Some new factor screening designs using the search linear model (Q1073514)

Factor screening designs can be effectively put into the framework of a particular type of search linear models. This can be easily seen as follows: Suppose k important factors are to be screened out of m factors each at two levels. Then under the usual assumptions of no-interactions and negligible amounts of random errors the search linear model for this is \[ E[Y]=A_ 1\xi_ 1+A_ 2\xi_ 2,\quad V[Y]=\sigma^ 2I,\quad and\quad Rank(A_ 1)=\nu_ 1, \] where \(A_ 1\) is an \(N\times 1\) column vector of all 1's, \(A_ 2\) is an \(N\times m\) (-1,1)-matrix, \(\xi_ 1\) is the general mean and \(\xi_ 2\) consists of the main effects. A factor screening design is an \(N\times m\) (0,1)-matrix T whose rows consist of m treatment combinations of a \(2^ m\) factorial. The matrix \(A_ 2\) is obtained from T by replacing 0's by -1's. It is assumed that the treatments (1,1,...,1) and (0,0,...,0) are always present in the design. This paper is concerned with the construction of factor screening designs (with a small number of treatments) which have the property \(P_{2t}\) (no 2t columns are linearly dependent) for \(t=2\) and 3. In this connection a ''packing problem'' is introduced. A list of useful designs is also presented.