Analysis of exprimental designs with outliers
Keywords:
Central Composite Designs, Robust Designs, Outliers, Minimax.
Abstract
Primary purpose of the article is to develop outlier robust designs. As a matter of fact, negative effect of outliers in any experimental settings is established where the outliers at any specific design point can destroy the features of the design for which it is being developed. It is attempted here in this article to develop a version of robustness for central composite designs which may protect it for outliers by introducing the idea of minimax outlying effect. This involves the calculation of the degree of outlying effect(s) outlier(s) may produce and then minimize the maximum of these outlying effects in an attempt to equalize the influence of all design points. On comparison, these outlier robust designs are proved to be more optimal, on the scales of A, D, and E optimalities, against existing conventional rotatable, orthogonal, and other such designs. The outlier robust designs, developed here, are suitable for settings prone to outliers where conventional designs fail to represent and analyze the processes and systems.
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References
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Chauvenet, W. (1960). A Manual of Spherical and Practical Autonomy (Vol. II.
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Cousineau, D., & Chartier, S. (2010). Outliers detection and treatment: a review. International Journal of Psychological Research, 3(1), 58-67.
Daniell, P. J. (1920). Observations weighted according to order. American Journal of Mathematics, 42(4), 222-236.
Dixon, W. J. (1950). Analysis of extreme values. The Annals of Mathematical Statistics, 21(4), 488-506.
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Eubank, R. L. (1984). The hat matrix for smoothing splines. Statistics & probability letters, 2(1), 9-14.
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Grubbs, F. E. (1950). Sample criteria for testing outlying observations. The Annals of Mathematical Statistics, 21(1), 27-58.
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Jauffret, C. (2007). Observability and Fisher information matrix in nonlinear regression. IEEE Transactions on Aerospace and Electronic Systems, 43(2), 756-759.
Kiefer, J. (1985). Collected Papers III: design of experiments. Springer-Verlag.
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Montgomery, D. C., & Myers, R. H. (1995). Response surface methodology: process and product optimization using designed experiments, Raymond H. Meyers, Douglas C. Montgomery, A Wiley-Interscience Publications.
Mukerjee, R., & Huda, S. (1985). Minimax second-and third-order designs to estimate the slope of a response surface. Biometrika, 72(1), 173-178.
Newcomb, S. (1886). A generalized theory of the combination of observations so as to obtain the best result. American journal of Mathematics, 343-366.
Peirce, B. (1852). Criterion for the rejection of doubtful observations. The Astronomical Journal, 2, 161-163.
Peirce, B. (1877, May). On Peirce's criterion. In Proceedings of the American Academy of Arts and Sciences (Vol. 13, pp. 348-351). American Academy of Arts & Sciences.
Rao, C. R., & Mitra, S. K. (1973). Theory and application of constrained inverse of matrices. SIAM Journal on Applied Mathematics, 24(4), 473-488.
Rao, C. R., & Rao, M. B. (1998). Matrix algebra and its applications to statistics and econometrics. World Scientific.
Rao, C. R., & Toutenburg, H. (1995). Linear models. In Linear models (pp. 3-18). Springer, New York, NY.
Saunder, S. A. (1903). Note on the use of Peirce's criterion for the rejection of doubtful observations. Monthly Notices of the Royal Astronomical Society, 63, 432-436.
Shannon, C. E. (1948). A mathematical theory of communication. Bell system technical journal, 27(3), 379-423.
Siddiqi, A. F. (2003). Outliers in Designed Experiments; Classical Robust & Resistant Methods. Journal of Statistics, 10, 49-65.
Siddiqi, A. F. (2008). Outlier robust designs. International Journal of Applied Mathematics and Statistics™, 13(M08), 64-77.
Siddiqi, A. F. (2010). Outlier Robust Draper & Lin Designs. Pakistan Journal of Statistics and Operation Research, 7(1), 87-99.
Sitter, R. R. (1992). Robust designs for binary data. Biometrics, 1145-1155.
STUDENT 1927. Errors of routine analysis. Biometrika, 151-164.
Thompson, W. R. (1935). On a criterion for the rejection of observations and the distribution of the ratio of deviation to sample standard deviation. The Annals of Mathematical Statistics, 6(4), 214-219.
Wiens, D. P. (1990). Robust minimax designs for multiple linear regression. Linear algebra and its applications, 127, 327-340.
Wiens, D. P. (1992). Minimax designs for approximately linear regression. Journal of Statistical Planning and Inference, 31(3), 353-371.
Winlock, J. (1856). On Professor Airy's objections to Peirce's criterion. The Astronomical Journal, 4, 145-147.
Atkinson, A., Donev, A., & Tobias, R. (2007). Optimum experimental designs, with SAS (Vol. 34). Oxford University Press.
Atkinson, A. C., Fedorov, V. V., Herzberg, A. M., & Zhang, R. (2014). Elemental information matrices and optimal experimental design for generalized regression models. Journal of Statistical Planning and Inference, 144, 81-91.
Barnett, V., & Lewis, T. (1964). Outliers in statistical data, Chichester: John Wiley, 1995. 584p.
Bay, S. D., & Schwabacher, M. (2003, August). Mining distance-based outliers in near linear time with randomization and a simple pruning rule. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 29-38). ACM.
Beckman, R. J., & Cook, R. D. (1983). Outlier………. s. Technometrics, 25(2), 119-149.
Box, G. E., & Draper, N. R. (1975). Robust designs. Biometrika, 347-352.
Box, G. E., & Draper, N. R. (1987). Empirical model-building and response surfaces. John Wiley & Sons.
Box, G. E., & Hunter, J. S. (1957). Multi-factor experimental designs for exploring response surfaces. The Annals of Mathematical Statistics, 28(1), 195-241.
Chang, L. C., Jones, D. K., & Pierpaoli, C. (2005). RESTORE: robust estimation of tensors by outlier rejection. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, 53(5), 1088-1095.
Chatterjee, S., & Hadi, A. S. (1986). Influential observations, high leverage points, and outliers in linear regression. Statistical Science, 379-393.
Chauvenet, W. (1960). A Manual of Spherical and Practical Autonomy (Vol. II.
Cook, R. D., & Weisberg, S. (1980). Characterizations of an empirical influence function for detecting influential cases in regression. Technometrics, 22(4), 495-508.
Cook, R. D., & Weisberg, S. (1982). Residuals and influence in regression. New York: Chapman and Hall.
Cousineau, D., & Chartier, S. (2010). Outliers detection and treatment: a review. International Journal of Psychological Research, 3(1), 58-67.
Daniell, P. J. (1920). Observations weighted according to order. American Journal of Mathematics, 42(4), 222-236.
Dixon, W. J. (1950). Analysis of extreme values. The Annals of Mathematical Statistics, 21(4), 488-506.
Edgeworth, F. Y. (1887). Xli. on discordant observations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23(143), 364-375.
Eubank, R. L. (1984). The hat matrix for smoothing splines. Statistics & probability letters, 2(1), 9-14.
Festing, M. F., & Altman, D. G. (2002). Guidelines for the design and statistical analysis of experiments using laboratory animals. ILAR journal, 43(4), 244-258.
RA Fisher, M. A. (1922). On the mathematical foundations of theoretical statistics. Phil. Trans. R. Soc. Lond. A, 222(594-604), 309-368.
Gao, G., & Yang, J. (2015, June). Matrix Based Regression with Local Position-Patch and Nonlocal Similarity for Face Hallucination. In International Conference on Intelligent Science and Big Data Engineering (pp. 110-117). Springer, Cham.
Geramita, A. V., Geramita, J. M., & Wallis, J. S. (1976). Orthogonal desingns. Linear and Multilinear Algebra, 3(4), 281-306.
Glaisher, J. W. L. (1873). On the rejection of discordant observations. Monthly Notices of the Royal Astronomical Society, 33, 391-402.
GLAISHER, J. 1874. Note on a Paper by Mr. Stone; On the Rejection of Discordanr Observations. Monthly Notices of the Royal Astronomical Society, 24, 251.
Gould, B. A. (1855). On Peirce's Criterion for the Rejection of Doubtful Observations, with tables for facilitating its application. The Astronomical Journal, 4, 81-87.
Grubbs, F. E. (1950). Sample criteria for testing outlying observations. The Annals of Mathematical Statistics, 21(1), 27-58.
Heo, G., Schmuland, B., & Wiens, D. P. (2001). Restricted minimax robust designs for misspecified regression models. Canadian Journal of Statistics, 29(1), 117-128.
Hoaglin, D. C., & Welsch, R. E. (1978). The hat matrix in regression and ANOVA. The American Statistician, 32(1), 17-22.
Irwin, J. O. (1925). On a criterion for the rejection of outlying observations. Biometrika, 238-250.
Jauffret, C. (2007). Observability and Fisher information matrix in nonlinear regression. IEEE Transactions on Aerospace and Electronic Systems, 43(2), 756-759.
Kiefer, J. (1985). Collected Papers III: design of experiments. Springer-Verlag.
Knorr, E. M., Ng, R. T., & Tucakov, V. (2000). Distance-based outliers: algorithms and applications. The VLDB Journal—The International Journal on Very Large Data Bases, 8(3-4), 237-253.
Montgomery, D. C., & Myers, R. H. (1995). Response surface methodology: process and product optimization using designed experiments, Raymond H. Meyers, Douglas C. Montgomery, A Wiley-Interscience Publications.
Mukerjee, R., & Huda, S. (1985). Minimax second-and third-order designs to estimate the slope of a response surface. Biometrika, 72(1), 173-178.
Newcomb, S. (1886). A generalized theory of the combination of observations so as to obtain the best result. American journal of Mathematics, 343-366.
Peirce, B. (1852). Criterion for the rejection of doubtful observations. The Astronomical Journal, 2, 161-163.
Peirce, B. (1877, May). On Peirce's criterion. In Proceedings of the American Academy of Arts and Sciences (Vol. 13, pp. 348-351). American Academy of Arts & Sciences.
Rao, C. R., & Mitra, S. K. (1973). Theory and application of constrained inverse of matrices. SIAM Journal on Applied Mathematics, 24(4), 473-488.
Rao, C. R., & Rao, M. B. (1998). Matrix algebra and its applications to statistics and econometrics. World Scientific.
Rao, C. R., & Toutenburg, H. (1995). Linear models. In Linear models (pp. 3-18). Springer, New York, NY.
Saunder, S. A. (1903). Note on the use of Peirce's criterion for the rejection of doubtful observations. Monthly Notices of the Royal Astronomical Society, 63, 432-436.
Shannon, C. E. (1948). A mathematical theory of communication. Bell system technical journal, 27(3), 379-423.
Siddiqi, A. F. (2003). Outliers in Designed Experiments; Classical Robust & Resistant Methods. Journal of Statistics, 10, 49-65.
Siddiqi, A. F. (2008). Outlier robust designs. International Journal of Applied Mathematics and Statistics™, 13(M08), 64-77.
Siddiqi, A. F. (2010). Outlier Robust Draper & Lin Designs. Pakistan Journal of Statistics and Operation Research, 7(1), 87-99.
Sitter, R. R. (1992). Robust designs for binary data. Biometrics, 1145-1155.
STUDENT 1927. Errors of routine analysis. Biometrika, 151-164.
Thompson, W. R. (1935). On a criterion for the rejection of observations and the distribution of the ratio of deviation to sample standard deviation. The Annals of Mathematical Statistics, 6(4), 214-219.
Wiens, D. P. (1990). Robust minimax designs for multiple linear regression. Linear algebra and its applications, 127, 327-340.
Wiens, D. P. (1992). Minimax designs for approximately linear regression. Journal of Statistical Planning and Inference, 31(3), 353-371.
Winlock, J. (1856). On Professor Airy's objections to Peirce's criterion. The Astronomical Journal, 4, 145-147.
Published
2019-02-27
How to Cite
Shabbir, M., Siddiqi, A., Kassim, N., Mustafa, F., & Abbas, M. (2019). Analysis of exprimental designs with outliers. Amazonia Investiga, 8(18), 53-68. Retrieved from https://amazoniainvestiga.info/index.php/amazonia/article/view/258
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