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#REDIRECT [[Blocking (statistics)#Randomized block design]] {{R from merge}} {{R to section}}
{{more footnotes|date=January 2013}}
In the [[statistical]] theory of the [[design of experiments]], [[blocking (statistics)|blocking]] is the arranging of [[experimental unit]]s in groups (blocks) that are similar to one another. Typically, a blocking factor is a source of [[statistical dispersion|variability]] that is not of primary interest to the experimenter. An example of a blocking factor might be the sex of a patient; by blocking on sex, this source of variability is controlled for, thus leading to greater accuracy.
==Blocking to "remove" the effect of nuisance factors==
For randomized block designs, there is one factor or variable that is of primary interest. However, there are also several other [[nuisance variable|nuisance factors]].
Nuisance factors are those that may affect the measured result, but are not of primary interest. For example, in applying a treatment, nuisance factors might be the specific operator who prepared the treatment, the time of day the experiment was run, and the room temperature. All experiments have nuisance factors. The experimenter will typically need to spend some time deciding which nuisance factors are important enough to keep track of or control, if possible, during the experiment.
==Blocking used for nuisance factors that can be controlled==
When we can control nuisance factors, an important technique known as blocking can be used to reduce or eliminate the contribution to experimental error contributed by nuisance factors. The basic concept is to create homogeneous blocks in which the nuisance factors are held constant and the factor of interest is allowed to vary. Within blocks, it is possible to assess the effect of different levels of the factor of interest without having to worry about variations due to changes of the block factors, which are accounted for in the analysis.
==Definition of blocking factors==
A nuisance factor is used as a blocking factor if every level of the primary factor occurs the same number of times with each level of the nuisance factor. The analysis of the experiment will focus on the effect of varying levels of the primary factor within each block of the experiment.
==Block for a few of the most important nuisance factors==
The general rule is:
:"Block what you can; randomize what you cannot."
Blocking is used to remove the effects of a few of the most important nuisance variables. Randomization is then used to reduce the contaminating effects of the remaining nuisance variables. For important nuisance variables, blocking will yield higher significance in the variables of interest than randomizing.
==Table of randomized block designs==
One useful way to look at a randomized block experiment is to consider it as a collection of [[completely randomized design|completely randomized]] experiments, each run within one of the blocks of the total experiment.
{| class="wikitable"
|+ Randomized Block Designs (RBD)
|-
! Name of Design
! Number of Factors ''k''
! Number of Runs ''n''
|-
| 2-factor RBD
| 2
| ''L''<sub>1</sub> * ''L''<sub>2</sub>
|-
| 3-factor RBD
| 3
| ''L''<sub>1</sub> * ''L''<sub>2</sub> * ''L''<sub>3</sub>
|-
| 4-factor RBD
| 4
| ''L''<sub>1</sub> * ''L''<sub>2</sub> * ''L''<sub>3</sub> * ''L''<sub>4</sub>
|-
| <math>\vdots</math>
| <math>\vdots</math>
| <math>\vdots</math>
|-
| ''k''-factor RBD
| ''k''
| ''L''<sub>1</sub> * ''L''<sub>2</sub> * <math>\cdots</math> * ''L<sub>k</sub>''
|}
with
:''L''<sub>1</sub> = number of levels (settings) of factor 1
:''L''<sub>2</sub> = number of levels (settings) of factor 2
:''L''<sub>3</sub> = number of levels (settings) of factor 3
:''L''<sub>4</sub> = number of levels (settings) of factor 4
::<math>\vdots</math>
:''L<sub>k</sub>'' = number of levels (settings) of factor ''k''
==Example of a Randomized Block Design==
Suppose engineers at a semiconductor manufacturing facility want to test whether different water implant material dosages have a significant effect on resistivity measurements after a diffusion process taking place in a furnace. They have four different dosages they want to try and enough experimental wafers from the same lot to run three wafers at each of the dosages.
The nuisance factor they are concerned with is "furnace run" since it is known that each furnace run differs from the last and impacts many process parameters.
An ideal way to run this experiment would be to run all the 4x3=12 wafers in the same furnace run. That would eliminate the nuisance furnace factor completely. However, regular production wafers have furnace priority, and only a few experimental wafers are allowed into any furnace run at the same time.
A non-blocked way to run this experiment would be to run each of the twelve experimental wafers, in random order, one per furnace run. That would increase the experimental error of each resistivity measurement by the run-to-run furnace variability and make it more difficult to study the effects of the different dosages. The blocked way to run this experiment, assuming you can convince manufacturing to let you put four experimental wafers in a furnace run, would be to put four wafers with different dosages in each of three furnace runs. The only randomization would be choosing which of the three wafers with dosage 1 would go into furnace run 1, and similarly for the wafers with dosages 2, 3 and 4.
===Description of the experiment===
Let ''X''<sub>1</sub> be dosage "level" and ''X''<sub>2</sub> be the blocking factor furnace run. Then the experiment can be described as follows:
:''k'' = 2 factors (1 primary factor ''X''<sub>1</sub> and 1 blocking factor ''X''<sub>2</sub>)
:''L''<sub>1</sub> = 4 levels of factor ''X''<sub>1</sub>
:''L''<sub>2</sub> = 3 levels of factor ''X''<sub>2</sub>
:''n'' = 1 replication per cell
:''N'' = ''L''<sub>1</sub> * ''L''<sub>2</sub> = 4 * 3 = 12 runs
Before randomization, the design trials look like:
{| class="wikitable"
|-
! ''X''<sub>1</sub>
! ''X''<sub>2</sub>
|-
| 1
| 1
|-
| 1
| 2
|-
| 1
| 3
|-
| 2
| 1
|-
| 2
| 2
|-
| 2
| 3
|-
| 3
| 1
|-
| 3
| 2
|-
| 3
| 3
|-
| 4
| 1
|-
| 4
| 2
|-
| 4
| 3
|}
===Matrix Representation===
An alternate way of summarizing the design trials would be to use a 4x3 matrix whose 4 rows are the levels of the treatment ''X''<sub>1</sub> and whose columns are the 3 levels of the blocking variable ''X''<sub>2</sub>. The cells in the matrix have indices that match the ''X''<sub>1</sub>, ''X''<sub>2</sub> combinations above.
By extension, note that the trials for any K-factor randomized block design are simply the cell indices of a ''k'' dimensional matrix.
==Model for a Randomized Block Design==
The model for a randomized block design with one nuisance variable is
:<math>
Y_{ij} = \mu + T_i + B_j + \mathrm{random\ error}
</math>
where
:''Y''<sub>ij</sub> is any observation for which ''X''<sub>1</sub> = ''i'' and ''X''<sub>2</sub> = ''j''
:''X''<sub>1</sub> is the primary factor
:''X''<sub>2</sub> is the blocking factor
:μ is the general ___location parameter (i.e., the mean)
:''T''<sub>i</sub> is the effect for being in treatment ''i'' (of factor ''X''<sub>1</sub>)
:''B''<sub>j</sub> is the effect for being in block ''j'' (of factor ''X''<sub>2</sub>)
==Estimates for a Randomized Block Design==
:Estimate for μ : <math>\overline{Y}</math> = the average of all the data
:Estimate for ''T''<sub>i</sub> : <math>\overline{Y}_{i\cdot} - \overline{Y}</math> with <math>\overline{Y}_{i\cdot}</math> = average of all ''Y'' for which ''X''<sub>1</sub> = ''i''.
:Estimate for ''B''<sub>j</sub> : <math>\overline{Y}_{\cdot j} - \overline{Y}</math> with <math>\overline{Y}_{\cdot j}</math> = average of all ''Y'' for which ''X''<sub>2</sub> = ''j''.
==Generalizations of randomized block designs==
*[[Generalized randomized block design]]s (GRBD) allow tests of block-treatment interaction, and has exactly one blocking factor like the RCBD.
*[[Latin square]]s (and other row-column designs) have two blocking factors that are believed to have no interaction.
*[[Latin hypercube sampling]]
*[[Graeco-Latin square]]s
*[[Hyper-Graeco-Latin square design]]s
==See also==
*[[Algebraic statistics]]
*[[Combinatorial design]]
*[[Generalized randomized block design]]
*[[Glossary of experimental design]]
*[[Optimal design]]
==References==
*{{cite journal
|doi=10.2307/2681737
|title=The Generalized Randomized Block Design <!-- |authorlink=Sidney Addelman -->
|first=Sidney
|last=Addelman
|journal=The American Statistician
|volume=23
|issue=4
|date=Oct 1969
|pages=35–36
|jstor=2681737
}}
*{{cite journal
|doi=10.2307/2284277
|title=Variability of Treatments and Experimental Units in the Design and Analysis of Experiments
|first=Sidney
|last=Addelman
|journal=Journal of the American Statistical Association
|volume=65
|issue=331
|date=Sep 1970
|pages=1095–1108
|jstor = 2284277
}}
*{{cite journal|doi=10.2307/2984159|title=The Validity of Comparative Experiments|authorlink=Francis J. Anscombe|first=F. J.|last=Anscombe|journal=[[Journal of the Royal Statistical Society]]. Series A (General)|volume=111|issue=3|year=1948|pages=181–211|jstor=2984159}} | mr = 30181
*{{cite book |author=[http://www.maths.qmw.ac.uk/~rab/ Bailey, R. A]|title=Design of Comparative Experiments|url=http://www.maths.qmul.ac.uk/~rab/DOEbook/|publisher=[http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521683579 Cambridge University Press]|year=2008 |isbn=978-0-521-68357-9}} Pre-publication chapters are available on-line.
*{{cite book |author=[http://www.isid.ac.in/~rbb/ Bapat, R. B.]|title=Linear Algebra and Linear Models|url=http://books.google.se/books?id=T5dsExIP3aAC|edition=Second|publisher=[http://www.springer.com/math/algebra/book/978-0-387-98871-9?cm_mmc=Google-_-Book%20Search-_-Springer-_-0 Springer] |year=2000 |isbn=978-0-387-98871-9}}
*{{cite book
|author=Caliński, Tadeusz and Kageyama, Sanpei
|title=Block designs: A Randomization approach, Volume '''I''': Analysis
|series=Lecture Notes in Statistics
|volume=150
|publisher=Springer-Verlag
|___location=New York
|year=2000
|isbn=0-387-98578-6
}}
*{{cite book
|author=Caliński, Tadeusz and Kageyama, Sanpei
|title=Block designs: A Randomization approach, Volume '''II''': Design
|series=Lecture Notes in Statistics
|volume=170
|publisher=Springer-Verlag
|___location=New York
|year=2003
|isbn=0-387-95470-8|mr=1994124
}}
*{{cite journal
|doi=10.2307/2684574
|title=What Really Is Experimental Error in Block Designs?
|first=Charles E.
|last=Gates
|journal=The American Statistician
|volume=49
|issue=4
|date=Nov 1995
|pages = 362–363
|jstor = 2684574
}}
*{{cite book
|author=[[Oscar Kempthorne|Kempthorne, Oscar]]
|year=1979
|title=The Design and Analysis of Experiments
|edition=Corrected reprint of (1952) Wiley
|publisher=Robert E. Krieger
|isbn=0-88275-105-0
}}
*{{cite book
|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
|year=2008
|title=Design and Analysis of Experiments
|volume=I and II
|edition=Second
|publisher=Wiley
|isbn=978-0-470-38551-7}}
**{{cite book
|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
|year=2008
|title=Design and Analysis of Experiments, Volume I: Introduction to Experimental Design
|edition=Second
|publisher=Wiley
|isbn=978-0-471-72756-9
}}
**{{cite book
|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
|year=2005
|title=Design and Analysis of Experiments, Volume 2: Advanced Experimental Design
|edition=First
|publisher=Wiley
|isbn=978-0-471-55177-5
}}
*{{cite book
|first=Marvin
|last=Lentner
|author2=Thomas Bishop
|chapter=The Generalized RCB Design (Chapter 6.13)
|pages=225–226
|title=Experimental design and analysis
|edition=Second
|publisher=Valley Book Company
|___location=P.O. Box 884, Blacksburg, VA 24063
|year=1993
|isbn=0-9616255-2-X
}}
*{{cite book
|title=Constructions and Combinatorial Problems in Design of Experiments
|author=[[Damaraju Raghavarao|Raghavarao, Damaraju]]
|___location=New York
|year=1988
|edition=corrected reprint of the 1971 Wiley
|publisher=Dover
|isbn=0-486-65685-3
}}
*{{cite book
|title=Block Designs: Analysis, Combinatorics and Applications
|author=[[Damaraju Raghavarao|Raghavarao, Damaraju]] and Padgett, L.V.
|year=2005
|edition=
|publisher=World Scientific
|isbn=981-256-360-1
}}
*{{cite book |author=Shah, Kirti R. and Sinha, Bikas K. |title=Theory of [[Optimal design|Optimal Designs]] |series=[http://www.springer.com/series/694 Lecture Notes in Statistics]| volume=54 | publisher=Springer-Verlag | year=1989 | pages=171+viii |isbn=0-387-96991-8 |doi=}}
*{{cite book
|author=Street, Anne Penfold and Street, Deborah J.
|title=Combinatorics of Experimental Design
|publisher=Oxford U. P. [Clarendon]
|year=1987
|pages=400+xiv
|isbn=0-19-853256-3
}}
*{{cite journal
|title=The Randomization Analysis of a Generalized Randomized Block Design
|first=M. B.
|last=Wilk
|journal=Biometrika
|volume=42
|issue=1–2
|date=June 1955
|pages=70–79
| jstor = 2333423
}}
*{{cite journal
|title=Some Consequences of randomization in a Generalization of the Balanced Incomplete Block Design
|first=George
|last=Zyskind
|journal=The Annals of Mathematical Statistics
|volume=34
|issue=4
|date=Dec 1963
|pages=1569–1581
|jstor = 2238364
|doi = 10.1214/aoms/1177703889
}}
==External links==
*[http://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]
*[http://www.itl.nist.gov/div898/handbook/pri/section3/pri332.htm Randomized Block Designs]
{{Experimental design|state=expanded}}
{{Statistics|state=collapsed}}
{{NIST-PD|state=collapsed}}
[[Category:Design of experiments]]
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