### … look the same !!

The following tables show:

#### A Classical 2^{3} design

to investigate the effects of factors A,B, and C, and also the 3 associated two way interactions, and the single three way interaction.

##### Classical 2^{3} Design

factor | interactions | ||||||

run | A |
B |
C |
AB | AC | BC | ABC |

1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 |

2 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |

3 | -1 | 1 | -1 | -1 | 1 | -1 | 1 |

4 | 1 | 1 | -1 | 1 | -1 | -1 | -1 |

5 | -1 | -1 | 1 | 1 | -1 | -1 | 1 |

6 | 1 | -1 | 1 | -1 | 1 | -1 | -1 |

7 | -1 | 1 | 1 | -1 | -1 | 1 | -1 |

8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Taguchi col no |
4 |
2 |
1 |
6 |
5 |
3 |
7 |

#### A Taguchi L_{8} Array

to investigate the effects of up to 7 factors in 8 runs. Use of Linear Graphs would select columns 1,2 and 4 to identify the effects of 3 factors – and this corresponds to the same classical design above.

##### Taguchi L8 Array

run | 1 |
2 |
3 |
4 |
5 |
6 |
7 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |

3 | 1 | 2 | 2 | 1 | 1 | 2 | 2 |

4 | 1 | 2 | 2 | 2 | 2 | 1 | 1 |

5 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |

6 | 2 | 1 | 2 | 2 | 1 | 2 | 1 |

7 | 2 | 2 | 1 | 1 | 2 | 2 | 1 |

8 | 2 | 2 | 1 | 2 | 1 | 1 | 2 |

classical col no | C | B | BC | A | AC | AB | ABC |

For each design, each row represents a run of the experiment – here, each design has 8 runs. Each column represents the settings of the factor at the top of the column. In the classical design, the levels are (-1,+1); in the Taguchi design, the levels are (1,2) – each means (low, high) for each factor.At the bottom of each design is the corresponding column number for the alternative design – for example, column 1 in the Taguchi design corresponds to the column C in the classical design, and vice versa.

The classical design has the runs in “standard” or “Yates” (a statistician who invented an algorithm to quickly calculate the effects from a designed experiment) order. Here (because there are 3 factors), the first three columns are the factors; the first column is a series of -1’s and +1’s; the next column is pairs of (-1,-1) then (+1,+1); the third column has the pattern (-1,-1,-1,-1) then (+1,+1,+1,+1). The interaction column elements are formed by calculating the product of the values in the first three columns – easy to calculate! All 7 columns are independent (not equal to each other or a multiple of another single column) – hence, all 7 effects (of the main factors and interactions) can be independently estimated.

The Taguchi design has the same components as the classical design, but in a different order. However, the columns for the settings of the factors, are chosen according to the interactions that the investigator assumes may or may not be present in the process. The investigator consults a Interaction Table, and/or Linear Graphs, to determine which columns to choose in the design.

In the classical methodology, the experimenter makes no assumptions about the presence or absence of interactions before the experiment is run. If the experimenter knows for sure that there are no interactions present, then the classical design above could be used to investigate the effects of up to 7 factors, in 8 runs. (Incidentally, that is the most information you’ll ever get from an experiment – in n runs, you will get n-1 effects, and the average; that’s all, no more and no less! So, if you run an experiment and vary 7 factors in 8 runs, you will still end up with 7 effects at the end of the experiment!)

See Why learn classical DOE? and also an overview of Taguchi vs Classical DOE.

Learn how to set up, run, analyse and present designed experiments, at our intensive hands on Design of Experiments Workshop.