Two-Level Factorial Designs

Introduction

Design of experiments in analytical chemistry is a term associated to the establishment of adequate process conditions in order to: improve the instrumental conditions and/or extract the highest information from the experimental data (Otto, 2017). Before apply a design of experiments strategy it is appropriate to consider the following questions (Miller & Miller, 2010): a) the initial knowledge of the system, what variables or factors (and their levels) are important to consider in our study? b) the weight of each variable in the system, what variable has a higher contribution? and are there interaction between variables? c) optimize the response, what is the response associated to the highest quality of the product/process? and d) evaluate the system robustness, what is the effect of uncontrolled variables?

A system can be considered as a conjunct of elements interdependent forming and integrating whole (Deming & Morgan, 1993). The system has an entrance (generally the analyte) expressed as a quantity or quality, the factors involved in the response of the system, an output variable (signal) influenced by the processes and chemical reactions which took place during the transformation of entrance into response.

The main drawback in all cases is the existence of factors so-called uncontrolled which are included in all the experiments, these factors cannot be eliminated and affect the response by adding a variability or variance to the output variable. The variance related to the response has then two contributions: variance of the signal and a second one known as uncertainty or noise. In this sense, the design of experiments can be applied to estimate and quantify the contributions of the controlled factors and minimize the uncontrolled (Figure 1).

Figure 1.

Scheme of a system

The proposed chapter is planned to be a tutorial for application and resolution of experimental problems based on the use of full and fractional (2-level) designs of experiments. Taking into account the proposed aim, we must consider the following steps commonly employed when it is applied design of experiments (Goupy, 1993):

• 1.

  Problem statement

  • a.

    Definition of the response or output variable which must describe adequately the process.

  • b.

    Identification of the factors (continue or discrete)

  • c.

    Selection of the levels and experimental domain

  • d.

    Identification of uncontrolled factors (noise)

  • e.

    Definition of signal/noise ratio

  • f.

    Evaluation of the system robustness

  • g.

    Selection of the experimental methodology (experimental arrangement)

• 2.

  Experimentation

  • a.

    Randomization

  • b.

    Blocks

  • c.

    Replicates

• 3.

  Results and analysis (interpretation)

  • a.

    Analysis of variance (ANOVA)

• 4.

  Optimization

  • a.

    Selection of the experimental conditions which generate the adequate signal/noise ratio

• 5.

  Conclusions

  • a.

    Feed-back