Response Surface

A Response surface model is
a special type of multiple
regression model with:
Explanatory variables
 Interaction variables
 Squared variables

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Response Surface

A response surface model is
often used to approximate a
complicated relationship
between a response variable
and several explanatory
variables.
2
Response Surface

In order to get reliable data for
a response surface model, a
designed experiment is often
used to collect the data on the
explanatory variables and the
response.
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Designed Experiment

In a designed experiment, the
experimenter chooses values of
the explanatory variables to
investigate and measures the
response for the chosen
combinations of explanatory
variables.
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Tennis Ball Experiment
In the manufacture of tennis
balls certain additives are
thought to affect the
bounciness of the tennis ball.
 Response: A measure of
bounce.

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Tennis Ball Experiment

Explanatory variables
Amount of silica
 Amount of sulfur
 Amount of silane

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Tennis Ball Experiment

Each explanatory variable has
three levels
Silica: 0.7, 1.2, 1.7
 Sulfur: 1.8, 2.3, 2.8
 Silane: 40, 50, 60

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Tennis Ball Experiment
A total of 15 combinations of
silica, sulfur and silane are
examined and the bounce
response is measured for each
combination.
 The target bounce response is
450.

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Silica
Sulfur
Silane
Bounce
0.7
0.7
1.7
1.7
1.2
1.8
2.8
1.8
2.8
1.8
50
50
50
50
40
570
285
260
433
422




1.2
2.3
50
396
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Response Surface Model
Y   0  1 x1   2 x2   3 x3
 12 x1 x2  13 x1 x3   23 x2 x3
2
2
2
 11 x1   22 x2   33 x3  
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JMP – Fit Model
Put Bounce in for the Y
response.
 Highlight silica, sulfur and silane
in Select Columns.
 Macros – Response Surface

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Summary

The model is useful.


F=2488.146, P-value < 0.0001
R2=0.999777, virtually all of the
variation in Bounce is explained
by the model.
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15
Statistical Significance
The interaction between Silica
and Silane is not statistically
significant.
 The squared term for Silane is
not statistically significant.

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Reduced Model
Remove the interaction term:
Silica*Silane.
 Remove the squared term:
Silane*Silane.

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Summary

The model is useful.


F=4372.207, P-value < 0.0001
R2=0.999771, virtually all of the
variation in Bounce is explained
by the model. Only slightly
lower than R2 for the full model.
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Statistical Significance
All variables in the model are
statistically significant.
 This is the best response
surface model.

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Prediction

What combinations of silica,
sulfur and silane will give you
the target bounce of 450, on
average?
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22
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Prediction

There are several combinations
that will give a predicted
bounce of 450.
Silica = 1.0, Sulfur = 1.948,
Silane = 50
 Silica = 0.8, Sulfur = 2.251,
Silane = 40

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