Introduction
Fractional planes make it possible to drastically reduce the number of tests compared to a complete plan. Based on the theory of “aliases “, they allow to reduce the number of tests by 2, 4, 8… The number of tests is reduced and is calculated as follows: N^{K-Q}
- N: The number of factor levels. Usually 2.
- K: The number of factors in total.
- Q: the number of factors aliased in the initial model
Historical
Building on the work of Yates^{1}, Box and Hunter^{2} develop fractional planes at 2 levels (screening) at the end of the Second World War.
Aliase and contrast concept
The theory of aliases is the basis of the implementation of a fractional plan. Mastering this theory is essential to constructing fractional planes and above all knowing how to interpret the results.
This technique consists of bringing together the unknown from the experiment plan (the effects of different factors and interactions) in groups. Groups that are then analyzed using approximation assumptions.
They are grouped in such a way that there is n unknown. So we solve a system of n equations to n groups of coefficients called aliases. For example, the factor C for a table 2^{4-1} is aliased with the ABD interaction. In other words, the coefficient that is actually calculated is a coefficient that will demonstrate the effect of C and ABD. So we do not have a ” pure ” result. This phenomenon of addition is called contrast.
We understand immediately that, indeed, the results cannot be interpreted directly without making a number ofassumptions that we detail a little further.
Building The matrix of experiments
Fractional planes use predesign tables that are chosen based on what we want or can do and the degree of precision we want.
For example, we want to study seven factors. If we had to make a complete experience plan, we would have to do 2^{7}experiments or 128. For reasons of cost and delay, we cannot do the same. We look in the summary table of fractional planes, the plan that best fits our situation.
For 7 factors, we have the choice between 4 plans of experiments that need respectively 8, 16, 32 or 64 experiments. This according to their ” resolution “. Obviously, the higher the resolution, the better the quality of the result will be.
Construction of the effects matrix
The construction of the Matrix also responds to the Yates algorithm and is similar to the construction of the complete plan matrix.
Step 1: Take into account the constant a_{ }
First we add a column to the left of the matrix with only + 1. This column is the to take into account in the calculation, the constant has_{ }.
Step 2: Identify the aliase generator
The difference from a complete plan is based on the allocation of columns for interactions. We explain this through the following example:
For 3 factors, x_{1}, x_{2} and x_{3}, a complete plan will require 2^{3} tests or 8. We want to set up only 4 tests for cost reasons. We’re going to achieve a fractional plane at 2^{3-1}. To build our matrix of effects, we start from the matrix of a complete plan 2^{2} :
Test Matrix 2^{2} or 2^{3-1}
Test number |
A_{ } |
X_{1} |
X_{2} |
X_{1}x_{2} |
1 |
+1 |
-1 |
-1 |
+1 |
2 |
+1 |
+1 |
-1 |
-1 |
3 |
+1 |
-1 |
+1 |
-1 |
4 |
+1 |
+1 |
+1 |
+1 |
Since we have 3 main factors, we replace the x_{1}x_{2} interaction with the X_{3}factor. Names the aliase generator ” Q “. The following matrix is obtained:
Trial number |
A_{ } |
X_{1} |
X_{2} |
X_{1}x_{2} + x_{3} |
1 |
+1 |
-1 |
-1 |
+1 |
2 |
+1 |
+1 |
-1 |
-1 |
3 |
+1 |
-1 |
+1 |
-1 |
4 |
+1 |
+1 |
+1 |
+1 |
It should be noted that a complete plan is therefore ” cut ” in 2^{Q} times. For example, if we only have one generator, we will have 2 times less experience than a complete plan. If we have 2 generators, we will have 4 times less experience…
Step 3: Identify aliases
We still have to find the location of the other 3 interactions x_{1}x_{2}x_{3},_{x 2}x_{3} and x_{1}x_{3}. It has been seen in the complete plans, the construction of the signs of the interactions is calculated by making the product of the 2 variables. Thus, if A =-1 and B =-1, on the same line AB interaction will have the + sign. Thus, to build our “aliases”, we will try to find out which columns of the main effects correspond to this or that interaction. By following this logic, we will build our aliases and deduce the matrix of effects:
Effects Matrix 2^{3-1}
Trial number |
A_{ } + x_{1}x_{2}x_{3} |
X_{1} + x_{2}x_{3} |
X_{2} + x_{1}x_{3} |
X_{1}x_{2} + x_{3} |
1 |
+1 |
-1 |
-1 |
+1 |
2 |
+1 |
+1 |
-1 |
-1 |
3 |
+1 |
-1 |
+1 |
-1 |
4 |
+1 |
+1 |
+1 |
+1 |
Thus, to have the column X_{2} of the traditional table 2^{2} , one can make the product of the signs of effects x_{1} with effects x_{3}…
Interpret the results
Step 11 of a plan of experiments, all fractional planes pose the same problem of interpretation of the results. Since not all the experiments of the complete plan are carried out, the value of all the effects of the main factors and the interactions can not be obtained independently.
To interpret the results, work hypotheses are ” introduced ” that will have to be verified before concluding on the results:
- hypothesis 1: Interactions of order 3 (interaction between 3 factors) or more are considered to be negligible.
- hypothesis 2: If an aliase -related effect is zero, it can mean two things:
That the effects of the factors and the interactions of the aliase are all null. This is the most probable hypothesis and it is the one we will hold in the general case.
That the effects of the factors and the interactions of the aliase are compensating. This hypothesis is unlikely and we will not hold it.
- hypothesis 3: If two major effect coefficients are weak, it will be assumed that their interaction is also low.
- hypothesis 4: If the effect of a major factor is strong and another weak, then it will be assumed that their interaction is low.
- hypothesis 5: If two effects are strong, one will be wary of their interaction that can also be.
In the case of fractional plans, the conclusions are still somewhat “ uncertain “. The construction of this type of plan based on a number of assumptions, there is always a probability, that there are approximations. It is therefore highly recommended to validate the model with other tests that come out of the experimental field.
Example of interpretation
1. Neglecting high-order effects
At first, and this is called hypothesis 1, we will neglect the interactions of the highest orders. For a plan 2^{4-1}, it is considered that the interaction of order 4, X_{1} X_{2} X_{3} X_{4}, and the 4 interactions of order 3, X_{2} X_{3} X_{4}, X_{1} X_{3} X_{4}, X_{1} X_{2} X_{4} and X_{1} X_{2} X_{3} are null. Take the following example:
We have the results |
Applications of hypothesis 1 |
A_{ } + a_{1234} = 2.4125 A_{1} + a_{234} = 0.1125 A_{2} + a_{134} =-1.4125 A_{3} + a_{124} =-0.2625 A_{4} + a_{123 }= 0.3125 A_{12} + a_{34} =-0.1625 A_{13} + a_{24} = 0.0875 A_{23} + a_{14} =-0.4875 |
By neglecting the interactions of order 3 and 4, we deduce: A_{ } = 2.4125 A_{1} = 0.1125 A_{2} =-1.4125 A_{3} =-0.2625 A_{4} = 0.3125 A_{12} + a_{34} =-0.1625 A_{13} + a_{24} = 0.0875 A_{23} + a_{14} =-0.4875 |
2. Interpretation of the effects
We will now interpret the effects and implement hypotheses 2, 3, 4 and 5 of work.
- for hypothesis 2: We have no zero effect, so we do not have to use it.
- for hypothesis 3: The effects of 1, 3, and 4 are observed to be low. So it is assumed that their associated interactions are weak. We then neglect interaction 13, 14, and 34.
- for hypothesis 4: interactions 12, 23 and 24 are neglected, since only the effect of factor 2 is strong.
- Finally, in our case, we can only apply hypothesis 5 since we do not have 2 strong effects (one does not consider a_{ })
We were following step 1 |
Applications of Assumptions 2, 3, 4 and 5 |
A_{ } + a_{1234} = 2.4125 A_{1} + a_{234} = 0.1125 A_{2} + a_{134} =-1.4125 A_{3} + a_{124} =-0.2625 A_{4} + a_{123 }= 0.3125 A_{12} + a_{34} =-0.1625 A_{13} + a_{24} = 0.0875 A_{23} + a_{14} =-0.4875 |
In conclusion of the above elements, we get the following model: A_{ } = 2.4125 A_{1} = 0.1125 A_{2} =-1.4125 A_{3} =-0.2625 A_{4} = 0.3125 |
3. A priori Conclusion
By applying the different working assumptions, we realize that we ” neglect ” all the interactions between factor. So our model is simplified to the maximum.
To validate it, we can at first put in place some additional tests. In our case, we want to validate that interactions 1 and 4 are negligible. We will therefore perform tests by varying successively 1 and 4 then 1 and 4 at the same time and compare the results with our prediction model. If there is no gap, then the interaction is very negligible.
Complete plan/Fractional plan comparison
Let’s take the case where we have three factors that we want to study. Here is the complete table with the associated answers:
Full factorial Plan
Fractional plan
If we had achieved a fractional plan, we would have achieved only 4 tests, the 4 corresponding tests, in order, in the tests 5, 2, 3 and 8 of the complete plan:
Results table
Full factorial Plan |
Fractional plan |
A_{ } = 27.25 A_{1} =-1 A_{2} =-6 A_{3} =-4 A_{12} =-0.25 A_{13} =-0.25 A_{23} = 0.25 A_{123} = 0 |
A_{ } + a_{123} = 27.25 A_{1} + a_{23} =-0.75 A_{2} + a_{13} =-6.25 A_{3} + a_{12} =-4.25 |
Comparing with the results of the complete plan, one observes well the fact that in the case of a fractional plane the effect of the interaction adds to the main effect of the phenomenon of aliases. This is called a contrast. So we do not have a pure effect. Without mastery of aliases and without knowledge of the phenomenon, it is very difficult to interpret the results.
The concept of resolution
It is understood that the only way to set up the aliases to reduce the number of tests is not sufficient. There is also a ” resolution ” system in place that identifies how to set up aliases. The principle is: at the highest resolution is high, at least the main effects and low-order interactions are aliased. In other words, at the most the resolution is high, at the most the accuracy is great, but on the contrary, the more the test number increases.
So we call^{3} :
- resolution III: The main effects are not confused with other effects but only with interactions of order 2 and more. This is the least ” accurate ” resolution system in the sense that we have here the main effects that are aliased with many interactions. example, for a plan factorial 2^{7-4}, resolution III, we have a which is aliased with BD + CE + FG + BCG + CDF + DEG + ABCF + ABEG + ACDG + ADEF + abcde + ABDFG + ACEFG + BCDEFG.
- resolution IV: The main effects are not confused with other main effects or double interactions. They are simply aliased with 3-and more-order interactions. example, for a plan factorial 2^{7-3} Resolution IV, we have a which is aliased with BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG.
- resolution V: the main effects and the double interactions are not confused. For example, for a plan 2^{5-1} resolution V, one has a which is aliased with BCDE and AB with CDE.
It should be noted that setting up a fractional plan is ” profitable ” from a resolution of III. Below, resolution II and I, the accuracy is statically no longer sufficient to have significant results. This is why there are no such proposals in the table or in the software.
Source
1 – Yates (1937) – The design and analysis of factorial experiments
2 – G. E. P. Box, W. G. Hunter, J. S. Hunter (1978) – Statistics for experimenters, an introduction to design, data analysis and model building
3 – ISO Standard 3534-3 (2013) – Plans of experiments
S. Vivier (2009) – Method of experience plans