Quality Progress - March 2017 - 46


Statistics Spotlight
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of the mean density from 9.56
to 10.28. Christine M. Anderson-Cook, Yongtao Cao and Lu
Lu provide suggestions about
how to provide a summary of
the uncertainty for identifying
the ideal concentration value.3
If, however, we consider the
possibility that the quadratic
model is the true model (after
all, the science suggests that
this might be right one), the
minimum density is estimated
to occur at a concentration of
10.8 with a value of 10.05 (95%
CI is [9.59,10.51]). Figure 3
shows the two estimated curves
overlaid with the quadratic
curve with CI shown in red and
the cubic model in blue. The
solid colored circles on each
estimated line provide the best
indication of where the minimum density lies for each curve.
In this case, the values of
the minimum as well as the
locations of the minima both
differ. If we were going to
select where to set our process
to optimize, ignoring the differences suggested by the two

TA B L E   1

Quadratic and cubic
model comparison
Summary

Quadratic model

Cubic model

R²

88.6%

93.5%

Adjusted R²

87.8%

92.8%

PRESS

32.038

32.302

βˆ₀

10.684

11.133

βˆ₁

-0.118

-0.275

βˆ₂

0.0054

0.0182

βˆ₃

-0.00028

PRESS = predicted residual error sum of squares

46 QP March 2017 ❘ qualityprogress.com

FIGURE 3

Quadratic and cubic models fit to
data with 95% confidence intervals
Estimated minimum density shown with colored filled circles

12

Density

estimating that curve.
There is another source
of uncertainty that we also
should acknowledge and
account for in our reporting.
Did we, in fact, choose the
right model? This idea is captured by model uncertainty,
and reflects a potentially bigger contributor to the outcome
of our study, the interpretation of results and our overall
confidence in reported results.
Its source lies in the process
that we use for selecting the
final model on which to report,
and in some cases, may play
a bigger role in affecting our
predictions than the model
parameter uncertainty.
For our engineer, the main
goal was to identify at what
concentration the minimum
density occurs, and the
expected value of the density at
that location. If we just consider
the cubic model, the minimum
density is estimated to occur
at a concentration of 9.7 with
a value of 9.92. The CI at that
concentration suggests a range

11

10
0

5

10

15

20

25

30

Concentration

models could lead to artificially
high confidence in the results.

Quantity of interest

The quantity of interest also
can influence the relative contributions of model and model
parameter uncertainty. If the
goal was to determine the estimated density as a function of
concentration for explanatory
variable values between five
and 25, the estimated curves
and the associated CIs are
relatively close-with quite a
bit of overlap. Hence, model
parameter uncertainty likely
contributes more than the
model uncertainty (the widths
of the CIs at a given concentration for each model are wider
than the differences between
the two sets of colored lines).
Things change, however, if
we are interested in the curves
near the extreme end of the
data set range-for example,

for concentrations near zero
or near 30. In these cases, the
relative contributions of model
and model parameter uncertainty reverse, with model
uncertainty contributing more
to the overall uncertainty (the
difference between the two
colored curves becomes larger
relative to the width of the CIs
at a given concentration). Of
course, extrapolation beyond
the range of the data with
polynomials has well documented dangers,4 and the
burden of having the model
correct increases if the model
is used to estimate outside of
the observed data.
So, what is best in terms of
reporting results to take into
account model uncertainty
as well as model parameter
uncertainty?
First, it is important to
acknowledge the process by
which a model was chosen.


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Table of Contents for the Digital Edition of Quality Progress - March 2017

Seen and Heard
Expert Answers
Progress Report
Mr. Pareto Head
Field Notes
Innovation Imperative
Hard Wired
Propel Forward
Life After Disruption
Work Smarter, Not Harder
Statistics Spotlight
Standard Issues
Marketplace
Footnotes
Back to Basics
Quality Progress - March 2017 - cover1
Quality Progress - March 2017 - cover2
Quality Progress - March 2017 - 1
Quality Progress - March 2017 - 2
Quality Progress - March 2017 - 3
Quality Progress - March 2017 - 4
Quality Progress - March 2017 - 5
Quality Progress - March 2017 - Seen and Heard
Quality Progress - March 2017 - 7
Quality Progress - March 2017 - Expert Answers
Quality Progress - March 2017 - 9
Quality Progress - March 2017 - Progress Report
Quality Progress - March 2017 - 11
Quality Progress - March 2017 - 12
Quality Progress - March 2017 - Mr. Pareto Head
Quality Progress - March 2017 - Field Notes
Quality Progress - March 2017 - 15
Quality Progress - March 2017 - 16
Quality Progress - March 2017 - 17
Quality Progress - March 2017 - Innovation Imperative
Quality Progress - March 2017 - 19
Quality Progress - March 2017 - 20
Quality Progress - March 2017 - 21
Quality Progress - March 2017 - Hard Wired
Quality Progress - March 2017 - 23
Quality Progress - March 2017 - 24
Quality Progress - March 2017 - 25
Quality Progress - March 2017 - 26
Quality Progress - March 2017 - 27
Quality Progress - March 2017 - Propel Forward
Quality Progress - March 2017 - 29
Quality Progress - March 2017 - 30
Quality Progress - March 2017 - 31
Quality Progress - March 2017 - 32
Quality Progress - March 2017 - 33
Quality Progress - March 2017 - Life After Disruption
Quality Progress - March 2017 - 35
Quality Progress - March 2017 - 36
Quality Progress - March 2017 - 37
Quality Progress - March 2017 - 38
Quality Progress - March 2017 - 39
Quality Progress - March 2017 - Work Smarter, Not Harder
Quality Progress - March 2017 - 41
Quality Progress - March 2017 - 42
Quality Progress - March 2017 - 43
Quality Progress - March 2017 - Statistics Spotlight
Quality Progress - March 2017 - 45
Quality Progress - March 2017 - 46
Quality Progress - March 2017 - 47
Quality Progress - March 2017 - Standard Issues
Quality Progress - March 2017 - 49
Quality Progress - March 2017 - Marketplace
Quality Progress - March 2017 - 51
Quality Progress - March 2017 - Footnotes
Quality Progress - March 2017 - 53
Quality Progress - March 2017 - 54
Quality Progress - March 2017 - 55
Quality Progress - March 2017 - Back to Basics
Quality Progress - March 2017 - cover3
Quality Progress - March 2017 - cover4
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