DECOMPOSITION FOR THE MODEL AND GENERALIZED ORTHOGONAL DESIGN

The concept of orthogonal design of experiments is generalized and some new ideas are proposed in this paper. Let us consider the model Y=η(X)+εwhere Y can be observed and ε is the random error. One always makes assumption that η（X） is the linear combination of known function φ_i（X）, i=1, …, p, for classical expe- rimental design. But now, η（X） is assumed to be a function defined on R^m. We consider X as a random vector taking value in R^m. Any distribution of X is called a design, and if its components are independent of each other, then it is called orthogonal. Let X_a=(x_i2, x_i2, …, x_iα) with 1≤i₁≤i₂…＜i_α≤m, 1≤α≤m, and $\mu\left(X_a\right)=\sum_{\beta=0}^{\infty}(-1)^{a-\beta} \sum_{X_a \subset X_\theta} M\left(X_\beta\right)$ where M （X_β）=E （η（X）/X_β） then μ（X_α） is called interactive effective function among x_i1, x_i2…, x_iα. In this paper, we give the decomposition $\eta(X)=\sum_{\beta=0}^m \sum_{X_A \subset X} \mu\left(X_B\right)$ and discuss the properties of effective function for generalized orthogonal experimental designs. Finally, some numerical examples are given.