Design of experiments

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The first statistician to consider a methodology for the design of experiments was Sir Ronald A. Fisher. He described how to test the hypothesis that a certain lady could distinguish by flavor alone whether the milk or the tea was first placed in the cup. While this sounds like a frivolous application, it allowed him to illustrate the most important means of experimental design:

Randomization - The process of making something random
Replication - repeating the creation of a phenomenon, so that the variability associated with the phenomenon can be estimated
Blocking - the arranging of experimental units in groups (blocks) which are similar to one another
Orthogonality - Means perpendicular, at right angles or statistically normal.
Use of factorial experiments instead of the one-factor-at-a-time method

Analysis of the design of experiments was built on the foundation of the analysis of variance, a collection of models in which the observed variance is partitioned into components due to different factors which are estimated and/or tested.

Some efficient designs for estimating several main effects simultaneously were found by Raj Chandra Bose and K. Kishen in 1940 at the Indian Statistical Institute, but remained little known until the Plackett-Burman designs were published in Biometrika in 1946.

In 1950, Gertrude Mary Cox and William Cochran published the book Experimental Design which became the major reference work on the design of experiments for statisticians for years afterwards.

Developments of the theory of linear models have encompassed and surpassed the cases that concerned early writers. Today, the theory rests on advanced topics in abstract algebra and combinatorics.

As with all other branches of statistics, there is both classical and Bayesian experimental design.

[edit] Example

This example is attributed to Harold Hotelling in ^[1]. Although very simple, it conveys at least some of the flavor of the subject.

The weights of eight objects are to be measured using a pan balance that measures the difference between the weight of the objects in the two pans. Each measurement has a random error. The average error is zero; the standard deviations of the probability distribution of the errors is the same number σ on different weighings; and errors on different weighings are independent. Denote the true weights by

We consider two different experiments:

Weigh each object in one pan, with the other pan empty. Call the measured weight of the ith object X_i for i = 1, ..., 8.

Do the eight weighings according to the following schedule and let Y_i be the measured difference for i = 1, ..., 8:

Then the estimated value of the weight θ₁ is

The question of design of experiments is: which experiment is better?

The variance of the estimate X₁ of θ₁ is σ² if we use the first experiment. But if we use the second experiment, the variance of the estimate given above is σ^{2</sub>/8. Thus the second experiment gives us 8 times as much precision.}

Many problems of the design of experiments involve combinatorial designs, as in this example.

[edit] See also

[edit] References

↑ Herman Chernoff, Sequential Analysis and Optimal Design, SIAM Monograph, 1972.

[edit] External links

This page uses content from the English-language version of Wikipedia. The original article was at Design_of_experiments. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License.

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[ref1-0] Herman Chernoff, Sequential Analysis and Optimal Design, SIAM Monograph, 1972.

[1]