Javier Garcia - Verdugo Sanchez Six Sigma Training. Week 4. Multiple Regression.

1. Page 1/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
50403020100
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Input
Output
Multiple
Regression
Week 4
Page 2/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
We will extend the experience in regression to
applications with more than one predictor.
You will learn additional functions in Minitab
Evaluation of more complex examples
About this Module

2. Page 3/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Results
Outputs
Y1, Y2 …
ResultsResults
OutputsOutputs
Y1, Y2 …Y1, Y2 …
Examples
• Ambient temperature
• Air pressure
• Relative Humidity
• Raw material condition
Example
• Different operators.
• Different machines
• Different shifts
ProcessProcess
Noise inputs
(Continuous)
Noise inputsNoise inputs
(Continuous)(Continuous)
Noise inputs
(Discrete)
Noise inputsNoise inputs
(Discrete)(Discrete)
Tools
• Scatter plots
• Correlation
• Regression
Tools
• Scatter plots
• Correlation
• Regression
Example
• Temperature
• Pressure
• Time
Controllable
Inputs
ControllableControllable
InputsInputs
Tools
• Correlation
• Regression
• ANOVA
Tools
• Correlation
• Regression
• ANOVA
Tools
• Measurements
system
• Process
capability
Tools
• Measurements
system
• Process
capability
A View into the Tool Box
General tools
C & E - Matrix
FMEA
Tools
• Box plots
• Diagram of main
effects & interaction
• ANOVAs, T-test
Tools
• Box plots
• Diagram of main
effects & interaction
• ANOVAs, T-test
Page 4/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Factor X = Input
Discrete / Attributive Continuous / Variable
ResponseY=Output
Discrete
Attributive
Continuous
Variable
Chi - Square
Logistic
Regression
T - Test
ANOVA ( F - Test)
Median Tests
Regression
Statistical techniques for all combination of data types are availableStatistical techniques for all combination of data types are available
Validation of Factors Y = f(x)

3. Page 5/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
During a reaction a loss of
ammoniac is observed.
During a study following inputs
are manipulated or observed:
Water temperature
Amount of airflow
Nitric acid concentration
File: Multiple1.mtw
An Example
Goal:
We want to control the loss of
ammoniac between 20 – 30%!
Experiment Air amount Water temp Acid conc NH3 loss
1 80 27 89 42
2 80 27 88 37
3 75 25 90 37
4 62 24 87 28
5 62 22 87 18
6 62 23 87 18
7 62 24 93 19
8 62 24 93 20
9 58 23 87 15
10 58 18 80 14
11 58 18 89 14
12 58 17 88 13
13 58 18 82 11
14 58 19 93 12
15 50 18 89 8
16 50 18 86 7
17 50 19 72 8
18 50 19 79 8
19 50 20 80 9
20 56 20 82 15
21 70 20 91 15
Page 6/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Stat
>Basic Statistics
>Correlation…
Stat
>Basic Statistics
>Correlation…
Air amount and water temperature have a
strong relation to the NH3 loss.
On the other side both of them have a
correlation to each other
Step 1: Correlation
Results for: MULTIPLE1.MTW
Correlations: NH3 loss; Acid concentration; Water temp; Air amount
NH3 loss Acid concent Water temp
Acid concent 0,400
0,073
Water temp 0,876 0,391
0,000 0,080
Air amount 0,920 0,500 0,782
0,000 0,021 0,000
Cell Contents: Pearson correlation
P-Value

4. Page 7/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
>Stat
>Regression
>Best Subsets…
>Stat
>Regression
>Best Subsets…
Minitab shows the best combination of the
factors.
Air amount and water temperature explain 90%
of the variation.
Step 2: Best Combination
Best Subsets Regression: NH3 loss versus Air amount; Water temp; ...
Response is NH3 loss
A W A
i a c
r t i
e d
a r
m c
o t o
Vars R-Sq R-Sq(adj) C-p S u e n
1 84,6 83,8 13,3 4,0982 X
1 76,7 75,4 28,9 5,0427 X
2 90,9 89,9 2,9 3,2386 X X
2 85,1 83,4 14,4 4,1442 X X
3 91,4 89,8 4,0 3,2434 X X X
Page 8/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Step 3: Fitted Line Plot
Water temp
Airamount
28262422201816
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75
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65
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55
50
S 5,86458
R-Sq 61,1%
R-Sq(adj) 59,1%
Fitted Line Plot
Air amount = 12,59 + 2,268 Water temp
Air amount
NH3loss
80757065605550
45
40
35
30
25
20
15
10
5
S 4,09824
R-Sq 84,6%
R-Sq(adj) 83,8%
Fitted Line Plot
NH3 loss = - 44,13 + 1,020 Air amount
Water temp
NH3loss
28262422201816
45
40
35
30
25
20
15
10
5
S 5,04272
R-Sq 76,7%
R-Sq(adj) 75,4%
Fitted Line Plot
NH3 loss = - 41,91 + 2,817 Water temp
Acid concentration
NH3loss
959085807570
45
40
35
30
25
20
15
10
5
S 9,56540
R-Sq 16,0%
R-Sq(adj) 11,6%
Fitted Line Plot
NH3 loss = - 47,96 + 0,7590 Acid concentration
>Stat
>Regression
>Fitted Line Plot…
>Stat
>Regression
>Fitted Line Plot…

5. Page 9/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Is there an indication of an quadratic effect?
We may receive better information from the residual plots.
How would the quadratic term be indicated in the residual plots?
Step 3: Fitted Line Plot
Air amount
NH3loss
80757065605550
45
40
35
30
25
20
15
10
5
S 4,09824
R-Sq 84,6%
R-Sq(adj) 83,8%
Fitted Line Plot
NH3 loss = - 44,13 + 1,020 Air amount
Water temp
NH3loss
28262422201816
45
40
35
30
25
20
15
10
5
S 5,04272
R-Sq 76,7%
R-Sq(adj) 75,4%
Fitted Line Plot
NH3 loss = - 41,91 + 2,817 Water temp
Page 10/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
>Stat
>Regression
>Regression…
>Stat
>Regression
>Regression…
Minitab gives us the equation to calculate the
NH3 loss.
The water temperature in the cooling tower is
known and fix.
Step 4: Presentation of the Regression
Regression Analysis: NH3 loss versus Air amount; Water temp
The regression equation is
NH3 loss = - 50,4 + 0,671 Air amount + 1,30 Water temp
Predictor Coef SE Coef T P
Constant -50,359 5,138 -9,80 0,000
Air amou 0,6712 0,1267 5,30 0,000
Water te 1,2954 0,3675 3,52 0,002
S = 3,239 R-Sq = 90,9% R-Sq(adj) = 89,9%
Analysis of Variance
Source DF SS MS F P
Regression 2 1880,44 940,22 89,64 0,000
Residual Error 18 188,80 10,49
Total 20 2069,24
Source DF Seq SS
Air amou 1 1750,12
Water te 1 130,32

6. Page 11/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Step 5: Residual Analysis
840-4-8
99
90
50
10
1
Residual
Percent
N 21
AD 0,175
P-Value 0,912
40302010
6
3
0
-3
-6
Fitted Value
Residual
6420-2-4-6-8
4,8
3,6
2,4
1,2
0,0
Residual
Frequency
2018161412108642
6
3
0
-3
-6
Observation Order
Residual
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for NH3 loss
Page 12/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
RSM Results
Lets remember: RSM is an application of the multiple regression.
If we can describe the factors than we can also use the RSM to
get a contour plot for further evaluation.
At a given water temperature the air amount can be determined to
control the NH3 loss between 20 an 30 %.
At a given water temperature the air amount can be determined to
control the NH3 loss between 20 an 30 %.
Water temp
Airamount
2624222018
80
75
70
65
60
55
50
NH3
22
24
26
28
30
loss
32
34
36
16
18
20
Contour Plot of NH3 loss vs Air amount; Water temp

7. Page 13/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Step 2: Polymerization
Actions
1. Calculate limit evaporation temperature to control the supplier process
2. Describe root causes for consumption peaks.
3. Redefine measurement and control techniques
Define
Measure
Analyze
Improve
Control
Define
Measure
Analyze
Improve
Control
Polymerization
Output
Water consumption
Input
Prod. Volume
Outside temp.
Humidity
Flow temp.
Lactam recovery
Vacuum
Temperature
Volume
Control valve
Example: Cooling Water Cost
Supplier provides conditioned cool
water. What are the possibilities to
reduce the consumption?
Page 14/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
530490450410370
95%Confidence Interval for Mu
525515505495485475465
95%Confidence Interval for Median
Verbrauch Poly 2-12.2000
472,052
24,825
467,151
Maximum
3rd Quartile
Median
1stQuartile
Minimum
N
Kurtosis
Skewness
Variance
StDev
Mean
P-Value:
A-Squared:
519,896
59,500
511,683
547,000
520,250
484,500
472,000
420,000
12
0,335590
-1,0E-01
1228,08
35,044
489,417
0,552
0,289
95%Confidence Interval for Median
95%Confidence Interval for Sigma
95%Confidence Interval for Mu
Anderson-Darling Normality Test
C2:2
Descriptive Statistics
Baseline 2000: 490 m³/h
Goal Poly 2001: 400 m³/h
Impact Comp + Poly = 50 K€
Define
Measure
Analyze
Improve
Control
Define
Measure
Analyze
Improve
Control
Example: Cooling Water Cost

8. Page 15/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
8
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25201510
100
90
80
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60
50
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30
20
10
Outside temperature
Humidity
Contour Plot of Evaporation temperature
Regression Evaporation temp. = - 9,52 + 0,130 Humidity + 0,825 Outside temp
Predictor Coef StDev T P
Constant -9,5238 0,3079 -30,93 0,000
Humidity 0,129522 0,002565 50,50 0,000
Outside temp 0,82498 0,01474 55,98 0,000
S = 0,6988 R-Sq = 98,5% R-Sq(adj) = 98,5%
Analysis of Variance
Source DF SS MS F P
Regression 2 2776,2 1388,1 2842,21 0,000
Residual Error 87 42,5 0,5
Total 89 2818,7
Define
Measure
Analyze
Improve
Control
Define
Measure
Analyze
Improve
Control
Example: Cooling Water Cost
Ambient temperature + humidity are defining the
evaporation temperature!
This results in a defined cool water flow
temperature.
Page 16/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Outputs for local
focus:
Width
Length
Unbalanced
Inputs: Measurements of key parameters at
different process steps.
Correlation between inputs is possible
Example:X-Ray Tube
The focal spot is one of the critical X-ray tube parameters. 8 X-ray
tubes has been produced and measured under controlled conditions.
The results are listed in the worksheet. File: Diabolo.mtw
How can we describe the connections between inputs and outputs?
Diabolo PL1 RL2 RL3 Para3.1 RL4 PL5 PL6 PL7 Dis8 Dis9 Width Length Unbal.
V 07 0,0059 0,0053 0,0562 0,0586 0,0657 0,052 0,09 0,0085 -5,0143 -19,372 0,87 0,92 0,3
S 08 0,0207 0,0435 0,1548 0,0474 0,0244 0,0607 0,1721 0,0359 -5,0017 -19,468 * * 0,38
D 06 0,01 0,0375 0,0364 0,1382 0,1194 0,0553 0,34 0,0159 -4,9977 -19,535 0,91 1,12 0,3
W 06 0,0144 0,0265 0,0878 0,0175 0,1932 0,2755 0,0357 0,0155 -5,0401 * 0,88 0,88 0,17
H 04 0,0151 0,0528 0,059 0,2724 0,0694 0,0432 0,6126 0,0204 -5,2124 -19,105 1,19 0,91 0,56
T 03 0,0167 0,0081 0,1257 0,0692 0,0394 0,0906 0,1996 0,0248 -5,2065 -19,308 1,31 1,06 0,65
G 05 0,0087 0,0399 0,1423 0,02 0,0916 0,3755 0,0842 0,0079 -5,2101 -19,119 1,1 0,84 0,46
M 04 0,0052 0,0278 0,0417 0,0039 0,0646 0,1568 0,0749 0,0107 -5,2089 -19,376 0,95 1,06 0,11

9. Page 17/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
File:
CorrelationCorrelation Xraytube.mtwXraytube.mtw
There is a high amount of customer complaints because of noisenoise.
Can we describe a connection between noise and the inputs?
The following data has been collected over a period of 3 months.
Example: Noise and Vibration
Tube Numb Run noise cold Noise Tube cold Noise Tube warm Plate unbal Rotor unbal Axial end play Radial run out Axial run out
4224 89 69,4 74,5 0,2 0,3 180 0,06 0,04
4217 84 69,3 62,1 0,3 0,3 180 0,04 0,04
4276 90 72,6 72,9 0,4 0,4 180 0,05 0,02
4272 90 74,8 73,2 0,4 0,4 195 0,03 0,03
4277 88 73,7 68,3 0,2 0,3 175 0,02 0,04
4279 90 70,8 66,7 0,3 0,3 185 0,03 0,03
4278 90 72,6 65,6 0,4 0,3 195 0,03 0,03
4262 88 71,9 66,8 0,3 0,2 200 0,03 0,04
4250 88 69,5 67,5 0,3 0,3 180 0,04 0,04
4263 88 69,1 70,7 0,2 0,2 180 0,04 0,02
4251 85 69 68,2 0,5 0,3 170 0,03 0,01
4259 88 73,6 69,5 0,3 0,3 190 0,05 0,05
4248 83 67,6 63 0,2 0,3 190 0,03 0,02
4267 90 75,2 69,4 0,3 0,4 195 0,05 0,04
4218 88 74,1 65,5 0,2 0,3 200 0,02 0,05
Page 18/1802 BB W4 Multiple Regression 07, D. Szemkus/H. Winkler
Process output is the corrosion (surface oxidation)
Example: Zinc phosphating process
Best Subsets Regression: Corrosion % versus FS Ultratec; Fe Phosph.; ...
Response is Corrosion
F F G F F F T P A S T
S e S S S e E E e I h n c e
n n m s o t h m
U P P P A A t t p t s i i p
l h h h k k f f . z p c c
t o o o t t e e e h o h U
r s s s i i t t P i a r t l
Vars R-Sq R-Sq(adj) C-p S a p p p v v t t h t t i d t
1 14,7 11,3 4,2 31,351 X
1 13,3 9,8 4,7 31,614 X
2 25,0 18,8 3,0 30,009 X X
2 22,0 15,5 3,9 30,613 X X
3 36,6 28,4 1,2 28,181 X X X
3 32,2 23,4 2,7 29,147 X X X
4 44,5 34,4 0,7 26,965 X X X X
4 43,4 33,1 1,1 27,236 X X X X
5 49,6 37,6 1,1 26,306 X X X X X
5 48,5 36,2 1,5 26,587 X X X X X
6 56,7 43,8 0,8 24,966 X X X X X X
6 52,0 37,7 2,3 26,289 X X X X X X
7 58,7 43,5 2,2 25,030 X X X X X X X
7 57,7 42,1 2,5 25,325 X X X X X X X
8 60,2 42,5 3,7 25,251 X X X X X X X X
8 59,6 41,7 3,9 25,428 X X X X X X X X
9 61,3 40,8 5,4 25,607 X X X X X X X X X
9 60,9 40,2 5,5 25,742 X X X X X X X X X
10 61,9 38,1 7,2 26,192 X X X X X X X X X X
10 61,8 37,9 7,2 26,238 X X X X X X X X X X
11 62,3 34,6 9,1 26,929 X X X X X X X X X X X
11 62,1 34,4 9,1 26,970 X X X X X X X X X X X
12 62,4 30,2 11,0 27,813 X X X X X X X X X X X X
12 62,3 30,0 11,0 27,864 X X X X X X X X X X X X
13 62,4 24,9 13,0 28,858 X X X X X X X X X X X X X
13 62,4 24,8 13,0 28,863 X X X X X X X X X X X X X
14 62,4 18,6 15,0 30,035 X X X X X X X X X X X X X X