Completely randomized design
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In the
design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
, completely randomized designs are for studying the effects of one primary factor without the need to take other
nuisance variable In the theory of stochastic processes in probability theory and statistics, a nuisance variable is a random variable that is fundamental to the probabilistic model, but that is of no particular interest in itself or is no longer of any interest: o ...
s into account. This article describes completely randomized designs that have one primary factor. The experiment compares the values of a
response variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
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 unit In statistics, a unit is one member of a set of entities being studied. It is the main source for the mathematical abstraction of a " random variable". Common examples of a unit would be a single person, animal, plant, manufactured item, or countr ...
s.


Randomization

To
randomize Randomization is the process of making something random. Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern, but follow an evolution d ...
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 table Random number tables have been used in statistics for tasks such as selected randomness, random samples. This was much more effective than manually selecting the random samples (with dice, cards, etc.). Nowadays, tables of random numbers have been ...
s 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
sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...
(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 (''X''1) * ''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_ = \mu + T_i + \mathrm with * ''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 In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
* ''T''i is the effect of having treatment level ''i''


Estimates and statistical tests


Estimating and testing model factor levels

* Estimate for μ : \bar = the
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
of all the data * Estimate for ''T''i : \bar_i - \bar with \bar_i = average of all ''Y'' for which ''X''1 = ''i''. Statistical tests for levels of ''X''1 are those used for a
one-way ANOVA In statistics, one-way analysis of variance (abbreviated one-way ANOVA) is a technique that can be used to compare whether two sample's means are significantly different or not (using the F distribution). This technique can be used only for numeric ...
and are detailed in the article on
analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
.


Bibliography

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See also

*
Randomized block design In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) that are similar to one another. Blocking can be used to tackle the problem of pseudoreplication. Use Blocking reduces un ...


External links


Completely randomized designsCompletely randomized design (CRD)
{{NIST-PD Design of experiments Analysis of variance Statistical models