添加된 要因實驗計劃法 = Augmented Factorial Designs

다국어 초록 (Multilingual Abstract)

In most plans, especially in the farm experiments, it is necessary to use block of six or fewer experimental units. Such a design as the Table ⑴ may be suitable in the sense of reducing of the size of block. This may be called a “Augmented 2^3 factorial Design,” or may be called a “Fractional Replication of the Factorial Experiments with Block of 6 Units.”
Let the i th one of the v treatments be replicated r_i times in the b incomplete blocks of size n._i_h. Let the yield of the ijh th observation be expressed by the equation ⑴, where i＝1,2,…v＝number of treatments: j＝1,2,…r＝number of complete blocks; h＝1,2,…b＝number of incomplete blocks in the j th complete block; n_i_j_h＝1 if i th treatment occurs in the h th incomplete block of the jth complete block and zero otherwise.
Intrablock Analysis
The least square estimates of effects for the linear model 1 are obtained by minimizing the residual sum of squares and equating to zero each of the partial derivatives of the above residual sum of spuares with respect to μ, T_i, p_j and β_j_h results in the normal equations ⑵.
In matrix notation in the augmented 2^3 factorial design, the v＋1 equations from ⑶ plus the equation ∑T＾_i＝0 is the equation ⑷. The solutions for T＾_g is obtained such as expression ⑸.
The solutions for the p_i and β_j_h must be obtained jointly since they are not orthogonal. From the equation ⑺ solutions for p＾_j＋β＾_j_h are obtained such as expression ⑻.
Recovery of Interblock Information
The sum of squares to be minimized is the expression ⑾, where the true weights are w＝σ_e^2 and w´＝mean of w´_j_h, w_j_h＝1／（σ_e^2＋n_j_hσ_β^2）From the resulting normal equations, I obtained w=1/E_e, w'=2.7/6E_b - 3.3E_e from the Table ⑴.
Analysis for Repetitions of the Design
In this case, the block ss contains two components. One （component b） is the component that is present even when there are no repetitions （E_b）, and the other one （component a） is a new component. The sum of squares for the component a is obtained by same method of the lattice design （see Table 3）.
Factorial Analysis （with adjusted data）
We can obtain the Table ⑷ of three 3×3 table from the Table ⑴. And, from these we can calculate the main effects of N, P, k and two factor interactions with the method for fitting constants. But, if interaction becomes obvious, different coefficients must be calculated for each component in each effect. But, it is a bit tedious. I think, the following approximate method would have several advantage, including a familiar form of calulation. ⑴. We can test the main effect （say N）only for standards as a 2_3 factorial design, and test the same effect for the new treatments in the second and third replication. ⑵. We can test the main effect of N for the first replication as a 3×2^2 factorial design.