## Reducing the Size of Bundle Maps

NAEP instrument developers found that bundle maps could be cut in half or even in quarters, reducing the number of bundles accordingly, without significant damage to the distribution parameters. To cut a map in half (which requires an even number of bundles as well as an even bundle length), the second half of column 1 of the map becomes column 2 of the reduced map; similarly, the original column 2 forms columns 3 and 4 of the reduced map, etc., proceeding through half of the columns of the original map to generate a new map with half the rows. Consider the following bundle map. Since the spiral length, 6, and bundle length, 4, are even numbers, it is possible to reduce the full design to a half design.

 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6

The original bundle length of n (here, 4) is retained in a half design, since the right most n/2 (4/2 = 2) columns of the full design can be eliminated. To create the half design, the second half of the spiral cycle for each column of the full design becomes a new, adjacent column. In the figure below, the first and second halves of the original columns are denoted by A and B, respectively. The number of bundles has been cut in half, from 6 to 3:

 A B A B 1 1 4 1 4 2 2 5 2 5 3 3 6 3 6

If the order of the columns remained ABAB, the first halves (the As) would be distributed in the field more often than the second halves (the Bs). This is because booklet usage decreases as one progresses through a bundle. To protect against this unequal distribution, the columns of the half design are re-arranged so that they alternate according to the pattern of a 2 by 2 Latin Square:

 A B B A 1 1 4 4 1 2 2 5 5 2 3 3 6 6 3

Finally, to apply the characteristics of a cyclic vertical design, the starting position of each column is determined by the first-row forward differences:

 A B B A 1 1 5 4 3 2 2 6 5 1 3 3 4 6 2

To cut a map to quarter-length (both the number of rows and columns must be a multiple of 4), a similar procedure is followed, where each original column forms four columns in the reduced map. In either the halving or quartering of a design, one additional requirement must be met: The parts must be distributed according to a Latin square. For example, in the case of a half-map, if the first half of an original column is labeled "A," and the second half  "B," the sequence of halves in the new design must be ABBAABBA…, in other words, alternating the placement of the halves according to the rows of this Latin square:

 A B B A

This corrects the unevenness that would result from an arrangement of ABABAB. In the quarter-map, with the four quarters of the spiral cycle called ABCD, NAEP instrument developers use ABCDBADCCDABDCBA for a bundle map where the length of bundles is 16. This follows the rows of the Latin square:

 A B C D B A D C C D A B D C B A

This is required to regain the evenness of distribution which would be lost by an arrangement of ABCDABCDABCD…, which would favor the earlier quarters at the expense of the later ones.

Last updated 28 October 2008 (GF)