chi square control charts - the institute for perception...
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www.ifpress.com
Chi Square Control Charts
The Institute for Perception
+001 804 675 2980
Rotterdam, The NetherlandsJuly 26th, 2010
Presented By:
Dr. John M. Ennis
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An Image with a Defect
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Mean Control Chart
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UCL 21.5
Mean 20.5
LCL 19.5
Mean: 20.5
Standard Error of the Mean: 0.5
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Multiple Sensory Descriptors
Descriptive and consumer panel ratings data
Highly multivariate
Correlated assessments
Assumption of homogenous assessors
Test with Dirichlet Multinomial model
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Soup Example
Existing data for current production soup ratings and
ratings for a test sample
Product Thickness Saltiness
Current 4.1 4.5
Current 5.3 5.0
Current 6.6 3.8
… … …
Current 4.2 5.3
Test Sample 2.0 8.0
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Thickness Rating
Salt
ines
s R
ati
ng
0 2 4 6 8 10 12
02
46
810
12
T
Bivariate Normal Control Chart
(o = current production, T = test sample)
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Comments
Product X rated low on thickness and high on saltiness
95% contour for the bivariate normal suggests it not typical
Can we get a measure of departure from the current process?
Thickness Rating
Sa
ltin
es
s R
atin
g
0 2 4 6 8 10 12
02
46
81
01
2
XXT
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Beverage Example
Ratings on sweetness, sourness and carbonation
available for current production and a test product
Is the test product within specification?
Construct a star plot with confidence limits
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Star Plot Control Chart
Sweetness
Sourness
Carbonation
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Comments
Star plots are univariate plots of more than one attribute
No account made of attribute correlations or the
multivariate nature of the data
Questions:
Product inside acceptance region means it should be accepted?
Product outside acceptance region means it should be rejected?
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Conversion to Standard Form
For univariate normal variables,
The analogous Cholesky transformation exists for
multivariate normal variables
ZX
z A x
AA V
1 ( )t
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Why Transform the Data?
Standard distributions desired for testing
Similar to calculating z-scores and using normal tables
In standard form results are more meaningful and
easier to interpret
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Equal Likelihood Contours
For a non standard normal, contours of equal likelihood
are ellipses
For a standard normal, contours are circles
Points falling outside these contours are not typical
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Acceptance Regions
Uni: Accept
Multi: Reject
Uni: Reject
Multi: Accept
Uni: Accept
Multi: Accept
Uni: Reject
Multi: Reject
Uni CL Multi CL
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Bivariate Normal Control Chart
Thickness Rating
Salt
ines
s R
ati
ng
0 2 4 6 8 10 12
02
46
810
12
T
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Becomes...
Thickness Rating
Sa
ltin
ess
Rati
ng
-6 -4 -2 0 2 4 6
-6-4
-20
24
6
T
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Statistics: Multivariate vs Univariate
Products may fail a multivariate QC test and pass a
univariate test or the opposite
Need to take into account correlations among attributes
Need to take into account accumulated evidence of
deviance from a standard
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Beverage Example
Sweetness
Sourness
Carbonation
Although acceptable on
each attribute, 2 test
rejects this sample
www.ifpress.com
Presented By:
Chi Square Control Charts
Dr. John M. Ennis
The Institute for Perception
+001 804 675 2980
Rotterdam, The NetherlandsJuly 26th, 2010