Editing Design of experiments (section)

==Example==
[[File:Balance à tabac 1850.JPG|right|240px]]
This example of design experiments is attributed to [[Harold Hotelling]], building on examples from [[Frank Yates]].<ref>{{cite journal| last = Hotelling| first = Harold | title = Some Improvements in Weighing and Other Experimental Techniques| journal = Annals of Mathematical Statistics| volume = 15 | issue = 3| pages = 297–306 | date = 1944 | doi = 10.1214/aoms/1177731236 | url=https://projecteuclid.org/euclid.aoms/1177731236| doi-access = free}}</ref><ref>{{cite book | last1 = Giri | first1 = Narayan C.  | last2 = Das | first2 = M. N. | title = Design and Analysis of Experiments | publisher = Wiley | location = New York, N.Y | year = 1979 | isbn = 9780852269145 | url = https://books.google.com/books?id=-vGlnx-ZVvEC | pages=350–359 }}</ref><ref name="ref3">[[Herman Chernoff]], ''Sequential Analysis and Optimal Design'', [[Society for Industrial and Applied Mathematics|SIAM]] Monograph, 1972.</ref> The experiments designed in this example involve [[combinatorial design]]s.<ref name="yout_Howt">{{Cite web
| title = How to Use Design of Experiments to Create Robust Designs With High Yield
| author = Jack Sifri
| work = youtube.com
| date = 8 December 2014
| access-date = 2015-02-11
| url = https://www.youtube.com/watch?v=hfdZabCVwzc
}}</ref>

Weights of eight objects are measured using a [[pan balance]] and set of standard weights.  Each weighing measures the weight difference between objects in the left pan and any objects in the right pan by adding calibrated weights to the lighter pan until the balance is in equilibrium. Each measurement has a [[errors and residuals in statistics|random error]].  The average error is zero; the [[standard deviation]]s of the [[probability distribution]] of the errors is the same number σ on different weighings; errors on different weighings are [[statistical independence|independent]].  Denote the true weights by

:<math>\theta_1, \dots, \theta_8.\,</math>

We consider two different experiments:

# Weigh each object in one pan, with the other pan empty.  Let ''X''<sub>''i''</sub> be the measured weight of the object, for ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;8.
# Do the eight weighings according to the following schedule—a [[weighing matrix]]:

:: <math>
\begin{array}{lcc}
& \text{left pan} & \text{right pan} \\
\hline
\text{1st weighing:} & 1\ 2\ 3\ 4\ 5\ 6\ 7\ 8 & \text{(empty)} \\ 
\text{2nd:} & 1\ 2\ 3\ 8\ & 4\ 5\ 6\ 7 \\
\text{3rd:} & 1\ 4\ 5\ 8\ & 2\ 3\ 6\ 7 \\
\text{4th:} & 1\ 6\ 7\ 8\ & 2\ 3\ 4\ 5 \\
\text{5th:} & 2\ 4\ 6\ 8\ & 1\ 3\ 5\ 7 \\
\text{6th:} & 2\ 5\ 7\ 8\ & 1\ 3\ 4\ 6 \\
\text{7th:} & 3\ 4\ 7\ 8\ & 1\ 2\ 5\ 6 \\
\text{8th:} & 3\ 5\ 6\ 8\ & 1\ 2\ 4\ 7
\end{array}
</math>

: Let ''Y''<sub>''i''</sub> be the measured difference for ''i'' = 1, ..., 8. Then the estimated value of the weight ''θ''<sub>1</sub> is

:: <math>\widehat{\theta}_1 = \frac{Y_1 + Y_2 + Y_3 + Y_4 - Y_5 - Y_6 - Y_7 - Y_8}{8}. </math>

:Similar estimates can be found for the weights of the other items:

:: <math>
\begin{align}
\widehat{\theta}_2 & = \frac{Y_1 + Y_2 - Y_3 - Y_4 + Y_5 + Y_6 - Y_7 - Y_8} 8. \\[5pt]
\widehat{\theta}_3 & = \frac{Y_1 + Y_2 - Y_3 - Y_4 - Y_5 - Y_6 + Y_7 + Y_8} 8. \\[5pt]
\widehat{\theta}_4 & = \frac{Y_1 - Y_2 + Y_3 - Y_4 + Y_5 - Y_6 + Y_7 - Y_8} 8. \\[5pt]
\widehat{\theta}_5 & = \frac{Y_1 - Y_2 + Y_3 - Y_4 - Y_5 + Y_6 - Y_7 + Y_8} 8. \\[5pt]
\widehat{\theta}_6 & = \frac{Y_1 - Y_2 - Y_3 + Y_4 + Y_5 - Y_6 - Y_7 + Y_8} 8. \\[5pt]
\widehat{\theta}_7 & = \frac{Y_1 - Y_2 - Y_3 + Y_4 - Y_5 + Y_6 + Y_7 - Y_8} 8. \\[5pt]
\widehat{\theta}_8 & = \frac{Y_1 + Y_2 + Y_3 + Y_4 + Y_5 + Y_6 + Y_7 + Y_8} 8.
\end{align}
</math>

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

The variance of the estimate ''X''<sub>1</sub> of ''θ''<sub>1</sub> is ''σ''<sup>2</sup> if we use the first experiment.  But if we use the second experiment, the variance of the estimate given above is ''σ''<sup>2</sup>/8.  Thus the second experiment gives us 8 times as much precision for the estimate of a single item, and estimates all items simultaneously, with the same precision. What the second experiment achieves with eight would require 64 weighings if the items are weighed separately. However, note that the estimates for the items obtained in the second experiment have errors that correlate with each other.

Many problems of the design of experiments involve [[combinatorial design]]s, as in this example and others.<ref name="yout_Howt"/>