Desings (1/5)(5³)

The statistical solutions for quadratic and square root polynomials for a group of special 1/5 (53) fractional factorial, aiming, primarily, its application to fertilizer experiments are reported. These factorial designs were originated by the superposition of three of the four existing orthogonal 5x5 latin squares. Three basic designs are obtained: I-II-III, I-II-IV, and I-III-IV; the last one is presented below. 111 245 324 453 532 222 351 435 514 143 333 412 541 125 254 444 523 152 231 315 555 134 213 342 421 The quadratic model of second order with ten parameters is: Yijk = b0x0 +b li x li + b lj x lj + b lk x lk + b2i x2i + b2j x2j + b2k x2k + + b lilj x lilj + b lilk x lilk + b ljlk x ljlk + x ijk where x lm = a1 + Xm, x2m = a2 + g2Xm + X²m , m= i,j,k; each factor varying from 1 to 5, with the orthogonality conditions: Sx lm=0, Sx2m=0, Sx lm x2m=0, giving Sxlm=-3+Xm e Sx2m= 7-6Xm+X²m, so: x l1=-2; x l2=-1; x l3=0; x l4=1; x l5=2; x21=2; x22=-1; x23=-2; x24=-1; x25=2 The linear regression coefficient for each factor can be estimated independently; the quadratic and the linear x linear interaction coefficients are estimated from a 6x6 full matrix. Consequently in the analysis of variance the linear sums of squares for each factor are independent but the quadratic and interactions sums of squares for all factors are entangled and should be jointly tested. If the contribution of a factor and its respective interaction with the others are negligible, independent estimators of the linear and quadratic regression of the other two factors and the correspondent interaction can be calculated, with correspondent parallelism in the analysis of variance. On the other hand, if the factors are important but its interactions are negligible, the linear and quadratic coefficients for each factor are estimated independently. The square root polynomial model may be represented as in (1) with the values: x lm= a1+ (Xm)½ and x2m= a2+ g2(Xm)½ +Xm, where m= i,j,k; Sx lm=0, Sx2m=0, Sx lm x2m=0, giving: x lm= -1,67646 + (Xm)½, x2m= 2,41157-3,22798 (Xm)½ +Xm; x l1=-0,67646; x l2=-0,26226; x l3=0,05554; x l4=0,32354; x l5=0,55964; x21=0,18359; x22=-0,15342; x23=-0,17928; x24=-0,04438 and x25=0,19349, for each factor i,j,k. Regarding this model, with the exclusion of b0, the coefficients for each factor and the square-root interactions are estimated from a full 9x9 symetric matrix. In consequence; with the exception of b0; the sums of squares correspondent to the other coefficients are tested together. Equivalent properties to the quadratic model hold true for the square root model, when the interactions or when one main factor and its interactions are negligible. Assuming no interaction, the Mitscherlich model Y=A [l _ 10 _ c(x+b) ], can be used for evaluation of each factor response from the corresponding marginal totals. An extra evaluation from the main diagonal of the design can be obtained, representing the response to increasing amounts of the three factors at equal levels. In case of fertilizer experiments, treatments like 000 should be added as extra points to the 25 used in this design, in order to allow the determination of the increment due to the use of macronutrient combinations and their costs. Using proper range of dosages (avoiding plateau responses), as it should be in npk fertilizer experiments, this group of designs allows a more efficient analysis of the curvature of the surface functions on the area of economical decision. If the models are used without interactions, the independent estimation of the parameters for the quadratic, square-root and Mitscherlich models can be very easily achieved. These properties are of great interest in the economical studies of fertilization programs for developing countries. With the help of a net of experiments of this type, economical studies of fertility nature with macronutrients as nitrogen, phosphorus, and potassium, for example, can be obtained with five different levels of each factor, with experiments of medium size.