples. The theory that underlies the jackknife variance estimators used in NAEP studies is discussed in the section Replicate Variance Estimation.
For each sample, replicates were formed in two steps. First, each school was assigned to one or more of 62 replicate strata. In the next step, a random subset of schools (or, in some cases, students within schools) in each replicate stratum was excluded. The remaining subset and all schools in the other replicate strata then constituted one of the 62 replicates.
A replicate weight was calculated for each of the 62 replicates using weighting procedures similar to those used for the full-sample weight. Each replicate base weight contains an additional component, known as a replicate factor, to account for the subsetting of the sample to form the replicate. By repeating the various weighting procedures on each set of replicate base weights, the impact of these procedures on the sampling variance of an estimate is appropriately reflected in the variance estimate.
Each of the 62 replicate weights for student k in school s and stratum j can be expressed as follows:
STU_BWTjsk(r) is the replicate student base weight for replicate r,
SCH_NRAFjs(r) is the school-level nonresponse adjustment factor for replicate r,
STU_NRAFjsk(r) is the student-level nonresponse adjustment factor for replicate r,
SCH_TRIMjs is the school-level weight trimming adjustment factor, and
STU_TRIMjsk is the student-level weight trimming adjustment factor.
Specific school and student nonresponse adjustment factors were calculated separately for each replicate, thus the use of the index (r), and applied to the replicate student base weights. Computing separate school and student nonresponse adjustment factors for each replicate is generally known as replicating the school and student nonreponse adjustment factors. This allows resulting variances from the use of the final student replicate weights to reflect components of variance due to the school and student nonresponse adjustments.
School and student weight trimming adjustments were not replicated, that is, not calculated separately for each replicate. Instead, each replicate used the school and student trimming adjustment factors derived for the full sample. Statistical theory for replicating trimming adjustments under the jackknife approach has not been developed in the literature. Due to the absence of a statistical framework and since relatively few school and student weights in NAEP require trimming, the weight trimming adjustments were not replicated.