Desempenho do delineamento composto central em experimentos com alto coeficiente de variação
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2012-02-17
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Universidade Federal de Viçosa
Resumo
Esse trabalho teve como objetivo avaliar o desempenho do delineamento composto central rotacional (DCCR) em relação às estimativas dos parâmetros da superfície de resposta estimada, sob condições de erros experimentais simulados que proporcionam altos coeficientes de variação. O grande impulso da metodologia de resposta foi dado por Box e Wilson (1951), que desenvolveram métodos de otimização de processos em pesquisas industriais. Dentre esses métodos, pode-se citar o DCCR como um delineamento econômico para a superfície de resposta, devido ao número reduzido de combinações entre os níveis dos fatores estudados, quando comparado ao fatorial completo. No entanto, sabe-se que estes delineamentos são eficientes onde normalmente os erros experimentais são pequenos e as condições do experimento são mais facilmente controláveis. Portanto, dada a economia do número de ensaios pelo DCCR, tornou-se importante verificar o seu potencial em delinear tratamentos visando o ajuste de superfícies de respostas para experimentos ligados às ciências agrárias, que apresentam, naturalmente, maiores erros aleatórios. Para tanto, os delineamentos de tratamentos, fatorial completo e composto central rotacional foram utilizados para planejar as combinações entre os níveis codificados de dois fatores (A e B). Foi estabelecido um modelo de segunda ordem para dois fatores A e B sem interação entre eles, denominada de superfície de resposta verdadeira. Foi estabelecido um fatorial completo 5x5, com as combinações entre os níveis codificados dos fatores A e B e com 40 repetições por tratamento. No total, foram constituídas 1000 unidades experimentais. Posteriormente, foram feitas 100 simulações para os erros experimentais presentes no experimento sob distribuição normal com média zero e variância σε2. O parâmetro σε foi especificado em 32, 48, 64 e 80, para fornecer os coeficientes de variação residuais (CV) iguais a 25, 37, 50 e 62 %. A escolha de tais coeficientes de variação foi feita de modo a abranger as classificações criadas por Ferreira (1991), citado por Silva et al. (2011), e Pimentel Gomes (1985) para as áreas agrárias. Os valores observados de Y foram obtidos a partir da soma dos valores verdadeiros de Y ( ) obtidos a partir da superfície de resposta verdadeira, com os resíduos , gerados pela simulação. Em cada tipo de delineamento (DCCR e fatorial completo), foram estabelecidas três, seis, nove e doze repetições por tratamento. De acordo com as combinações entre os tipos de delineamentos, coeficientes de variação e número de repetições por tratamento, foram realizados 100 ajustes da superfície de resposta dos quais foram obtidas as médias do coeficiente de determinação, EQM, a distância média entre o ponto crítico verdadeiro e o estimado (DPC) e a diferença média entre os coeficientes de variação estimados e simulados (DCV), além da porcentagem de acerto e intervalo de confiança de cada parâmetro e a porcentagem de acerto da superfície de resposta. Posteriormente foi feita uma regressão dessas medidas avaliadas em função do delineamento, coeficiente de variação e número de repetições por tratamento. A superioridade do desempenho do fatorial completo em relação DCCR aumentou em função do aumento do CV e da diminuição do número de repetições por tratamento. Recomendou-se o DCCR sob condições experimentais mais bem controladas, por ser um delineamento de tratamento mais econômico. No entanto, sabendo do difícil controle do erro aleatório em experimentos das áreas agrárias, em experimentos dessa natureza recomendou-se o fatorial completo ou o DCCR com um número maior de repetições por tratamento. Conclui-se também que o aumento do CV prejudica a qualidade de ajuste do fatorial completo e principalmente a do DCCR e esse prejuízo pode ser compensado com o aumento do número de repetições por tratamento. Percebeu-se que a qualidade de ajuste proporcionada pelo delineamento de tratamento não depende só da quantidade dos mesmos, mas principalmente da quantidade de unidades experimentais suficientes para proporcionar estimativas adequadas dos efeitos dos fatores conhecidos e desconhecidos.
The present work had the purpose of evaluating the performance of the rotational central composite design (RCCD) in relation to the estimated response surface parameters, under conditions of simulated experimental errors that provide high coefficients of variation. The big impulse of the response methodology was given by Box and Wilson (1951), who developed methods of process optimization in industrial researches. Within these methods, the RCCD may be mentioned as an economic design for the response surface, due to the reduced number of level combinations in the studied factors, when compared to the full factorial. However, it is known that these designs are efficient where experimental mistakes are usually small and the conditions of the experiment are easily controlled. However, given the reduced number of tests by the RCCD, it has become important to verify its potential in designing treatments to adjust response surfaces on experiments related to agrarian sciences, that present, naturally, larger random error. For such, the treatment designs, full factorial and rotational central composite have been used to plan the combinations between the coded levels of two factors (A and B). A second-order model has been established for two factors A and B without interaction among them, named true response surface. A full 5x5 factorial has been established, with the combinations between the coded levels of factors A and B and 40 repetitions per treatment. In total, 1000 experimental units have been built. Afterwards, 100 simulations have been made for the experimental errors ε present in the experiment under normal conditions with average zero and variations σε2. The parameter σε has been specified in 32, 48, 64 and 80, to provide the residual coefficients of variation (CV) equal to 25, 37, 50 and 60 %. The choice of these coefficients of variation has been made in order to include the classifications created by Ferreira (1991), quoted by Silva et al. (2011), and Pimentel Gomes (1985) for the agrarian areas. The observed values of Y have been obtained from the sum of the true values of Y ( ) obtained from the true response surface, with residuals ( ), generated by the simulation. In each type of design (RCCD and full factorial), three, six, nine and twelve repetitions per treatment have been established. According to the combinations between the types of treatment, coefficients of variation and number of repetitions per treatment, 100 adjustments of the response surface have been realized, from which the averages of the coefficient of determination, the average distances between the true and the estimated critical point (CPD) and between the coefficients of variation estimated and simulated (CVD), besides the success percentage and confidence interval of each parameter and the percentage of success from the response surface. Afterwards, a regression of these measures as been made, evaluated based on the design, coefficient of variation and number of repetitions per treatment. The superiority of performance of the full factorial in relation to the RCCD increase in relation to the increase of the CV and the reducing of the number of repetitions per treatment. The RCCD has been recommended under more controlled experimental conditions, for being a more economic treatment design. However, knowing the difficulty of controlling the random error on experiments for the agrarian areas, in experiments of this nature the full factorial or the RCCD were recommended with a larger number of repetitions per treatment. It has also been concluded that the increase of the CV prejudices the adjustment quality of the full factorial and mainly of the RCCD and this prejudice can be compensated with the increase in the number of repetitions per treatment. It has been seen that the adjustment quality provided by the treatment design does not depend only of their quantity, but mostly on the sufficient amount of experimental units to provide proper estimates of the effects of the known and unknown factors.
The present work had the purpose of evaluating the performance of the rotational central composite design (RCCD) in relation to the estimated response surface parameters, under conditions of simulated experimental errors that provide high coefficients of variation. The big impulse of the response methodology was given by Box and Wilson (1951), who developed methods of process optimization in industrial researches. Within these methods, the RCCD may be mentioned as an economic design for the response surface, due to the reduced number of level combinations in the studied factors, when compared to the full factorial. However, it is known that these designs are efficient where experimental mistakes are usually small and the conditions of the experiment are easily controlled. However, given the reduced number of tests by the RCCD, it has become important to verify its potential in designing treatments to adjust response surfaces on experiments related to agrarian sciences, that present, naturally, larger random error. For such, the treatment designs, full factorial and rotational central composite have been used to plan the combinations between the coded levels of two factors (A and B). A second-order model has been established for two factors A and B without interaction among them, named true response surface. A full 5x5 factorial has been established, with the combinations between the coded levels of factors A and B and 40 repetitions per treatment. In total, 1000 experimental units have been built. Afterwards, 100 simulations have been made for the experimental errors ε present in the experiment under normal conditions with average zero and variations σε2. The parameter σε has been specified in 32, 48, 64 and 80, to provide the residual coefficients of variation (CV) equal to 25, 37, 50 and 60 %. The choice of these coefficients of variation has been made in order to include the classifications created by Ferreira (1991), quoted by Silva et al. (2011), and Pimentel Gomes (1985) for the agrarian areas. The observed values of Y have been obtained from the sum of the true values of Y ( ) obtained from the true response surface, with residuals ( ), generated by the simulation. In each type of design (RCCD and full factorial), three, six, nine and twelve repetitions per treatment have been established. According to the combinations between the types of treatment, coefficients of variation and number of repetitions per treatment, 100 adjustments of the response surface have been realized, from which the averages of the coefficient of determination, the average distances between the true and the estimated critical point (CPD) and between the coefficients of variation estimated and simulated (CVD), besides the success percentage and confidence interval of each parameter and the percentage of success from the response surface. Afterwards, a regression of these measures as been made, evaluated based on the design, coefficient of variation and number of repetitions per treatment. The superiority of performance of the full factorial in relation to the RCCD increase in relation to the increase of the CV and the reducing of the number of repetitions per treatment. The RCCD has been recommended under more controlled experimental conditions, for being a more economic treatment design. However, knowing the difficulty of controlling the random error on experiments for the agrarian areas, in experiments of this nature the full factorial or the RCCD were recommended with a larger number of repetitions per treatment. It has also been concluded that the increase of the CV prejudices the adjustment quality of the full factorial and mainly of the RCCD and this prejudice can be compensated with the increase in the number of repetitions per treatment. It has been seen that the adjustment quality provided by the treatment design does not depend only of their quantity, but mostly on the sufficient amount of experimental units to provide proper estimates of the effects of the known and unknown factors.
Descrição
Palavras-chave
Delineamento composto central, Superfície de resposta, Delineamentos econômicos, Central composite design, Surface parameters, Economic designs
Citação
MENDONÇA, Layanne Andrade. Performance of the Rotacional Central Composite Design in experiments with high coefficientes of variation. 2012. 80 f. Dissertação (Mestrado em Estatística Aplicada e Biometria) - Universidade Federal de Viçosa, Viçosa, 2012.