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Resumos

Simulation was used to evaluate the properties of (1/5) (5³) designs, obtained by the superposition of three orthogonal Latin squares. The following basic quadratic equation was used: Yijk = 3500 + 180 x li+ 250 x lj + 120 x lk - 42 x 2i - 55 x2j - - 28 x2k - 25 x lilj - 18 x lilk- 12 x ljlk (A) The values of the linear and second order polynomials for the quadratic model and the correspondent polynomials for the square root model were, respectively: Quadratic: g1=-3+X, g2 =7- 6X+X² Square root: x?= a???+(X)½ e x2=a2+ g2 (X + X,)½ where a?=1.67646; a2=2.41157; g2=-3.22798, obtained with the restrictions for orthogonality: Sx?= Sx2= Sx?Sx2,=0, as described in the "Designs (1/5) (5x5x5)", given by the authors. The coefficients given in (A) represent the mean value, the linear, quadratic, and interactions effects of three factors npk from a surface response expressed in kg of corn per hectare; their values were chosen in such a way that main effects were significantly different from zero, the quadratic coefficients and interactions allowing a maximum point of the function to be allocated between the fourth and fifth levels, so inside the range of the dosages used. This is the situation that should be looked for when choicing the levels of the dosages in planning field fertilizer experiment programmes: we must avoid dosages that give origin to plateau response because they are far apart from the area of economical decision. A histogram of normal shape for the errors, using a coefficient of variation of 8.5%, was built, and the errors were distributed at random for the 25 treatments in the equation (A), creating 60 sets of data. The quadratic and square root polynomials were fitted to each set of 25 treatments and also to the mean of the treatments obtained from groups of n experiments, n equal to 2, 3, 4, 5, 6, 10, 12, 15, 20 or 60. A good convergence of the estimators of the parameters was obtained for the two models. The coefficients of determination were practically of the same value. The percentage of maximum, obtained through the canonical form for both models for the different groups, is: MODEL GROUP MODEL 1 2 3 4 5 6 10 12 15 20 30 60 Quadratic 35 70 70 67 92 90 100 100 100 100 100 100 Square root 13 27 35 47 50 50 67 80 75 100 100 100 The coefficients of the variance of the treatments on the main diagonal were: MODEL TREATMENT 1 2 3 4 5 6 10 12 15 20 30 60 Quadratic 33 70 65 67 92 90 100 100 100 100 100 100 Square root 15 27 35 47 50 50 67 80 75 100 100 100 The results indicated that the grouping of 10 experiments of the (1/5) (5x5x5) design was sufficient to obtain 100 percent maximum, when quadratic model was used, whereas 20 experiments are necessary when the square root model was utilized. Another basic equation (C), related to (A) but of square root nature, was used for a similar simulation. The results obtained for bP and the percentage of maximum points were analogous. Independently of the basic equation (A) or (C), with a coefficient of variation of 8.5%, the grouping of 10 experiments was the first one that assured the obtaining of 100 percent maximum, when quadratic response surface was fitted, whereas 20 experiments were necessary when the square root model was used.

Estudos em um grupo especial de delineamentos (1/5)(53)1 1 Trabalho apresentado na IX Conf. int. de Biometria, Boston, Ma., USA, em agosto de 1976.

Studies in a special group of (1/5)(53) designs

Joassy de Paula Neves Jorge2 2 Com bolsas de suplementação do C.N.Pq. ; Armando Conagin2 2 Com bolsas de suplementação do C.N.Pq.

Divisão de Plantas Alimentícias Básicas, Instituto Agronômico, IAC

SINOPSE

MODELO GRUPAMENTO

1 2 3 4 5 6 10 12 15 20 30 60

Quadrático 35 70 70 67 92 90 100 100 100 100 100 100

Raiz quadrada 13 27 35 47 50 50 67 80 75 100 100 100

Também foi feita simulação tendo por base uma equação fundamental (C) de tipo raiz quadrada. Conclusões análogas foram obtidas em relação aos estimadores bP. Em relação à porcentagem de pontos de máximo, para esta nova equação fundamental utilizada, foram obtidos os seguintes resultados:

MODELO GRUPAMENTO

1 2 3 4 5 6 10 12 15 20 30 60

Quadrático 33 70 65 67 92 90 100 100 100 100 100 100

Raiz quadrada 15 27 35 47 50 50 67 80 75 100 100 100

Os valores indicam que, independentemente da equação fundamental utilizada, quando o modelo é de natureza quadrática, o grupamento de 10 experimentos do delineamento (1/5) (53) é suficiente para obter 100% de pontos de máximo, enquanto são necessários 20 experimentos quando se utiliza o modelo com raiz quadrada. É desenvolvida, para os dados da simulação com (A) e (C), a partir das equações em X, a análise econômica para os dois modelos, com apresentação sucinta dos principais resultados obtidos.

SUMMARY

Simulation was used to evaluate the properties of (1/5) (53) designs, obtained by the superposition of three orthogonal Latin squares. The following basic quadratic equation was used: Yijk = 3500 + 180 xli+ 250 xlj + 120 xlk - 42 x 2i - 55 x2j - - 28 x2k - 25 xlilj - 18 xlilk- 12 xljlk (A) The values of the linear and second order polynomials for the quadratic model and the correspondent polynomials for the square root model were, respectively: Quadratic: g1=-3+X, g2 =7- 6X+X2 Square root: x?= a???+(X)1/2 e x2=a2+ g2 (X + X,)1/2 where a?=1.67646; a2=2.41157; g2=-3.22798, obtained with the restrictions for orthogonality: Sx?= Sx2= Sx?Sx2,=0, as described in the "Designs (1/5) (5x5x5)", given by the authors. The coefficients given in (A) represent the mean value, the linear, quadratic, and interactions effects of three factors npk from a surface response expressed in kg of corn per hectare; their values were chosen in such a way that main effects were significantly different from zero, the quadratic coefficients and interactions allowing a maximum point of the function to be allocated between the fourth and fifth levels, so inside the range of the dosages used. This is the situation that should be looked for when choicing the levels of the dosages in planning field fertilizer experiment programmes: we must avoid dosages that give origin to plateau response because they are far apart from the area of economical decision. A histogram of normal shape for the errors, using a coefficient of variation of 8.5%, was built, and the errors were distributed at random for the 25 treatments in the equation (A), creating 60 sets of data. The quadratic and square root polynomials were fitted to each set of 25 treatments and also to the mean of the treatments obtained from groups of n experiments, n equal to 2, 3, 4, 5, 6, 10, 12, 15, 20 or 60. A good convergence of the estimators of the parameters was obtained for the two models. The coefficients of determination were practically of the same value. The percentage of maximum, obtained through the canonical form for both models for the different groups, is:

MODEL GROUP MODEL

1 2 3 4 5 6 10 12 15 20 30 60

Quadratic 35 70 70 67 92 90 100 100 100 100 100 100

Square root 13 27 35 47 50 50 67 80 75 100 100 100

The coefficients of the variance of the treatments on the main diagonal were:

MODEL TREATMENT

1 2 3 4 5 6 10 12 15 20 30 60

Quadratic 33 70 65 67 92 90 100 100 100 100 100 100

Square root 15 27 35 47 50 50 67 80 75 100 100 100

The results indicated that the grouping of 10 experiments of the (1/5) (5x5x5) design was sufficient to obtain 100 percent maximum, when quadratic model was used, whereas 20 experiments are necessary when the square root model was utilized. Another basic equation (C), related to (A) but of square root nature, was used for a similar simulation. The results obtained for bP and the percentage of maximum points were analogous. Independently of the basic equation (A) or (C), with a coefficient of variation of 8.5%, the grouping of 10 experiments was the first one that assured the obtaining of 100 percent maximum, when quadratic response surface was fitted, whereas 20 experiments were necessary when the square root model was used.

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Recebido para publicação em 20 de julho de 1976.

• 1
Trabalho apresentado na IX Conf. int. de Biometria, Boston, Ma., USA, em agosto de 1976.
• 2
Com bolsas de suplementação do C.N.Pq.
• 1
ALVAREZ, R.; ARRUDA, H. V. & GARGANTINI, H. Adubação da cana de açúcar. Ensaio preliminar de adubação NPK em terra-roxa. Bragantia 19:361-368, 1960.
• 2
CONAGIN, A.; JORGE, J. P. N. & VENTURINI, W. R. Delineamentos experimentais utilizáveis na experimentação de campo. In: Reynaert, E. E. La investigación de fertilidad de suelos para la producción agrícola en la zona templada. Montevideo, IICA Zona Sur, 1969.
• 3
__________ & JORGE, J. P. N. Delineamentos (1/5) (53). Bragantia 36:23-58, 1977.
• 4
FUZATTO, M. G.; VENTURINI, W. R. & CAVALERI, P. A. Estudo técnico-econômico na adubaçáo do algodoeiro no Estado de São Paulo. Campinas, Instituto Agronômico, 1970. 15 p. (Projeto BNDE/ANDA/CIA-1)
• 5
IGUE, T.; MASCARENHAS, H. A. A. & MIYASAKA, S. Estudo comparativo dos métodos de Mitscherlich e do trinômio do segundo grau, na determinação das doses mais econômicas de fertilizantes, na adubação do feijoeiro (Phaseolus vulgaris L.). Campinas, Instituto Agronômico, 1971. 15 p. (Projeto BNDE/ ANDA/CIA-4)
• 6
MIRANDA, L. T. Resultados de experimentos de adubaçáo e sugestões para interpretação, baseada na análise química do solo. In: Cultura e adubação do milho. São Paulo, Instituto Brasileiro de Potassa, 1966. p. 451-471.
• 7
MOOD, A. M. & GRAYBILL, F. A. Introduction to the Theory of Statistics. Tokyo, Kogakusha Company Ltd., 1963.
• 8
MYERS, R. H. Response Surface Methodology. Boston, Allyn and Bacon, 1971.
• 9
PACITT, T. Fortran Monitor. Princípios. Rio de Janeiro, Livro Técnico, 1967.
• 10
TEJEDA, H. Consideraciones sobre disenos experimentales en la investigación de campo en fertilidad de suelo. In: Reynaert, E. E. La investigación de fertilidad de suelos para la producción agrícola en la zona templada. Montevideo, HCA Zona Sur, 1969. p. 171-182.
• 11
TRAMEL, T. E. A suggested procedure for agronomic-economic fertilizer experiments. In: Baum, E. L. e outros. Economic and Technical analysis of fertilizer innovations and resource use. Ames, Iowa, State College Press, 1957. p. 168-175.
• 12
VOSS, R. & PESEK, J. T. Yield of corn as affected by fertilizer rates and environmental factors. Agronomy 59:567-572, 1967. (Original não consultado; extraído de Tejeda. H.
1 Trabalho apresentado na IX Conf. int. de Biometria, Boston, Ma., USA, em agosto de 1976. 2 Com bolsas de suplementação do C.N.Pq.

Datas de Publicação

• Publicação nesta coleção
19 Dez 2007
• Data do Fascículo
1977
Instituto Agronômico de Campinas Avenida Barão de Itapura, 1481, 13020-902 Campinas SP - Brazil, Tel.: +55 19 2137-0653, Fax: +55 19 2137-0666 - Campinas - SP - Brazil
E-mail: bragantia@iac.sp.gov.br