Analysis of Covariates in DOE Experiments to Improve Process Understanding

/1503 BB W4 Covariates 07, D. Szemkus/H. Winkler
2520151050
10
5
0
Introduction into the
Analysis of Covariates
Advanced DOE Tool
for Black Belts Week 4
Page 2/1503 BB W4 Covariates 07, D. Szemkus/H. Winkler
The analysis of process variation is often determined by effects
which are hardly to detect with classical designs. These factors are
difficult to control or to manipulate. We call them noise factors.
The following possibilities are already discussed:
• Blocking
Mostly attributive factors can be included in the design. (e. g.
supplier; A vs. B; shift: 1 vs. 2)
•Randomization of experimental runs
Here we try to avoid a miss interpretation from noise effects showing
up over time.
Now we will to discuss how to include continuous measurable but
uncontrollable factors into the design.
Example: Measure material characteristics incoming inspection.
Factors like temperature and humidity cannot be manipulated. But it
is easy to measure these factors and also to evaluate their effects.
About this Module

4321
70
65
60
55
Factor
Response The “classical” ANOVA
2520151050
10
5
0
Covariate
Response
Effects of Covariates
Slope = b1
Graphical Display
Y = bo + b1x
Null Hypotheses Ho : b1 is equal or close to 0
Alternative Hypotheses Ha : b1 is unequal 0 or larger than
determined by the probability of error
The Mathematical Model
The procedure to receive information about non manipulated variables
is to measure and to include them as a factor in the analysis. This is
called the analysis of covariance (ANCOVA).
Covariates can be included as factors in fractional and full factorial
experiments.
For each covariate we need one degree of freedom to calculate the
effect. The degree of freedom is taken from the unexplained Variation
(error). In the case that the covariate has no significant effect we should
remove it from the analysis.
Several covariates can be included in the analysis.

An engineer likes to investigate the influence from the production line to
the tensile strength of the produced fiber filaments. He knows that the
tensile strength is strong effect by the filament diameter. Therefore he
measured the filament diameter and include the diameter as a covariate
into the experiment.
Please determine the effect of the lines with and without the covariate.
What is your conclusion?
Example: Tensile Strength
File: mo57417.mtw
Stat
>ANOVA
>General Linear Model…
Stat
>ANOVA
General Linear Model: Tensile strength versus Line
Factor Type Levels Values
Line fixed 3 1; 2; 3
Analysis of Variance for Tensile strength, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Line 2 140,40 140,40 70,20 4,09 0,044
Error 12 206,00 206,00 17,17
Total 14 346,40
S = 4,14327 R-Sq = 40,53% R-Sq(adj) = 30,62%
General Linear Model: Tensile strength versus Line
Factor Type Levels Values
Line fixed 3 1; 2; 3
Analysis of Variance for Tensile strength, using Adjusted SS for Tests
Diameter 1 303,35 175,24 175,24 62,68 0,000
Line 2 12,30 12,30 6,15 2,20 0,157
Error 11 30,76 30,76 2,80
Total 14 346,40
S = 1,67214 R-Sq = 91,12% R-Sq(adj) = 88,70%
Example: Tensile Strength
Without covariate
With covariate
Attention!

A more complex example of PCB manufacturing
A device of an automatic manufacturing line places components on a
PCB. The position of the components is controlled by a locator pin.
This pin is included by the supplier into the PCB in accordance to our
requirements.
Considerable variation in the positioning (x or y) has been detected in
the past time.
Two factors (A and B) of the positioning device are known, as easy to
manipulated. We have decided to perform a a small experiment with
two factors. The target is to redefine the optimum device settings to
improve the placement of the components.
Example: Printed Circuit Board Assembly
Our design is completely randomized and
includes 4 replications
The factors A & B of the devise have been
manipulated
The result is the position of the mounted
component.
The positions of the locator pin (covariate)
have been documented for every trial.
Assignment: Calculate the effect of the
factors A & B with and without the covariate
(Can be done under GLM and DOE)
Comment the results, is there a significant
effect of the factors or there are other
sources for the observed variation (locator
pin, noise)?
A B covar response
-1 -1 192,278 312,928
-1 -1 197,866 366,014
-1 -1 195,834 347,726
-1 -1 195,326 327,66
-1 1 197,612 327,66
-1 1 199,39 361,442
-1 1 194,056 311,912
-1 1 196,088 317,754
1 -1 197,612 313,69
1 -1 195,58 295,148
1 -1 198,882 325,12
1 -1 197,104 297,434
1 1 193,548 330,2
1 1 191,77 328,676
1 1 194,056 352,552
1 1 195,58 363,982
File:
ANCOVA.mtw

The evaluation without covariate
Fractional Factorial Fit
Estimated Effects and Coefficients for response
Term Effect Coef StDev Coef T P
Constant 329.994 4.886 67.53 0.000
A -8.287 -4.143 4.886 -0.85 0.413
B 13.557 6.779 4.886 1.39 0.191
A*B 22.447 11.224 4.886 2.30 0.040
Analysis of Variance for response
Main Effects 2 1009.88 1009.88 504.9 1.32 0.303
2-Way Interactions 1 2015.52 2015.52 2015.5 5.28 0.040
Residual Error 12 4584.17 4584.17 382.0
Pure Error 12 4584.17 4584.17 382.0
Total 15 7609.56
Stat
>DOE
>Factorial
>Analyze Factorial Design…
Stat
>DOE
>Factorial
Attention!
R-Sq = 39,76%
B
Mean
1-1
345
340
335
330
325
320
315
310
A
-1
1
Interaction Plot (data means) for response

Interval plots for the
evaluations of the co
variable vs. factor level
over the distribution of the
covariate.
Stat
>ANOVA
>Interval Plot…
>One Y with groups
Stat
>ANOVA
>Interval Plot…
>One Y with groups
response
A
B
1-1
1-11-1
380
360
340
320
300
280
Interval Plot of response vs A; B
95% CI for the Mean
General Linear Model
Factor Levels Values
A 2 -1 1
B 2 -1 1
covar 1 305.9 3892.2 3892.2 61.88 0.000
A 1 210.5 40.7 40.7 0.65 0.438
B 1 1015.2 1987.5 1987.5 31.60 0.000
A*B 1 5386.0 5386.0 5386.0 85.62 0.000
Error 11 691.9 691.9 62.9
Total 15 7609.6
Term Coef StDev T P
Constant -1503.1 233.0 -6.45 0.000
covar 9.363 1.190 7.87 0.000
Stat
>ANOVA
Stat
>ANOVA
Example: PCB, Evaluation with Covariate
R-Qd = 90,91%

Fractional Factorial Fit (Analyze Custom Design)
Estimated Effects and Coefficients for response
Term Effect Coef StDev Coef T P
Constant -1503 233.039 -6.45 0.000
covar 9 1.190 7.87 0.000
A -3 -2 2.009 -0.80 0.438
B 23 12 2.079 5.62 0.000
A*B 46 23 2.482 9.25 0.000
Covariates 1 305.9 3892.2 3892.23 61.88 0.000
Main Effects 2 1225.7 2060.3 1030.14 16.38 0.001
2-Way Interactions 1 5386.0 5386.0 5386.03 85.62 0.000
Residual Error 11 691.9 691.9 62.90
Total 15 7609.6
Stat
>DOE … Factorial
Stat
>DOE … Factorial
The portion of the explained variation has been significantly increased
due to the model with the covariate. The question, how can that
variable be controlled?
Example: PCB, Evaluation with Covariate
R-Qd = 90,91%
Interpretation of the Results
• 16 runs has been performed. Due to the inclusion of the covariate the error
term has 11 degrees of freedom instead of 12.
• The portion of the non explained variation has been reduced by the
covariate from 4584 to 692.
• Only if the variation of the locator pins can be reduced we have the
possibility to control the mounting process with the factors A and B.
• The experiment with 16 runs shows us clearly how to improve the mounting
process.
• For the discussion with the supplier and/or the designer we may have
discovered important facts.
• If the evaluations don’t include the major sources of the variation we can
get misleading results. This could cause high cost.

Summary
The evaluation of covariates offers us an
additional possibility to describe models for
special applications.
These adapted models increases the certainty of
our conclusions.
Here again 3 important hints for your
experiments:
• Maximize randomization of your experiments
• Include variables as a block factor
• Use this technique to evaluate the covariates

Analysis of Covariates in DOE Experiments to Improve Process Understanding

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (20)

Similar to Analysis of Covariates in DOE Experiments to Improve Process Understanding

Similar to Analysis of Covariates in DOE Experiments to Improve Process Understanding (20)

More from J. García - Verdugo

More from J. García - Verdugo (16)

Recently uploaded

Recently uploaded (20)

Analysis of Covariates in DOE Experiments to Improve Process Understanding