Response surface methodology

<h2 id="basic-approach-of-response-surface-methodology">Basic approach of response surface methodology</h2>
<p>An easy way to estimate a first-degree polynomial model is to use a <a href="/facts/Factorial_experiment/lPlCA847">factorial experiment</a> or a <a href="/facts/Fractional_factorial_design/BeP1flsB">fractional factorial design</a>.  This is sufficient to determine which explanatory variables affect the response variable(s) of interest.  Once it is suspected that only significant explanatory variables are left, then a more complicated design, such as a <a href="/facts/Central_composite_design/aECtW6xG">central composite design</a> can be implemented to estimate a second-degree polynomial model, which is still only an approximation at best.  However, the second-degree model can be used to optimize (maximize, minimize, or attain a specific target for) the response variable(s) of interest.
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<h2 id="important-rsm-properties-and-features">Important RSM properties and features</h2>
Orthogonality
The property that allows individual effects of the k-factors to be estimated independently without (or with minimal) confounding. Also orthogonality provides minimum variance estimates of the model coefficient so that they are uncorrelated.
Rotatability
The property of rotating points of the design about the center of the factor space. The moments of the distribution of the design points are constant.
Uniformity
A third property of CCD designs used to control the number of center points is uniform precision (or Uniformity).
<h2 id="special-geometries">Special geometries</h2>
<h3>Cube</h3>
<p>Cubic designs are discussed by Kiefer, by Atkinson, Donev, and Tobias and by Hardin and Sloane.
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<h3>Sphere</h3>
<p><a href="/facts/Spherical_design/ETfwFLly">Spherical designs</a> are discussed by Kiefer and by Hardin and Sloane.
</p>
<h3>Simplex geometry and mixture experiments</h3>
<p>Mixture experiments are discussed in many books on the <a href="/facts/Design_of_experiments/VMrNftbr">design of experiments</a>, and in the response-surface methodology textbooks of Box and Draper and of Atkinson, Donev and Tobias. An extensive discussion and survey appears in the advanced textbook by John Cornell.
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<h2 id="extensions">Extensions</h2>
<h3>Multiple objective functions</h3>
<p class="note">See also: <a href="/facts/Multiobjective_optimization/1Hb4gjGJ">Multiobjective optimization</a> and <a href="/facts/Pareto_efficiency/nMTdVPqy">Pareto efficiency</a></p>
<p>Some extensions of response surface methodology deal with the multiple response problem.  Multiple response variables create difficulty because what is optimal for one response may not be optimal for other responses.  Other extensions are used to reduce variability in a single response while targeting a specific value, or attaining a near maximum or minimum while preventing variability in that response from getting too large.
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<h2 id="practical-concerns">Practical concerns</h2>
<p>Response surface methodology uses statistical models, and therefore practitioners need to be aware that even the best statistical model is an approximation to reality. In practice, both the models and the parameter values are unknown, and subject to uncertainty on top of ignorance. Of course, an estimated optimum point need not be optimum in reality, because of the errors of the estimates and of the inadequacies of the model.
</p><p>Nonetheless, response surface methodology has an effective track-record of helping researchers improve products and services: For example, Box's original response-surface modeling enabled chemical engineers to improve a process that had been stuck at a saddle-point for years. The engineers had not been able to afford to fit a cubic three-level design to estimate a quadratic model, and their <a href="/facts/Bias_(statistics)/Pql981K3">biased</a> linear-models estimated the gradient to be zero. Box's design reduced the costs of experimentation so that a quadratic model could be fit, which led to a (long-sought) ascent direction.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Box%E2%80%93Behnken_design/IBcMvLJq">Box–Behnken design</a></li>
<li><a href="/facts/Central_composite_design/aECtW6xG">Central composite design</a></li>
<li><a href="/facts/Gradient-enhanced_kriging/gT2nCkdY">Gradient-enhanced kriging</a> (GEK)</li>
<li><a href="/facts/IOSO/PjVhsptY">IOSO</a> method based on response-surface methodology</li>
<li><a href="/facts/Optimal_design/qCH9pPt5">Optimal designs</a></li>
<li><a href="/facts/Plackett%E2%80%93Burman_design/uHhn9RS8">Plackett–Burman design</a></li>
<li><a href="/facts/Polynomial_and_rational_function_modeling/eLvaPzIl">Polynomial and rational function modeling</a></li>
<li><a href="/facts/Polynomial_regression/5nL7tBoc">Polynomial regression</a></li>
<li><a href="/facts/Probabilistic_design/aHwxDAIn">Probabilistic design</a></li>
<li><a href="/facts/Surrogate_model/6uNjvd8B">Surrogate model</a></li>
<li><a href="/facts/Bayesian_Optimization/gF5LpjzB">Bayesian Optimization</a></li></ul>

<ul><li>Box, G.E.P.; Wilson, K.B. (1951). "On the Experimental Attainment of Optimum Conditions". <i>Journal of the Royal Statistical Society, Series B</i>. 13 (1): 1–45. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1111%2Fj.2517-6161.1951.tb00067.x">10.1111/j.2517-6161.1951.tb00067.x</a>.</li>
<li>Box, G. E. P. and Draper, Norman. 2007. <i>Response Surfaces, Mixtures, and Ridge Analyses</i>, Second Edition [of <i>Empirical Model-Building and Response Surfaces</i>, 1987], Wiley.</li>
<li>Atkinson, A.C.; Donev, A.N.; Tobias, R.D. (2007). <a href="https://books.google.com/books?id=oIHsrw6NBmoC"><i>Optimum Experimental Designs, with SAS</i></a>. Oxford University Press. pp. 511+xvi. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-929660-6.</li>
<li>Cornell, John (2002). <i>Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data</i> (third ed.). Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-07916-3.</li></ul>
<ul><li>Goos, Peter] (2002). <a href="http://users.telenet.be/peter.goos/springer.htm"><i>The Optimal Design of Blocked and Split-plot Experiments</i></a>. Lecture Notes in Statistics. Vol. 164. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95515-5.</li>
<li><a href="/facts/Jack_Kiefer_(mathematician)/WksfJ7By">Kiefer, Jack Carl</a> (1985). <a href="/facts/Lawrence_D._Brown/aEU0IRMG">L. D. Brown</a>; et al. (eds.). <a href="/facts/Jack_Kiefer_(mathematician)/WksfJ7By"><i>Jack Carl Kiefer Collected Papers III Design of Experiments</i></a>. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-96004-3.</li>
<li>Pukelsheim, Friedrich (2006). <a href="https://books.google.com/books?id=5ZcfDZUJ4F8C"><i>Optimal Design of Experiments</i></a>. <a href="/facts/Society_for_Industrial_and_Applied_Mathematics/2HY4T5WY">SIAM</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-89871-604-7.</li>
<li>Hardin, R.H.; <a href="/facts/Neil_Sloane/kLWw29Me">Sloane, N.J.A.</a> (1993). <a href="http://neilsloane.com/doc/Me175.pdf">"A New Approach to the Construction of Optimal Designs"</a> (PDF). <i>Journal of Statistical Planning and Inference</i>. 37 (3): 339–369. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0378-3758%2893%2990112-J">10.1016/0378-3758(93)90112-J</a>.</li>
<li>Hardin, R.H.; Sloane, N.J.A. <a href="http://neilsloane.com/doc/doeh.pdf">"Computer-Generated Minimal (and Larger) Response Surface Designs: (I) The Sphere"</a> (PDF).</li>
<li>Hardin, R.H.; Sloane, N.J.A. <a href="http://neilsloane.com/doc/meatball.pdf">"Computer-Generated Minimal (and Larger) Response Surface Designs: (II) The Cube"</a> (PDF).</li>
<li>Ghosh, S.; <a href="/facts/Calyampudi_Radhakrishna_Rao/SxnbA5YV">Rao, C. R.</a>, eds. (1996). <i>Design and Analysis of Experiments</i>. Handbook of Statistics. Vol. 13. North-Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-444-82061-7.
<ul><li>Draper, Norman; Lin, Dennis K.J. (1996). <a href="https://www.sciencedirect.com/science/article/pii/S0169716196130133">"11 Response surface designs"</a>. <i>Response surface designs</i>. Handbook of Statistics. Vol. 13. pp. 343–375. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0169-7161%2896%2913013-3">10.1016/S0169-7161(96)13013-3</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780444820617.</li>
<li>Gaffke, Norbert; Heiligers, Berthold (1996). <a href="https://www.sciencedirect.com/science/article/pii/S0169716196130327">"30 Approximate designs for polynomial regression: Invariance, admissibility, and optimality"</a>. <i>Approximate designs for polynomial regression: Invariance, admissibility, and optimality</i>. Handbook of Statistics. Vol. 13. pp. 1149–99. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0169-7161%2896%2913032-7">10.1016/S0169-7161(96)13032-7</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780444820617.</li></ul></li></ul>
<h3>Historical</h3>
<ul><li><a href="/facts/Joseph_Diaz_Gergonne/B8ggHgVo">Gergonne, J. D.</a> (1974) [1815]. <a href="https://doi.org/10.1016%2F0315-0860%2874%2990034-2">"The application of the method of least squares to the interpolation of sequences"</a>. <i>Historia Mathematica</i>. 1 (4) (Translated by Ralph St. John and <a href="/facts/Stephen_M._Stigler/VyCKqbS6">S. M. Stigler</a> from the 1815 French ed.): 439–447. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0315-0860%2874%2990034-2">10.1016/0315-0860(74)90034-2</a>.</li>
<li><a href="/facts/Stephen_M._Stigler/VyCKqbS6">Stigler, Stephen M.</a> (1974). <a href="https://doi.org/10.1016%2F0315-0860%2874%2990033-0">"Gergonne's 1815 paper on the design and analysis of polynomial regression experiments"</a>. <i>Historia Mathematica</i>. 1 (4): 431–9. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0315-0860%2874%2990033-0">10.1016/0315-0860(74)90033-0</a>.</li></ul>
<ul><li><a href="/facts/Charles_Sanders_Peirce/z76dNqhM">Peirce, C. S.</a> (1876). <a href="http://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0025.PDF#page=222">"Note on the Theory of the Economy of Research"</a> (PDF). <i>Coast Survey Report</i>. Appendix No. 14: 197–201.
<ul><li>Reprinted in <a href="/facts/Charles_Sanders_Peirce_bibliography/HqaDGjpW"><i>Collected Papers of Charles Sanders Peirce</i></a>. Vol. 7. 1958. paragraphs 139–157,</li>
<li>and in Peirce, C. S. (July–August 1967). "Note on the Theory of the Economy of Research". <i>Operations Research</i>. 15 (4): 643–8. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1287%2Fopre.15.4.643">10.1287/opre.15.4.643</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/168276">168276</a>.</li></ul></li>
<li>Smith, Kirstine (1918). <a href="https://zenodo.org/record/1431591">"On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance They Give Towards a Proper Choice of the Distribution of the Observations"</a>. <i>Biometrika</i>. 12 (1/2): 1–85. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2331929">10.2307/2331929</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2331929">2331929</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.itl.nist.gov/div898/handbook/pri/section3/pri336.htm">Response surface designs</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Karmoker, J.R.; Hasan, I.; Ahmed, N.; Saifuddin, M.; Reza, M.S. (2019). "Development and Optimization of Acyclovir Loaded Mucoadhesive Microspheres by Box -Behnken Design". Dhaka University Journal of Pharmaceutical Sciences. 18 (1): 1–12. doi:10.3329/dujps.v18i1.41421. <a href="https://doi.org/10.3329%2Fdujps.v18i1.41421" target="_blank">https://doi.org/10.3329%2Fdujps.v18i1.41421</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Asadi, Nooshin; Zilouei, Hamid (March 2017). "Optimization of organosolv pretreatment of rice straw for enhanced biohydrogen production using Enterobacter aerogenes". Bioresource Technology. 227: 335–344. Bibcode:2017BiTec.227..335A. doi:10.1016/j.biortech.2016.12.073. PMID 28042989. <a href="https://www.researchgate.net/publication/311881656" target="_blank">https://www.researchgate.net/publication/311881656</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Box & Wilson 1951 - Box, G.E.P.; Wilson, K.B. (1951). "On the Experimental Attainment of Optimum Conditions". Journal of the Royal Statistical Society, Series B. 13 (1): 1–45. doi:10.1111/j.2517-6161.1951.tb00067.x. <a href="https://doi.org/10.1111%2Fj.2517-6161.1951.tb00067.x" target="_blank">https://doi.org/10.1111%2Fj.2517-6161.1951.tb00067.x</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Improving Almost Anything: Ideas and Essays, Revised Edition (Wiley Series in Probability and Statistics) George E. P. Box <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Response surface methodology open-in-new

Response surface methodology