Completely randomized design
In the
Randomization
To randomize is to determine the run sequence of the experimental units 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 (e.g., 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
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
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 : = the average of all the data
- Estimate for Ti :
with = average of all Y for which X1 = i.
Statistical tests for levels of X1 are those used for a
Bibliography
- Caliński, Tadeusz; Kageyama, Sanpei (2000). Block designs: A Randomization approach, Volume I: Analysis. Lecture Notes in Statistics. Vol. 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.
- ISBN 0-88275-105-0.
- Hinkelmann, Klaus and ISBN 978-0-470-38551-7.)
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See also
- Randomized block design
External links
This article incorporates public domain material from the National Institute of Standards and Technology