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In essence, complete plans are the most effective in establishing a causal relationship. Very used, they quickly find their limit as soon as the number of factors increases.

## Introduction

The complete factorial plans are the first test planning tools that were developed at the beginning of the 20th century. Very effective because the only ones able to take into account all the interactions between factor, it finds on the contrary quickly their limit with regard to the number of tests to be carried out.

The number of experiments is calculated using the following formula:

n = nk

• N: number of level by factors
• K: Number of factors

To illustrate the rapid growth in the number of experiments, here is a comparison taking into account the fact that it takes 5 minutes per experiment:

• 3 Factors at 2 levels: 8 experiments be 40mn.
• 5 Factors at 2 levels: 32 experiments more than 2hr.
• 7 Factors at 2 levels: 128 experiments more than 10hr.
• 9 Factors at 2 levels: 512 experiments more than 42hr.
• Finally, if we want to try these same 9 factors but at 3 levels: 19 683 experiences more than 69 days.

And just for example, if you want to study 15 factors on 3 levels, we would need 14 348 907 experiments, which would take us over 136 years. Might as well say unachievable.

## Building The matrix of experiments

In the context of use for screening, the construction of the table follows the Yates algorithm and is developed as follows:

• The first column is an alternate sequence of – and +, starting with a –
• The second column is an alternating suite of 2 – and 2 +, starting with –
• The third column is an alternating suite of 4 – and 4 +, starting with –

Example: The matrix for 4 factors

 Test No. Factor 1 Factor 2 Factor 3 Factor 4 1 -1 -1 -1 -1 2 +1 -1 -1 -1 3 -1 +1 -1 -1 4 +1 +1 -1 -1 5 -1 -1 +1 -1 6 +1 -1 +1 -1 7 -1 +1 +1 -1 8 +1 +1 +1 -1 9 -1 -1 -1 +1 10 +1 -1 -1 +1 11 -1 +1 -1 +1 12 +1 +1 -1 +1 13 -1 -1 +1 +1 14 +1 -1 +1 +1 15 -1 +1 +1 +1 16 +1 +1 +1 +1

## Construction of the effects 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 there to take into account in the calculation, the constant has .

### Step 2: Integrating interactions

We add as many columns as there are interactions of order 2, 3… For 3 factors, we will add 4 columns.

The most and least of each of the columns of the interactions are the result of the product of the + and the-associated factors.

In the end, we get the following matrix for a plan 23:

 a A1X1 A2X2 A3X3 A12x1x2 A13x1x3 A23x2x3 A123x1x2x3 +1 -1 -1 -1 +1 +1 +1 -1 +1 +1 -1 -1 -1 -1 +1 +1 +1 -1 +1 -1 -1 +1 -1 +1 +1 +1 +1 -1 +1 -1 -1 -1 +1 -1 -1 +1 +1 -1 -1 +1 +1 +1 -1 +1 -1 +1 -1 -1 +1 -1 +1 +1 -1 -1 +1 -1 +1 +1 +1 +1 +1 +1 +1 +1

## Source

R. A. Fisher (1925) – Statistical Methods for research workers

R. A. Fisher (1935) – The design of experiments

W. G. Cochran, G. M. Cox (1957) – Experimental Design

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