Análise de um delineamento box para três fatores, com níveis não eqüidistantes

Analysis of a central composite box design for three factors, with non-equidistant levels

Resumos

Procedeu-se à análise estatística para dados provenientes do uso de um delineamento de tratamentos central composto Box para três fatores, com níveis não eqüidistantes, através do estudo de uma superfície de resposta, utilizando um modelo quadrático em X, com dez parâmetros. São dadas as estimativas dos parâmetros beta, suas variâncias e covariâncias e o desenvolvimento da análise da variância, chegando-se ainda à equação em X, que possibilita a análise econômica dos ensaios. Resultados de um experimento de capim-swannee-bermuda, conduzido durante três anos em solo de cerrado, são analisados estatisticamente seguindo o esquema apresentado.

The response of swannee bermuda grass (Cynodon dactylon (L.) Pers.) to nitrogen, phosphate and sulfur fertilization was studied. during three years, on a leached and phosphorus deficient "cerrado" soil. The experiment was conducted in four randomized blocks, using the treatment combinations specified in the Box composite design, but at non-equidistant levels (0, 1, 2, 4, 8); an extra point 000 was added, for economical reasons; the basic rates were 150, 100 and 20 kg/ha of N, P2O5, and S, respectively. The main objective of the present paper is to determine the response surface fitted to the data of the experiment, allowing, later on, the economical analysis of the results. A quadratic model with ten parameters was fitted to the annual data. Yijk= b0x0+ b1i x1i+ b1j x1j+ b1k x1k+ b2i x2i+ b2j x2j+ b2k x2k+ b1i1j x1i1j+ b1i1k x1i1k+ b1j1k x1j1k+ e ijk, where e ijk~ N(0, s²); b0= h0+ (b1i1j+b1i1k+b1j1k ); x1mt= -+Xmt and x2mt=- Xmt+ + X²mt, with m= i,j,k and t= 0, 1, 2, 4, 8. (The linear and quadratic components x1 and x2 are orthogonal.) The beta parameters were estimated through the least squares procedures; as the dosages were not equidistant, the sum of squares due to the regression was not available independently for each component. The coefficients of variation for the three years were 10,8%, 14,1%, and 9,1%, and the coefficients of determination wore 93,8%, 91,7% and 86,7% respectively. The general equation for the total of the period, including linear and quadratic year coefficients and their interactions with the b coefficients, was determined. The canonical equations showed negative signs for the l values; the point of maximum was outside the experimental regions. The X equations were given; so that the economical analysis of the experiment can be easily performed.

NOTAS

Análise de um delineamento box para três fatores, com níveis não eqüidistantes¹

Analysis of a central composite box design for three factors, with non-equidistant levels

Joassy de Paula Neves JorgeI, ²; Luiz M. M. de Freitas^II

^IDivisão de Plantas Alimentícias Básicas, Instituto Agronômico, IAC

^IIInstituto de Pesquisas IRI, Matão, IAC

SINOPSE

Procedeu-se à análise estatística para dados provenientes do uso de um delineamento de tratamentos central composto Box para três fatores, com níveis não eqüidistantes, através do estudo de uma superfície de resposta, utilizando um modelo quadrático em X, com dez parâmetros.

São dadas as estimativas dos parâmetros b, suas variâncias e covariâncias e o desenvolvimento da análise da variância, chegando-se ainda à equação em X, que possibilita a análise econômica dos ensaios.

Resultados de um experimento de capim-swannee-bermuda, conduzido durante três anos em solo de cerrado, são analisados estatisticamente seguindo o esquema apresentado.

SUMMARY

The response of swannee bermuda grass (Cynodon dactylon (L.) Pers.) to nitrogen, phosphate and sulfur fertilization was studied. during three years, on a leached and phosphorus deficient "cerrado" soil. The experiment was conducted in four randomized blocks, using the treatment combinations specified in the Box composite design, but at non-equidistant levels (0, 1, 2, 4, 8); an extra point 000 was added, for economical reasons; the basic rates were 150, 100 and 20 kg/ha of N, P₂O₅, and S, respectively.

The main objective of the present paper is to determine the response surface fitted to the data of the experiment, allowing, later on, the economical analysis of the results.

A quadratic model with ten parameters was fitted to the annual data.

Y_ijk= b₀x₀+ b_1ix_1i+ b_1jx_1j+ b_1kx_1k+ b_2ix_2i+ b_2jx_2j+ b_2kx_2k+ b_1i1jx_1i1j+ b_1i1kx_1i1k+ b_1j1kx_1j1k+ e_ijk, where e_ijk~ N(0, s²); b₀= h₀+ (b_1i1j+b_1i1k+b_1j1k ); x_1mt= -+X_mt and x_2mt=- X_mt+ + X²_mt, with m= i,j,k and t= 0, 1, 2, 4, 8. (The linear and quadratic components x₁ and x₂ are orthogonal.)

The b parameters were estimated through the least squares procedures; as the dosages were not equidistant, the sum of squares due to the regression was not available independently for each component. The coefficients of variation for the three years were 10,8%, 14,1%, and 9,1%, and the coefficients of determination wore 93,8%, 91,7% and 86,7% respectively.

The general equation for the total of the period, including linear and quadratic year coefficients and their interactions with the b coefficients, was determined.

The canonical equations showed negative signs for the l values; the point of maximum was outside the experimental regions.

The X equations were given; so that the economical analysis of the experiment can be easily performed.

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LITERATURA CITADA

Recebido para publicação a 20 de setembro de 1979.

• 1. ANDERSON, R. L. & BANCROFT, T. A. Statistical theory in research. New York, McGraw-Hill, 1952. 399p.
• 2. BOX, G. E. P. & WILSON, K. B. On the experimental attainment of optimum conditions. J. R. statist Soc., B, 13:1-45, 1951.
• 3. CONAGIN. A. & JORGE, J. de P. N. Delineamentos (1/5) (5Ş). Bragantia, Campinas, 36:23-58, 1977
• 4. DAVIES, O. L., ed. Design and analysis of industrial experiments. Edinburgh, Oliver and Boyd, 1954. 636p.
• 5. DRAPER, N. R. & SMITH, H. Applied regression analysis. New York. J. Wiley, 1966. 407p.
• 6. GRAYBILL, F. A. An introduction to linear statistical models. New York, McGraw-Hill, 1961. 463p.
• 7. MYERS, R. H. Response surface methodology. Boston, Allyn and Bacon, 1971. 243p.

1

Trabalho apresentado na 10.a Conferência Internacional de Biornetria, Guarujá (SP), em agosto de 1979.

2

Com bolsa de suplementação do CNPq.

Datas de Publicação

• Publicação nesta coleção
  15 Dez 2007
• Data do Fascículo
  1979

Histórico

• Recebido
  20 Set 1979