statistics and doe me 470 spring 2012. we will use statistics to make good design decisions! we will...
TRANSCRIPT
We will use statistics to make good design decisions!
We will categorize populations by the mean, standard deviation, and use control charts to determine if a process is in control.
We may be forced to run experiments to characterize our system. We will use valid statistical tools such as Linear Regression, DOE, and Robust Design methods to help us make those characterizations.
What is DOE
Sometimes we would like to model or optimize a design or process, but the “first principles” analytic approach (Conservation laws, etc.) is too challenging.Empirical (experimental) models are the best next choice. The organized approach using DOE provides a good modeling tool for optimization of designs and processes
Example
Suppose you are trying to get faster cycle time on an injection molder and are looking for the best settings for several temperatures and pressures.The transient heat transfer and transient fluid mechanics (with changing viscosity) problem is challenging. Instead we change the inputs in an organized way, measure the outputs, and develop an experimental model that can be optimized.
Here is an email that made my day!
I'm working on a project that is nearing data collection. The study is focused on Thumb-Tip force resulting from muscle/tendon force. We're working with cadaveric specimens so this is awesome lab work. Some of the relationships are expected to be nonlinear so we're looking 10 levels of loading for each tendon. We also wish to document first order and possible second order interactions between tendons if they are significant.
Last year with the human powered vehicle team we used minitab to create a test procedure for testing power output resulting from chain ring shape, crank length, and rider. There were 3 chain ring shapes, 3 different crank lengths. If possible, in this current study I would like to run a preliminary factorial experiments to determine which interactions are significant before exhaustively testing every combination at every level of loading. If such a method is appropriate it could save us a lot of time. Could you suggest a reference that I might be able to find at the library or on amazon?
DOE Objectives
At the end of this module, the participant will be able to :
Understand the basic concepts and advantages of designed experiments
Understand key terminology used in experimental design
Use different techniques to deal with noise in an experiment
Make good design decisions!!!
Statistical Design of Experiments
Chapter 13 of your text. Your text only looks at DOE in the context of Robust Design. This is one of several important uses of DOE: Determining Significant Factors Constructing a Model of System Performance
Design and Analysis of Experiments, Montgomery and Hines is an excellent reference.
Look at the Frisbee Thrower from Catapult
What Experiments Can Do
Characterize a Process/Product determines which X’s most affect the Y’s includes controllable and uncontrollable X’s identifies critical X’s and noise variables identifies those variables that need to be carefully
controlled provides direction for controlling X’s rather than control
charting the Y’s Optimize a Process/Product
determines where the critical X’s should be set determines “real” specification limits provides direction for “robust” designs
Definition of Terms
Factor - A controllable experimental variable thought to influence response (example air flow rate, or in the case of the Frisbee thrower: angle, tire speed, tire pressure)
Response - The outcome or result; what you are measuring (cycle time to produce one bottle, distance Frisbee goes)
Levels - Specific value of the factor (fast flow vs. slow flow, 15 degrees vs. 30 degrees)
Interaction - Factors may not be independent, therefore combinations of factors may be important. Note that these interactions can easily be missed in a straight “hold all other variables constant” scientific approach. If you have interaction effects you can NOT find the global optimum using the “OFAT” (one factor at a time) approach!
Replicate – performance of the basic experiment
How Can DOE Help?
run a relatively small number of tests to isolate the most important factors (screening test).
determine if any of the factors interact (combined effects are as important as individual effects) and the level of interaction.
predict response for any combination of factors using only empirical results
optimize using only empirical results determine the design space for simulation models
Trial and Error Problem : Current gas mileage is 23 mpg. Want 30
mpg. But what is the best we can get?
We might do the following : Change brand of gas Change octane rating Drive slower Tune-up car Wash and wax car Buy new tires Change tire pressure
Terminology :response variable / dependent variable - what is being measured/optimized ex. gas mileagefactor / main effect / independent variable - a controlled variable being studied at 2 or more levels during the experiment ex. brand of gas
From the previous example, here were some potential factors to investigate (factors can be either continuous or discrete) :
brand of gas (discrete)octane rating (continuous)driving speed (continuous)tires (discrete)tire pressure (continuous)
Terminology :factor levels / main effect levels - the values or settings of the variable being manipulated
ex. levels for brand of gas are Shell & Texacoex. levels for speed are 55 & 60ex. levels for octane are 85 & 90
One Factor at a Time
Can you explain the results? How many more runs would you need to figure out the
best combination of variables? If there were more variables, how many runs would it
take to get an optimized solution? What if there is a specific combination of two or more
variables that leads to the best mpg?
Problem: Gas mileage is 23 mpg
Speed Octane Tire Pressure Y = MPG55 85 30 2360 85 30 2455 90 30 2255 85 35 20
Baseline =>
Full Factorial Experiment
OFAT Runs
Problem: Gas Mileage is 23 mpg
What conclusion do you make now?
Speed Octane Tire Pressure Y = MPG55 85 30 2360 85 30 2455 90 30 2260 90 30 2855 85 35 2060 85 35 2155 90 35 2760 90 35 25
Full Factorial Experiment Full Factorial experiment consists of all possible combinations of the
levels of the factors Design Matrix is the complete specification of the experimental test runs,
as seen in the example below Treatment Combination is a specific test run set-up, consisting of a
specific combination of the factor levels
Design Matrix
A treatment combination
Response Variable
The variable that is measured and the object of the characterization or optimization (the Y)
Defining the response variable can be difficult Often selected due to ease of measurement Some questions to ask :
How will the results be quantified/analyzed? How good is the measurement system? What are the baseline mean and standard deviation? How big of a change do we care about? Are there several response variables of interest?
Factor
A variable which is controlled or varied in a systematic way during the experiment (the X)
Tested at 2 or more levels to observe its effect on the response variable(s) (Ys)
Some questions to ask : what are reasonable ranges to ensure a change in Y? knowledge of relationship, i.e. linear or quadratic, etc?
Examples material, supplier, EGR rate, injection timing can you think of others?
Randomization Randomization can be done in several ways :
run the treatment combinations in random order assign experimental units to treatment combinations
randomly an experimental unit is the entity to which a specific
treatment combination is applied Advantage of randomization is to “average out” the effects of
extraneous factors (called noise) that may be present but were not controlled or measured during the experiment spread the effect of the noise across all runs these extraneous factors (noise) cause unexplained
variation in the response variable(s)
Repetition and Replication
Repetition : Running several samples during one experimental setup (short-term variability)
Replication : Repeating the entire experiment (long-term variability)
You can use both in the same experiment
Repetition and Replication provide an estimate of the experimental error this estimate will be used to determine whether observed
differences are statistically significant
252015105130
120
110
100
Index
Yie
ldPressure : HHHH LLLL HHHH LLLL HHHH LLLL
Temp: HHLL HHLL HHLL HHLL HHLL HHLL
Pressure : HHHH LLLL HHHH LLLL HHHH LLLL
Temp: HHLL HHLL HHLL HHLL HHLL HHLL
Test Sequence
Repetition
Replication
252015105130
120
110
100
Index
Yie
ld
Pressure : HHHH LLLL HHHH LLLL HHHH LLLL
Temp: HHLL HHLL HHLL HHLL HHLL HHLL
Pressure : HHHH LLLL HHHH LLLL HHHH LLLL
Temp: HHLL HHLL HHLL HHLL HHLL HHLL
Test Sequence
3rdReplicate2nd
Replicate
1stReplicate
Steps in DOE
1. Statement of the Problem2. Selection of Response Variable 3. Choice of Factors and Levels
Factors are the potential design parameters, such as angle or tire pressure
Levels are the range of values for the factors, 15 degrees or 30 degrees
4. Choice of Design screening tests response prediction factor interaction
5. Perform Experiment6. Data Analysis
23 Factorial Design Example
Problem Statement: A soft drink bottler is interested in obtaining more uniform heights in the bottles produced by his manufacturing process. The filling machine theoretically fills each bottle to the correct target height, but in practice, there is variation around this target, and the bottler would like to understand better the sources of this variability and eventually reduce it.
Choice of Factors: The process engineer can control three variables during the filling process: (A) Percent Carbonation (B) Operating Pressure (C) Line Speed
Pressure and speed are easy to control, but the percent carbonation is more difficult to control during actual manufacturing because it varies with product temperature.
23 Factorial Design Example
Choice of Levels – Each test will be performed for both high and low levels
Selection of Response Variable – Variation of height of liquid from target
Choice of Design – Interaction effects Perform Experiment
Determine what tests are required using tabular data Determine the order in which the tests should be
performed
Determine which experiments should be performedRun A(% C) B(Pressure) C(Speed) Treatment
combinations
1 - - - (1)
2 + - - a
3 - + - b
4 + + - ab
5 - - + c
6 + - + ac
7 - + + bc
8 + + + abc
Determine Order of Experiments
Decided to run two replicates Requires 16 tests Put 16 numbers in a hat and draw out the numbers
in a random order Assume that the number 7 is pulled out first, then
run test 7 first. (% C low, Pressure high, line speed high)
What happens when you draw a 10? Minitab can do this for you automatically!!
Data for the Fill Height Problem(Average deviation from target in tenths of an inch)
Operating Pressure (B)
25 psi
Line speed (C)
30 psi
Line Speed (C)
(A) Percent
Carbonation
200
(bpm)
250
(bpm)
200
(bpm)
250
(bpm)
10 -3
-1
-1
0
-1
0
1
1
12 0
1
2
1
2
3
6
5
Data Gathered on Each Run
DOE Run A B C Data
1 (1) - - - -3,-1
2 (a) + - - 0,1
3 (b) - + - -1,0
4 (ab) + + - 2,3
5 (c) - - + -1,0
6 (ac) + - + 2,1
7 (bc) - + + 1,1
8 (abc) + + + 6,5
Low, Low, Low
Algebraic Signs for Calculating Effects
Treatment
Combination
Factorial Effect
(1) A B AB C AC BC ABC
(1) + - - + - + + -
a + + - - - - + +
b + - + - - + - +
ab + + + + - - - -
c + - - + + - - +
ac + + - - + + - -
bc + - + - + - + -
abc + + + + + + + +
I now know how this one is done!
Calculate Contrast for A
Procedure1. Add all runs where A
is positive.
2. Subtract all runs where A is negative.
3. The difference is called the contrast.
LowHigh AAContrast
Definition of terms in formulas
A,B,C are factors. a, b, c are the levels of factors A, B, C. In
our example we are only testing high and low, so a = b = c = 2.
n = the number of replicates which happens to be 2 in our example
ANOVA TABLESource of variation
Sum of squares
Degrees of freedom
Mean
Square
Fo Who cares?
A (a-1) Num = DOFA
Den = DOFMSE
B (b-1) Num = DOFB
Den = DOFMSE
C (c-1) Num = DOFC
Den = DOFMSE
AB (a-1)(b-1) Num = DOFAB
Den = DOFMSE
AC (a-1)(c-1) Num = DOFAC
Den = DOFMSE
BC (b-1)(c-1) Num = DOFBC
Den = DOFMSE
ABC (a-1)(b-1)(c-1) Num = DOFABC
Den = DOFMSE
Error abc(n-1)
Total abcn-1
n
contrastSS A
A 8
)( 2
A
AA DOF
SSMS
TSS
n
contrastSS B
B 8
)( 2
n
contrastSS BC
BC 8
)( 2
n
contrastSS AC
AC 8
)( 2
n
contrastSS ABC
ABC 8
)( 2
n
contrastSS C
C 8
)( 2
n
contrastSS AB
AB 8
)( 2
B
BB DOF
SSMS
C
CC DOF
SSMS
BA
ABAB DOF
SSMS
AC
ACAC DOF
SSMS
BC
BCBC DOF
SSMS
ABC
ABCABC DOF
SSMS
E
EE DOF
SSMS EffectsTE SSSSSS
E
A
MS
MS
E
B
MS
MS
E
C
MS
MS
E
AB
MS
MS
E
AC
MS
MS
E
BC
MS
MS
E
ABC
MS
MS
The effect is significant if Fo is greater than the value from the table.
a
i
b
j
c
k
n
lijkl
a
i
b
j
c
k
n
lijklT
yy
abcn
yySS
1 1 1 1...
2...
1 1 1 1
2
In English, y… is the sum of all data points. So SST is the (sum of the square of each data point) - (sum of all data points)2/(abcn)
Carbonation Example
Source of variation
Sum of squares
Degrees of freedom
Mean
Square
Fo Who cares?
% Carbon. (A)
36.00 1 36.00 57.60 5.32 for =.05 (95%)
Pressure (B) 20.25 1 20.25 32.40 11.26 for 99%
Line speed © 12.25 1 12.25 19.60
AB 2.25 1 2.25 3.6
AC 0.25 1 0.25 0.4
BC 1.00 1 1.00 1.6
ABC 1.00 1 1.00 1.6
Error 5.00 8 0.625
Total 78.00 15
>Stat>ANOVA>General Linear ModelSelect the response by double clicking
Select model terms by double clicking or by typing label,
c5*c6
>Stat>DOE>Factorial>Analyze Factorial Design
Select the Graphs tab to get the next screen
Select “Deviation”As Response
Term
Standardized Effect
AC
BC
ABC
AB
C
B
A
876543210
2.306Factor NameA % CarbonationB Pressure
C Line Speed
Pareto Chart of the Standardized Effects(response is Deviation from Target, Alpha = .05)
The Pareto Chart shows the significant effects. Anything to the right of the red line is significant at a (1-) level. In our case so we are looking for significant effects at the 0.95 or 95% confidence level. So what is significant here?
Residual
Perc
ent
10-1
99
90
50
10
1
Fitted Value
Resi
dual
6420-2
1.0
0.5
0.0
-0.5
-1.0
Residual
Fre
quency
1.00.50.0-0.5-1.0
6.0
4.5
3.0
1.5
0.0
Observation Order
Resi
dual
16151413121110987654321
1.0
0.5
0.0
-0.5
-1.0
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for Deviation from Target
Estimated Effects and Coefficients for Deviation from Target (coded units)
Term Effect Coef SE Coef T PConstant 1.0000 0.1976 5.06 0.001%Carbonation 3.0000 1.5000 0.1976 7.59 0.000Pressure 2.2500 1.1250 0.1976 5.69 0.000Line Speed 1.7500 0.8750 0.1976 4.43 0.002%Carbonation*Pressure 0.7500 0.3750 0.1976 1.90 0.094%Carbonation*Line Speed 0.2500 0.1250 0.1976 0.63 0.545Pressure*Line Speed 0.5000 0.2500 0.1976 1.26 0.242%Carb*Press*Line Speed 0.5000 0.2500 0.1976 1.26 0.242
S = 0.790569 PRESS = 20R-Sq = 93.59% R-Sq(pred) = 74.36% R-Sq(adj) = 87.98%
We could construct an equation from this to predict Deviation from Target. Deviation = 1.00 + 1.50*(%Carbonation) +1.125*(Pressure) + 0.875*(Line Speed) + 0.375*(%Carbonation*Pressure) + 0.125*(%Carbonation*Line Speed) + 0.250*(Pressure*Line Speed) + 0.250*(%Carbonation*Pressure*Line Speed)We can actually get a better model, which we will discuss in a few slides.
Mean o
f Devia
tion fro
m T
arg
et
1210
2
1
0
3025
250200
2
1
0
%Carbonation Pressure
Line Speed
Main Effects Plot (data means) for Deviation from Target
Practical Application
Carbonation has a large effect, so try to control the temperature more precisely
There is less deviation at low pressure, so use the low pressure
Although the slower line speed yields slightly less deviation, the process engineers decided to go ahead with the higher line speed - WHY???
We can also use Minitab to construct a predictive model!!
Estimated Effects and Coefficients for Deviation from Target (coded units)
Term Effect Coef SE Coef T PConstant 1.0000 0.1976 5.06 0.001%Carbonation 3.0000 1.5000 0.1976 7.59 0.000Pressure 2.2500 1.1250 0.1976 5.69 0.000Line Speed 1.7500 0.8750 0.1976 4.43 0.002%Carbonation*Pressure 0.7500 0.3750 0.1976 1.90 0.094%Carbonation*Line Speed 0.2500 0.1250 0.1976 0.63 0.545Pressure*Line Speed 0.5000 0.2500 0.1976 1.26 0.242%Carb*Press*Line Speed 0.5000 0.2500 0.1976 1.26 0.242
S = 0.790569 PRESS = 20R-Sq = 93.59% R-Sq(pred) = 74.36% R-Sq(adj) = 87.98%
It is recommended to delete items with P > 0.200
>Stat>DOE>Factorial>Analyze Factorial Design
Select this arrow to remove the 3-way interaction term.
Estimated Effects and Coefficients for Deviation from Target (coded units)
Term Effect Coef SE Coef T PConstant 1.0000 0.2041 4.90 0.001%Carbonation 3.0000 1.5000 0.2041 7.35 0.000Pressure 2.2500 1.1250 0.2041 5.51 0.000Line Speed 1.7500 0.8750 0.2041 4.29 0.002%Carbonation*Pressure 0.7500 0.3750 0.2041 1.84 0.099%Carbonation*Line Speed 0.2500 0.1250 0.2041 0.61 0.555Pressure*Line Speed 0.5000 0.2500 0.2041 1.22 0.252
S = 0.816497 PRESS = 18.9630R-Sq = 92.31% R-Sq(pred) = 75.69% R-Sq(adj) = 87.18%
Next term to remove
Here is the final model from Minitab with the appropriate terms.
Estimated Effects and Coefficients for Deviation from Target (coded units)
Term Effect Coef SE Coef T PConstant 1.0000 0.2030 4.93 0.000%Carbonation 3.0000 1.5000 0.2030 7.39 0.000Pressure 2.2500 1.1250 0.2030 5.54 0.000Line Speed 1.7500 0.8750 0.2030 4.31 0.001%Carbon*Press 0.7500 0.3750 0.2030 1.85 0.092
S = 0.811844 PRESS = 15.3388R-Sq = 90.71% R-Sq(pred) = 80.33% R-Sq(adj) = 87.33%
Deviation from Target = 1.000 + 1.5*(%Carbonation) + 1.125*(Pressure) + 0.875*(Line Speed) + 0.375*(%Carbonation*Pressure)
Estimated Effects and Coefficients for Deviation from Target (coded units).The term coded units means that the equation uses a -1 for the low value and a +1 for the high value of the data.
Deviation from Target = 1.000 + 1.5*(%Carbonation) + 1.125*(Pressure) + 0.875*(Line Speed) + 0.375*(%Carbonation*Pressure)
Let’s check this for %Carbonation = 10, Pressure = 30 psi, and Line Speed = 200 BPM%Carbonation is at its low value, so it gets a -1. Pressure is at its high value, so it gets +1, Line Speed is at its low value, so it gets a -1.
Deviation from Target = 1.000 + 1.5*(-1) + 1.125*(1)+ 0.875*(-1)+ 0.375*(-1*-1)
Deviation from Target = -0.625 tenths of an inch
How does this compare with the actual runs at those settings?
NEVER GIVE THIS SETTING TO PRODUCTION UNTIL YOU HAVE VERIFIED THE MODEL!
The engineer wants the higher line speed and
decides to put the target slightly negative. Why??