Completely randomized design: Difference between revisions

Revision as of 19:16, 14 June 2008

Latest revision as of 09:34, 14 June 2021

(29 intermediate revisions by 21 users not shown)

Line 1:

In the [[design of experiments]], '''completely randomized designs''' are for studying the effects of one primary factor without the need to take other [[nuisance factorvariable]]s into account. This article describes completely randomized designs that have one primary factor. The experiment compares the values of a [[response variable]] based on the different levels of that primary factor. For completely randomized designs, the levels of the primary factor are [[random assignment|randomly assigned]] to the [[experimental unit]]s.

==Randomization==

ByTo [[randomization|randomize]], that is to saydetermine the run sequence of the experimental units is determined randomly. For example, if there are 3 levels of the primary factor with each level to be run 2 times, then there are 6! (where ! denotes [[factorial]]) possible run sequences (or ways to order the experimental trials). Because of the [[replication (statistics)|replication]], the number of unique orderings is 90 (since 90 = 6!/(2!*2!*2!)). An example of an unrandomized design would be to always run 2 replications for the first level, then 2 for the second level, and finally 2 for the third level. To randomize the runs, one way would be to put 6 slips of paper in a box with 2 having level 1, 2 having level 2, and 2 having level 3. Before each run, one of the slips would be drawn blindly from the box and the level selected would be used for the next run of the experiment.

In practice, the randomization is typically performed by a computer program. However, the randomization can also be generated from [[random number table]]s or by some [[random number generation#Physical methods|physical mechanism]] (e.g., drawing the slips of paper).

Line 11:

* ''k'' = number of factors (= 1 for these designs)

* ''L'' = number of levels

* ''n'' = number of replications

and the total [[sample size]] (number of runs) is ''N'' = ''k'' × ''L'' × ''n''. Balance dictates that the number of replications be the same at each level of the factor (this will maximize the sensitivity of subsequent statistical ''t''- (or ''F''-) tests).

Line 20:

* ''L'' = 4 levels of that single factor (called "1", "2", "3", and "4")

* ''n'' = 3 replications per level

* ''N'' = 4 levels × 3 replications per level = 12 runs

===Sample randomized sequence of trials===

The randomized sequence of trials might look like: X₁: 3, 1, 4, 2, 2, 1, 3, 4, 1, 2, 4, 3

Note that in this example there are 12!/(3!*3!*3!*3!) = 369,600 ways to run the experiment, all equally likely to be picked by a randomization procedure.

==Model for a completely randomized design==

Line 36:

* ''Y''_i,j being any observation for which ''X''₁ = ''i'' (''i'' and ''j'' denote the level of the factor and the replication within the level of the factor, respectively)

* μ (or mu) is the general [[___location parameter]]

* ''T''_i is the effect of having treatment level ''i''

==Estimates and Statisticalstatistical Teststests==

==Estimates and Statistical Tests==

===Estimating and testing model factor levels===

* Estimate for &mu; : <math>\bar{Y}</math> = the [[average]] of all the data

Line 44 ⟶ 45:

with <math>\bar{Y}_i</math> = average of all ''Y'' for which ''X''₁ = ''i''.

Statistical tests for levels of ''X''₁ are shownthose used for a [[one-way ANOVA]] and are detailed in the article on [[one-wayanalysis ANOVAof variance]].

==Bibliography==

* {{cite book

|author1=Caliński, Tadeusz

|author2=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

|url-access=registration

|url=https://archive.org/details/blockdesignsrand0002cali

}}

*{{cite book

|title=Plane Answers to Complex Questions: The Theory of Linear Models|last=Christensen|first=Ronald|___location=New York|publisher=Springer|year=2002| edition=Third|isbn=0-387-95361-2}}

*{{cite book

|author=Kempthorne, Oscar

|author-link=Oscar Kempthorne

|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

}}

==See also==

Line 51 ⟶ 106:

==External links==

*[http://www.itl.nist.gov/div898/handbook/pri/section3/pri331.htm Completely randomized designs]

*[http://itfeature.com/design-of-experiment-doe/completely-randomized-design-crd Completely randomized design (CRD)]

[[Category:{{Experimental design]]}}

{{Statistics}}

{{NIST-PD}}

[[Category:ExperimentalDesign designof experiments]]

[[Category:Analysis of variance]]

[[Category:Statistical models]]

[[Category:Experimental design]]