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Statistically the most interesting, the fractional plans were the first generation of the plans of experiments. They allow to associate a good quality of the results while limiting the number of tests.

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: NK-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 Yates1, Box and Hunter2 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 24-1 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 27experiments 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, x1, x2 and x3, a complete plan will require 23 tests or 8. We want to set up only 4 tests for cost reasons. We’re going to achieve a fractional plane at 23-1. To build our matrix of effects, we start from the matrix of a complete plan 22 :

Test Matrix 22 or 23-1

Test number

A

X1

X2

X1x2

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 x1x2 interaction with the X3factor. Names the aliase generator ” Q “. The following matrix is obtained:

Trial number

A

X1

X2

X1x2 + x3

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 2Q 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 x1x2x3,x 2x3 and x1x3. 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 23-1

Trial number

A + x1x2x3

X1 + x2x3

X2 + x1x3

X1x2 + x3

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 X2 of the traditional table 22 , one can make the product of the signs of effects x1 with effects x3

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 24-1, it is considered that the interaction of order 4, X1 X2 X3 X4, and the 4 interactions of order 3, X2 X3 X4, X1 X3 X4, X1 X2 X4 and X1 X2 X3 are null. Take the following example:

We have the results

Applications of hypothesis 1

A + a1234 = 2.4125

A1 + a234 = 0.1125

A2 + a134 =-1.4125

A3 + a124 =-0.2625

A4 + a123 = 0.3125

A12 + a34 =-0.1625

A13 + a24 = 0.0875

A23 + a14 =-0.4875

By neglecting the interactions of order 3 and 4, we deduce:

A = 2.4125

A1 = 0.1125

A2 =-1.4125

A3 =-0.2625

A4 = 0.3125

A12 + a34 =-0.1625

A13 + a24 = 0.0875

A23 + a14 =-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 + a1234 = 2.4125

A1 + a234 = 0.1125

A2 + a134 =-1.4125

A3 + a124 =-0.2625

A4 + a123 = 0.3125

A12 + a34 =-0.1625

A13 + a24 = 0.0875

A23 + a14 =-0.4875

In conclusion of the above elements, we get the following model:

A = 2.4125

A1 = 0.1125

A2 =-1.4125

A3 =-0.2625

A4 = 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

A1 =-1

A2 =-6

A3 =-4

A12 =-0.25

A13 =-0.25

A23 = 0.25

A123 = 0

A + a123 = 27.25

A1 + a23 =-0.75

A2 + a13 =-6.25

A3 + a12 =-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 call3 :

  • 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 27-4, 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 27-3 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 25-1 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

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