1. Design of Experiments Lecture I

2. Topics Today Review of Error Analysis
Theory & Experimentation in Engineering
Some Considerations in Planning Experiments
Review of Statistical formulas and theory
Begin Statistical Design of experiments (“DOE” or “DOX”)

3. Part 1: Review of Error Analysis
Uncertainty or “random error” is inherent in all measurements
Statistical basis
Unavoidable- seek to estimate and take into account
Can minimize with better instruments, measurement techniques, etc.

4. Review of Error Analysis
Systematic errors (or “method errors”) are mistakes in assumptions, techniques etc. that lead to non-random bias
Careful experimental planning and execution can minimize
Difficult to characterize; can only look at evidence after the fact, troubleshoot process to find source and eliminate

5. Graphical Description of Random and Systematic Error

6. Why do we need to estimate uncertainty and include in stated experimental values?
Probability of being wrong will influence process and/or financial decisions
Cost / benefit of accepting result as “fact”?
What would be the effect downstream as the uncertainty propagates through the process?
When comparing two values and determining if they are different
Overlap of uncertainty?
What is the probability that the difference is significant?

7. Stating Results +/- Uncertainty
Rule for Stating Uncertainties
Experimental uncertainties should almost always be rounded to one significant figure.
Rule for Stating Answers
The last significant figure in any stated answer should usually be of the same order of magnitude (in the same decimal position) as the uncertainty.
Express Uncertainty as error bars and confidence interval for graphical data and curve fits (regressions) respectively

8. Determining Propagated Error: Non-statistical Method
Compute from total differential

9. Propagated error
OR Can do sensitivity analysis in spreadsheet of other software program
Compute possible uncertainty in calculated result based on varying values of inputs according to the uncertainty of each input
Example: Use “Solver” optimization tool in Excel to find maximum and minimum values of computed value in a cell by varying the value of each input cell
Set constraint that the input values lie in the range of uncertainty of that value

10. Or Can Use repeat measurements to estimate uncertainty in a result using probability and statistics for random errors:
mean
standard deviation of each measurement
standard deviation of the mean of the measurements
Confidence intervals on dependant variable
Confidence intervals on regression parameters
1)Mean all familiar with- average or “best” value
2)std. dev. Is a good meas. Of variation in repeat meas.- tells how far apart individual measurements are from each other.
3) SDOM gives estimate of how much ind. Measurements vary form the mean value.
4) one std. dev defines a 66% CI. . ..if you take more meas., then 66% should lie within 1 std. dev.

11. Statistical Formulas from chapter 4 of Taylor

12. Relationship of standard deviation to confidence intervals

13. Confidence intervals on non-linear regression coefficients
Can be complex- use software but understand theory of how calculated for linear case

14. Error bars that represent uncertainty in the dependant variable

15. How measurements at a given x,y would be distributed for multiple measurements

16. Determining Slope and Intercept In Linear Regression

17. Confidence intervals (SD) on slope B and Intercept A

18. Regression Output in Excel
Simple ANOVA- we will be looking at more complex cases in DOE
Slope and intercept
Confidence limits (+/-) om slope & intercept

19. Confidence Intervals in TableCurve

20. Confidence Intervals in TableCurve

21. Regression in Polymath

22. Statistical Process Control
Very Widely Used
Used for quality control and in conjunction with DOE for process improvement
Control Charts provide statistical evidence
That a process is behaving normally or if something wrong
Serve as data output (dependant variable )from process in designed statistical experiments

23. Variation from expected behavior in control charts- similar to regression and point statistics
Control Limit is the mean of a well behaved process output (i.e. product)
Upper and lower Control Limits represent confidence limit on mean of “well behaved” process ouptut
Expect random deviations form mean just like in regression

24. Part 2: Theory and Experimentation in Engineering

25. Theory and Experimentation in Engineering
Two fundamental approaches to problem solving problems in the discovery of knowledge:
Theoretical (physical/mathematical modeling)
Experimental measurement
(Most often a combination is used)
Each application is different
But in General:
-math models: ALWAYS only an approximation of the “real world”
Good? Fair? Poor?
-Engineering as opposed to pure science: more money & time constraints
Theoretical: apply to whole class of problems
Experimental; more specific to application at hand
(Note example of correlations based on theoretical development- ie. Correlated data based on dimensionless groups, etc.)

26. Example of combination of theory and experimentation to get semi-empirical correlation
Here, the researchers have correlated data of several experimenters:
1) Have used similarity principles / Dimensional Analysis ( Buckingham Pi method) to reduce the number of variables
-theory, reason to determine the variables that the phenomena or process depends on
- Construct dimensionless “Pi variables
2) This process is somewhat complex: heat transfer in a fluidized bed (recall you should have done fluid part- model as many straight tubes, find a type of friction factor from model= Ergun Eqn, Then too the theoretical model needed to be curve fit to data to get coefficeints- LOOK UP)
-Did regression analysis of log transformed data to find exponents on dimensionless groups

27. Features of alternative methods
Theoretical Models
Simplifying assumptions needed
General results
Less facilities usually needed
Can start study immediately
Experimental approach
Study the “real world”-no simplifying assumptions needed
Results specific to apparatus studied
High accuracy measurements need complex instruments
Extensive lab facilities maybe needed
Time delays from building apparatus, debugging

28. Functional Types of Engineering Experiments
Determine material properties
Determine component or system performance indices
Evaluate/improve theoretical models
Product/process improvement by testing
Exploratory experimentation
Acceptance testing
Teaching/learning through experimentation
Mixing a chem E example, or polymer properties, etc.
Specifications for product: ie if you buy a stereo: signal/noise, frequency response, total harmonic distortion, etc
Self ex.
**VERY IMPORTANT- most applicable to what you might do
ie, what research faculty, grad students do
Large of Expensive pieces of equipment- before you pay, test it out (do your tests meet performance indices in #2?)
What we do in this class. . .

29. Some important classes of Experiments
Estimation of parameter mean value
Estimate of parameter variability
Comparison of mean values
Comparison of variability
Modeling the dependence of dependant Variable on several quantitative and/or qualitative variables

30. Practical Experimental Planning
Experimental design:
Consider goals
Consider what data can be collected.
Difficulty of obtaining data
What data is most important
What measurements can be ignored
Type of data: Categorical? Quantitative?
Test to make sure that measurements/apparatus are reliable
Collect data carefully and document fully in ink using bound notebooks. Make copies and keep separately
PLANNING SAVES YOU TIME!!!

31. Preview of Uses for DOE Lab experiments for research
Industrial process experiments
-first is similar to what we have been doing
-second is likely more relevant to what you may do in future

32. Four engineering problem classes to which DOE is applied in manufacturing
1. Comparison
2. Screening/ characterization
3. Modeling
4. Optimization
-will talk briefly about each and try to present examples
Ie two means the same for tow diff. processes?
Which factors important: ID the significant few from the many insignificant- complex manufacturing, etc.
Model to predict output given values of significant few inputs (response = dependant variable)
Make process give best product possible, or least expensive process, combination of the two, etc.

33. Comparison
Compares to see if a change in a single “factor” (variable) has resulted in a process change (ideally an improvement)
Ties in with what we discussed about SPC. Often SPC would be used as output data from a process to tell us whether the product or intermediate is better, etc.

34. Screening/Characterization
Used when you want to see the effect of a whole range of factors so as to know which one(s) are most important.
Two factorial experiments usually used
Recall when we talked about SPC we said that it was important to “understand your process”

35. Modeling
Used when you want to be able to construct a mathematical model that will predict the effect on a process of manipulating a variables or multiple variables
-example would be our Kla experiment
-continuous variables where in many process applications we are often talking about discreet of even binary factors

36. Optimization
When you want to determine the optimal settings for all factors to give an optimal process response.
For example in an industrial process, the optimal response might be the best product spec or the least cost to manufacture, or some optimal combination of those 2 responses, etc.