Factorial designs

A design is called factorial when its set of potential treatments consists of all combinations of the levels of two or more treatment factors. For example, there might be six fertilizer treatments, consisting of three different quantities each applied early or late.

If there is also a zero quantity, then it does not make sense to say that this is applied early or late, so it makes a seventh treatment, often called control. The designs discussed on this page do not include such control treatments.

The number of potential treatments in a factorial experiment can be very large. Confounding is a technique for dividing them into blocks, or into rows and columns, etc., in such a way that the most important treatment effects can be estimated orthogonally to blocks.

Partial confounding occurs when the treatment subspace cannot be decomposed orthogonally in such a way that each subspace is in one of the strata defined by the block structure.

Even more extreme are fractional factorial designs, in which only a carefully chosen subset of the potential treatments is used. The fraction is regular if the treatment effects are divided into collections which are aliased with each other: this means that the sum of the effects in a collection can be estimated but the individual effects cannot be. It is hoped that at most one effect in each collection is non-zero.

Sometimes practical constraints mean that, in each block, only a subset of the levels of each factor can be used, but all combinations of such levels can be used in that block. Such designs are called multi-part designs.

Topics Some of my publications

General background
R. A. Bailey: Confounding. In Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson), J. Wiley, New York, Volume 2, 1982, pp. 128–134. doi: 10.1002/0471667196.ess0401.pub2
R. A. Bailey: Interaction. In Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson), J. Wiley, New York, Volume 4, 1983, pp. 176–181. doi: 10.1002/0471667196.ess1255.pub2
R. A. Bailey: Inference from randomized (factorial) experiments. Statistical Science, 32 (2017), 352–355. doi: 10.1214/16-STS600

How can we identify confounding and aliasing in a given factorial design?
R. A. Bailey, F. H. L. Gilchrist and H. D. Patterson: Identification of effects and confounding patterns in factorial designs. Biometrika 64 (1977), 347–354. doi: 10.1093/biomet/64.2.347
R. A. Bailey: Patterns of confounding in factorial designs. Biometrika 64 (1977), 597–603. doi: 10.1093/biomet/64.3.597 [Maths Reviews 0501643 (58 #18945)]
H. D. Patterson and R. A. Bailey: Design keys for factorial experiments. Applied Statistics 27 (1978), 335–343. doi: 10.2307/2347170
R. A. Bailey: Design keys for multiphase experiments. In mODa 11—Advances in Model-Oriented Design and Analysis (eds. Joachim Kunert, Christine H. Müller and Anthony C. Atkinson), Springer International Publishing, 2016, pp. 27–35. doi: 10.1007/978-3-319-31266-8_4

How can we construct factorial designs so as to achieve specified confounding of factorial treatment effects in specified strata, or so that designated treatment effects are estimable?
M. F. Franklin and R. A. Bailey: Selection of defining contrasts and confounded effects in two-level experiments. Applied Statistics 26 (1977), 321–326. doi: 10.2307/2346973
H. D. Patterson and R. A. Bailey: Design keys for factorial experiments. Applied Statistics 27 (1978), 335–343. doi: 10.2307/2347170
R. A. Bailey: Design keys for multiphase experiments. In mODa 11—Advances in Model-Oriented Design and Analysis (eds. Joachim Kunert, Christine H. Müller and Anthony C. Atkinson), Springer International Publishing, 2016, pp. 27–35. doi: 10.1007/978-3-319-31266-8_4
André Kobilinsky, Hervé Monod and R. A. Bailey: Automatic generation of generalised regular factorial designs. Computational Statistics and Data Analysis, 113 (2017), 311–329. doi: 10.1016/j.csda.2016.09.003

How can we use character theory of Abelian groups to identify confounding and aliasing in factorial designs, or to construct designs with specified confounding and aliasing?
R. A. Bailey: Patterns of confounding in factorial designs. Biometrika 64 (1977), 597–603. doi: 10.1093/biomet/64.3.597 [Maths Reviews 0501643 (58 #18945)]
R. A. Bailey: Dual Abelian groups in the design of experiments. In Algebraic Structures and Applications (eds. P. Schultz, C. E. Praeger and R. P. Sullivan), Marcel Dekker, New York, 1982, pp. 45–54. [Maths Reviews 0647165 (83m: 62135)]
R. A. Bailey: Factorial design and Abelian groups. Linear Algebra and its Applications 70 (1985), 349–368. doi: 10.1016/0024-3795(85)90064-3 [Maths Reviews 0808552 (87c: 62151)]

In non-orthogonal designs, when does the partial confounding correspond to the factorial decomposition?
R. A. Bailey: Balance, orthogonality and efficiency factors in factorial design. Journal of the Royal Statistical Society, Series B 47 (1985), 453–458. doi: 10.1111/j.2517-6161.1985.tb01374.x [Maths Reviews 0844475 (87k: 62124)]
R. A. Bailey and C. A. Rowley: General balance and treatment permutations. Linear Algebra and its Applications 127 (1990), 183–225. doi: 10.1016/0024-3795(90)90343-B [Maths Reviews 1048802 (91d: 05014)]
R. A. Bailey: Cyclic designs and factorial designs. In Probability, Statistics and Design of Experiments (proceedings of the R. C. Bose Symposium on Probability, Statistics and Design of Experiments, Delhi, 27–30 December, 1988) (ed. R. R. Bahadur), Wiley Eastern, New Delhi, 1990, pp. 51–74.
H. Monod and R. A. Bailey: Pseudofactors: normal use to improve design and facilitate analysis. Applied Statistics 41 (1992), 317–336. doi: 10.2307/2347564
C. J. Brien and R. A. Bailey: Decomposition tables for experiments. I. A chain of randomizations. Annals of Statistics 37 (2009), 4184–4213. doi: 10.1214/09-AOS717 [Maths Reviews 2572457 (2010k: 62294)]
R. A. Bailey: Symmetric factorial designs in blocks. Journal of Statistical Theory and Practice 5 (2011), 13–24. doi: 10.1080/15598608.2011.10412047 [Maths Reviews 2829819]

How do the above methods apply if some treatment factors are quantitative?
R. A. Bailey: The decomposition of treatment degrees of freedom in quantitative factorial experiments. Journal of the Royal Statistical Society, Series B 44 (1982), 63–70. doi: 10.1111/j.2517-6161.1982.tb01188.x [Maths Reviews 0655375 (83e: 62103) and 0721758 (85d: 62076)]
R. A. Bailey: Cyclic designs and factorial designs. In Probability, Statistics and Design of Experiments (proceedings of the R. C. Bose Symposium on Probability, Statistics and Design of Experiments, Delhi, 27–30 December, 1988) (ed. R. R. Bahadur), Wiley Eastern, New Delhi, 1990, pp. 51–74.
H. Monod and R. A. Bailey: Pseudofactors: normal use to improve design and facilitate analysis. Applied Statistics 41 (1992), 317–336. doi: 10.2307/2347564

If the treatments are applied and investigated consecutively, how can we ensure that important factorial effects are orthogonal to a low-order polynomial in the time-trend?
R. A. Bailey, Ching-Shui Cheng and Patricia Kipnis: Construction of trend-resistant factorial designs. Statistica Sinica 2 (1992), 393–411. url: http://www.jstor.org/stable/24304867 [Maths Reviews 1187950 (93i: 62197)]

Regular fractions
Ulrike Grömping and Rosemary A. Bailey: Regular fractions of factorial arrays. In mODa 11—Advances in Model-Oriented Design and Analysis (eds. Joachim Kunert, Christine H. Müller and Anthony C. Atkinson), Springer International Publishing, 2016, pp. 143–151. doi: 10.1007/978-3-319-31266-8_17

Irregular fractions
R. A. Bailey: A note on loosely balanced incomplete block designs. Computational Statistics and Data Analysis 3 (1985), 115–117 and 121–122. doi: 10.1016/0167-9473(85)90068-4 and 10.1016/0167-9473(85)90070-2
R. A. Bailey: Contribution to the discussion of `Detection of interactions in experiments on large numbers of factors' by S. M. Lewis and A. M. Dean. Journal of the Royal Statistical Society, Series B 63 (2001), 662.
R. A. Bailey: Contribution to the discussion of `Optimum design of experiments for statistical inference' by Steven G. Gilmour and Luzia A. Trinca. Applied Statistics 61 (2012), 374–375.

Multi-part designs
R. A. Bailey and Peter J. Cameron: Multi-part balanced incomplete-block designs. Statistical Papers, 60 (2019), 55–76. doi: 10.1007/s00362-018-01071-x [Maths Reviews 3969037] Appendage: doi: 10.1007/s00362-019-01137-4 [Maths Reviews 4262205]

Designs for estimating variance components R. A. Bailey, Célia Fernandes and Paulo Ramos: Sparse designs for estimating variance components of nested random factors. Journal of Statistical Planning and Inference, 214 (2021), 76–88. doi: 10.1016/j.jspi.2021.01.002

Topics	Some of my publications
General background	R. A. Bailey: Confounding. In Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson), J. Wiley, New York, Volume 2, 1982, pp. 128–134. doi: 10.1002/0471667196.ess0401.pub2 R. A. Bailey: Interaction. In Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson), J. Wiley, New York, Volume 4, 1983, pp. 176–181. doi: 10.1002/0471667196.ess1255.pub2 R. A. Bailey: Inference from randomized (factorial) experiments. Statistical Science, 32 (2017), 352–355. doi: 10.1214/16-STS600
How can we identify confounding and aliasing in a given factorial design?	R. A. Bailey, F. H. L. Gilchrist and H. D. Patterson: Identification of effects and confounding patterns in factorial designs. Biometrika 64 (1977), 347–354. doi: 10.1093/biomet/64.2.347 R. A. Bailey: Patterns of confounding in factorial designs. Biometrika 64 (1977), 597–603. doi: 10.1093/biomet/64.3.597 [Maths Reviews 0501643 (58 #18945)] H. D. Patterson and R. A. Bailey: Design keys for factorial experiments. Applied Statistics 27 (1978), 335–343. doi: 10.2307/2347170 R. A. Bailey: Design keys for multiphase experiments. In mODa 11—Advances in Model-Oriented Design and Analysis (eds. Joachim Kunert, Christine H. Müller and Anthony C. Atkinson), Springer International Publishing, 2016, pp. 27–35. doi: 10.1007/978-3-319-31266-8_4
How can we construct factorial designs so as to achieve specified confounding of factorial treatment effects in specified strata, or so that designated treatment effects are estimable?	M. F. Franklin and R. A. Bailey: Selection of defining contrasts and confounded effects in two-level experiments. Applied Statistics 26 (1977), 321–326. doi: 10.2307/2346973 H. D. Patterson and R. A. Bailey: Design keys for factorial experiments. Applied Statistics 27 (1978), 335–343. doi: 10.2307/2347170 R. A. Bailey: Design keys for multiphase experiments. In mODa 11—Advances in Model-Oriented Design and Analysis (eds. Joachim Kunert, Christine H. Müller and Anthony C. Atkinson), Springer International Publishing, 2016, pp. 27–35. doi: 10.1007/978-3-319-31266-8_4 André Kobilinsky, Hervé Monod and R. A. Bailey: Automatic generation of generalised regular factorial designs. Computational Statistics and Data Analysis, 113 (2017), 311–329. doi: 10.1016/j.csda.2016.09.003
How can we use character theory of Abelian groups to identify confounding and aliasing in factorial designs, or to construct designs with specified confounding and aliasing?	R. A. Bailey: Patterns of confounding in factorial designs. Biometrika 64 (1977), 597–603. doi: 10.1093/biomet/64.3.597 [Maths Reviews 0501643 (58 #18945)] R. A. Bailey: Dual Abelian groups in the design of experiments. In Algebraic Structures and Applications (eds. P. Schultz, C. E. Praeger and R. P. Sullivan), Marcel Dekker, New York, 1982, pp. 45–54. [Maths Reviews 0647165 (83m: 62135)] R. A. Bailey: Factorial design and Abelian groups. Linear Algebra and its Applications 70 (1985), 349–368. doi: 10.1016/0024-3795(85)90064-3 [Maths Reviews 0808552 (87c: 62151)]
In non-orthogonal designs, when does the partial confounding correspond to the factorial decomposition?	R. A. Bailey: Balance, orthogonality and efficiency factors in factorial design. Journal of the Royal Statistical Society, Series B 47 (1985), 453–458. doi: 10.1111/j.2517-6161.1985.tb01374.x [Maths Reviews 0844475 (87k: 62124)] R. A. Bailey and C. A. Rowley: General balance and treatment permutations. Linear Algebra and its Applications 127 (1990), 183–225. doi: 10.1016/0024-3795(90)90343-B [Maths Reviews 1048802 (91d: 05014)] R. A. Bailey: Cyclic designs and factorial designs. In Probability, Statistics and Design of Experiments (proceedings of the R. C. Bose Symposium on Probability, Statistics and Design of Experiments, Delhi, 27–30 December, 1988) (ed. R. R. Bahadur), Wiley Eastern, New Delhi, 1990, pp. 51–74. H. Monod and R. A. Bailey: Pseudofactors: normal use to improve design and facilitate analysis. Applied Statistics 41 (1992), 317–336. doi: 10.2307/2347564 C. J. Brien and R. A. Bailey: Decomposition tables for experiments. I. A chain of randomizations. Annals of Statistics 37 (2009), 4184–4213. doi: 10.1214/09-AOS717 [Maths Reviews 2572457 (2010k: 62294)] R. A. Bailey: Symmetric factorial designs in blocks. Journal of Statistical Theory and Practice 5 (2011), 13–24. doi: 10.1080/15598608.2011.10412047 [Maths Reviews 2829819]
How do the above methods apply if some treatment factors are quantitative?	R. A. Bailey: The decomposition of treatment degrees of freedom in quantitative factorial experiments. Journal of the Royal Statistical Society, Series B 44 (1982), 63–70. doi: 10.1111/j.2517-6161.1982.tb01188.x [Maths Reviews 0655375 (83e: 62103) and 0721758 (85d: 62076)] R. A. Bailey: Cyclic designs and factorial designs. In Probability, Statistics and Design of Experiments (proceedings of the R. C. Bose Symposium on Probability, Statistics and Design of Experiments, Delhi, 27–30 December, 1988) (ed. R. R. Bahadur), Wiley Eastern, New Delhi, 1990, pp. 51–74. H. Monod and R. A. Bailey: Pseudofactors: normal use to improve design and facilitate analysis. Applied Statistics 41 (1992), 317–336. doi: 10.2307/2347564
If the treatments are applied and investigated consecutively, how can we ensure that important factorial effects are orthogonal to a low-order polynomial in the time-trend?	R. A. Bailey, Ching-Shui Cheng and Patricia Kipnis: Construction of trend-resistant factorial designs. Statistica Sinica 2 (1992), 393–411. url: http://www.jstor.org/stable/24304867 [Maths Reviews 1187950 (93i: 62197)]
Regular fractions	Ulrike Grömping and Rosemary A. Bailey: Regular fractions of factorial arrays. In mODa 11—Advances in Model-Oriented Design and Analysis (eds. Joachim Kunert, Christine H. Müller and Anthony C. Atkinson), Springer International Publishing, 2016, pp. 143–151. doi: 10.1007/978-3-319-31266-8_17
Irregular fractions	R. A. Bailey: A note on loosely balanced incomplete block designs. Computational Statistics and Data Analysis 3 (1985), 115–117 and 121–122. doi: 10.1016/0167-9473(85)90068-4 and 10.1016/0167-9473(85)90070-2 R. A. Bailey: Contribution to the discussion of `Detection of interactions in experiments on large numbers of factors' by S. M. Lewis and A. M. Dean. Journal of the Royal Statistical Society, Series B 63 (2001), 662. R. A. Bailey: Contribution to the discussion of `Optimum design of experiments for statistical inference' by Steven G. Gilmour and Luzia A. Trinca. Applied Statistics 61 (2012), 374–375.
Multi-part designs	R. A. Bailey and Peter J. Cameron: Multi-part balanced incomplete-block designs. Statistical Papers, 60 (2019), 55–76. doi: 10.1007/s00362-018-01071-x [Maths Reviews 3969037] Appendage: doi: 10.1007/s00362-019-01137-4 [Maths Reviews 4262205]
Designs for estimating variance components	R. A. Bailey, Célia Fernandes and Paulo Ramos: Sparse designs for estimating variance components of nested random factors. Journal of Statistical Planning and Inference, 214 (2021), 76–88. doi: 10.1016/j.jspi.2021.01.002

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