The fractional factorial design for experiments is used when you have too many factors to do a complete tutorial design.

In the fractional factorial you have only lk-p combination of factor where:

• l is the number of levels in each factor. Typically is run with only 2 levels (- for low and + for high).
• k is the number of factor;
• p is the number of interactions that are confounded.

The number of sample needed is so 1/lp

But what’s a confounding effect?

A confounding effect is a factor that we associate with the level of two other factors. So, for example, if we start with a full factorial design, with two levels and with a, b,c factor, we have this combination:

1. – – – that means a=low, b=low, c=low;
2. + – – that means a=high, b=low, c=low. It can be also rappresented as a;
3. – + – that means a=low, b=high, c=low. It can be also rappresented as b;
4. + + – that means a=high, b=high, c=low. It can be also rappresented as a,b;
5. – – + that means a=low, b=low, c=high. It can be also rappresented as c;
6. + – + that means a=high, b=low, c=high. It can be also rappresented as a,c;
7. – + + that means a=low, b=high, c=high. It can be also rappresented as b,c;
8. + + + that means a=high, b=high, c=high. It can be also rappresented as a,b,c;

Now if we want to add D we can say that D=AxBxC, where we now that:

• – x – = +
• – x + = –
• + x + = +

So we still have 8 combination but with this result:

1. – – – – that means a=low, b=low, c=low.
2. + – – + that means a=high, b=low, c=low. It can be also rappresented as a,d;
3. – + – + that means a=low, b=high, c=low. It can be also rappresented as b,d;
4. + + – – that means a=high, b=high, c=low. It can be also rappresented as a,b;
5. – – + + that means a=low, b=low, c=high. It can be also rappresented as c,d;
6. + – + – that means a=high, b=low, c=high. It can be also rappresented as a,c;
7. – + + – that means a=low, b=high, c=high. It can be also rappresented as b,c;
8. + + + + that means a=high, b=high, c=high. It can be also rappresented as a,b,c,d;

The dowbrack of this method is that we can miss some critical interaction.

The resolution describe the degree with the main effect are confounded with the other level (2-level, 3-level, and so on). The level of resolution is one more than he smallest order interaction that some main effect is confounded.

So for instance:

• Resolution = 3 – Main effect and a 2-factor interaction may be confounded;
• Resolution = 4 – No main effect are confounded with a 2-factor interaction. At least two 2-factor interactions may be confounded with one another.
• Resolution = 5 – No main or 2-factor interactions are confounded with one another. Two-factor interactions may be confounded with 3-factor.
• Resolution = 6 – No main or 2-factor or 3-factor interactions are confounded with one another. Three-factor interactions may be confounded with one another.

So in our example we have a resolution of 4.

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