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| 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.
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| ==Blocking to "remove" the effect of nuisance factors==
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| 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]].
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|
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| 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.
| | Among the most difficult instances of your life will come up out from time spent in the courtroom. Make points less difficult on on your own by deciding on someone to represent you that can create a solid scenario in your safeguard. Look at this article to decide on the proper legal professional for just about any scenario you may find yourself in.<br><br>If there's a particular legal representative truly gunning for yourself, don't be motivated to employ him. This can be common training amongst crooks and dishonest authorized kinds it is usually seen if you find a big funds arrangement up for grabs. Do cautious investigation to get a legal professional who can work with terms you define.<br><br>Locate a attorney that is certainly a specialist with your kind of case. Many attorneys are designed for a variety of situations, but if you want the very best final result possible, you need a legal professional that bargains only with instances the exact same nature as yours. The very best lawyer for you will get numerous situations like yours and knows legal requirements the best.<br><br>When you engage a attorney, make sure they're easy to communicate with. You should be confident you can get in touch when you really need them one of the most. I needed a lawyer once who vanished for a month! Looking at evaluations will help you locate a lawyer, who is a great communicator.<br><br>Make sure you know how a lot you will commit prior to engage a lawyer. Legal professionals can be very expensive. Their by the hour charges can crack your financial budget prior to they even can begin to assist you. Be sure you be aware of expenses in advance and if you cannot afford to pay for a specific legal professional, seek out one particular that you could pay for.<br><br>Search for an attorney that includes a great number of situations with the scenario that you are currently working with. For instance, if you are planning to court for taxation scam, you are going to want someone who focuses on this industry or at least has a lot of encounter beneath his buckle. This will aid improve [http://www.youtube.com/watch?v=TOFAmHkvdkM http://youtube.com/watch?v=TOFAmHkvdkM] your chance of victory.<br><br>Even though you is going to be continually stressing regarding your circumstance, it is important to keep in mind that your attorney offers quite a bit on her or his platter, and maybe more urgent matters sometimes. [http://www.youtube.com/watch?v=Obn-lfHhI4g morris bart - personal injury lawyer] Your lawyer may possibly have other more emergency situations. Keep in mind that your urgency is your personal, so realize that.<br><br>If you fail to afford to pay for a lawyer, consider symbolizing oneself. Get a attorney that is willing that will help you prepare a good shield and give use some beneficial guidance. Some lawyers will allow you to create a solid shield and monthly bill you for the the couple of hours they invested utilizing you.<br><br>When you have a the courtroom circumstance that the standard attorney is not familiar with, you almost certainly should work with a consultant. While you might feel that professionals can be more expensive, this isn't always true. Also, should your circumstance entails a lot of cash, you need to help make certain you win. Using a lawyer who is not really acquainted with your distinct litigation will likely be a significant problem for yourself. Alternatively, work with a specialist that is employed to dealing with your circumstance.<br><br>Question a potential attorney while they are open to response your calls or reply to your expections. Your preferences can vary, depending on your scenarios,so you should make certain that your lawyer's standards meet these requirements. Just because a single attorney only operates in the daytime does not mean that we now have not fantastic lawyers that are offered at any time throughout the day.<br><br>There are lawyers for most particular types of instances, and regardless of any humor, lawyers ought to be interested in justice. Rather than boosting the anxiety encompassing your situation, make things go efficiently. Keep in mind tips on this page in order to select the best attorney for you personally and get on top! |
| | |
| ==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.
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| | |
| ==Definition of blocking factors==
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| 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.
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| | |
| ==Block for a few of the most important nuisance factors==
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| The general rule is:
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| :"Block what you can; randomize what you cannot."
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| 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.
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| | |
| ==Table of randomized block designs==
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| 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.
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| {| class="wikitable"
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| |+ Randomized Block Designs (RBD)
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| |-
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| ! Name of Design
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| ! Number of Factors ''k''
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| ! Number of Runs ''n''
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| |-
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| | 2-factor RBD
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| | 2
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| | ''L''<sub>1</sub> * ''L''<sub>2</sub>
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| |-
| |
| | 3-factor RBD
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| | 3
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| | ''L''<sub>1</sub> * ''L''<sub>2</sub> * ''L''<sub>3</sub>
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| |-
| |
| | 4-factor RBD
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| | 4
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| | ''L''<sub>1</sub> * ''L''<sub>2</sub> * ''L''<sub>3</sub> * ''L''<sub>4</sub>
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| |-
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| | ''k''-factor RBD
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| | ''k''
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| | ''L''<sub>1</sub> * ''L''<sub>2</sub> * <math>\cdots</math> * ''L<sub>k</sub>''
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| |}
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| | |
| with
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| :''L''<sub>1</sub> = number of levels (settings) of factor 1
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| :''L''<sub>2</sub> = number of levels (settings) of factor 2
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| :''L''<sub>3</sub> = number of levels (settings) of factor 3
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| :''L''<sub>4</sub> = number of levels (settings) of factor 4
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| ::<math>\vdots</math>
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| :''L<sub>k</sub>'' = number of levels (settings) of factor ''k''
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| | |
| ==Example of a Randomized Block Design==
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| Suppose engineers at a semiconductor manufacturing facility want to test whether different wafer 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.
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| 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.
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| 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.
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| 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.
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| | |
| ===Description of the experiment===
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| HMLet ''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:
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| :''k'' = 2 factors (1 primary factor ''X''<sub>1</sub> and 1 blocking factor ''X''<sub>2</sub>)
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| :''L''<sub>1</sub> = 4 levels of factor ''X''<sub>1</sub>
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| :''L''<sub>2</sub> = 3 levels of factor ''X''<sub>2</sub>
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| :''n'' = 1 replication per cell
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| :''N'' = ''L''<sub>1</sub> * ''L''<sub>2</sub> = 4 * 3 = 12 runs
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| | |
| Before randomization, the design trials look like:
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| {| class="wikitable"
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| |-
| |
| ! ''X''<sub>1</sub>
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| ! ''X''<sub>2</sub>
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| |-
| |
| | 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===
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| 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.
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| | |
| By extension, note that the trials for any K-factor randomized block design are simply the cell indices of a ''k'' dimensional matrix.
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| | |
| ==Model for a Randomized Block Design==
| |
| The model for a randomized block design with one nuisance variable is
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| :<math>
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| Y_{ij} = \mu + T_i + B_j + \mathrm{random\ error}
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| </math> | |
| where
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| :''Y''<sub>ij</sub> is any observation for which ''X''<sub>1</sub> = ''i'' and ''X''<sub>2</sub> = ''j''
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| :''X''<sub>1</sub> is the primary factor
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| :''X''<sub>2</sub> is the blocking factor
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| :μ is the general location parameter (i.e., the mean)
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| :''T''<sub>i</sub> is the effect for being in treatment ''i'' (of factor ''X''<sub>1</sub>)
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| :''B''<sub>j</sub> is the effect for being in block ''j'' (of factor ''X''<sub>2</sub>)
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| | |
| ==Estimates for a Randomized Block Design==
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| :Estimate for μ : <math>\overline{Y}</math> = the average of all the data
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| :Estimate for ''T''<sub>i</sub> : <math>\overline{Y}_i - \overline{Y}</math> with <math>\overline{Y}_i</math> = average of all ''Y'' for which ''X''<sub>1</sub> = ''i''.
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| :Estimate for ''B''<sub>j</sub> : <math>\overline{Y}_j - \overline{Y}</math> with <math>\overline{Y}_j</math> = average of all ''Y'' for which ''X''<sub>2</sub> = ''j''.
| |
| | |
| ==Generalizations of randomized block designs==
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| * [[Generalized randomized block design]]s (GRBD) allow tests of block-treatment interaction, and has exactly one blocking factor like the RCBD.
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| * [[Latin square]]s (and other row-column designs) have two blocking factors that are believed to have no interaction.
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| * [[Latin hypercube sampling]]
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| * [[Graeco-Latin square]]s
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| * [[Hyper-Graeco-Latin square design]]s
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| | |
| ==See also==
| |
| * [[Algebraic statistics]]
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| * [[Combinatorial design]]
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| * [[Generalized randomized block design]]
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| * [[Glossary of experimental design]]
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| * [[Optimal design]]
| |
| | |
| ==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]
| |
| | |
| ==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
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| |journal=Journal of the American Statistical Association
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| |volume=65
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| |issue=331
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| |date=Sep 1970
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| |pages=1095–1108
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| |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
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| | |
| * {{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
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| |publisher=Springer-Verlag
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| |location=New York
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| |year=2003
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| |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
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| |issue=4
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| |date=Nov 1995
| |
| |pages = 362–363
| |
| |jstor = 2684574
| |
| }}
| |
| | |
| *{{cite book
| |
| |author=[[Oscar Kempthorne|Kempthorne, Oscar]]
| |
| |year=1979
| |
| |title=The Design and Analysis of Experiments
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| |edition=Corrected reprint of (1952) Wiley
| |
| |publisher=Robert E. Krieger
| |
| |isbn=0-88275-105-0
| |
| }}
| |
| *{{cite book
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| |author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
| |
| |year=2008
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| |title=Design and Analysis of Experiments
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| |volume=I and II
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| |edition=Second
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| |publisher=Wiley
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| |isbn=978-0-470-38551-7}}
| |
| **{{cite book
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| |author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
| |
| |year=2008
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| |title=Design and Analysis of Experiments, Volume I: Introduction to Experimental Design
| |
| |edition=Second
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| |publisher=Wiley
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| |isbn=978-0-471-72756-9
| |
| }}
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| **{{cite book
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| |author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
| |
| |year=2005
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| |title=Design and Analysis of Experiments, Volume 2: Advanced Experimental Design
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| |edition=First
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| |publisher=Wiley
| |
| |isbn=978-0-471-55177-5
| |
| }}
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| | |
| * {{cite book
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| |first=Marvin
| |
| |last=Lentner
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| |coauthor=Thomas Bishop
| |
| |chapter=The Generalized RCB Design (Chapter 6.13)
| |
| |pages=225–226
| |
| |title=Experimental design and analysis
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| |edition=Second
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| |publisher=Valley Book Company
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| |location=P.O. Box 884, Blacksburg, VA 24063
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| |year=1993
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| |isbn=0-9616255-2-X
| |
| }}
| |
| | |
| * {{cite book
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| |title=Constructions and Combinatorial Problems in Design of Experiments
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| |author=[[Damaraju Raghavarao|Raghavarao, Damaraju]]
| |
| |location=New York
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| |year=1988
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| |edition=corrected reprint of the 1971 Wiley
| |
| |publisher=Dover
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| |isbn=0-486-65685-3
| |
| }}
| |
| * {{cite book
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| |title=Block Designs: Analysis, Combinatorics and Applications
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| |author=[[Damaraju Raghavarao|Raghavarao, Damaraju]] and Padgett, L.V.
| |
| |year=2005
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| |edition=
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| |publisher=World Scientific
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| |isbn=981-256-360-1
| |
| }}
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| *{{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.
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| |title=Combinatorics of Experimental Design
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| |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
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| |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
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| |date=Dec 1963
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| |pages=1569–1581
| |
| |jstor = 2238364
| |
| |doi = 10.1214/aoms/1177703889
| |
| }}
| |
| | |
| {{Experimental design|state=expanded}}
| |
| {{Statistics|state=collapsed}}
| |
| {{NIST-PD|state=collapsed}}
| |
| | |
| [[Category:Design of experiments]]
| |
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Make sure you know how a lot you will commit prior to engage a lawyer. Legal professionals can be very expensive. Their by the hour charges can crack your financial budget prior to they even can begin to assist you. Be sure you be aware of expenses in advance and if you cannot afford to pay for a specific legal professional, seek out one particular that you could pay for.
Search for an attorney that includes a great number of situations with the scenario that you are currently working with. For instance, if you are planning to court for taxation scam, you are going to want someone who focuses on this industry or at least has a lot of encounter beneath his buckle. This will aid improve http://youtube.com/watch?v=TOFAmHkvdkM your chance of victory.
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