Restricted randomization: Difference between revisions

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In [[statistics]], '''restricted randomization''' occurs in the [[design of experiments]] and in particular in the context of [[randomized experiment]]s and [[randomized controlled trial]]s. Restricted randomization allows intuitively poor allocations of treatments to experimental units to be avoided, while retaining the theoretical benefits of randomization.<ref>{{cite book|last1=Dodge|first1=Y.|title=The Oxford Dictionary of Statistical Terms|publisher=OUP|year=2006|isbn=978-0-19-920613-1|url-access=registration|url=https://archive.org/details/oxforddictionary0000unse}}</ref><ref>{{cite journal|last1=Grundy|first1=P.M.|last2=Healy|first2=M.J.R.|author-link2=Michael Healy (statistician)|title=Restricted randomization and quasi-Latin squares|journal=Journal of the Royal Statistical Society, Series B|date=1950 |volume=12|issue=2 |pages=286–291 |doi=10.1111/j.2517-6161.1950.tb00062.x }}</ref> For example, in a [[clinical trial]] of a new proposed treatment of obesity compared to a control, an experimenter would want to avoid outcomes of the randomization in which the new treatment was allocated only to the heaviest patients.

In statistics, '''restricted randomization''' occurs in [[experimental design]]s with more than one source of [[statistical dispersion|variability]]. The concept was introduced by [[Frank Yates]] (1948) and [[William J. Youden]] (1972) "as a way of avoiding bad spatial patterns of treatments in designed experiments."<ref name="ref1">Bailey, R. A. [http://www.jstor.org/discover/10.2307/2288775?uid=3739808&uid=2&uid=4&uid=3739256&sid=21100687318461 Restricted Randomization: A Practical Example], ''Journal of the American Statistical Association'', Vol. 82, No. 399 (Sep., 1987), pp. 712–719, at 712</ref>

The concept was introduced by [[Frank Yates]] (1948){{full citation needed|date=November 2012}} and [[William J. Youden]] (1972){{full citation needed|date=November 2012}} "as a way of avoiding bad spatial patterns of treatments in designed experiments."<ref name="ref1">{{Cite journal |jstor = 2288775|title = Restricted Randomization: A Practical Example|last1 = Bailey|first1 = R. A.|journal = Journal of the American Statistical Association|year = 1987|volume = 82|issue = 399|pages = 712–719|doi = 10.1080/01621459.1987.10478487}}</ref>

In order to [[variance reduction|reduce variation]] in processes, these multiple sources must be understood, and that often leads to the concept of nested or hierarchical data structures. For example, in the [[semiconductor]] industry, a [[batch production|batch process]] may operate on several [[wafer (electronics)|wafers]] at a time (wafers are said to be '''nested''' within batch). Understanding the input variables that control variation among those wafers, as well as understanding the variation across each wafer in a run, is an important part of the strategy for minimizing the total variation in the system.

==Example of nested data==

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There are 16 subplot experimental units for this experiment. Solution temperature and current are the subplot factors in this experiment. There are four whole-plot experimental units in this experiment. Solution concentration is the whole-plot factor in this experiment. Since there are two sizes of experimental units, there are two error terms in the model, one that corresponds to the whole-plot error or whole-plot experimental unit and one that corresponds to the subplot error or subplot experimental unit.

The [[ANOVA]] table for this experiment would look, in part, as follows:

{| class="wikitable"

|+ Partial ANOVA table

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|1

|-

|Error (Wholewhole plot) = Rep* × Conc

|1

|-

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|1

|-

|Rep* × Temp

|1

|-

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|1

|-

|Rep* × Current

|1

|-

|Temp* × Conc

|1

|-

|Rep* × Temp* × Conc

|1

|-

|Temp* × Current

|1

|-

|Rep* × Temp* × Current

|1

|-

|Current* × Conc

|1

|-

|Rep* × Current* × Conc

|1

|-

|Temp* × Current* × Conc

|1

|-

|Error (Subplot) = Rep* × Temp* × Current* × Conc

|1

|}

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|1

|-

|Error (Wholewhole plot) = Conc* × Temp

|1

|-

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|1

|-

|Conc* × Current

|1

|-

|Temp* × Current

|1

|-

|Conc* × Temp* × Current

|1

|-

|Error (Subplotsubplot)

|8

|}

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|1

|-

|Error (Wholewhole plot)

|4

|-

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|1

|-

|Conc* × Current

|1

|-

|Temp* × Current

|1

|-

|Conc* × Temp* × Current

|1

|-

|Error (Subplotsubplot)

|4

|}

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==See also==

{{Portal|Mathematics}}

* [[Hierarchical linear modeling]]

* [[Mixed-design analysis of variance]]

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==References==

{{reflist}}

* {{cite web | url=http://www.itl.nist.gov/div898/handbook/pri/section5/pri55.htm | title=How can I account for nested variation (restricted randomization)? | publisher=(U.S.) National Institute of Standards and Technology: Information Technology Laboratory | accessdateaccess-date=March 26, 2012}}

==Further reading==

For a more detailed discussion of these designs and the appropriate analysis procedures, see:

* {{cite book

|authorauthor1 = Milliken, G. A.

|coauthorsauthor2 = Johnson, D. E.

|year = 1984

|title = Analysis of Messy Data

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|volume = 39

|issue = 2

|page = 153

|doi = 10.2307/1270903

|publisher = Technometrics, Vol. 39, No. 2

|jstor = 1270903

|pages = 153–161}}

==External links==

* [https://www.southampton.ac.uk/~cpd/anovas/datasets/index.htm Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R]

{{Statistics}}

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[[Category:Analysis of variance]]

[[Category:Design of experiments]]

[[Category:Statistical models]]