Measuring and data collection

Choosing a statistical test

Minimum sample size

Not normally distributed

Statistical process control (SPC)

How to conduct a DOE

Setting up a Response surface test (RSM)

For commercial use

75 EURO

Factorial Array

Edit factorial array

Rename Levels of DOE Factors

Interactions

Adding repetitions

Block Levels

Sort data in subgroups

Anova DOE

Confirmation run

Response graphs

Response surface methodology

For commercial use

75 EURO

A DOE (Design Of Experiments) is structured, planned method, which is used to find the relationship between different factors that affect a tested subject and the different outputs. This is done with multiple factors on various levels combined in one experiment. Instead of testing each factor individually in a DOE multiple factors are variated at once to reduce the amount of test with the possibility to analyze interactions between factors.

In other words reduce cost and time.

As visible in the below table with a classical approach with 16 tests 1 factor is examined and with a DOE approach 4. With an other matrix this could be even more!

Classical | DOE | |||||

Test | Factor 1 | Factor 2 | Factor 3 | Factor 4 | ||

1 | High | High | High | High | High | |

2 | High | High | High | High | Low | |

3 | High | High | High | Low | High | |

4 | High | High | High | Low | Low | |

5 | High | High | Low | High | High | |

6 | High | High | Low | High | Low | |

7 | High | High | Low | Low | High | |

8 | High | High | Low | Low | Low | |

9 | Low | Low | High | High | High | |

10 | Low | Low | High | High | Low | |

11 | Low | Low | High | Low | High | |

12 | Low | Low | High | Low | Low | |

13 | Low | Low | Low | High | High | |

14 | Low | Low | Low | High | Low | |

15 | Low | Low | Low | Low | High | |

16 | Low | Low | Low | Low | Low |

Of course it is possible to do the classical test with only 1 High and 1 Low test but then the statistical power is lower as with the DOE approach.

With more than 2 factors it is wise to use a DOE approach.

- The output can be measured, preferable in continuous scale!!
- The influencing factors are known
- Important Factors can be controlled (variated on a desired level or fixed on a constant level)
- Try to control Noise (uncontrollable factors) or record them (Environment temperature, Air pressure, different operators, ect)
- Keep the DOE simple as possible
- DO the Confirmation Run!

But the bigness of a lamp can have more influences from different factors with it own levels:

Factor | Levels |

Shape of lamp | Ball, Cone, Candle |

Power | 1, 2, 9, 30, 40 60 100Watt |

Setting on the dimmer | 0%…100% |

Input current | 0...230V |

Color of glass | Clear, White, Silver, Green, Red |

Type of lamp | Light bulb, LED, TL |

Armature | Silver reflector, White reflector, No reflector |

- Try to chose realistic values for the levels (not impractical high or low)
- Avoid impossible combinations of the levels with other factors in the experiment

Name | Full factorial | Response surface | Orthogonal |

Type | Full Factorial Array | Box-Behnken design | Orthogonal Array |

Central Composite design | Plackett–Burman Array | ||

Amount of tests | High (all combinations) | Medium | Small |

No interactions | |||

Usage | Simple | More complex | Simple no interactions |

Complex with interactions | |||

Interactions | All interactions | First level interactions | Yes possible |

Response Surface design | No | Yes | No |

- Full factorial with a small amount of factors or Orthogonal without interactions.

Test | Full Factorial Array | Box-Behnken design BB3 | Orthogonal Array L9 | |||||||||

Factor 1 | Factor 2 | Factor 3 | Factor 1 | Factor 2 | Factor 3 | Factor 1 | Factor 2 | Factor 3 | Factor 4 | |||

1 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | ||

2 | 0 | -1 | -1 | -1 | -1 | 0 | -1 | 0 | 0 | 0 | ||

3 | 1 | -1 | -1 | 1 | -1 | 0 | -1 | 1 | 1 | 1 | ||

4 | -1 | 0 | -1 | -1 | 1 | 0 | 0 | -1 | 0 | 1 | ||

5 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 1 | -1 | ||

6 | 1 | 0 | -1 | 0 | -1 | -1 | 0 | 1 | -1 | 0 | ||

7 | -1 | 1 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | ||

8 | 0 | 1 | -1 | 0 | -1 | 1 | 1 | 0 | -1 | 1 | ||

9 | 1 | 1 | -1 | 0 | 1 | -1 | 1 | 1 | 0 | -1 | ||

10 | -1 | -1 | 0 | 0 | 1 | 1 | ||||||

11 | 0 | -1 | 0 | -1 | 0 | -1 | ||||||

12 | 1 | -1 | 0 | -1 | 0 | 1 | ||||||

13 | -1 | 0 | 0 | 1 | 0 | -1 | ||||||

14 | 0 | 0 | 0 | 1 | 0 | 1 | ||||||

15 | 1 | 0 | 0 | 0 | 0 | 0 | ||||||

16 | -1 | 1 | 0 | |||||||||

17 | 0 | 1 | 0 | |||||||||

18 | 1 | 1 | 0 | |||||||||

19 | -1 | -1 | 1 | |||||||||

20 | 0 | -1 | 1 | |||||||||

21 | 1 | -1 | 1 | |||||||||

22 | -1 | 0 | 1 | |||||||||

23 | 0 | 0 | 1 | |||||||||

24 | 1 | 0 | 1 | |||||||||

25 | -1 | 1 | 1 | |||||||||

26 | 0 | 1 | 1 | |||||||||

27 | 1 | 1 | 1 |

Of course this small amount of test is not for free.... With the L9 orthogonal array interactions and Response Surface can't be calculated and the power is lower. The power can easily increased by adding repetitions.

2 Levels | 3 Levels | |||||

Factors | Full Factorial | Orthogonal | Full Factorial | Orthogonal | Box-Behnken | |

Test Runs | Test Runs | Test Runs | Test Runs | Test Runs | ||

2 | 4 | 4 | 9 | 9 | ||

3 | 8 | 4 | 27 | 9 | 16 | |

4 | 16 | 8 | 81 | 9 | 26 | |

5 | 32 | 8 | 243 | 18 | 45 | |

6 | 64 | 8 | 729 | 18 | 54 | |

7 | 128 | 8 | 2187 | 18 | ||

8 | 256 | 12 | 6561 | 27 |

Above 4 factors with 2 levels go for a Orthogonal array, and above 3 factors with 3 levels the advise is to go for an Orthogonal array. If interactions are needed a Box-Behnken design or Central Composite design. When building the samples for an Orthogonal array a reduced amount of different samples need to be build. If the power is to low the array can easily be replicated to increase the sample size.

When using an Orthogonal array interactions are not included. The interactions have to be included in the array this can be calculated with Develve see.

Be careful with adding interactions in a design:

- A interaction is not a common thing
- Adding interactions will increase the array size especially in a 3 level array (2 extra test runs)
- If it is not logic that an interaction between factors do not include it

Interaction with the typical crossing line | No interaction |

2 Levels | 3 Levels | |

Amount of tests | Low | Higher |

Non linear response | No | Yes |

Factor | Level 1 | Level 2 |

Size | 10mm | 20mm |

Weight | 1kg | 10kg |

Color | White | Black |

Factor | Level 1 | Level 2 |

Size | 10mm | 20mm |

Weight | 1kg | 10kg |

Color | White | Black |

Interaction 1 | Size | Weight |

Interaction 2 | Size | Color |

Interaction 3 | Weight | Color |

Open this array.

Now add the first interaction. Column C is now used for the interaction calculation. | The second interaction. Column E is also used for an interaction calculation. |

The third interaction. Column F is now also used for a interaction calculation. | The name of the factors and levels are added. |

As you can see the result is the same as a Full factorial array this is because all the interactions between the factors are tested.

Fist sample 10mm, 1kg and white etc.

Some important points

- Always build and test the complete array.
- When adding repetitions it is preferred is not to test replicates on a row (samples with the same setting) but first finish the first replicate then the next. This is to randomize the order to prevent drift over time in the result.
- When testing try not to test all factors on the first level than on the second level but try to randomize.
- Do not add an extra test in the array except Center points this will create an unbalanced array, and can lead to wrong results.

Data file

Looking to the data

**Size**20mm gives a higher output this factor is also significant according the Anova**Weight**Small influence not significant But it looks that it has a big influence on the variation! This can be seen in the Response graph Column D and E.**Color**Small not significant influence- The interaction between Size and Weight is significant. So if Weight is combined with Size Weight is significant (see Anova)! In the Response graph Column H and I the typical crossing lines that is an indication for an interaction.
- The interaction is the cause of the bigger variation of
**Weight**at 1 kg!

- The interaction is the cause of the bigger variation of

Weight and color are not significant.

After pooling Color (the least significant factor) and Weight are still not significant.

When a interaction is significant only set the level of the interaction do not set them individual. If you do both you set the influence of the factor twice!