Design of experiments
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Experimental Design
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.
Example: Full Factorial Design
|| ||<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 a survey set of every possible combination of questions one can ask. It is both balanced and orthogonal. In the case of example shown above, a full factorial design would compose of 2x2x2 = 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 the effects of any attribute cancels out the effects of the other attributes.
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;"> + ||
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 unecessary 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"]
