The **2 ^{k} full factorial designs of the experiment** is a specific case of the full factorial where each k factor can assume only two values. In this case, 2

^{k}is all the possible combinations of factors.

For the execution of this test, we can use the **Yates algorithm**. For setup, this test, the first step is write the combination of factor in this way:

- Assuming that each factor of the test can assume high or low value, so we are + for high and – for low;
- Now write all the combination of value, for example for a three factor (a,b,c) we have:
- – – – that means a=low, b=low, c=low;
- + – – that means a=high, b=low, c=low. It can be also rappresented as a;
- – + – that means a=low, b=high, c=low. It can be also rappresented as b;
- + + – that means a=high, b=high, c=low. It can be also rappresented as a,b;
- – – + that means a=low, b=low, c=high. It can be also rappresented as c;
- + – + that means a=high, b=low, c=high. It can be also rappresented as a,c;
- – + + that means a=low, b=high, c=high. It can be also rappresented as b,c;
- + + + that means a=high, b=high, c=high. It can be also rappresented as a,b,c;

Now we need to make the calc in **image1**:

where:

**Effect**: represent the combination of high value of theindependent variable as explained first;*n=3***Run1 and Run2**: is two-runof test and represent the value observed at the end of the test (our Y);*(r=2)***Sum**: is the sum of run1 and run2 for each effet;**Col1:**for the first 4 value is the sum of adjacent pair of summed responses, for example (E2) 82 = (D2) 41 + (D3) 41; for the last 4 value is the difference of adjacent pair of response, for example (E9) 11 = (D9) 37 – (D8) 26;**Col2**: is made in the same way of Col1 but on the value of col1;**Col3**: is made in the same way of Col1 but on the value of col2;**Estimate****(of impact**): is Col3 / r*2^n, where n is the number of indipendent variable, r is the number of run, 2 is the possibile value (+ or -)

In this example, it seems that C has a high estimated impact on the dependant variable. To assess if this impact has a statistical significance, we can make an ANOVA hypothesis test for each combination of values.

In ** image2 **we can look at the calc for the ANOVA test:

**where:**

**SS**is the sum of squared for all combinations. Is calculated as**(COL3^2)/(r*(2^n))**;**Response squared run 1**is simply the square of Run1, the same for the run2;**Df**is the degree of freedom, that is 1 for each vale except for the error that is 2^n *(r-1)**Sum of response squarred**is only the sum of the two respons squared column (it’s a single value in A15);**MS = SS / DF**exepct for the MS of the error that is**MS**_{error}= sum of all SS / DF- F (the statistics) if finally calculated as:
**F=MS/MS**_{Error}

Finally we have our F statistics and we need to get the critical value from the F statistics table. In this case we have df1 =1 and df2 = 8 that is **2^n*(r-1)** and we select alpha = 0,05. So **F(df=1; df2=8; alpha= 0,05) = 5,3177** so the test denote that *only the factor C have a significant impact.*