In the NAEP 2006 assessment, replicates were formed separately for each sample indicated by grade (4, 8, or 12), school type (public or private), and session type (history or other). The first step in forming replicates was to assign each first-stage sampling unit in a primary stratum to a replicate stratum. In 2006, the formation of replicate strata varied by noncertainty and certainty primary sampling units (PSUs). For noncertainty PSUs, the first-stage units were PSUs, and the primary stratum was the combination of region and metropolitan statistical area (MSA, non-MSA). For certainty PSUs, the first-stage units were schools, and the primary stratum was school type (public, private).
For noncertainty PSUs, where only one PSU was selected per PSU stratum, replicate strata were formed by pairing sampled PSUs with similar stratum characteristics within the same primary stratum (region by metropolitan status). This was accomplished by first sorting the 76 sampled PSUs by PSU stratum number and then grouping adjacent PSUs into 38 pairs. The values for a PSU stratum number reflect region and metropolitan status, as well as socioeconomic characteristics such as percent Black and percent children below poverty. The formation of these 38 replicate strata in this manner models a design of selecting two PSUs with probability proportional to size with replacement from each of 38 strata.
For certainty PSUs, the first stage of sampling is at the school level, and the formation of replicate strata must reflect the sampling of schools within the certainty PSUs. Replicate strata were formed by sorting the sampled schools in the 24 certainty PSUs by their order of selection within a primary stratum (school type) so that the sort order reflected the implicit stratification (region, locality type, race/ethnicity stratification, and median household income for public schools; and region, private school type, and student enrollment size for private schools) and systematic sampling features of the sample design.
The first-stage units were then paired off into preliminary replicate strata. Within each primary stratum with an even number of first-stage units, all of the preliminary replicate strata were pairs, and within primary strata with an odd number of first-stage units, one of the replicate strata was a triplet (the last one), and all others were pairs.
If there were more than 24 preliminary replicate strata within a primary stratum, the preliminary replicate strata were grouped to form 24 replicate strata. This grouping effectively maximized the distance in the sort order between grouped preliminary replicate strata. The first 24 preliminary replicate strata, for example, were assigned to 24 different final replicate strata in order (1 through 24), with the next 24 preliminary replicate strata assigned to final replicate strata 1 through 24, so that, for example, preliminary replicate stratum 1, preliminary replicate stratum 25, preliminary replicate stratum 49 (if there were that many), etc., were all assigned to the first final replicate stratum. The final replicate strata for the schools in the certainty PSUs were 1 through 24.
Within each pair of preliminary replicate stratum, the first first-stage unit was assigned as the first variance unit and the second first-stage unit as the second variance unit. Within each triplet preliminary replicate stratum, the three schools were assigned variance units 1 through 3.