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Completely randomized design

completely randomized design, completely randomized design in statistics
In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take other nuisance variables 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 randomly assigned to the experimental units

Contents

  • 1 Randomization
  • 2 Three key numbers
  • 3 Example
    • 31 Sample randomized sequence of trials
  • 4 Model for a completely randomized design
  • 5 Estimates and statistical tests
    • 51 Estimating and testing model factor levels
  • 6 Bibliography
  • 7 See also
  • 8 External links

Randomization

By randomization, that is to say 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, 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 tables or by some physical mechanism eg, drawing the slips of paper

Three key numbers

All completely randomized designs with one primary factor are defined by 3 numbers:

  • 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

Example

A typical example of a completely randomized design is the following:

  • k = 1 factor X1
  • 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

The model for the response is Y i , j = μ + T i + r a n d o m   e r r o r =\mu +T_+\mathrm }

with

  • Yi,j being any observation for which X1 = 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
  • Ti is the effect of having treatment level i

Estimates and statistical tests

Estimating and testing model factor levels

  • Estimate for μ : Y ¯ }} = the average of all the data
  • Estimate for Ti : Y ¯ i − Y ¯ }_-}}

with Y ¯ i }_} = average of all Y for which X1 = i

Statistical tests for levels of X1 are those used for a one-way ANOVA and are detailed in the article on analysis of variance

Bibliography

  • Caliński, Tadeusz; Kageyama, Sanpei 2000 Block designs: A Randomization approach, Volume I: Analysis Lecture Notes in Statistics 150 New York: Springer-Verlag ISBN 0-387-98578-6 
  • Christensen, Ronald 2002 Plane Answers to Complex Questions: The Theory of Linear Models Third ed New York: Springer ISBN 0-387-95361-2 
  • Kempthorne, Oscar 1979 The Design and Analysis of Experiments Corrected reprint of 1952 Wiley ed Robert E Krieger ISBN 0-88275-105-0 
  • Hinkelmann, Klaus and Kempthorne, Oscar 2008 Design and Analysis of Experiments I and II Second ed Wiley ISBN 978-0-470-38551-7 CS1 maint: Multiple names: authors list link
    • Hinkelmann, Klaus and Kempthorne, Oscar 2008 Design and Analysis of Experiments, Volume I: Introduction to Experimental Design Second ed Wiley ISBN 978-0-471-72756-9 CS1 maint: Multiple names: authors list link
    • Hinkelmann, Klaus and Kempthorne, Oscar 2005 Design and Analysis of Experiments, Volume 2: Advanced Experimental Design First ed Wiley ISBN 978-0-471-55177-5 CS1 maint: Multiple names: authors list link

See also

  • Randomized block design

External links

  • Completely randomized designs
  • Completely randomized design CRD

 This article incorporates public domain material from the National Institute of Standards and Technology website http://wwwnistgov

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    29.10.2014


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