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=978-0-19-920613-91|url-access=registration|url=https://archive.org/details/oxforddictionary0000unse}}</ref><ref>{{cite journal|last1=Grundy|first1=P.M.|last2=Healy|first2=M.J.R.|authorlink2author-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.

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">Bailey,{{Cite R.journal A.|jstor (1987)= [http://www.jstor.org/discover/10.2307/2288775?uid|title =3739808&uid=2&uid=4&uid=3739256&sid=21100687318461 "Restricted Randomization: A Practical Example"],|last1 = Bailey|first1 = R. A.|journal = ''Journal of the American Statistical Association'',|year Vol.= 1987|volume = 82,|issue No.= 399|pages (Sep., 1987), pp.= 712–719,|doi at= 71210.1080/01621459.1987.10478487}}</ref>

==Example of nested data==

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==Split-plot designs==

Split-plot designs result when a particular type of restricted randomization has occurred during the experiment. A simple [[factorial experiment]] can result in a split-plot type yerwof design because of the way the experiment was actually executed.

of design because of the way the experiment was actually executed.

In many industrial experiments, three situations often occur:

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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==

{{portalPortal|StatisticsMathematics}}

* [[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}}

{{Experimental design}}

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[[Category:Analysis of variance]]

[[Category:Design of experiments]]

[[Category:Statistical models]]