Characterization of Six Categories of Systematic 2^{<i>n</i>－(<i>n</i>－<i>k</i>)} Fractional Factorial Designs

Abstract

Six categories of systematic 2 n －( n － k ) designs derivable from the full 2 k factorial experiment by the interactions-main effects assignment are available for carrying out 2 n －( n － k ) factorial experiments sequentially run after the other such that main effects are protected against the linear/quadratic time trend and/or such that the number of factor level changes ( i.e. cost) between the runs is minimal . Three of these six categories are of resolution at least III and three are of resolution at least IV. The three categories of designs within each resolution are: 1) minimum cost 2 n －( n － k ) designs , 2) minimum cost linear trend free 2 n －( n － k ) designs and 3) minimum cost linear and quadratic trend free 2 n －( n － k ) designs. This paper characterizes these six categories and documents their differences with regard to either time trend resistance of factor effects and/or the number of factor level changes . The paper introduces the last category of systematic 2 n －( n － k ) designs ( i.e. the sixth) for the purpose of extending the design resolution from III into IV and also for raising the level of protection of main effects from the linear time trend into the quadratic, where a catalog of minimum cost linear and quadratic trend free 2 n －( n － k ) designs (of resolution at least IV) will be proposed. The paper provides for each design in any of the six categories: 1) the sequence of its runs in minimum number of factor level changes 2) the defining relation or its 2 n －( n － k ) alias structure and 3) the k independent generators needed for sequencing the 2 n －( n － k ) runs by the generalized foldover scheme. A comparison among these six categories of designs reveals that when the polynomial degree of the time trend increase s from linear into quadratic and/or when the design ’ s resolution increases from III to IV, the number of factor level changes between the 2 n －( n － k ) runs increases. Also as the number of factors ( i.e. n ) increases, the design ’ s resolution decreases.