{ "id": "ept1dq5fz2", "slug": "central-composite-design", "title": "Central composite design", "updated": "2026-10-21", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Non-geographic · 1946 to 2000: Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" } ] } ], "excerpt": "The central composite design (CCD), also called the Box–Wilson design, is a response surface design of experiments, introduced by Box and Wilson in 1951 and widely used to fit quadratic models.", "snippet": "The central composite design (CCD), also called the Box–Wilson design, is a response surface design of experiments, introduced by Box and Wilson in 1951 and widely used to fit quadratic models.", "node": "physical.math.statistics.stats_statistical_inference", "markdown": "# Central composite design\n\nA central composite design (CCD) is a [response surface design](https://www.edgepedia.org/response-surface-design) of experiments that combines factorial points, axial (star) points, and center points to fit a second-order (quadratic) model efficiently. Also called the Box–Wilson design, it has been described as the most popular response surface method (RSM) design[1](https://www.statease.com/docs/v22.0/contents/response-surface-designs/central-composite-design/) and has been described as perhaps the most popular class of second-order designs since its introduction by Box and Wilson in 1951.[2](https://link.springer.com/chapter/10.1007/978-3-7908-2064-5_17) For a model with n factors, a CCD provides enough runs to estimate the \\( (n+2)(n+1)/2 \\) coefficients of a full quadratic model[3](https://www.mathworks.com/help/stats/response-surface-designs.html), avoiding the much larger three-level factorial experiment that would otherwise be needed.[4](https://akjournals.com/view/journals/066/52/4/article-p521.xml)\n\n| Key fact | Detail |\n|---|---|\n| Purpose | Fitting full quadratic response surface models with \\( (n+2)(n+1)/2 \\) coefficients[3](https://www.mathworks.com/help/stats/response-surface-designs.html) |\n| Structure | \\( 2^k \\) (or fractional) factorial points, \\( 2k \\) axial points at \\( \\pm\\alpha \\), and center points[5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1) |\n| Total runs | \\( n_F + 2k + n_0 \\), where \\( n_F \\) is the number of factorial or fractional-factorial runs (\\( n_F = 2^k \\) for a full factorial) and \\( n_0 \\) is the number of center runs[6](https://2024.help.altair.com/2024.1/hwdesktop/hst/topics/design_exploration/method_central_composite_design_r.htm) |\n| Rotatable axial distance | \\( \\alpha = (n_F)^{1/4} \\), e.g., 1.414 (k = 2), 1.682 (k = 3), 2.000 (k = 4)[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm) |\n| Center points | Usually 4–6 replicated runs for pure error and uniform precision[1](https://www.statease.com/docs/v22.0/contents/response-surface-designs/central-composite-design/) |\n| Main varieties | Circumscribed (CCC), inscribed (CCI), face-centered (CCF, \\( \\alpha = 1 \\))[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm) |\n| Origin | Box and Wilson, Journal of the Royal Statistical Society Series B, 1951[8](https://doi.org/10.1111/j.2517-6161.1951.tb00067.x) |\n\n## How it works\n\nThe design augments a two-level factorial or fractional factorial array (with center points) with a group of star points that allow estimation of curvature.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm) Each part of the design has a distinct role. Factorial points, coded −1 and +1, estimate main effects and two-factor interactions.[9](https://www.jmp.com/en/statistics-knowledge-portal/design-of-experiments/response-surface-methodology/central-composite-design) Axial points, coded \\( -\\alpha \\) and \\( +\\alpha \\) on each factor axis, estimate the quadratic effects[9](https://www.jmp.com/en/statistics-knowledge-portal/design-of-experiments/response-surface-methodology/central-composite-design); a design with k factors always has 2k star points.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm) Center points, coded 0, detect curvature, increase precision, and test lack of fit.[9](https://www.jmp.com/en/statistics-knowledge-portal/design-of-experiments/response-surface-methodology/central-composite-design)\n\nThe design is built for the quadratic model only: it estimates linear, quadratic, and first-order interaction terms, but cannot estimate cubic terms or higher-order interactions[5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1), and its matrix provides too few unique points to determine all cubic-model terms.[10](https://www.statease.com/docs/v23.0/tutorials/multifactor-rsm/)\n\n## How it is done\n\nThe practitioner chooses the factorial portion (full \\( 2^k \\), or a half or quarter fraction of at least resolution V[5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1) • [11](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/305596ec-45cd-4a68-9946-91e2aada437d/content)), the axial distance α, and the number of center runs. Total runs follow \\( n_F + 2k + n_0 \\), where \\( n_F \\) is the number of factorial or fractional-factorial runs (\\( n_F = 2^k \\) for a full factorial).[6](https://2024.help.altair.com/2024.1/hwdesktop/hst/topics/design_exploration/method_central_composite_design_r.htm)\n\n**Choosing α.** Three basic choices exist[11](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/305596ec-45cd-4a68-9946-91e2aada437d/content):\n\n- Rotatable: \\( \\alpha = (n_F)^{1/4} \\), the fourth root of the number of factorial points.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm) • [12](https://cdn.standards.iteh.ai/samples/53479/9e4eeebf9f71434d9a4b5429c3e91e59/ISO-TR-13195-2015.pdf) Values are 1.414 for \\( k = 2 \\), 1.682 for \\( k = 3 \\), 2.000 for \\( k = 4 \\), and 2.378 for \\( k = 5 \\).[13](https://pydoe.github.io/pydoe/theory/choosing-design/central-composite/)\n- Spherical: \\( \\alpha = \\sqrt{k} \\), a near-rotatable choice.[5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1) • [14](https://www.qualitydigest.com/static/magazine/june01/html/doe.html)\n- Face-centered: \\( \\alpha = 1 \\), placing star points at the center of each face of the factorial space, requiring only 3 levels per factor.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm)\n\nThe rotatable α is most D-efficient for spherical regions of interest, while \\( \\alpha = 1 \\) is most D-efficient for cuboidal regions.[11](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/305596ec-45cd-4a68-9946-91e2aada437d/content) Under some circumstances α can be chosen so the design is simultaneously rotatable and orthogonally blocked (for example, \\( k = 2 \\) with \\( \\alpha = 1.414 \\)).[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm)\n\n**Center points.** Center points are usually repeated 4–6 times to obtain a good estimate of pure experimental error.[1](https://www.statease.com/docs/v22.0/contents/response-surface-designs/central-composite-design/) Choosing five to six center runs makes the prediction variance at the middle of the design approximately the same as at the edge, the uniform-precision property.[5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1) A three-factor CCD with uniform precision uses 20 runs: 8 factorial, 6 axial, and 6 center points.[9](https://www.jmp.com/en/statistics-knowledge-portal/design-of-experiments/response-surface-methodology/central-composite-design)\n\n**Sequential use.** A CCD can be run sequentially: the first subset of points estimates linear and two-factor interaction effects, and the second subset estimates curvature effects.[14](https://www.qualitydigest.com/static/magazine/june01/html/doe.html) In practice, one runs the factorial design with center points, tests for lack of fit, and adds the axial points only if the first-order model proves inadequate.[9](https://www.jmp.com/en/statistics-knowledge-portal/design-of-experiments/response-surface-methodology/central-composite-design) • [5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1)\n\n**Run counts and analysis.** Run counts grow as \\( 2^k + 2k + n_0 \\): 13 runs for \\( k = 2 \\), 20 for \\( k = 3 \\) (with 6 center points), 30 for \\( k = 4 \\), and 52 (full factorial) for \\( k = 5 \\).[15](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3363.htm) A three-level factorial needs 9, 27, 81, and 243 runs for \\( k = 2, 3, 4, \\) and \\( 5 \\).[5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1) Analysis uses least squares to fit candidate models in a hierarchy (linear, two-factor interaction, quadratic), ANOVA to assess terms, and lack-of-fit tests that compare residual error with pure error from replicated design points; contour and 3D response surface plots then locate optima.[10](https://www.statease.com/docs/v23.0/tutorials/multifactor-rsm/) Software implementations include MATLAB's ccdesign function[3](https://www.mathworks.com/help/stats/response-surface-designs.html), Stat-Ease[1](https://www.statease.com/docs/v22.0/contents/response-surface-designs/central-composite-design/), JMP[9](https://www.jmp.com/en/statistics-knowledge-portal/design-of-experiments/response-surface-methodology/central-composite-design), R's DoE.wrapper[16](https://search.r-project.org/CRAN/refmans/DoE.wrapper/html/CentralCompositeDesigns.html), and PyDOE.[13](https://pydoe.github.io/pydoe/theory/choosing-design/central-composite/)\n\n## Origin\n\nThe design was reported by G. E. P. Box and K. B. Wilson, both of [Imperial Chemical Industries](https://www.edgepedia.org/imperial-chemical-industries), in \"On the Experimental Attainment of Optimum Conditions,\" Journal of the Royal Statistical Society Series B, Volume 13, Issue 1, pages 1–38, published January 1951.[8](https://doi.org/10.1111/j.2517-6161.1951.tb00067.x) This paper introduced composite designs, adding a star portion to a two-level factorial array to allow efficient estimation of quadratic terms in the second-order model.[17](https://www.stat.cmu.edu/technometrics/80-89/VOL-31-02/v3102137.pdf) G. E. P. Box and J. S. Hunter's 1957 paper \"Multi-Factor Experimental Designs for Exploring Response Surfaces\" (The Annals of Mathematical Statistics, 28(1), 195–241) introduced rotatability, the uniform-precision recommendation for center runs, and orthogonal blocking conditions for composite designs.[18](https://doi.org/10.1214/aoms/1177707047) G. E. P. Box and Norman R. Draper's 1959 JASA paper formalized design selection by minimizing J, the expected mean squared error over the region of interest, split into variance and bias components.[19](https://doi.org/10.1080/01621459.1959.10501525) R. J. Hader and Sung H. Park later proposed slope-rotatable central composite designs (Technometrics, 1978).[20](https://doi.org/10.1080/00401706.1978.10489695)\n\n## Variants\n\nThree varieties are distinguished[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm):\n\n- **Circumscribed (CCC)**, the original form, with star points outside the factorial cube; it uses 5 levels per factor and explores the largest process space.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm)\n- **Inscribed (CCI)**, a scaled-down CCC that uses the specified factor limits as star points; it uses only points within the original factor ranges.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm)\n- **Face-centered (CCF)**, with \\( \\alpha = 1 \\), so the star points have coordinates \\( \\pm 1 \\) on each factor axis, and 3 levels per factor; it is not rotatable.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm)\n\nCCC and CCI are rotatable; CCF is not.[7](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm) The factorial part can also be reduced with regular fractions of resolution III*, giving small CCDs that are sometimes saturated and serve as an alternative when minimizing runs is the priority.[21](https://hal.science/hal-04438106/document) CCDs can also be arranged in orthogonal blocks.[5](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1)\n\n## Applications\n\nCCDs are used across industrial and scientific process optimization. In pharmacy, the CCD is the most commonly used design in response surface modeling, allowing quick estimation of first-order and second-order terms[22](https://www.intechopen.com/chapters/74955); CCD and Box–Behnken are the two most widely applied RSM methods for optimizing self-emulsifying drug delivery system formulations.[23](https://www.mdpi.com/2673-3501/6/1/4) A 2024 study used a CCD to optimize solid self-nanoemulsifying drug delivery systems of quetiapine fumarate via hot-melt extrusion.[24](https://doi.org/10.3390/pharmaceutics16030324)\n\n## Limitations and alternatives\n\nThe CCC requires factor settings outside the range of the factorial part, which can be unsafe or nonallowable; the CCI avoids this but, for \\( k = 3 \\), prediction error at the excluded extreme points is 36% higher (95% prediction interval ±1.074 vs ±0.788 for face-centered).[15](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3363.htm) • [14](https://www.qualitydigest.com/static/magazine/june01/html/doe.html) Practitioner guidance is to use an inscribed design when two or more factor extremes are nonallowable, and a face-centered design when the operability region encompasses the region of interest or only one extreme is nonallowable.[14](https://www.qualitydigest.com/static/magazine/june01/html/doe.html) The face-centered design's weakness is poor accuracy in estimating pure quadratic coefficients, because the axial points lie in the planes of the factorial points.[15](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3363.htm) • [4](https://akjournals.com/view/journals/066/52/4/article-p521.xml) The CCD also cannot estimate individual linear-by-quadratic or quadratic-by-quadratic interaction terms[22](https://www.intechopen.com/chapters/74955), and its run count increases substantially with additional factors because it includes both axial and factorial points.[23](https://www.mdpi.com/2673-3501/6/1/4) The Box–Behnken design avoids cube corners, so measurements at combined factor extremes are unnecessary[4](https://akjournals.com/view/journals/066/52/4/article-p521.xml), but the CCD gives better information within or beyond process limits and can be built in two steps from the \\( 2^k \\) design.[4](https://akjournals.com/view/journals/066/52/4/article-p521.xml) A 2024 comparison of the circumscribed, inscribed, and face-centered designs for \\( k = 4, \\) and \\( 5 \\) found the circumscribed design more efficient under D- and A-optimality when center points are replicated, and both CCCD and CCID superior to the face-centered design on G-optimality over reduced-model subsets; the study recommends replicating center runs to enable lack-of-fit testing.[25](https://doi.org/10.48048/tis.2024.8193)\n\n## References\n\n1. [Stat-Ease v22.0, Central Composite Design](https://www.statease.com/docs/v22.0/contents/response-surface-designs/central-composite-design/)\n2. [Optimal Central Composite Designs for Fitting Second Order Response Surface Linear Regression Models (Park, Kim & Cho, 2008)](https://link.springer.com/chapter/10.1007/978-3-7908-2064-5_17)\n3. [Response Surface Designs, MATLAB & Simulink (MathWorks)](https://www.mathworks.com/help/stats/response-surface-designs.html)\n4. [Methods for experimental design, central composite design and the Box–Behnken design, to optimise operational parameters: A review (Acta Alimentaria 52(4), 2023)](https://akjournals.com/view/journals/066/52/4/article-p521.xml)\n5. [11.2 - Response Surface Designs, Penn State STAT 503](https://online.stat.psu.edu/stat503/lesson/11/11.2/11.2.1)\n6. [Central Composite Design (CCD), Altair HyperStudy 2024 documentation](https://2024.help.altair.com/2024.1/hwdesktop/hst/topics/design_exploration/method_central_composite_design_r.htm)\n7. [5.3.3.6.1. Central Composite Designs (CCD), NIST/SEMATECH e-Handbook](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm)\n8. [G. E. P. Box, K. B. Wilson (1951). On the Experimental Attainment of Optimum Conditions. Journal of the Royal Statistical Society Series B (Statistical Methodology).](https://doi.org/10.1111/j.2517-6161.1951.tb00067.x)\n9. [Central Composite Design, JMP Statistics Knowledge Portal](https://www.jmp.com/en/statistics-knowledge-portal/design-of-experiments/response-surface-methodology/central-composite-design)\n10. [Stat-Ease v23.0, Response Surface Tutorial](https://www.statease.com/docs/v23.0/tutorials/multifactor-rsm/)\n11. [Recommendations for Design Parameters for Central Composite Designs with Restricted Randomization (dissertation, Virginia Tech)](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/305596ec-45cd-4a68-9946-91e2aada437d/content)\n12. [ISO/TR 13195:2015, Selected illustrations of response surface method, Central composite design](https://cdn.standards.iteh.ai/samples/53479/9e4eeebf9f71434d9a4b5429c3e91e59/ISO-TR-13195-2015.pdf)\n13. [Central Composite Designs (CCD), PyDOE documentation](https://pydoe.github.io/pydoe/theory/choosing-design/central-composite/)\n14. [Selecting the Right Central Composite Design (Quality Digest, June 2001)](https://www.qualitydigest.com/static/magazine/june01/html/doe.html)\n15. [5.3.3.6.3. Comparisons of response surface designs, NIST/SEMATECH e-Handbook](https://www.itl.nist.gov/div898/handbook/pri/section3/pri3363.htm)\n16. [R DoE.wrapper documentation: Statistical background of central composite designs](https://search.r-project.org/CRAN/refmans/DoE.wrapper/html/CentralCompositeDesigns.html)\n17. [Response Surface Methodology: 1966-1988 (Myers et al., Technometrics, May 1989, Vol. 31, No. 2)](https://www.stat.cmu.edu/technometrics/80-89/VOL-31-02/v3102137.pdf)\n18. [G. E. P. Box, J. S. Hunter (1957). Multi-Factor Experimental Designs for Exploring Response Surfaces. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177707047)\n19. [G. E. P. Box, Norman R. Draper (1959). A Basis for the Selection of a Response Surface Design. Journal of the American Statistical Association.](https://doi.org/10.1080/01621459.1959.10501525)\n20. [R. J. Hader, Sung H. Park (1978). Slope-Rotatable Central Composite Designs. Technometrics.](https://doi.org/10.1080/00401706.1978.10489695)\n21. [Tinsson (2024), Analysis of small central composite designs](https://hal.science/hal-04438106/document)\n22. [Central Composite Design for Response Surface Methodology and Its Application in Pharmacy (IntechOpen)](https://www.intechopen.com/chapters/74955)\n23. [The Use of Design of Experiments (DoE) Approaches for the Development of Self-Emulsifying Drug Delivery Systems (SEDDS)](https://www.mdpi.com/2673-3501/6/1/4)\n24. [Prateek Uttreja and colleagues (2024). Formulation Development of Solid Self-Nanoemulsifying Drug Delivery Systems of Quetiapine Fumarate via Hot-Melt Extrusion Technology: Optimization Using Central Composite Design. Pharmaceutics.](https://doi.org/10.3390/pharmaceutics16030324)\n25. [Chawanee Suphirat, Wasinee Pradubsri (2024). Comparison of the Three Types of Central Composite Designs Over Subsets of Reduced Models by Design Optimality Criteria. Trends in Sciences.](https://doi.org/10.48048/tis.2024.8193)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026; Oct 21, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/central-composite-design", "markdown_url": "https://www.edgepedia.org/central-composite-design.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Central composite design\", Edgepedia (EdgeChat), https://www.edgepedia.org/central-composite-design. Edgepedia Community License 1.0.", "credit_md": "\"[Central composite design](https://www.edgepedia.org/central-composite-design)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/central-composite-design](https://www.edgepedia.org/central-composite-design). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Central composite design\", Edgepedia (EdgeChat), https://www.edgepedia.org/central-composite-design. Edgepedia Community License 1.0.", "speakable": "The central composite design, also called the Box–Wilson design, is a response surface design of experiments, introduced by Box and Wilson in 1951 and widely used to fit quadratic models." }