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.
What is a Factor
A factor is a input for a experiment that can change the output when variating. Its like a dimmer (a factor) of a lamp when turning the knob the brightness of the lamp changes.
Why use a complex DOE instead of the standard approach?
With the classical approach only one input factor is changed to determine the influence of it on the output with a DOE more than one input is changed to see the influence of multiple factors on the output.
So the answer is to reduce the amount of experiments.
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!
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.
What is important for a successful DOE?
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!
The Level of the factor
The level of a factor is the input setting of test. For the lamp dimmer the setting of the dimmer is the level (0%, 25%, 50% ect).
But the bigness of a lamp can have more influences from different factors with it own levels:
Shape of lamp
Ball, Cone, Candle
1, 2, 9, 30, 40 60 100Watt
Setting on the dimmer
Color of glass
Clear, White, Silver, Green, Red
Type of lamp
Light bulb, LED, TL
Silver reflector, White reflector, No reflector
Settings of levels
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
The test Arrays
For a DOE there are various types of Arrays each with it own pros and cons. See table below.
When starting for the first time with a DOE start with a simple and small DOE
Full factorial with a small amount of factors or Orthogonal without interactions.
Amount of test with different arrays
There are various types of arrays the simple one is the Full Factorial array, this array consisted of all the combinations of the levels. The amount of test with a Full Factorial is high, therefor there are arrays designed to reduce this.
As visible for the orthogonal array the smallest amount of test is needed. With this small amount of test even 4 factors can be tested on 3 levels (-1,0 and 1) for the other arrays types only 3 factors can be tested on 3 levels!
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.
The choice between arrays for a DOE is between Full Factorial and a Fractional array (Orthogonal) if you not want to draw a Response surface. Looking to the table below it is clear that quite fast a Full Factorial array is not economical.
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.
A interaction is when the result is not the sum of two factors. With an interaction it can occur that the result of the to factors is lower or higher as the sum of the result.
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
with the typical crossing line
2 or 3 level design
Amount of tests
Non linear response
There are Orthogonal arrays with 2 or 3 levels and even with different amount of levels in one array. With Orthogonal Arrays it is possible to combining columns to get factors with more levels. When generating Full Factorial array for each factor an dedicated amount of levels can be chosen.
Example without interactions
We want to create a DOE with the following factors
There are 3 factors and in a L4 is space for 3 factors.
Example with interactions
We want to create a DOE with the following factors
There are 3 factors, in a L4 array is space for 3 Factors on 2 levels but we also want to include interactions so a bigger array is needed L8 is space for 7 factors.
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.
With this array all the factors and first order interaction can be tested. The not used columns can be removed (C, E, F and G).
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.
Building and testing the samples
Now build the samples according the array.
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.
Analyzing the result
Put the measurement data in the input table. In this case there is one replicate. See sort data in subgroups for how it is calculated.
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!