Design and Analysis

of Experiments Part I: Introduction

Prof. Dr. Anselmo E de Oliveira

anselmo.quimica.ufg.br

anselmo.disciplinas@gmail.com

DOE Course

• Classes

– Theory and Lab

– CA A, 105

• Syllabus – anselmo.quimica.ufg.br

• Softwares

• Project

– 05/07 • Course topics

• Calculations

• Article – Full reference

– 2015 to 2017

– Qualis: A1... B3

The Analytical Process

Tools: Exploratory data analysis

Data mining

Calibration

Information/control theory

Optimization

Experimental design

Sampling theory

Luck

Information: chemical concentrations...

Measurements: voltages, currents, volumes...

Samples

System

Knowledge of properties of system

Fonte: M.A. Sharaf; D.L. Illman; B.R. Kowalski, Chemical Analysis: Chemometrics

Mechanistic and Empirical Models

• Mechanistic models – Scientific phenomena are so well

understood that useful results including mathematical models can be developed directly by applying these well-understood principles • Ex: Perfect gas law: 𝑃𝑉 = 𝑛𝑅𝑇

• Empirical models – Observation of the system at work

and experimentation are required to elucidate information about why and how it works

– Well-designed experiments can often lead to a model of system performance

General Model of a Process

Inputs Output

Uncontrollable factors

. . .

𝑧1 𝑧2 𝑧𝑞

𝒚, or 𝐘

. . .

𝑥1 𝑥2 𝑥𝑝

Process

Controllable factors

Strategy of Experimentation

• The objectives of the experiment may include the following – Determining which variables are most influential

on the response y

– Determining where to set the influential x’s so that y is almost near the desired nominal value

– Determining where to set the influential x’s so that the variability in y is small

– Determining where to set the influential y’s so that the effects of the uncontrollable variables z1, z2, ..., zq are minimized

• Usually, an objective of the experimenter is to determine the influence that these factors have on the output response of the system

• Strategy of experimentation – Analytical measurements

• sampling

• number of replicates

• pH

• solvent

• GC, MS, HPLC

• ...

• Best-guess approach – Selecting an arbitrary

combination of the factors, test them, and see what happens

• One-factor-at-a-time (OFAT) – Selecting a starting point, or

baseline set of levels, for each factor, and then successively varying each factor over its range with the other factors held constant at the baseline level

• Factorial experiment – All factors are varied togheter

Modeling

• All models are approximations – Mechanistic

– Empirical

• Sometimes an empirical model can suggest a mechanism – 𝑦 = 𝑓 𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑘

– or: 𝐗 = 𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑘

𝑦 = 𝑓 𝐗

Example: reaction inside two chemical reactors

• Yielding 170 oC < T < 190 oC

• Suposition #1

reactor 1

reactor 2 170 190 Temperature /oC

Yie

ldin

g

Reactor 1

Reactor 2 𝒚 = 𝜶𝟏 + 𝜷𝟏𝒙

𝒚 = 𝜶𝟐 + 𝜷𝟐𝒙

Example: reaction inside two chemical reactors

• Suposition #2: quadractic model

suposition #1

parallel curves

identical results for both reactors

• 𝜸𝟏 e 𝜸𝟐 = 𝟎

• 𝜷𝟏 = 𝜷𝟐 ; 𝜸𝟏 = 𝜸𝟐 ; 𝜶𝟏 𝜶𝟐

• 𝜶𝟏 = 𝜶𝟐 ; 𝜷𝟏 = 𝜷𝟐 ;

𝜸𝟏 = 𝜸𝟐

𝒚 = 𝜶𝟏 + 𝜷𝟏𝒙 + 𝜸𝟏𝒙𝟐

𝒚 = 𝜶𝟐 + 𝜷𝟐𝒙 + 𝜸𝟐𝒙𝟐

Graphical Representation: 2D and 3D Plots

𝒚 = 𝒇 𝒙𝟏

𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐

Graphical Representation: Contourn Plots

𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐

𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐

𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐, 𝒙𝟑

Graphical Representation: Surface Plots

𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐

Graphical Representation: 4D Plots

𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐, 𝒙𝟑

Gerlon de A.R. Oliveira, Anselmo E. de Oliveira, Edemílson C. da Conceição, Maria I.G. Leles “Multiresponse optimization of an extraction procedure of carnosol and rosmarinic and carnosic acids from rosemary” Food Chemistry 2016, 211, 465-476.

Some Applications of Experimental Design

• Evaluation and comparison of basic design configurations

Fonte: D.C. Montgomery, Design and Analysis of Experiments, 8th ed.

• Evaluation of material alternatives

Fonte: D.C. Montgomery, Design and Analysis of Experiments, 8th ed.

• Selection of design parameters so that the product will work well under a wide variety of field conditions, that is, so that the product is robust

Fonte: D.C. Montgomery, Design and Analysis of Experiments, 8th ed.

• Determination of key product design parameters that impact product performance

Fonte: D.C. Montgomery, Design and Analysis of Experiments, 8th ed.

• Formulation of new products

Fonte: D.C. Montgomery, Design and Analysis of Experiments, 8th ed.

Some Results of Experimental Design

The application of experimental design techniques early in process development can result in

• Improved process yields

• Reduced variability and closer conformance to nominal or target requirements

• Reduced development time

• Reduced overall costs

Guidelines for Designing Experiments

1. Recognition of and statement of the problem (a team approach to designing experiments is recommended)

– Factor screening or characterization Which factors have the most influence on the response(s) of interest?

– Optimization Find the settings or levels of the important factors that result in desirable values of the response

– Confirmation

– Discovery New materials

New factors, or

New ranges for factors

– Robustness Under what conditions do the response variables of interest seriously degrade?

What conditions would lead to unacceptable variability in the response variables?

2. Selection of the response variable – Average or standard deviation (or both)

– Decide how each response will be measured

– The gauge or measurement system capability (or measurement error)

– Identify issues related to defining the responses of interest and how they are to be measured before conducting the experiment

3. Choice of factors, levels, and range – Potential design factors

Design (selected), Held-constant, and Allowed-to-vary factors

– Nuisance factors Controllable, Uncontrollable (analysis of variance), and Noise factors

– Choose the ranges over which these factors will be varied and the specific levels at which runs will be made (process knowledge)

Factor screening or process characterization: keep the number of factor levels low

4. Choice of experimental design – Sample size (number of replicates)

– Selection of a suitable run order for the experimental trials

– Determination of whether or not blocking or other randomization restrictions are involved

– Empirical model First-order model: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝜀

Interaction term: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝛽12𝑥1𝑥2 + 𝜀

second-order model: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝛽12𝑥1𝑥2 + 𝛽11𝑥112 + 𝛽22𝑥22

2 + 𝜀

– Some of the factor levels will result in different values for the response. Identify which factors cause this difference and estimate the magnitude of the response change

5. Performing the experiment – Prior to conducting the experiments a few

trial runs or pilot runs are often helpful

6. Statistical analysis of the data – Results and conclusions must be objective

– Graphical methods

– Empirical model

7. Conclusions and recommendations – Follow-up runs and confirmation testing

A very efficient way to use statistical Design of Experiments (DoE) is to follow a sequential approach

A good rule of thumb

It is usually a major mistake to design a single, large, comprehensive experiment at the start of a study. As a general rule, no more than 25% of the available resources should be invested in the first experiment

Mude,

mas começe devagar,

porque a direção

é mais importante

do que a velocidade.

Clarice Lispector