Agricultural Response Surface Experiments Based on Four-Level Factorial Designs

Treatment factors with four equally spaced quantitative levels can be expressed as direct products of pairs of pseudo-factors, each at two levels, where the interaction between the two pseudo-factors of a pair represents the quadratic trend component of a treatment factor. Assuming no cubic trend component in the response model, the two pseudo-factors of the pair each represent components of the pure linear trend term of the factor. Interactions between pseudo-factors in different pairs represent linear-by-linear treatment interaction effects. Further, the variances of the two independent estimates of a linear trend component derived from a pair of pseudo-factor effects are in the ratio 1:4 so one pseudo-factor of a pair is of major and the other of minor importance. These relationships are exploited to construct quadratic response surface designs based on fourlevel factorial designs using the theory of aliasing and confounding applied to the equivalent twolevel pseudo-factorial designs. The advantages of these designs for agricultural experiments are discussed and a set of example designs is listed. The design and analysis of a glasshouse experiment are outlined using this approach.