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Locally D-Optimal Designs with Heteroscedasticity: A Comparison between Two Methodologies
Diseños D-óptimos locales con heterocedasticidad: una comparación entre dos metodologías
Keywords:
D-efficiency, D-optimal design, Box-Cox transformations, Heteroscedasticity. (en)D-eficiencia, Diseños D-óptimos, heterocedasticidad, transformación de Box-Cox (es)
The classic theory of optimal experimental designs assumes that the errors of the model are independent and have a normal distribution with constant variance. However, the assumption of homogeneity of variance is not always satisfied. For example when the variability of the response is a function of the mean, it is probably that a heterogeneity model be more adequate than a homogeneous one. To solve this problem there are two methods: The first one consists of incorporating a function which models the error variance in the model, the second one is to apply some of the Box-Cox transformations to both sides on the nonlinear regression model to achieve a homoscedastic model (Carroll & Ruppert 1988, Chapter 4). In both cases it is possible to find the optimal design but the problem becomes more complex because it is necessary to find an expression for the Fisher information matrix of the model. In this paper we present the two mentioned methodologies for the D-optimality criteria and we show a result which is useful to find D- optimal designs for heteroscedastic models when the variance of the response is a function of the mean. Then we apply both methods with an example, where the model is nonlinear and the variance is not constant. Finally we find the D-optimal designs with each methodology, calculate the efficiencies and evaluate the goodness of fit of the obtained designs via simulations.
La teoría clásica de los diseños experimentales óptimos supone que los errores del modelo son independientes y tienen una distribución normal con varianza constante. Sin embargo, el supuesto de homogeneidad de varianza no siempre se satisface. Por ejemplo, cuando la variabilidad de la respuesta es una función de la media, es probable que un modelo heterocedástico sea más adecuado que uno homogéneo. Para solucionar este problema hay dos métodos: el primero consiste en incorporar una función que modele la varianza del error en el modelo; el segundo consiste en aplicar alguna de las transformaciones de Box-Cox en el modelo de regresión no lineal (Carroll & Ruppert 1988, Capítulo 4). En ambos casos es posible hallar el diseño óptimo, pero el problema se vuelve más complejo porque es necesario encontrar una expresión de la matriz de información de Fisher del modelo. En este artículo se presentan las dos metodologías mencionadas para el criterio D-optimalidad y se muestra un resultado que es útil para encontrar diseños D-óptimos para modelos heterocedásticos cuando la varianza de la respuesta es una función de la media. Luego, se aplican ambos métodos en un ejemplo donde el modelo es no lineal y la varianza no constante. Finalmente se encuentra el diseño D-óptimo con cada metodología, se calculan las eficiencias y se evalúa la bondad del ajuste de los diseños obtenidos a través de simulaciones.
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