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Response Surface Optimization in Growth Curves Through Multivariate Analysis
Optimización de superficies de respuesta en curvas de crecimiento a través de análisis multivariado
Keywords:
Growth curves, Multiple optimization, Response surfaces, Second order models (en)curvas de crecimiento, optimización múltiple, superficies de respuesta, modelos de segundo orden (es)
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En este artículo se propone una metodología para modelar conjuntamente tratamientos con niveles cuantitativos medidos en el tiempo, mediante la combinación de técnicas de superficies de respuesta con curvas de crecimiento. Se estiman los parámetros del modelo, los cuales miden el efecto en el tiempo de los factores relacionados con el modelo de superficie de respuesta de segundo orden. Estas estimaciones se realizan a través de una transformación que permite expresar el modelo como un modelo clásico de MANOVA; de esta manera, se expresan y juzgan las hipótesis tradicionales. En este artículo se propone una metodología para modelar conjuntamente tratamientos con niveles cuantitativos medidos en el tiempo, mediante la combinación de técnicas de superficies de respuesta con curvas de crecimiento. Se estiman los parámetros del modelo, los cuales miden el efecto en el tiempo de los factores relacionados con el modelo de superficie de respuesta de segundo orden. Estas estimaciones se realizan a través de una transformación que permite expresar el modelo como un modelo clásico de MANOVA; de esta manera, se expresan y juzgan las hipótesis tradicionales.
1Universidad Santo Tomás, Facultad de Estadística, Bogotá, Colombia. Lecturer. Email: andresortiz@usantotomas.edu.co
2Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Bogotá, Colombia. MsC in Statistics. Email: jcriverar@unal.edu.co
3Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Bogotá, Colombia. Associate professor. Email: oomelom@unal.edu.co
A methodology is proposed to jointly model treatments with quantitative levels measured throughout time by combining the response surface and growth curve techniques. The model parameters, which measure the effect throughout time of the factors related to the second-order response surface model, are estimated. These estimates are made through a suitable transformation that allows to express the model as a classic MANOVA model, so the traditional hypotheses are formulated and tested. In addition, the optimality conditions throughout time are established as a set of specific combination factors by the fitted model. As a final step, two applications are analyzed using our proposed model: the first was previously analyzed with growth curves in another paper, and the second involves two factors that are optimized over time.
Key words: Growth curves, Multiple optimization, Response surfaces, \linebreak Second order models.
En este artículo se propone una metodología para modelar conjuntamente tratamientos con niveles cuantitativos medidos en el tiempo, mediante la combinación de técnicas de superficies de respuesta con curvas de\linebreak crecimiento. Se estiman los parámetros del modelo, los cuales miden el efecto en el tiempo de los factores relacionados con el modelo de superficie de respuesta de segundo orden. Estas estimaciones se realizan a través de una transformación que permite expresar el modelo como un modelo clásico de MANOVA; de esta manera, se expresan y juzgan las hipótesis tradicionales. Además, las condiciones de optimización a través del tiempo son establecidas para un conjunto de factores específicos por medio del modelo ajustado. Como paso final, se analizan dos aplicaciones utilizando el modelo propuesto: la primera fue analizada mediante curvas de crecimiento en otro artículo, y la segunda consiste en dos factores que son optimizados a lo largo del tiempo.
Palabras clave: curvas de crecimiento, optimización múltiple, superficies de respuesta, modelos de segundo orden.
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References
1. Box, G. E. P. & Draper, N. R. (1982), 'Measures of lack of fit for response surface designs and predictor variable transformations', Technometrics 24, 1-8.
2. Box, G. E. P. & Wilson, K. B. (1951), 'On the experimental attainment of the optimum conditions', Journal of the Royal Statistical Society 13, 1-45.
3. Chiou, J., Müller, H. & Wang, J. (2004), 'Functional response models', Statistica Sinica 14, 675-693.
4. Chiou, J., Müller, H., Wang, J. & Carey, J. (2003), 'A functional multiplicative effects model for longitudinal data, with application to reproductive histories of female medflies', Statistica Sinica 13, 1119-1133.
5. Davidian, M. (2005), Applied Longitudinal Data Analysis, Chapman and hall, North Carolina state university.
6. Davis, C. S. (2002), Statistical Methods for the Analysis of Repeated Measurements, Springer-Verlag, New York.
7. Draper, N. & Ying, L. H. (1994), 'A note on slope rotatability over all directions', Journal of Statistical Planning and Inference 41, 113-119.
8. Frey, K. S., Potter, G. D., Odom, T. W., Senor, M. A., Reagan, V. D., Weir, V. H., Elsslander, R. V. T., Webb, M. S., Morris, E. L. & Smith, W. B. a. K. E. (1992), 'Plasma silicon and radiographic bone density on weanling quarter horses fed sodium zelolite A', Journal of Equine Veterinary Science 12, 292-296.
9. Grizzle, J. E. & Allen, D. M. (1969), 'Analysis of growth and dose response curves', Biometrics 25, 357-381.
10. Guerrero, S. C. & Melo, O. O. (2008), 'Optimization process of growth curves through univariate analysis', Revista Colombiana de Estadística 31(2), 193-209.
11. Heck, D. L. (1960), 'Charts of some upper percentage points of the distribution of the largest characteristic root', The Annals of Mathematical Statistics 31(3), 625-642.
12. Hill, W. J. & Hunter, W. G. (1966), 'A review of response surface methodology: A literature review', Technometrics 8, 571-590.
13. Kabe, D. G. (1974), 'Generalized Sverdrup's lemma and the treatment of less than full rank regression model', Canadian Mathematical Bulletin 17, 417-419.
14. Kahm, M., Hasenbrink, G., Lichtenberg-Fraté, H., Ludwig, J. & Kschischo, M. (2010), 'Grofit: fitting biological growth curves with R', Journal of Statistical Software 7, 1-21.
15. Khatri, C. A. (1966), 'A note on a MANOVA model applied to problems in growth curves', Annals of the Institute of Statistical Mathematics 18, 75-86.
16. Khatri, C. A. (1973), 'Testing some covariance structures under growth curve model', Journal Multivariate Analysis 3, 102-116.
17. Khatri, C. A. (1988), 'Robustness study for a linear growth model', Journal Multivariate Analysis 24, 66-87.
18. Kshirsagar, A. M. & Boyce, S. (1995), Growth Curves, Marcel Dekker, New York.
19. Lucas, J. M. (1976), 'Which response surfaces is best?', Technometrics 18, 411-417.
20. Magnus, J. R. (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley, New York.
21. Mead, R. & Pike, D. J. (1975), 'A review of responses surface methodology from a biometric viewpoint', Biometrics 31, 830-851.
22. Molenberghs, G. & Verbeke, G. (2005), Models for Discrete Longitudinal Data, Springer, New York.
23. Montoya, C. & Gallego, D. (2012), Modelo matemático que permita evaluar el cambio de la DBO_5 soluble debido a agentes inhibitorios en un proceso de lodos activados, Master's thesis, Facultad de minas. Universidad Nacional de Colombia.
24. Pan, J. & Fang, K. (2002), Growth Curve Models and Statistical Diagnostics, Springer Series in Statistics, New York.
25. Potthoff, R. & Roy, S. (1964), 'A generalized multivariate analysis of variance model useful especially for growth curve problems', Biometrika 51, 313-326.
26. Rao, C. R. (1959), 'Some problems involving linear hypothesis in multivariate analysis', Biometrika 46, 49-58.
27. Rao, C. R. (1967), Least squares theory using an estimated dispersion matrix and its applications to mesurement of signals, 'Proceeding of the Fifth Berkeley Symposium on Mathematical Statistics and Probability', Vol. I, University of California Press, Berkeley, p. 355-372.
28. Singer, J. M. & Andrade, D. F. (1994), 'On the choice of appropiate error terms in profile analysis', Royal of Statistical Society 43(2), 259-266.
29. Srivastava, M. S. (2002), 'Nested growth curves models', Sankhyã: The Indian Journal of Statistics, Series A, Selected Articles from San Antonio Conference in Honour of C. R. Rao 64(2), 379-408.
30. Verbyla, A. P. & Venables, W. N. (1988), 'An extension of the growth curve models', Biometrika 75, 129-138.
31. Wilks, S. S. (1932), 'Certain generalizations in the analysis of variance', Biometrika 24, 471-494.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv36n1a09,
AUTHOR = {Ortiz, Felipe and Rivera, Juan C. and Melo, Oscar O.},
TITLE = {{Response Surface Optimization in Growth Curves Through Multivariate Analysis}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2013},
volume = {36},
number = {1},
pages = {153-176}
}
References
Box, G. E. P. & Draper, N. R. (1982), ‘Measures of lack of fit for response surface designs and predictor variable transformations’, Technometrics 24, 1–8.
Box, G. E. P. & Draper, N. R. (2007), Response Surfaces, Mixtures, and Ridge Analyses, Wiley Series in Probability and Statistics, New York.
Box, G. E. P. & Wilson, K. B. (1951), ‘On the experimental attainment of the optimum conditions’, Journal of the Royal Statistical Society 13, 1–45.
Chiou, J., Müller, H. & Wang, J. (2004), ‘Functional response models’, Statistica Sinica 14, 675–693.
Chiou, J., Müller, H., Wang, J. & Carey, J. (2003), ‘A functional multiplicative effects model for longitudinal data, with application to reproductive histories of female medflies’, Statistica Sinica 13, 1119–1133.
Davidian, M. (2005), Applied Longitudinal Data Analysis, Chapman and hall, North Carolina state university.
Davis, C. S. (2002), Statistical Methods for the Analysis of Repeated Measurements, Springer-Verlag, New York.
Draper, N. & Ying, L. H. (1994), ‘A note on slope rotatability over all directions’, Journal of Statistical Planning and Inference 41, 113–119.
Frey, K. S., Potter, G. D., Odom, T. W., Senor, M. A., Reagan, V. D., Weir, V. H., Elsslander, R. V. T., Webb, M. S., Morris, E. L., Smith, W. B. & Weigand, K. E. (1992), ‘Plasma silicon and radiographic bone density on weanling quarter horses fed sodium zelolite A’, Journal of Equine Veterinary Science 12, 292–296.
Grizzle, J. E. & Allen, D. M. (1969), ‘Analysis of growth and dose response curves’, Biometrics 25, 357–381.
Guerrero, S. C. & Melo, O. O. (2008), ‘Optimization process of growth curves through univariate analysis’, Revista Colombiana de Estadística 31(2), 193–209.
Heck, D. L. (1960), ‘Charts of some upper percentage points of the distribution of the largest characteristic root’, The Annals of Mathematical Statistics 31(3), 625–642.
Hill, W. J. & Hunter, W. G. (1966), ‘A review of response surface methodology: A literature review’, Technometrics 8, 571–590.
Kabe, D. G. (1974), ‘Generalized Sverdrup’s lemma and the treatment of less tan full rank regression model’, Canadian Mathematical Bulletin 17, 417–419.
Kahm, M., Hasenbrink, G., Lichtenberg-Fraté, H., Ludwig, J. & Kschischo, M. (2010), ‘grofit: Fitting biological growth curves with R’, Journal of Statistical Software 7, 1–21.
Khatri, C. A. (1966), ‘A note on a MANOVA model applied to problems in growth curves’, Annals of the Institute of Statistical Mathematics 18, 75–86.
Khatri, C. A. (1973), ‘Testing some covariance structures under growth curve model’, Journal Multivariate Analysis 3, 102–116.
Khatri, C. A. (1988), ‘Robustness study for a linear growth model’, Journal Multivariate Analysis 24, 66–87.
Kshirsagar, A. M. & Boyce, S. (1995), Growth Curves, Marcel Dekker, New York.
Lucas, J. M. (1976), ‘Which response surfaces is best?’, Technometrics 18, 411– 417.
Magnus, J. R. (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley, New York.
Mead, R. & Pike, D. J. (1975), ‘A review of responses surface methodology from a biometric viewpoint’, Biometrics 31, 830–851.
Molenberghs, G. & Verbeke, G. (2005), Models for Discrete Longitudinal Data, Springer, New York.
Montoya, C. & Gallego, D. (2012), Modelo matemático que permita evaluar el cambio de la DBO5 soluble debido a agentes inhibitorios en un proceso de lodos activados, Master’s thesis, Facultad de minas. Universidad Nacional de Colombia.
Pan, J. & Fang, K. (2002), Growth Curve Models and Statistical Diagnostics, Springer Series in Statistics, New York.
Potthoff, R. & Roy, S. (1964), ‘A generalized multivariate analysis of variance model useful especially for growth curve problems’, Biometrika 51, 313–326.
Rao, C. R. (1959), ‘Some problems involving linear hypothesis in multivariate analysis’, Biometrika 46, 49–58.
Rao, C. R. (1967), Least squares theory using an estimated dispersion matrix and its applications to mesurement of signals, in ‘Proceeding of the Fifth Berkeley Symposium on Mathematical Statistics and Probability’, Vol. I, University of California Press, Berkeley, pp. 355–372.
Singer, J. M. & Andrade, D. F. (1994), ‘On the choice of appropiate error terms in profile analysis’, Royal of Statistical Society 43(2), 259–266.
Srivastava, M. S. (2002), ‘Nested growth curves models’, Sankhyã: The Indian Journal of Statistics, Series A, Selected Articles from San Antonio Conference in Honour of C. R. Rao 64(2), 379–408.
Verbyla, A. P. & Venables, W. N. (1988), ‘An extension of the growth curve models’, Biometrika 75, 129–138.
Wilks, S. S. (1932), ‘Certain generalizations in the analysis of variance’, Biometrika 24, 471–494.
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