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Revision as of 18:22, 14 December 2006

Experimental Design is the method of using statistical techniques to gather data, which can then be used as target or constraint values when optimizing models of interest. This method of systematically analyzing relationships between given choices and decisions made can be very useful in optimizing design processes.

Experimental Design

Experimental Design is the method of using statistical techniques to gather data efficiently. The same techniques can be used to design physical experiments that systematically vary physical parameters and measure the effect on relevant outputs or to design surveys (see also conjoint analysis) to efficiently measure the effect of product attributes on customer preference and choice.

Example: A Full Factorial Design for an experiment with three attributes (a, b, c), where each attribute can be set to one of two levels (+/-).

|| ||<style="background-color: #E0E0FF; width: 10%; text-align: center; font-weight: bold;">  a ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;">   b ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> c ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> y ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> name||
||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> 1 ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> -||<style="width: 10%; text-align: center;">y,,1,, ||<style="width: 10%; text-align: center;">I||
||<style="background-color: #E0E0FF; font-weight: bold; text-align: center;"> 2 ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> y,,2,, ||<style="text-align: center;">C||
||<style="background-color: #E0E0FF; font-weight: bold; text-align: center;"> 3 ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> y,,3,, ||<style="text-align: center;">B||
||<style="background-color: #E0E0FF; font-weight: bold; text-align: center;"> 4 ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> y,,4,, ||<style="text-align: center;">BC||
||<style="background-color: #E0E0FF; font-weight: bold; text-align: center;"> 5 ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> y,,5,, ||<style="text-align: center;">A||
||<style="background-color: #E0E0FF; font-weight: bold; text-align: center;"> 6 ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> y,,6,, ||<style="text-align: center;">AC||
||<style="background-color: #E0E0FF; font-weight: bold; text-align: center;"> 7 ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> y,,7,, ||<style="text-align: center;">AB||
||<style="background-color: #E0E0FF; font-weight: bold; text-align: center;"> 8 ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> y,,8,, ||<style="text-align: center;">ABC||

* a, b, c are the attributes

* y's are the dependent variables

Full Factorial

A Full Factorial experiment design contains every possible combination of attribute levels. It is both balanced and orthogonal. In the example shown above, a full factorial design would be composed of 2x2x2 = 2^3^ = 8 combinations. This is because there are three attributes, each with 2 levels. Generally, the number of combinations needed for a full factorial design of two-level attributes can be calculated by 2^n^, n being the number of attributes.

Balance

Having a balanced design is ensuring that each level of each attribute occurs an equal number of times.

Orthogonality

Orthogonality means that each combination pair of attributes occurs an equal number of times. An experimental design is orthogonal if all pairs of attribute levels appear an equal number of times.

Main Effects

|| ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> I ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> A ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> B ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> AB ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> C ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> AC ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> BC ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> ABC || ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> ME(a) ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + || ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> ME(b) ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + || ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> ME(c) ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + || ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> INT(ab) ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + || ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> INT(bc) ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + || ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> INT(ac) ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + || ||<style="background-color: #E0E0FF; font-weight: bold; width: 10%; text-align: center;"> INT(abc) ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> - ||<style="width: 10%; text-align: center;"> + ||

The main effect of a certain attribute is the difference between the average response to that attribute when it is low and the average response to that attribute when it is high.

The main effect of attribute "a" from level 1 to 2 as a function of the y data is calculated like this:

attachment:main.bmp

Interactions

The interaction effect between attribute "a" & "b" is calculated like this:

attachment:interaction.bmp

Fractional Factorial

Since there is often redundancy in full factorial designs, we can pick just enough number of question combinations to ask in a survey to get an adequate response. This design is called a fractional factorial design and by using this simplified design, we can avoid all the unnecessary questions that will only give us the same information. To make sure that the combinations chosen are enough to get a good cross-sectional sample of response, we must check if the selection is both balanced and orthogonal. Which combinations you should choose for a fractional factorial design can be determined by software such as SAS (developed by SAS Institute).

Aliasing

Otherwise known as confounding: when two or more effects cannot be distinguished.

See Also

*aliasing
*conjoint analysis

References

[1] Engineering Statistics Handbook