Mathematics and Statistics –
Design and Analysis of Experiments
• Where: Building 15, Level 4, Room 20 • Phone: 03 992 53158
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Course Objectives
• Develop a good working knowledge of the
basic concepts and models of experimental design.
• Learn how to properly analyse the results of a designed experiment in order to carry out the correct statistical analysis of the data.
• Develop the ability to interpret statistical results from an experiment and report them in non-technical language.
Course Resources
• Prescribed Textbook
– . Montgomery,
Design and Analysis of
Experiments, 8th Edition, Wiley. http://au.wiley.com/WileyCDA/ WileyTitle/productCd-EHEP002024.html
• Any scientific calculator is fine – Do not need graphical functions
Course Structure
• Twelve weeks
– Lectures – Labs
• Mondays 6:30pm – 8:30pm 056.03.095
• Laboratory Classes
– Computer based
– Need to have your laptop
– Mondays only weeks 2, 5, 7, 9, 11 – 8:30pm-9:30pm
• Room 056.04.086/089
Course Schedule
Course Assessments
• The 4 assignments will worth 10%, 10%, 15%, 15% respectively
• Total for all assignments : 50%
On line test worth 50%
Lab Assessments – R|SAS
Lab Assessment submission
• MATH1302 • Your name! • ID number
• Lecturer/teacher:
www.rmit.edu.au/ Students/forms
Design of Experiments
Part 1 – Introduction, Chapter 1, Text
• Why is this trip necessary?
• An abbreviated history of DOX
• Some basic principles and terminology
• The strategy of experimentation
• Guidelines for planning, conducting and analyzing experiments
Introduction to DOX
• An experiment is a test or a series of tests
• Experiments are used widely in the engineering
– Process characterization & optimization
– Evaluation of material properties
– Product design & development
– Component & system tolerance determination
• “All experiments are designed experiments, some are poorly designed, some are well-designed”
Engineering Experiments
• Reducetimeto design/develop new products & processes
• Improveperformanceof existing processes
• Improvereliabilityand performance of products
• Achieveproduct& process robustness
• Evaluation of materials, design alternatives, setting component & system tolerances, etc.
Four Eras in the History of DAE
The agricultural origins, 1908 – 1940s
– W.S. Gossett and the t-test (1908)
– R. A. Fisher & his co-workers
– Profound impact on agricultural science – Factorial designs, ANOVA
The first industrial era, 1951 – late 1970s
– Box & Wilson, response surfaces
– Applications in the chemical & process industries
The second industrial era, late 1970s – 1990
– Quality improvement initiatives in many companies
– Taguchi and robust parameter design, process robustness
The modern era, beginning circa 1990
Gosset (1876-1937)
Gosset’s interest in barley cultivation led him to speculate that design of experiments should aim, not only at improving the average yield, but also at breeding varieties whose yield was insensitive (robust) to variation in soil and climate.
Gosset was a friend of both and R.A. Fisher, an achievement, for each had a monumental ego and a loathing for the other.
Gosset was a modest man who cut short an admirer with the comment that “Fisher would have discovered it all anyway.”
R. A. Fisher (1890 – 1962)
The Basic Principles of DOX
• Randomization
– Running the trials in an experiment in random order – Notion of balancing out effects of “lurking” variables
• Replication
– Sample size (improving precision of effect estimation,
estimation of error or background noise)
– Replication versus repeat measurements? (see pages 12, 13)
• Blocking
– Dealing with nuisance factors – background variables
Strategy of Experimentation
• “Best-guess” experiments
– Used a lot
– More successful than you might suspect, but there are disadvantages…
• One-factor-at-a-time (OFAT) experiments
– Sometimes associated with the “scientific” or
“engineering” method
– Devastated by interaction, also very inefficient
• Statistically designed experiments
– Based on Fisher’s factorial concept
Experiment:
• Response variable – the measured outcome
• Factors (or variables) – deliberately and controllable changes to record the response
• Levels of a factor – conditions (values) of a factor
• Treatments–levels’combinationsofallfactors
• Effect on a response – measured changes on the response as conditions (treatments) are changed
• Experimental units – objects for which the response is measured.
• Background or Blocking variable – a variable that is of no interest to be examine but needs to be controlled
• Random error – uncontrollable variation of response
• Design matrix – a selected set of treatments.
• We perform an experiment to study the effect of four different sugar solutions on bacterial growth.
• Factors: Just one factor – The Sugar solution with different levels of contents.
• Treatments: (Level combinations) The four different sugar solutions.
• Experimental units: The bacterial.
• Response: Percentage increase of the
bacterial.
• Task: To compare the four different sugar solutions
In a agricultural experiment, we wish to determine the effects, if any, of four different varieties of wheat and three different fertilizers on the yield of wheat.
• Response variable: The yield (amount) of wheat harvest per square metre.
• Factors: Two factors – variety of wheat (with four levels), type of fertilizer (three levels).
• Experimental units: The wheat (square metre of planted wheat).
• Treatments: Combination of levels (4×3=12 different levels’ combination)
• Background variables:
• Task: We might wish to maximize the response.
Statistical Design of Experiment or Experimental Design
• The process of planning and applying the experiment so that “good” data to be collected. These data are expected to be easily analysed and give valid and objective results.
Factorial designs – Introduction
Experiment with two factors and six data
Factorial designs – Introduction
➢Experiment (one factor at a time) two factors and six data
An estimation of the effect of factor A is:
Factorial designs – Introduction
➢Experiment: Two factors with four measurements – data
An estimation of the effect of factor A is:
Factorial Designs
• In a factorial experiment, all possible combinations of factor levels are tested
• The golf experiment:
– Typeofdriver(oversized-regular)
– Type of ball (balata-three piece)
– Walking vs. riding
– Typeofbeverage(water-somethingelse)
– Timeofround(morning-afternoon)
– Weather (cool – hot)
– Type of golf shoe spike (metal-soft)
– Etc, etc, etc…
Factorial Design
Factorial Designs with Several Factors
Planning, Conducting & Analyzing an Experiment
1. Recognition of & statement of problem
2. Choice of factors, levels, and ranges
3. Selection of the response variable(s)
4. Choice of design
5. Conducting the experiment
6. Statistical analysis
7. Drawing conclusions, recommendations
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