Systematic enumeration of definitive screening designs

Abstract

Conference designs are $n \times k$ matrices, $k \le n$, with orthogonal columns, one zero in each column, at most one zero in each row, and $-1$ and $+1$ entries elsewhere. Conference designs with $k=n$ are called conference matrices. Definitive screening designs (DSDs) are constructed by folding over a conference design and adding a row vector of zeros. We propose methodology for the systematic enumeration of conference designs with a specified number of rows and columns, and thereby for the systematic enumeration of the corresponding DSDs. We demonstrate its potential by enumerating all conference designs with up to 24 rows and columns, and thus all DSDs with up to 49 runs. A large fraction of these DSDs cannot be obtained from conference matrices and is therefore new to the literature. We identify DSDs that minimize the correlation among contrast vectors of second-order effects and provide them in supplementary files.

Appendix: LM0 check function

The reduction algorithm in Sect. 3.3 calls the LM0 check function given in Algorithm 4, which itself calls a continuation function given in Algorithm 5. The purpose of the LM0 check function is to determine whether or not an input design is of LM0 form. To this end, it performs column permutations, row permutations, sign switches of rows, and sign switches of columns to check whether these operations result in a lexicographically larger design. As soon as a lexicographically larger design is found, the procedure stops and returns a ${ False}$ for the input design. The sign switches and a small portion of the permutations are addressed in Algorithm 4. All remaining column permutations are addressed in Algorithm 5.

In lines 1 and 2, Algorithm 4 considers all designs A in which column 1 is interchanged with any of the columns $1,\ldots ,(k+1)$, thereby including the option that the first column stays in its original position. Next, in lines 3 and 4, the rows are sorted and sign switches of the rows are applied until the first column is in the largest form possible: a zero entry in the first position $a_{1,1}$ and entries of 1 in all other positions $a_{i,1}$, where $i>1$. At this point, the design can only be made lexicographically larger by applying a sign switch to row 1 or to columns $2,\ldots ,(k+1)$ so that all columns other than the first start with a 1. These options are explored in lines 5–8 of Algorithm 4. At this point, all sign switches that might make the current design lexicographically larger are exhausted. However, there may still be column permutations that result in a lexicographically larger design. That part of the LM0 check is performed by Algorithm 5.

Algorithm 5 carries out additional column permutations as well as the actual LM0 check between the original input design K and the design modified by the sign switches and permutations of Algorithms 4 and 5. The input of Algorithm 5 is the original input design K, the modified design and the leftmost column c that is to be permuted with columns $c,c+1,\ldots ,k+1$. In its line 2, Algorithm 5 starts by conducting column permutations involving the current column c and sorting the rows such that earlier columns remain unchanged. This qualification is necessary because the ‘larger than’ operation works column by column. If the sorting would change the row order of the first $c-1$ columns, the current design would not be as large as possible.

After the sorting, the current column is tested against the corresponding column in the original design K (lines 4–9). If the current column is lexicographically larger, then the original design is not of LM0 form. If both columns are equal, further column permutations are invoked by calling the algorithm recursively using a new current column. Finally, if the current column it is smaller than the corresponding column in K, a new iteration of the for loop is started.