Stat 470-11 Stat 470-1 Introduction to the Design of Experiments Instructor: Derek Bingham, Office:...
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Transcript of Stat 470-11 Stat 470-1 Introduction to the Design of Experiments Instructor: Derek Bingham, Office:...
Stat 470-1 1
Stat 470-1Introduction to the Design of Experiments
Instructor: Derek Bingham,
Office: West Hall 451
Contact Information:Email: [email protected]
Phone: (734) 763-9294
Office hours: West Hall 443
Tuesday & Thursday, 12:30-2:00; others by appointment
Text: Experiments: Parameter Design and Optimization by Wu and Hamada
Stat 470-1 2
Stat 470 – Overview/Syllabus
• Coverage: Review of linear regression; most of Chapters 1-4, additional topics as needed, and Chapter 9 if time permits
• Course notes will be available on the web by 12:00 day of class…otherwise as handouts in class (www.stat.lsa.umich.edu/~dbingham/Stat470)
• Term Project– design, conduct, analyze, report on an experiment– Will be given out later
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Computing
• Computing– SPSS
– Any other package you like
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What is Experimental Design?
• Experiments performed in almost all fields of study
• Experiment is conducted to learn something about a process or system
• Designed experiment is a series of tests (experiments) where changes are made in the inputs to observe and identify the impact on the output
• Better understanding of how the factors impact the system allows the experimenter predict future values or optimize the process
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• Can consider a process as:
Inputs Process Output
• The input variables (usually called variables or factors) will be denoted will be denoted x1, x2,…, xp.
• The output variable (often called the response variable) will be denoted will be denoted by y.
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Example (Tomato Fertilizer)
• Experiment was conducted by a horticulturist• Has 2 types of fertilizer available for tomato production (A and B)
• Objective: Is one fertilizer better than the other – higher yield, on average?
• Has 11 tomato plots
• Experiment Procedure - specify fertilizer amounts each fertilizer; decide upon number of pots to receive each fertilizer; randomly assign fertilizer to pots
• Response: yield – pounds of tomatoes
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Some Definitions
• Factor: variable whose influence upon a response variable is being studied in the experiment
• Factor Level: numerical values or settings for a factor
• Treatment or level combination: set of values for all factors in a trial
• Experimental unit: object, to which a treatment is applied
• Trial: application of a treatment to an experimental unit
• Replicates: repetitions of a trial
• Randomization: using a chance mechanism to assign treatments to experimental units
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What is an Experiment Design?
• Suppose you are going to conduct an experiment with 8 factors
• Suppose each factor has only to possible settings
• How many possible treatments are there?
• Suppose you have enough resources for 32 trials. Which treatments are you going to perform?
• Design: specifies the treatments, replication, randomization, and conduct of the experiment
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Types of Experiments
• Treatment Comparisons: Purpose is to compare several treatments of a factor (have 3 diets and would like to see if they are different in terms of effectiveness)
• Variable Screening: Have a large number of factors, but only a few are important. Experiment should identify the important few. (we will focus on these!)
• Response Surface Exploration: After important factors have been identified, their impact on the system is explored to optimize response
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Types of Experiments
• System Optimization: Often interested in determining the optimum conditions (e.g., Experimenters often wish to maximize the yield of a process or minimize defects)
• System Robustness: Often wish to optimize a system and also reduce the impact of uncontrollable (noise) factors. (e.g., would like a fridge to cool to a set temperature…but the fridge must work in Florida, Alaska and Michigan!)
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Systematic Approach to Experimentation
• State the objective of the study
• Choose the response variable…should correspond to the purpose of the study
• Choose factors and levels
• Choose experiment design (purpose of this course)
• Perform the experiment
• Analyze data (design should be selected to meet objective and so analysis is efficient and easy)
• Draw conclusions
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Observation vs. Experimentation
• Data collection is not experimentation
• By observation, you can learn that lightning can cause fires
• By experimentation, you can learn that friction between certain materials can cause fires
• By more experimentation, we have learned how to make fire-starting reliable, cheap, easy, …
• By experimentation, we learn more and we learn faster
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The Need for Experimentation
• A doctor has impression that persons he gives Medicine A to recover more quickly than persons he gives Medicine B to.
• But, apparent difference could be due to:– luck, random variation, small sample sizes, … – bias in choice of medicine for patients– physical differences in people receiving A vs. B
• age, weight, sex, prior health, ….
• Clinical trials (experiments that control extraneous sources of variation and bias) are required to get a scientific assessment of Medicine A vs. Medicine B
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Three Principles
• Replication – each treatment is applied to experimental units that are representative of the population of interest– independent repetition of a trial
– provides a measure of “noise,” meaning:• experimental error -- the variability of experimental units which receive the
same treatment
– experimental error is the yardstick against which we compare different treatments
– increasing number of replicates decreases variance of treatment effects and increases the power to detect significant differences
– Replication provides a measure of experimental “noise” and the means for controlling the level of that noise. (More replication means less noise in averages.)
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Warning!Sometimes what looks like replication is not!
• Repeat measurements on one experimental unit is not replication• Measurements on multiple samples from one experimental unit is not
replication
• Example: two cake recipes. – Make one batch by recipe A; one batch by recipe B– Bake 12 cupcakes from each batch; measure fluffiness– The experimental unit is a batch;
• there has been no replication of either recipe A or recipe B• there is no valid comparison of recipe A to recipe B; apparent difference could
be random batch differences
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Principle 2 - Randomization
2. Randomization -- use of a chance mechanism (e.g., random number generator) to assign treatments to experimental units or to the sequence of experiments– provides protection against unknown lurking variables
– help justify the assumption of “independence” that will underlie many analyses
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Principle 3 - Blocking
3. Blocking -- run groups of treatments on homogenous units (block) to reduce variability of effect estimates and have more fair comparisons
– Example: To compare 4 varieties of corn, an experimenter could consider blocks of land of various soil types and terrain, subdivide each block into plots, and randomly assign the 4 varieties to plots in a block.
– Blocking:• controls variability due to soil types and terrain and allows varieties to be
compared within blocks
• broadens the scope of conclusions, e.g., by including variety of soil types and terrain in the experiment
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Case Study: Reliability of Wire Bonding on Integrated Circuits
• Process monitoring:– Sample ICs selected, pull-
tested
• Available data:– pull strengths from 1000s of
pull tests
• Success criterion:– pull strength > 2.5g
• Planned analysis:– fit a distribution to all the
data
– estimate reliability
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Designed Experiment to the Rescue
• Experimental Design:– 3 bonding operators
– 3 bonding machines
– 3 pull-test operators
– 2 IC packages per combination
– 48 wires per IC package
– All combinations = 2,592 observations (!)
Note: Unusually large experiment, but feasible in this case – many defective IC’s available and processing time is short
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Case Study (cont.)
• Analysis Findings:
– No appreciable difference among bonding machines
– Large and independent effects of bonding and pull-test operators. NOT GOOD!
Ave. Pull Strength - each the ave. of 288 observations.(grams)
Pull Test Operator
A B C
Bond. A 8.4 6.3 7.0
Op B 9.0 6.8 7.6 (noise std dev = 1.5 grams)
C 7.1 5.3 5.8
• Further conjectures and experiments led to improved consistency of manufacturing and testing techniques
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Case Study -- Messages
• People and procedures can have more of an influence on quality than machines.
• Think about possible sources of variability
• Use designed experiments to control and evaluate these sources of variability
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Assignment
• Review t-tests (2-sample and paired)
• Review Linear regression
• Review ANOVA
• These will be the fundamental analysis tools for this course.