Post on 10-Feb-2017
2k factorial DOECenter Points
Blocks+1
Blocks+1
F aa c t o
C
o r B
-1 Factor A +1
-1 Week 3
Knorr-Bremse Group
About this Module
We know two ways to make 2 level designsWe know two ways to make 2 level designs more robust and more informative.
Center points to check for linearity
Incorporate and evaluate additional input as aIncorporate and evaluate additional input as a block factor
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 2/27
Introduction of Center Points• We always are at the risk to overlook non linear relations
within the factor settings when using DOE’s with two factor levelslevels.
• The use of center points is an effective way to easily test for linearity (curvature)for linearity (curvature).
• An example:
– A chemical engineer wants to improve the yield. Two inputs are effecting the yield: reaction time and reaction temperaturetemperature.
– The chemical engineer decides to run an 2 x 2 design and adds center points in order to proof the linearity ofand adds center points in order to proof the linearity of this model.
Inputs:– Inputs:
• Reaction temperature: 150, 155 and 160 (°C)
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 3/27
• Reaction time: 30, 35 and 40 (minutes)
The DOE with 2 Factors and 3 Center Points
StdOrder CenterPt Temp. Time Yield 11 1 150 30 39,3
n = 4
mean = 40,425
1 1 150 30 39,32 1 160 30 403 1 150 40 40,94 1 160 40 41 5
,
n = 3
4 1 160 40 41,55 0 155 35 40,36 0 155 35 40,5
mean = 40,56 0 155 35 40,57 0 155 35 40,7
We want to know if there is a deviation between the actual center point and the theoretical value?
CenterPoints.mtw
( )2
centerpfactorcenterpfactor
Curvature nn
yynnSS
+−
=( )
345,40425,403*4
SS2
Curvature +−
=
centerpfactornn + 3
0096,0SSCurvature
=
L t thi l i Mi it b
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 4/27
Lets run this example in Minitab…
Create a DOE Stat>DOE>Factorial >Create factorial designs…
Open worksheet: center points mtw
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 5/27
Open worksheet: center points.mtw
Evaluation with Center PointsFactorial Fit: Yield 1 versus Temp.; Time
Estimated Effects and Coefficients for Yield 1 (coded units)
Stat>DOE>Factorial
Estimated Effects and Coefficients for Yield 1 (coded units)
Term Effect Coef SE Coef T PConstant 40,4250 0,1000 404,25 0,000
>Analyze Factorial Designs…
Temp. 0,6500 0,3250 0,1000 3,25 0,083Time 1,5500 0,7750 0,1000 7,75 0,016Temp.*Time -0,0500 -0,0250 0,1000 -0,25 0,826Ct Pt 0,0750 0,1528 0,49 0,672Ct Pt 0,0750 0,1528 0,49 0,672
S = 0,2 PRESS = *R-Sq = 97,26% R-Sq(pred) = *% R-Sq(adj) = 91,77%
Analysis of Variance for Yield 1 (coded units)
Source DF Seq SS Adj SS Adj MS F Pq j jMain Effects 2 2,82500 2,82500 1,41250 35,31 0,0282-Way Interactions 1 0,00250 0,00250 0,00250 0,06 0,826Curvature 1 0,00964 0,00964 0,00964 0,24 0,672
R id l E 2 0 08000 0 08000 0 04000Residual Error 2 0,08000 0,08000 0,04000Pure Error 2 0,08000 0,08000 0,04000
Total 6 2,91714
How do we decide?
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 6/27
How do we decide?
Main Effect Plot
Main Effects Plot for Yield 1Data Means
Stat>DOE>Factorial
41,25
41 00
Temp. TimeCornerCenter
Point Type
Data Means>Factorial Plots>Main Effect Plots…
41,00
40,75
n 40,50
40,25
Me
an
40,00
39,75
16015515039,50
403530
The actual center points don’t deviate from linearity significantly.
Therefore the null hypothesis is accepted
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 7/27
Therefore the null hypothesis is accepted.
Center Point Showing a Significant EffectRun the same experiment for the response column: yield 2.
Factorial Fit: Yield 2 versus Temp.; Time p ;
Estimated Effects and Coefficients for Yield 2 (coded units)
Term Effect Coef SE Coef T PConstant 40,4250 0,1000 404,25 0,000Temp. 0,6500 0,3250 0,1000 3,25 0,083Time 1 5500 0 7750 0 1000 7 75 0 016Time 1,5500 0,7750 0,1000 7,75 0,016Temp.*Time -0,0500 -0,0250 0,1000 -0,25 0,826Ct Pt 2,0750 0,1528 13,58 0,005
S = 0,2 PRESS = *R-Sq = 99,22% R-Sq(pred) = *% R-Sq(adj) = 97,67%
Analysis of Variance for Yield 2 (coded units)Analysis of Variance for Yield 2 (coded units)
Source DF Seq SS Adj SS Adj MS F PMain Effects 2 2,8250 2,82500 1,41250 35,31 0,028
i2-Way Interactions 1 0,0025 0,00250 0,00250 0,06 0,826Curvature 1 7,3811 7,38107 7,38107 184,53 0,005
Residual Error 2 0,0800 0,08000 0,04000Pure Error 2 0,0800 0,08000 0,04000
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 8/27
, , ,Total 6 10,2886
Main Effect Plots
42,5
Temp. TimeCornerCenter
Point Type
Main Effects Plot for Yield 2Data Means Stat
>DOEF t i l
42,0
41,5
an
Center >Factorial >Factorial Plots>Main Effect Plots…
41,0
40,5
40 0
Me
a
160155150
40,0
39,5403530
Temp Time P i t T
Main Effects Plot for Yield 1Data Means
41,25
41,00
40,75
Temp. TimeCornerCenter
Point Type
40,50
40,25
40 00
Me
an
160155150
40,00
39,75
39,50403530
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 9/27
Exercise with Real Data• Goal: Investigate the effects of concentration, ratio of B/A and
temperature on the yield
O t t ( ) Yi ld i %• Output (response): Yield in %
• Inputs:
C t ti l t hi h– Concentration → low; center; high
– Ratio B/A → low; center; high
T t l t hi h– Temperature → low; center; high
• Design: 2x2x2 factorial experiment with center points
P d• Procedure:
– Use the Minitab file: 3 fact. center points.mtw
A l f t i t ti d i ff t– Analyze for curvature, interaction and main effects
– Analyze the effects with the appropriate graphics
P f di ti– Perform diagnostics
– Calculate R² (How is the variation distributed)
P t lt d l i
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 10/27
– Present your results and conclusions
DOE Including BlocksThere are two ways to incorporate block factors!
1 The block factor is Advantage: fewer runs (effort)1. The block factor is confounded with the effect of the 3-way interaction
Advantage: fewer runs (effort)
Disadvantage: limited evaluation/information
A B C A*B*C Blocks-1 -1 -1 -1 11 -1 -1 1 2-1 1 -1 1 2
evaluation/information
1 1 -1 -1 1-1 -1 1 1 21 -1 1 -1 1-1 1 1 -1 11 1 1 1 2
A B C Blocks-1 -1 -1 11 -1 -1 1
1 1 1 1 2-1 1 -1 11 1 -1 1-1 -1 1 11 -1 1 11 1 1 1-1 1 1 11 1 1 1-1 -1 -1 21 -1 -1 21 1 1 22 Treat the block factor like -1 1 -1 21 1 -1 2-1 -1 1 21 -1 1 2-1 1 1 2
2. Treat the block factor like any independent factor.
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 11/27
1 1 1 21 1 1 2
Blocks Confounded with InteractionRun A B C A*B A*C B*C A*B*C Block
1 -1 -1 -1 1 1 1 -1 I2 1 -1 -1 -1 -1 1 1 II3 -1 1 -1 -1 1 -1 1 II3 -1 1 -1 -1 1 -1 1 II4 1 1 -1 1 -1 -1 -1 I5 -1 -1 1 1 -1 -1 1 II6 1 -1 1 -1 1 -1 -1 I7 -1 1 1 -1 -1 1 -1 I8 1 1 1 1 1 1 1 II
R A B C Bl kExample: 2 types of catalysts as block factor Run A B C Block1 -1 -1 -1 I4 1 1 -1 I6 1 -1 1 I
Example: 2 types of catalysts as block factorRule of thumb: High order interactions seldom contribute to the model.
7 -1 1 1 I
2 1 -1 -1 II
Therefore we overlay the 3-way interaction with an additional factor (catalyst)We assign catalyst A to the low level setting of
3 -1 1 -1 II5 -1 -1 1 II8 1 1 1 II
g y gthe 3-way interaction and catalyst B to the high level. We sacrificed the 3-way interaction to save 8 runs8 runs.
Be aware: in a real experiments we like to randomize the DOE within the blocks Minitab is taking care of that
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 12/27
blocks. Minitab is taking care of that.
Blocks Confounded with Interaction
1 Type I
212 Type II
12 12+1
12C
-11 2
-1 A +1
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 13/27
-1 A +1
Evaluation of an Example
• Goal: Analyze a DOE with 4 factors and 2 blocks
• Example: A chemical engineer wants to maximize the filtration rate of a p gchemical product which is produced in a pressure vessel. For the experiment he needs 16 runs. Only 8 runs per day are possible. The complete experiment requires 2 dayscomplete experiment requires 2 days.
• Output/response: Filtration rate in l/h
• Inputs:p
– Temperature
– Pressure
– Formaldehyde concentration
– Agitation speed
– Block variable day 1 vs. day 2
• Procedure:
U th Mi it b fil Bl k filt t DOE ith 4 f t 2 bl k– Use the Minitab file: Block filter.mtw, DOE with 4 factors, 2 blocks and 16 runs
– Analyze the data
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 14/27
a y e t e data
The Minitab Worksheet
Minitab file: Block filter.mtw
Blocks Temp Pressure F-Concentr. Agitator Filter rate1 1 -1 -1 -1 269 81 1 -1 -1 -1 269,81 -1 1 -1 -1 182,41 -1 -1 1 -1 258,41 1 1 1 -1 2471 1 1 1 1 2471 -1 -1 -1 1 163,41 1 1 -1 1 395,21 1 -1 1 1 326,81 -1 1 1 1 2662 -1 -1 -1 -1 1712 1 1 -1 -1 2472 1 -1 1 -1 2282 -1 1 1 -1 3042 1 -1 -1 1 3802 1 1 1 1 1712 -1 1 -1 1 1712 -1 -1 1 1 2852 1 1 1 1 364,8
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 15/27
The Evaluation
The factor pressure has A
31,82Factor Name
Pareto Chart of the Effects(response is Filtration rate, Alpha = ,05)
no significant effect.
We reduce the model
m B
ABD
C
D
AD
AC
A
A gitator
A TempB PressureC F -C oncentrationD
by this factor.Term
CD
ACD
ABC
BC
BCD
B
Effect
AB
BD
9080706050403020100
Lenth's PSE = 12,11252 26
Pareto Chart of the Standardized Effects(response is Filtration rate, Alpha = ,05)
AC
A
2,26Factor NameA TempC F -C oncentrationD A gitator
Term
D
AD
C
1086420
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 16/27
Standardized Effect
The EvaluationEstimated Effects and Coefficients for Filtration rate (coded units)
Term Effect Coef SE Coef T PConstant 266 24 4 337 61 39 0 000Constant 266,24 4,337 61,39 0,000Block -2,61 4,337 -0,60 0,562Temp 82,17 41,09 4,337 9,47 0,000F-Concentration 37,53 18,76 4,337 4,33 0,002Co ce t at o 3 ,53 8, 6 ,33 ,33 0,00Agitator 55,58 27,79 4,337 6,41 0,000Temp*F-Concentration -68,88 -34,44 4,337 -7,94 0,000Temp*Agitator 63,17 31,59 4,337 7,28 0,000
S = 17,3474 R-Sq = 96,73% R-Sq(adj) = 94,55%
Analysis of Variance for Filtration rate (coded units)
Source DF Seq SS Adj SS Adj MS F PBlocks 1 109 2 109 2 109 2 0 36 0 562Blocks 1 109,2 109,2 109,2 0,36 0,562Main Effects 3 44997,7 44997,7 14999,2 49,84 0,0002-Way Interactions 2 34939,4 34939,4 17469,7 58,05 0,000Residual Error 9 2708,4 2708,4 300,9Residual Error 9 2708,4 2708,4 300,9Total 15 82754,7
E l ti f ti l tti b f ll i h
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 17/27
Explanations for optimal settings by following graphs …
Interaction Plots
If the model shows significant effects of interactions we have to interpret these first.
Stat>DOE>Factorial
Interaction plots include all effects of the factors involved!
>Factorial Plots…>Interaction Plot
325 T
Interaction Plot for Filter rateData Means
380
Interaction Plot for Filter rateData Means
325
300
275
250
-11
Temp 380
360
340
320
-11
Temp
250
225
200
Me
an 300
280
260
240
Me
an
1-1
175
150
F-Concentr.1-1
220
200
Agitator
The optimal combination: High temperature and high agitation speed with a low formaldehyde concentration
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 18/27
with a low formaldehyde concentration
Multi Vari Chart
Stat>Quality Tools>Multi-Vari Chart>Multi Vari Chart…
Multi-Vari Chart for Filter rate by Temp - Agitator
400
1-1
-1 1-11
Temp
350
300ate
1
300
250Filt
er
ra
200
1-1F-Concentr.
Panel variable: Agitator
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 19/27
Residual DiagnosticsStat>DOE>Factorial >Analyze Fact…>Graph>Residual Plots 99
N 16 20
Normal Probability Plot Versus Fits
Residual Plots for Filter rate
>Four in one 90
50
10
Per
cent
N 16AD 0,369P-Value 0,384 10
0
-10Res
idua
l
40200-20-40
10
1
Residual400350300250200
-20
Fitted Value
Histogram Versus Order
4
3
2
requ
ency
20
10
0
Res
idua
l
20100-10-20
1
0
Residual
Fr
16151413121110987654321
-10
-20
Observation Order
R
The residuals don’t show any obvious pattern, the residuals are normal distributed Conclusion: The model is acceptable!
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 20/27
normal distributed. Conclusion: The model is acceptable!
Calculation of Components of VarianceStat> Anova> GLM…
Source DF Seq SS R2
Temp 1 27011 33%
F-concentr. 1 5633 7%
Agitator 1 12354 15%
Temp*F-Concentr 1 18975 23%Temp*F Concentr. 1 18975 23%
Temp*Agitator 1 15964 19%
E 10 2818 3%Error 10 2818 3%
Total 15 82755
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 21/27
Summary
In this module we have discussed how to use:
• Center points
Bl k• Blocks
This allows us to broaden the use of 2 factor levelThis allows us to broaden the use of 2 factor level designs. As a result we are more confident regarding our statementsstatements.
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 22/27
Appendix:Appendix:Evaluation of the Evaluation of the
examplep
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The Graphical Evaluation with Minitab
4,303Factor NameA C t ti
Pareto Chart of the Standardized Effects(response is Yield, Alpha = ,05)
File: 3 fact. Center points.mtw
m
AC
AB
A A C oncentrationB Rel. B/AC Temp.
Term
BC
B
C
Standardized Effect
ABC
543210
2 447
Pareto Chart of the Standardized Effects(response is Yield, Alpha = ,05)
A
2,447Factor NameA C oncentrationB Rel. B/A
Step 1: Reduce the model from the overview to get the
Term AB
gbest model. B
6543210
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 24/27
Standardized Effect
The Reduced ModelF t i l Fit Yi ld C t ti R l B/AFactorial Fit: Yield versus Concentration; Rel. B/A
Estimated Effects and Coefficients for Yield (coded units)
Term Effect Coef SE Coef T PConstant 87,382 0,2573 339,65 0,000Concentration -3,500 -1,750 0,3017 -5,80 0,001Co ce t at o 3,500 , 50 0,30 5,80 0,00Rel. B/A 0,700 0,350 0,3017 1,16 0,284Concentration*Rel. B/A -0,800 -0,400 0,3017 -1,33 0,226
S = 0,853260 PRESS = 13,6690R-Sq = 84,00% R-Sq(pred) = 57,09% R-Sq(adj) = 77,15%
Analysis of Variance for Yield (coded units)
Source DF Seq SS Adj SS Adj MS F PSource DF Seq SS Adj SS Adj MS F PMain Effects 2 25,4800 25,4800 12,7400 17,50 0,0022-Way Interactions 1 1,2800 1,2800 1,2800 1,76 0,226Residual Error 7 5,0964 5,0964 0,7281Curvature 1 0,0297 0,0297 0,0297 0,04 0,857Pure Error 6 5,0667 5,0667 0,8444
Total 10 31,8564 File:
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 25/27
3 fact. Center points.mtw
Residual DiagnosticsStat>DOE>Factorial >Analyze Factorial Designs
File: 3 fact. Center points.mtw
>Analyze Factorial Designs>Graphs…>Residual Plots >Four in one Residual Plots for Yield
or
99
90
ent
N 11AD 0,465P-Value 0,203
1,0
0,5
ual
Normal Probability Plot Versus Fits
S
or
210-1-2
50
10
1
R id l
Per
ce
9089888786
0,0
-0,5
-1,0
Fitt d V l
Res
id
Stat>Regression>Regression…>Graphs… “Four in one”
Residual Fitted Value
6,0
4,5y
1,0
0 5
Histogram Versus Order
Graphs… Four in one
Store first the
4,5
3,0
1,5
0 0
Freq
uenc
y 0,5
0,0
-0,5
-1,0
Res
idua
l
Store first the residuals and fits
1,00,50,0-0,5-1,0-1,50,0
Residual1110987654321
Observation Order
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 26/27
Interpretation of ResultsStat>DOE>Factorial >F t i l Pl t
File: 3 fact. Center points.mtw
>Factorial Plots>Main Effect
Concentration Rel B/A P i t T
Main Effects Plot for YieldData Means
89
88
87
Concentration Rel. B/ACornerCenter
Point Type
10-1
87
86
10-1
Me
an
Temp.89
88
87
p
10-1
86
Only the concentration is significant. The position of the center points supports the assumption that the model is linear within the factor settings
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 27/27
supports the assumption that the model is linear within the factor settings.