Fractional Factorial Design

Question

In a $2^5$ design, it is believed that only the main effects $(A,B,C,D, E)$ and $AB,AC$ interaction effects are non-zero. I need to construct a fractional factorial with minimum number of runs which can be used to estimate all the main effects and $AB,AC$ interaction effects .

• i hoped this be a $2^{5-2}$ fractional factorial and to construct the design :

• First, i wrote down the basic design, which consists of the $8$ runsfor a full $2^{5-2}=2^3$ design in $A,B,C$

• Then the two factors $D$ an $E$ are added by associatingtheir plus and minus levels with the plus and minus signs of the interaction $BC$ and $ABC$.
• The complete defining relation for this design is $$I=BCD=ABCE=ADE$$

Table01: Construction of the $2^{5-2}$ Design with the Generators $I=BCD=ABCE=ADE$

But is it Principal Fraction? If not, how can i construct Principal Fraction in this way?

If i write down the data of Table01(where the data is in plus and minus notation) in treatment combination , this is

$$ \begin{array}{|c|} \hline d\\ ade\\ be\\ ab\\ ce\\ ac\\ bcd\\ abcde\\ \hline \end{array} $$

With this data how can i estimate effect $A,B,C$? There aren't treatment combinations $a,b,c$ in the fraction.

Table02. Alias structure for the $2^{5-2}$ Design with $I=BCD=ABCE=ADE$ $$ \begin{array}{C} A=ABCD=BCE=DE\\ B=CD=ACE=ABDE\\ C=BD=ABE=ACDE\\ D=BC=ABCDE=AE\\ E=BCDE=ABC=AD\\ AB=ACD=CE=BDE\\ AC=ABD=BE=CDE\\ \end{array} $$

The alias structure determines which effects are confounded with each other. But does it play role in construction of the design? If so,how?

And the final question is if the situation were as following:

In a $2^5$ design, it is believed that only the main effects $(A,B,C,D, E)$ and $AB,ABD$ interaction effects are non-zero. I need to construct a fractional factorial with minimum number of runs which can be used to estimate all the main effects and $AB,ABD$ interaction effects .

• Will it be a $2^{5-2}$ design ? As I need the $ABD$ interaction to be estimated and my basic design, $2^{5-2}=2^3$, has only effect $A,B,C$ .

Answer 1

For estimating the effects, every one of those 8 runs has a particular combination of all five factors' levels. That notation $d, ade,$ etc. does not mean that only $D$ is present in the first one. It means that only $D$ is at its high level, while the other four factors are at their low level. Note that for each of the factors, you have 4runs where it is $-$ and 4 runs where it is $+$. Look at your text for how to compute the estimates using those signs.

If you look at the confounding relations, you can figure out a way to construct the design. For example, it shows that $D=BC$ -- and in fact that's exactly how you determined the levels of $D$. Now an exercise for you: Write down a $2^3$ design in factors $C,D,E$. Then, using the alias structure you have displayed, figure out how to generate the levels of $A$ and $B$ using interactions of $C,D,E$.

C D E A=? B=?
- - -  ?  ?
+ - -  ?  ?
  ...
+ + +  ?  ?

Now do it-- and verify that you get the same factor combinations as before, only in a different order.

Your last sentence. I hope the exercise I gave you above helps convince you that the design is in all the factors, not just 3 of them.