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R代写-IEE 572
时间：2021-03-17
Maximizing Cake Height to Achieve Desired Texture IEE 572 Final Project
Cynthia Bennett – Online Student
12/6/2011
Bennett – IEE 572 Final Project Page 2
Table of Contents
Proposal:................................................................................................................................................................ 3
Pre-experimental Planning: .................................................................................................................................... 3
Recognition of and statement of the problem...................................................................................................... 3
Selection of the response variable ...................................................................................................................... 3
Choice of factors, levels, and ranges .................................................................................................................. 3
Choice of Experimental Design ......................................................................................................................... 3
Performing the Experiment and Analyzing the Data:............................................................................................... 5
Performing the Experiment ................................................................................................................................ 5
Statistical Analysis of the Data .......................................................................................................................... 6
Final Equation and Future Considerations .........................................................................................................23
Appendix ..............................................................................................................................................................25
List of Tables and Figures
Table 1 – Chosen Factors and Levels ...................................................................................................................... 3
Table 2 – Experimental Run Order ......................................................................................................................... 4
Figure 1 – Parameter Estimates for Full 24 Factorial Model..................................................................................... 6
Figure 2 – ANOVA and Lack of Fit for Full 24 Factorial Model .............................................................................. 7
Figure 3 – PRESS for the Full 24 Factorial Model ................................................................................................... 7
Figure 4 – Normal Probability Plot for Significant Factor Identification for Full 24 Factorial Model ........................ 8
Figure 5 - Parameter Estimates for Collapsed 23 Factorial Model ............................................................................ 9
Figure 6 - ANOVA and Lack of Fit for Collapsed 23 Factorial Model ..................................................................... 9
Figure 7 – PRESS for Collapsed 23 Factorial Model ............................................................................................... 9
Figure 8 - Normal Probability Plot for Significant Factor Identification for Collapsed 23 Factorial Model ...............10
Figure 9 – Interaction Plots for Collapsed 23 Factorial Model .................................................................................10
Figure 10 – Residual Predicted Plot for Collapsed 23 Factorial Model ....................................................................11
Figure 11 – Run Order Residual Plot for Collapsed 23 Factorial Model ..................................................................11
Figure 12 - Residual Normal Probability Plot for Collapsed 23 Factorial Model ......................................................12
Figure 13 – Box-Cox Transformation Plot for Residual Predicted Plot for Collapsed 23 Factorial Model ................13
Figure 14 - Parameter Estimates for Transformed Collapsed 23 Factorial Model .....................................................13
Figure 15 - ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model ..............................................14
Figure 16 – PRESS for Transformed Collapsed 23 Factorial Model ........................................................................14
Figure 17 - Normal Probability Plot for Significant Factor Identification for Transformed Collapsed 23 Factorial
Model ...................................................................................................................................................................15
Figure 18 – Interaction Plots for Transformed Collapsed 23 Factorial Model ..........................................................15
Figure 19 - Residual Predicted Plot for Transformed Collapsed 23 Factorial Model ................................................16
Figure 20 - Run Order Residual Plot for Transformed Collapsed 23 Factorial Model ..............................................17
Figure 21 - Residual Normal Probability Plot for Transformed Collapsed 23 Factorial Model .................................17
Figure 22 – Residual Plot by Factor Level for Flour for Transformed Collapsed 23 Factorial Model .......................18
Figure 23 - Residual Plot by Factor Level for Baking Powder for Transformed Collapsed 23 Factorial Model .........18
Figure 24 - Residual Plot by Factor Level for Temperature for Transformed Collapsed 23 Factorial Model .............19
Figure 25 - Parameter Estimates for Transformed Collapsed 23 Factorial Model with Significant Terms Only ........19
Figure 26 – Summary of Fit, ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model with
Significant Terms Only .........................................................................................................................................20
Figure 27 – PRESS for Transformed Collapsed 23 Factorial Model with Significant Terms Only............................20
Figure 28 – Interaction Plots for Transformed Collapsed 23 Factorial Model with Significant Terms Only ..............21
Figure 29 – Contour Plot for the Flour-Baking Powder Interaction .........................................................................22
Figure 30 – Contour Plot for the Flour-Temperature Interaction .............................................................................22
Bennett – IEE 572 Final Project Page 3
Proposal:
Most Americans love a delicious piece of cake. But what is the secret to a good
cake? While many may comment on flavor, frostings or other attributes, most people
agree on 1 thing: texture. The texture of the cake needs to be “fluffy”. While there are
exceptions to the traditional textures, one of the easiest ways to get the right texture of a
cake is by having the cake rise to a maximum height. Why do we want the cake at the
maximum height? By having the tallest cake with the same amount of starting material,
more air is allowed into the cake creating the “fluffy” texture that people desire in their
desserts. In this experiment, cake height will be evaluated to achieve the optimum cake
texture.
Pre-experimental Planning:
Recognition of and statement of the problem
The goal is to achieve maximum cake height to create a cake with a fluffy texture.
Selection of the response variable
Cake height is the response variable. Cake height will be a physical measurement
using a standard ruler (measurements will be taken in centimeters).
Choice of factors, levels, and ranges
Factor Low Center High
Amount of Flour 1/2 Cups 1 Cup 1 1/2 Cups
Amount of Baking Powder 1/2 teaspoon 1 teaspoon 1 1/2 teaspoons
Amount of Milk 1/4 Cup 1/2 Cup 3/4 Cup
Temperature of Oven 325 Degrees 350 Degrees 375 Degrees
Table 1 – Chosen Factors and Levels
The normal recipe is being tested as well as an increase and decrease in each
parameter of interest to make sure all scenarios are considered.
Choice of Experimental Design
There were several choices of experimental design that were considered for this
project. The original design was a full 34 factorial experiment, but that alone was 81
different cakes with no replication included. There were several concerns with this. The
available time to complete this design would not allow for 81 cakes to be made and
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measured. The second concern was the available materials and the nuisance variables
it would introduce. For example, it is not possible to bake 81 cakes from the same bag
of flour which then introduces batches of flour as a potential variable.
Instead, a 24 factorial design will be used with no replicates, but with the addition of
4 center points to the design. This will result in 20 cakes, which is much more
reasonable from a cost and time perspective. The design also allows for an error term to
be calculated due to the replication of the center points and it also allows us to test for
curvature. If curvature is found to be present, additional runs will be completed using
axial points. Also, it allows for 3 levels of each factor to be tested which was the original
interest point.
The experimental runs for the design are listed below. The design was created in the
JMP software package using a full 24 factorial design with 4 center points present. The
table was created randomly and saved, waiting for data input.
Pattern Flour Baking Powder Milk Temperature
+−++ 1 -1 1 1
−−+− -1 -1 1 -1
+−−− 1 -1 -1 -1
+++− 1 1 1 -1
−+−+ -1 1 -1 1
0000 0 0 0 0
+−+− 1 -1 1 -1
0000 0 0 0 0
−−++ -1 -1 1 1
−+−− -1 1 -1 -1
++−+ 1 1 -1 1
+−−+ 1 -1 -1 1
0000 0 0 0 0
−+++ -1 1 1 1
−−−+ -1 -1 -1 1
++++ 1 1 1 1
0000 0 0 0 0
++−− 1 1 -1 -1
−++− -1 1 1 -1
−−−− -1 -1 -1 -1
Table 2 – Experimental Run Order
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Other concerns and considerations for this design include baking time and cake
pan type. To prevent introducing new variables, all cakes will be baked for 28 minutes
(the recommended time from the recipe) and all cakes will be baked in the same 8 inch
round cake pan. Bulk ingredients will be purchased to insure all cakes are made from
the same materials.
Performing the Experiment and Analyzing the Data:
Performing the Experiment
The recipe used for baking all cakes is listed here:
Yellow Cake Recipe
• ½ cup butter
• ¾ cup white sugar
• 4 egg yolks
• X cup milk
• ¾ teaspoon vanilla extract
• X cup flour
• X teaspoon baking powder
• ¼ teaspoon salt
X will vary depending on the run of the experiment and the levels of each factor in
that run.
For every cake, the butter and sugar were first mixed together in the same
Kitchenaid mixer on the same speed setting. Egg whites were separated and then
added to the bowl and mixed at the same speed and then the vanilla extract was added.
The flour, baking powder and salt were sifted together and added with the specified
amount of milk and mixed at the same speed. Batter was then poured into the cake pan
and a spatula was used to scrape all batter into the pan.
All cakes were baked from the same batch of ingredients and the same cake pan
was used for each run. The temperature also varied for each cake depending on the
run, but all cakes were baked for 28 minutes to minimize baking time as a nuisance
factor.
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All cakes were measured using the same ruler and measurements were taken in
centimeters (cm). No unusual behavior was observed during the experimental runs.
Statistical Analysis of the Data
The cakes were run in the order previously given. Using a full factorial model to
identify factors that affect the cake height yielded the following results:
Figure 1 – Parameter Estimates for Full 24 Factorial Model
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Figure 2 – ANOVA and Lack of Fit for Full 24 Factorial Model
Figure 3 – PRESS for the Full 24 Factorial Model
In this table, the Lack of Fit SS is equal to the SScurvature and only has a value of
0.0632. This is verified by also calculating the value by hand:
"#$%&'#$(= +-".+-000 − "2 .3- + " "#$%&'#$(= (16 ∗ 4)(3.359 − 3.5)316 + 4
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"#$%&'#$(= 0.0633
The F-ratio for the curvature is 1.5187 and the associated p-value is 0.3056. This is
not significant at the 5% level, so the model does not have curvature that is statistically
significant. No further runs with axial points are needed, and the analysis will be
continued as a traditional factorial design.
Figure 4 – Normal Probability Plot for Significant Factor Identification for Full 24 Factorial Model
There are 4 significant terms in this model. The amount of flour, the amount of
baking powder, the flour*baking powder interaction and the flour*temperature
interactions are all significant. Although the half normal probability plot identifies the
flour*milk interaction as significant, it is not supported by the given p-value. Reviewing
the p-values and the half-normal probability plot it was determined that milk and its
interactions are not statistically significant at the 5% level. If milk and all related
interactions are removed, the design collapses into a full 23 factorial with 4 center points
and 1 replicate of the main 8 runs. Performing this analysis results in the following:
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Figure 5 - Parameter Estimates for Collapsed 23 Factorial Model
Figure 6 - ANOVA and Lack of Fit for Collapsed 23 Factorial Model
Figure 7 – PRESS for Collapsed 23 Factorial Model
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Figure 8 - Normal Probability Plot for Significant Factor Identification for Collapsed 23 Factorial Model
Figure 9 – Interaction Plots for Collapsed 23 Factorial Model
The same factors are showing as significant as before although the p-values have
decreased for each of the parameters. The RSquare and RSquare Adj values are both
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nice for this design with 0.9508 and 0.9221 respectively. The PRESS value also
decreased from 81.195 to 2.741 indicating that the exclusion of milk was an
improvement to the overall model. Next, the residuals were checked to verify that the
assumptions of a normal distribution and constant variance are both true. The data from
the collapsed 23 factorial was used for the residual analysis.
Figure 10 – Residual Predicted Plot for Collapsed 23 Factorial Model
There are no issues with the predicted plot indicating there is no issue with the
assumption of constant variance.
Figure 11 – Run Order Residual Plot for Collapsed 23 Factorial Model
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There are no issues with the run order residual plot indicating there is no issue with
the assumption of error independence.
Figure 12 - Residual Normal Probability Plot for Collapsed 23 Factorial Model
There is an issue with the normal probability plot of the residuals. The points do not
pass the fat pencil test, meaning there is an issue with the assumption of normality for
this design. A transformation will be utilized to complete the analysis.
The Box-Cox transformation tool in JMP was utilized. The graph indicates that the
ideal value for λ in a power transformation is in the 1.5 to 1.6 range, confirming that the
original analysis with λ equal to 1 was not ideal. The Box-Cox transformation graph is
listed below.
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Figure 13 – Box-Cox Transformation Plot for Residual Predicted Plot for Collapsed 23 Factorial Model
Using the Save Best Transformation feature, the transformed data was
generated in the original data table (data is available in the appendix). Reviewing the
formula column, a value of 1.6 was used for λ.
The analysis was repeated, this time using the transformed data.
Figure 14 - Parameter Estimates for Transformed Collapsed 23 Factorial Model
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Figure 15 - ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model
Figure 16 – PRESS for Transformed Collapsed 23 Factorial Model
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Figure 17 - Normal Probability Plot for Significant Factor Identification for Transformed Collapsed 23 Factorial Model
Figure 18 – Interaction Plots for Transformed Collapsed 23 Factorial Model
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As expected, the same factors are showing as significant. The RSquare and
RSquare Adj values both increased after the transformation with values of 0.9571 and
0.9320 respectively. The PRESS value also decreased (though not as significantly) with
the transformation from 2.741 to 2.569. These all suggest that the transformed model is
a better fit to the data. Next, the residuals were checked to verify that the transformation
corrected the previously seen issues with the normality assumption. From the chart
below in Figure 13, the predicted residual plot still has no issues meaning there is no
problem with the constant variance assumption. Figure 14, the residuals by run order
plot, shows there are still no issues with the independence assumption. Figure 15
shows the normal probability plot now passes the “fat pencil” test and is a significant
improvement on the original probability. The analysis can proceed using the
assumptions that the variance is constant and the population represents a normal
distribution.
Figure 19 - Residual Predicted Plot for Transformed Collapsed 23 Factorial Model
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Figure 20 - Run Order Residual Plot for Transformed Collapsed 23 Factorial Model
Figure 21 - Residual Normal Probability Plot for Transformed Collapsed 23 Factorial Model
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Figure 22 – Residual Plot by Factor Level for Flour for Transformed Collapsed 23 Factorial Model
Figure 23 - Residual Plot by Factor Level for Baking Powder for Transformed Collapsed 23 Factorial Model
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Figure 24 - Residual Plot by Factor Level for Temperature for Transformed Collapsed 23 Factorial Model
To check if further refining the model would be an improvement, all terms identified
as insignificant were excluded. The transformed data was still used for this analysis.
Residual plots show there are still no issues with the assumptions made for this model.
The residual plots for this model are in the appendix since there was no change in the
assumption satisfaction.
Figure 25 - Parameter Estimates for Transformed Collapsed 23 Factorial Model with Significant Terms Only
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Figure 26 – Summary of Fit, ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model with Significant
Terms Only
Figure 27 – PRESS for Transformed Collapsed 23 Factorial Model with Significant Terms Only
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Figure 28 – Interaction Plots for Transformed Collapsed 23 Factorial Model with Significant Terms Only
The Summary of Fit shows the RSquare and RSquare Adj values both decreased
when using only the significant terms. The previous RSquare value was 0.9571 and it
decreased to 0.9468. The previous RSquare Adj value was 0.9320 and it decreased to
0.9278. However, these numbers are still very high and the PRESS value decreased
with the new model. The previous PRESS value was 2.569 and it is now 2.014. Based
on the decreased PRESS value and the still very high and acceptable values for
RSqaure and RSqaure Adj, the model with only the significant terms will be the final
model used to derive the equation for predicting cake height.
The contour plots were analyzed for this model and the graphs are given below
in Figures 28 and 29. The contour plots were analyzed for the flour*baking powder
interaction and for the flour*temperature interaction. As seen below, the cake height
increases when the level of flour and the level of baking powder both increase. The
cake height also increases when the level of flour and the temperature both increase.
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Figure 29 – Contour Plot for the Flour-Baking Powder Interaction
Figure 30 – Contour Plot for the Flour-Temperature Interaction
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Final Equation and Future Considerations
Reviewing the parameter estimates from the final analysis, the following equation was
derived to predict the cake height. The equation is given in coded variables of each parameter
and is given with the cake height itself transformed (allowing for an easier conversion to natural
variables). Temperature by itself was also included in the equation to retain hierarchal order. ( ℎ)I.J = 3.388 + 0.922 ∗ + 0.297 ∗ − 0.141 ∗ + 0.172 ∗ ∗ + 0.172 ∗ ∗
In natural variables the equation would be: ()I.J = 3.388 + 0.922 R − 10.5 S + 0.297 R − 10.5 S − 0.141 R − 35025 S + 0.172 R − 10.5 S R − 10.5 S+ 0.172 R − 10.5 S R − 35025 S
The equation above is given with flour (F) in cups, baking powder (BP) in teaspoons and
temperature (T) in ᵒF.
This is a linear equation; however it is important to note that this is not an infinite
equation. This equation is to be used with the provided recipe and baking time. There is a fixed
amount of the other ingredients present in the cake that would affect the baking. This experiment
was conducted with a range of values for the ingredients that would be realistically used in
practical application. The maximum for flour was 1.5 cups tested, and while it would be possible
to make a cake with 3 cups of flour that would successfully bake, it would not be possible to
make a cake with 20 cups of flour. If the tested conditions were the only possible run conditions
allowed, the combination to achieve maximum cake height would be flour at the +1 level and
baking powder at the +1 level. The temperature is an interesting parameter to look at since its
behavior is dependent on the level of flour chosen. If there is a high level of flour, high
temperature should be used. If there is a low level of flour, low temperature should be used to
maximize the cake height. As previously stated, a high level of flour is desirable based on the
flour*baking powder interaction and this decision is supported by the positive value for the flour
effect estimate. Based on this information, temperature would also be set at the +1 level to
achieve maximum cake height.
If this experiment were to be repeated, only the amount of flour, baking powder and the
temperature would be investigated. It is suspected there is curvature present in the overall
behavior of the cake height (the reason center points were added) and there is a global maximum,
but it could not be seen or determined from the experiment performed due to the region of
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experimentation. Should the study be continued, the next step would be utilizing the method of
steepest ascension to identify the region where the global maximum occurs.
To complete the method of steepest ascension, a step size must be chosen in one of the
process variables. Flour was chosen since it was the factor with the largest regression coefficient
and the lowest p-value, indicating it was the most significant process variable investigated.
Therefore ΔxF = 1.0 and the other step sizes are listed below as calculated. ∆VW = XYZ[XY\ ∆]\^ ∆_ = XY`XY\ ∆]\^ ∆VW = 0.322 ∆_ = 0.153
To convert the coded step sizes to natural units, the natural variable step size used in the
experiment from the center point (origin) to the high and low runs in either direction will be
multiplied by the calculated coded step size. For the case of flour, the coded step size (∆a) is
equal to 1, and the distance from the center point was 0.5 cups of flour. Therefore, the steps
along the steepest path of ascension for flour are in 0.5 cup increments. The procedure was
repeated for baking powder and temperature and the calculated steps in natural units were 0.161
teaspoons for baking powder and 3.825 ᵒF for temperature. These calculated units are not
reasonably measured, so the closest measurements that can be accurately executed will be
utilized. This means ¼ teaspoon steps for baking powder and 5 ᵒF steps for temperature. Based
on the interaction plots and previous discussion about the levels needed to maximize cake height,
flour, baking powder and temperature will all be increased for each run performed. A proposed
experiment plan is listed in the appendix with 10 runs (maximum should be easily identified by
then). After this run is completed, analysis of the response surface would need to be executed to
identify the correct second-order equation for the model.
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Appendix
Raw Data in Standard Order
Pattern
Treatment
Combination
Run
Order
Flour
(A)
Baking
Powder
(B) Milk (C)
Temperature
(D)
Cake
Height (in
cm)
−−−− (1) 20 -1 -1 -1 -1 2.75
+−−− a 3 1 -1 -1 -1 4
−+−− b 10 -1 1 -1 -1 2.5
++−− ab 18 1 1 -1 -1 4.75
−−+− c 2 -1 -1 1 -1 2.5
+−+− ac 7 1 -1 1 -1 3.75
−++− bc 19 -1 1 1 -1 3.25
+++− abc 4 1 1 1 -1 4.5
−−−+ d 15 -1 -1 -1 1 2
+−−+ ad 12 1 -1 -1 1 4
−+−+ bd 5 -1 1 -1 1 2
++−+ abd 11 1 1 -1 1 4.75
−−++ cd 9 -1 -1 1 1 2
+−++ acd 1 1 -1 1 1 3.5
−+++ bcd 14 -1 1 1 1 2.5
++++ abcd 16 1 1 1 1 5
0000 0000 6 0 0 0 0 3.5
0000 0000 8 0 0 0 0 3.25
0000 0000 13 0 0 0 0 3.75
0000 0000 17 0 0 0 0 3.5
Bennett – IEE 572 Final Project Page 26
Raw and Transformed Data in Experimental Run Order
Pattern Flour
Baking
Powder Milk Temperature
Cake
Height (in
cm)
Residual
Cake Height
(in cm)
Cake
Height (in
cm) X
Residual Cake
Height (in cm)
X
+−++ 1 -1 1 1 3.5 -0.278125 1.97713983 -0.2823434
−−+− -1 -1 1 -1 2.5 -0.153125 1.02590294 -0.120072
+−−− 1 -1 -1 -1 4 0.096875 2.52140747 0.1285797
+++− 1 1 1 -1 4.5 -0.153125 3.1081455 -0.1645449
−+−+ -1 1 -1 1 2 -0.278125 0.62543701 -0.2104426
0000 0 0 0 0 3.5 0.1125 1.97713983 0.03803975
+−+− 1 -1 1 -1 3.75 -0.153125 2.24382887 -0.1489989
0000 0 0 0 0 3.25 -0.1375 1.72164538 -0.2174547
−−++ -1 -1 1 1 2 -0.028125 0.62543701 -0.0102096
−+−− -1 1 -1 -1 2.5 -0.403125 1.02590294 -0.3580808
++−+ 1 1 -1 1 4.75 -0.153125 3.41681619 -0.1694981
+−−+ 1 -1 -1 1 4 0.221875 2.52140747 0.26192422
0000 0 0 0 0 3.75 0.3625 2.24382887 0.30472879
−+++ -1 1 1 1 2.5 0.221875 1.02590294 0.19002337
−−−+ -1 -1 -1 1 2 -0.028125 0.62543701 -0.0102096
++++ 1 1 1 1 5 0.096875 3.73539316 0.14907889
0000 0 0 0 0 3.5 0.1125 1.97713983 0.03803975
++−− 1 1 -1 -1 4.75 0.096875 3.41681619 0.14412574
−++− -1 1 1 -1 3.25 0.346875 1.72164538 0.33766162
−−−− -1 -1 -1 -1 2.75 0.096875 1.2456278 0.09965283
Residual Plots for the Final Model with Only Significant Terms
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Proposed Experimental Run Plan for Steepest Ascension Method
Coded Variables Natural Variables Response
Steps
Flour
(cup)
Baking
Powder
(tsp)
Temperature
(F) Flour (cup)
Baking
Powder
(tsp)
Temperature
(F)
Cake Height
(cm)
Origin 0 0 0 1.0 1.00 375
1.00 0.32 0.15 0.5 0.25 5
Origin + ∆ 1.00 0.32 0.15 1.5 1.25 380
Origin + 2∆ 2.00 0.64 0.30 2.0 1.50 385
Origin + 3∆ 3.00 0.96 0.45 2.5 1.75 390
Origin + 4∆ 4.00 1.28 0.60 3.0 2.00 395
Origin + 5∆ 5.00 1.60 0.75 3.5 2.25 400
Origin + 6∆ 6.00 1.92 0.90 4.0 2.50 405
Origin + 7∆ 7.00 2.24 1.05 4.5 2.75 410
Origin + 8∆ 8.00 2.56 1.20 5.0 3.00 415
Origin + 9∆ 9.00 2.88 1.35 5.5 3.25 420
Origin + 10∆ 10.00 3.20 1.50 6.0 3.50 425
学霸联盟
Cynthia Bennett – Online Student
12/6/2011
Bennett – IEE 572 Final Project Page 2
Table of Contents
Proposal:................................................................................................................................................................ 3
Pre-experimental Planning: .................................................................................................................................... 3
Recognition of and statement of the problem...................................................................................................... 3
Selection of the response variable ...................................................................................................................... 3
Choice of factors, levels, and ranges .................................................................................................................. 3
Choice of Experimental Design ......................................................................................................................... 3
Performing the Experiment and Analyzing the Data:............................................................................................... 5
Performing the Experiment ................................................................................................................................ 5
Statistical Analysis of the Data .......................................................................................................................... 6
Final Equation and Future Considerations .........................................................................................................23
Appendix ..............................................................................................................................................................25
List of Tables and Figures
Table 1 – Chosen Factors and Levels ...................................................................................................................... 3
Table 2 – Experimental Run Order ......................................................................................................................... 4
Figure 1 – Parameter Estimates for Full 24 Factorial Model..................................................................................... 6
Figure 2 – ANOVA and Lack of Fit for Full 24 Factorial Model .............................................................................. 7
Figure 3 – PRESS for the Full 24 Factorial Model ................................................................................................... 7
Figure 4 – Normal Probability Plot for Significant Factor Identification for Full 24 Factorial Model ........................ 8
Figure 5 - Parameter Estimates for Collapsed 23 Factorial Model ............................................................................ 9
Figure 6 - ANOVA and Lack of Fit for Collapsed 23 Factorial Model ..................................................................... 9
Figure 7 – PRESS for Collapsed 23 Factorial Model ............................................................................................... 9
Figure 8 - Normal Probability Plot for Significant Factor Identification for Collapsed 23 Factorial Model ...............10
Figure 9 – Interaction Plots for Collapsed 23 Factorial Model .................................................................................10
Figure 10 – Residual Predicted Plot for Collapsed 23 Factorial Model ....................................................................11
Figure 11 – Run Order Residual Plot for Collapsed 23 Factorial Model ..................................................................11
Figure 12 - Residual Normal Probability Plot for Collapsed 23 Factorial Model ......................................................12
Figure 13 – Box-Cox Transformation Plot for Residual Predicted Plot for Collapsed 23 Factorial Model ................13
Figure 14 - Parameter Estimates for Transformed Collapsed 23 Factorial Model .....................................................13
Figure 15 - ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model ..............................................14
Figure 16 – PRESS for Transformed Collapsed 23 Factorial Model ........................................................................14
Figure 17 - Normal Probability Plot for Significant Factor Identification for Transformed Collapsed 23 Factorial
Model ...................................................................................................................................................................15
Figure 18 – Interaction Plots for Transformed Collapsed 23 Factorial Model ..........................................................15
Figure 19 - Residual Predicted Plot for Transformed Collapsed 23 Factorial Model ................................................16
Figure 20 - Run Order Residual Plot for Transformed Collapsed 23 Factorial Model ..............................................17
Figure 21 - Residual Normal Probability Plot for Transformed Collapsed 23 Factorial Model .................................17
Figure 22 – Residual Plot by Factor Level for Flour for Transformed Collapsed 23 Factorial Model .......................18
Figure 23 - Residual Plot by Factor Level for Baking Powder for Transformed Collapsed 23 Factorial Model .........18
Figure 24 - Residual Plot by Factor Level for Temperature for Transformed Collapsed 23 Factorial Model .............19
Figure 25 - Parameter Estimates for Transformed Collapsed 23 Factorial Model with Significant Terms Only ........19
Figure 26 – Summary of Fit, ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model with
Significant Terms Only .........................................................................................................................................20
Figure 27 – PRESS for Transformed Collapsed 23 Factorial Model with Significant Terms Only............................20
Figure 28 – Interaction Plots for Transformed Collapsed 23 Factorial Model with Significant Terms Only ..............21
Figure 29 – Contour Plot for the Flour-Baking Powder Interaction .........................................................................22
Figure 30 – Contour Plot for the Flour-Temperature Interaction .............................................................................22
Bennett – IEE 572 Final Project Page 3
Proposal:
Most Americans love a delicious piece of cake. But what is the secret to a good
cake? While many may comment on flavor, frostings or other attributes, most people
agree on 1 thing: texture. The texture of the cake needs to be “fluffy”. While there are
exceptions to the traditional textures, one of the easiest ways to get the right texture of a
cake is by having the cake rise to a maximum height. Why do we want the cake at the
maximum height? By having the tallest cake with the same amount of starting material,
more air is allowed into the cake creating the “fluffy” texture that people desire in their
desserts. In this experiment, cake height will be evaluated to achieve the optimum cake
texture.
Pre-experimental Planning:
Recognition of and statement of the problem
The goal is to achieve maximum cake height to create a cake with a fluffy texture.
Selection of the response variable
Cake height is the response variable. Cake height will be a physical measurement
using a standard ruler (measurements will be taken in centimeters).
Choice of factors, levels, and ranges
Factor Low Center High
Amount of Flour 1/2 Cups 1 Cup 1 1/2 Cups
Amount of Baking Powder 1/2 teaspoon 1 teaspoon 1 1/2 teaspoons
Amount of Milk 1/4 Cup 1/2 Cup 3/4 Cup
Temperature of Oven 325 Degrees 350 Degrees 375 Degrees
Table 1 – Chosen Factors and Levels
The normal recipe is being tested as well as an increase and decrease in each
parameter of interest to make sure all scenarios are considered.
Choice of Experimental Design
There were several choices of experimental design that were considered for this
project. The original design was a full 34 factorial experiment, but that alone was 81
different cakes with no replication included. There were several concerns with this. The
available time to complete this design would not allow for 81 cakes to be made and
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measured. The second concern was the available materials and the nuisance variables
it would introduce. For example, it is not possible to bake 81 cakes from the same bag
of flour which then introduces batches of flour as a potential variable.
Instead, a 24 factorial design will be used with no replicates, but with the addition of
4 center points to the design. This will result in 20 cakes, which is much more
reasonable from a cost and time perspective. The design also allows for an error term to
be calculated due to the replication of the center points and it also allows us to test for
curvature. If curvature is found to be present, additional runs will be completed using
axial points. Also, it allows for 3 levels of each factor to be tested which was the original
interest point.
The experimental runs for the design are listed below. The design was created in the
JMP software package using a full 24 factorial design with 4 center points present. The
table was created randomly and saved, waiting for data input.
Pattern Flour Baking Powder Milk Temperature
+−++ 1 -1 1 1
−−+− -1 -1 1 -1
+−−− 1 -1 -1 -1
+++− 1 1 1 -1
−+−+ -1 1 -1 1
0000 0 0 0 0
+−+− 1 -1 1 -1
0000 0 0 0 0
−−++ -1 -1 1 1
−+−− -1 1 -1 -1
++−+ 1 1 -1 1
+−−+ 1 -1 -1 1
0000 0 0 0 0
−+++ -1 1 1 1
−−−+ -1 -1 -1 1
++++ 1 1 1 1
0000 0 0 0 0
++−− 1 1 -1 -1
−++− -1 1 1 -1
−−−− -1 -1 -1 -1
Table 2 – Experimental Run Order
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Other concerns and considerations for this design include baking time and cake
pan type. To prevent introducing new variables, all cakes will be baked for 28 minutes
(the recommended time from the recipe) and all cakes will be baked in the same 8 inch
round cake pan. Bulk ingredients will be purchased to insure all cakes are made from
the same materials.
Performing the Experiment and Analyzing the Data:
Performing the Experiment
The recipe used for baking all cakes is listed here:
Yellow Cake Recipe
• ½ cup butter
• ¾ cup white sugar
• 4 egg yolks
• X cup milk
• ¾ teaspoon vanilla extract
• X cup flour
• X teaspoon baking powder
• ¼ teaspoon salt
X will vary depending on the run of the experiment and the levels of each factor in
that run.
For every cake, the butter and sugar were first mixed together in the same
Kitchenaid mixer on the same speed setting. Egg whites were separated and then
added to the bowl and mixed at the same speed and then the vanilla extract was added.
The flour, baking powder and salt were sifted together and added with the specified
amount of milk and mixed at the same speed. Batter was then poured into the cake pan
and a spatula was used to scrape all batter into the pan.
All cakes were baked from the same batch of ingredients and the same cake pan
was used for each run. The temperature also varied for each cake depending on the
run, but all cakes were baked for 28 minutes to minimize baking time as a nuisance
factor.
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All cakes were measured using the same ruler and measurements were taken in
centimeters (cm). No unusual behavior was observed during the experimental runs.
Statistical Analysis of the Data
The cakes were run in the order previously given. Using a full factorial model to
identify factors that affect the cake height yielded the following results:
Figure 1 – Parameter Estimates for Full 24 Factorial Model
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Figure 2 – ANOVA and Lack of Fit for Full 24 Factorial Model
Figure 3 – PRESS for the Full 24 Factorial Model
In this table, the Lack of Fit SS is equal to the SScurvature and only has a value of
0.0632. This is verified by also calculating the value by hand:
"#$%&'#$(= +-".+-000 − "2 .3- + " "#$%&'#$(= (16 ∗ 4)(3.359 − 3.5)316 + 4
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"#$%&'#$(= 0.0633
The F-ratio for the curvature is 1.5187 and the associated p-value is 0.3056. This is
not significant at the 5% level, so the model does not have curvature that is statistically
significant. No further runs with axial points are needed, and the analysis will be
continued as a traditional factorial design.
Figure 4 – Normal Probability Plot for Significant Factor Identification for Full 24 Factorial Model
There are 4 significant terms in this model. The amount of flour, the amount of
baking powder, the flour*baking powder interaction and the flour*temperature
interactions are all significant. Although the half normal probability plot identifies the
flour*milk interaction as significant, it is not supported by the given p-value. Reviewing
the p-values and the half-normal probability plot it was determined that milk and its
interactions are not statistically significant at the 5% level. If milk and all related
interactions are removed, the design collapses into a full 23 factorial with 4 center points
and 1 replicate of the main 8 runs. Performing this analysis results in the following:
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Figure 5 - Parameter Estimates for Collapsed 23 Factorial Model
Figure 6 - ANOVA and Lack of Fit for Collapsed 23 Factorial Model
Figure 7 – PRESS for Collapsed 23 Factorial Model
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Figure 8 - Normal Probability Plot for Significant Factor Identification for Collapsed 23 Factorial Model
Figure 9 – Interaction Plots for Collapsed 23 Factorial Model
The same factors are showing as significant as before although the p-values have
decreased for each of the parameters. The RSquare and RSquare Adj values are both
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nice for this design with 0.9508 and 0.9221 respectively. The PRESS value also
decreased from 81.195 to 2.741 indicating that the exclusion of milk was an
improvement to the overall model. Next, the residuals were checked to verify that the
assumptions of a normal distribution and constant variance are both true. The data from
the collapsed 23 factorial was used for the residual analysis.
Figure 10 – Residual Predicted Plot for Collapsed 23 Factorial Model
There are no issues with the predicted plot indicating there is no issue with the
assumption of constant variance.
Figure 11 – Run Order Residual Plot for Collapsed 23 Factorial Model
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There are no issues with the run order residual plot indicating there is no issue with
the assumption of error independence.
Figure 12 - Residual Normal Probability Plot for Collapsed 23 Factorial Model
There is an issue with the normal probability plot of the residuals. The points do not
pass the fat pencil test, meaning there is an issue with the assumption of normality for
this design. A transformation will be utilized to complete the analysis.
The Box-Cox transformation tool in JMP was utilized. The graph indicates that the
ideal value for λ in a power transformation is in the 1.5 to 1.6 range, confirming that the
original analysis with λ equal to 1 was not ideal. The Box-Cox transformation graph is
listed below.
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Figure 13 – Box-Cox Transformation Plot for Residual Predicted Plot for Collapsed 23 Factorial Model
Using the Save Best Transformation feature, the transformed data was
generated in the original data table (data is available in the appendix). Reviewing the
formula column, a value of 1.6 was used for λ.
The analysis was repeated, this time using the transformed data.
Figure 14 - Parameter Estimates for Transformed Collapsed 23 Factorial Model
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Figure 15 - ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model
Figure 16 – PRESS for Transformed Collapsed 23 Factorial Model
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Figure 17 - Normal Probability Plot for Significant Factor Identification for Transformed Collapsed 23 Factorial Model
Figure 18 – Interaction Plots for Transformed Collapsed 23 Factorial Model
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As expected, the same factors are showing as significant. The RSquare and
RSquare Adj values both increased after the transformation with values of 0.9571 and
0.9320 respectively. The PRESS value also decreased (though not as significantly) with
the transformation from 2.741 to 2.569. These all suggest that the transformed model is
a better fit to the data. Next, the residuals were checked to verify that the transformation
corrected the previously seen issues with the normality assumption. From the chart
below in Figure 13, the predicted residual plot still has no issues meaning there is no
problem with the constant variance assumption. Figure 14, the residuals by run order
plot, shows there are still no issues with the independence assumption. Figure 15
shows the normal probability plot now passes the “fat pencil” test and is a significant
improvement on the original probability. The analysis can proceed using the
assumptions that the variance is constant and the population represents a normal
distribution.
Figure 19 - Residual Predicted Plot for Transformed Collapsed 23 Factorial Model
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Figure 20 - Run Order Residual Plot for Transformed Collapsed 23 Factorial Model
Figure 21 - Residual Normal Probability Plot for Transformed Collapsed 23 Factorial Model
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Figure 22 – Residual Plot by Factor Level for Flour for Transformed Collapsed 23 Factorial Model
Figure 23 - Residual Plot by Factor Level for Baking Powder for Transformed Collapsed 23 Factorial Model
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Figure 24 - Residual Plot by Factor Level for Temperature for Transformed Collapsed 23 Factorial Model
To check if further refining the model would be an improvement, all terms identified
as insignificant were excluded. The transformed data was still used for this analysis.
Residual plots show there are still no issues with the assumptions made for this model.
The residual plots for this model are in the appendix since there was no change in the
assumption satisfaction.
Figure 25 - Parameter Estimates for Transformed Collapsed 23 Factorial Model with Significant Terms Only
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Figure 26 – Summary of Fit, ANOVA and Lack of Fit for Transformed Collapsed 23 Factorial Model with Significant
Terms Only
Figure 27 – PRESS for Transformed Collapsed 23 Factorial Model with Significant Terms Only
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Figure 28 – Interaction Plots for Transformed Collapsed 23 Factorial Model with Significant Terms Only
The Summary of Fit shows the RSquare and RSquare Adj values both decreased
when using only the significant terms. The previous RSquare value was 0.9571 and it
decreased to 0.9468. The previous RSquare Adj value was 0.9320 and it decreased to
0.9278. However, these numbers are still very high and the PRESS value decreased
with the new model. The previous PRESS value was 2.569 and it is now 2.014. Based
on the decreased PRESS value and the still very high and acceptable values for
RSqaure and RSqaure Adj, the model with only the significant terms will be the final
model used to derive the equation for predicting cake height.
The contour plots were analyzed for this model and the graphs are given below
in Figures 28 and 29. The contour plots were analyzed for the flour*baking powder
interaction and for the flour*temperature interaction. As seen below, the cake height
increases when the level of flour and the level of baking powder both increase. The
cake height also increases when the level of flour and the temperature both increase.
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Figure 29 – Contour Plot for the Flour-Baking Powder Interaction
Figure 30 – Contour Plot for the Flour-Temperature Interaction
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Final Equation and Future Considerations
Reviewing the parameter estimates from the final analysis, the following equation was
derived to predict the cake height. The equation is given in coded variables of each parameter
and is given with the cake height itself transformed (allowing for an easier conversion to natural
variables). Temperature by itself was also included in the equation to retain hierarchal order. ( ℎ)I.J = 3.388 + 0.922 ∗ + 0.297 ∗ − 0.141 ∗ + 0.172 ∗ ∗ + 0.172 ∗ ∗
In natural variables the equation would be: ()I.J = 3.388 + 0.922 R − 10.5 S + 0.297 R − 10.5 S − 0.141 R − 35025 S + 0.172 R − 10.5 S R − 10.5 S+ 0.172 R − 10.5 S R − 35025 S
The equation above is given with flour (F) in cups, baking powder (BP) in teaspoons and
temperature (T) in ᵒF.
This is a linear equation; however it is important to note that this is not an infinite
equation. This equation is to be used with the provided recipe and baking time. There is a fixed
amount of the other ingredients present in the cake that would affect the baking. This experiment
was conducted with a range of values for the ingredients that would be realistically used in
practical application. The maximum for flour was 1.5 cups tested, and while it would be possible
to make a cake with 3 cups of flour that would successfully bake, it would not be possible to
make a cake with 20 cups of flour. If the tested conditions were the only possible run conditions
allowed, the combination to achieve maximum cake height would be flour at the +1 level and
baking powder at the +1 level. The temperature is an interesting parameter to look at since its
behavior is dependent on the level of flour chosen. If there is a high level of flour, high
temperature should be used. If there is a low level of flour, low temperature should be used to
maximize the cake height. As previously stated, a high level of flour is desirable based on the
flour*baking powder interaction and this decision is supported by the positive value for the flour
effect estimate. Based on this information, temperature would also be set at the +1 level to
achieve maximum cake height.
If this experiment were to be repeated, only the amount of flour, baking powder and the
temperature would be investigated. It is suspected there is curvature present in the overall
behavior of the cake height (the reason center points were added) and there is a global maximum,
but it could not be seen or determined from the experiment performed due to the region of
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experimentation. Should the study be continued, the next step would be utilizing the method of
steepest ascension to identify the region where the global maximum occurs.
To complete the method of steepest ascension, a step size must be chosen in one of the
process variables. Flour was chosen since it was the factor with the largest regression coefficient
and the lowest p-value, indicating it was the most significant process variable investigated.
Therefore ΔxF = 1.0 and the other step sizes are listed below as calculated. ∆VW = XYZ[XY\ ∆]\^ ∆_ = XY`XY\ ∆]\^ ∆VW = 0.322 ∆_ = 0.153
To convert the coded step sizes to natural units, the natural variable step size used in the
experiment from the center point (origin) to the high and low runs in either direction will be
multiplied by the calculated coded step size. For the case of flour, the coded step size (∆a) is
equal to 1, and the distance from the center point was 0.5 cups of flour. Therefore, the steps
along the steepest path of ascension for flour are in 0.5 cup increments. The procedure was
repeated for baking powder and temperature and the calculated steps in natural units were 0.161
teaspoons for baking powder and 3.825 ᵒF for temperature. These calculated units are not
reasonably measured, so the closest measurements that can be accurately executed will be
utilized. This means ¼ teaspoon steps for baking powder and 5 ᵒF steps for temperature. Based
on the interaction plots and previous discussion about the levels needed to maximize cake height,
flour, baking powder and temperature will all be increased for each run performed. A proposed
experiment plan is listed in the appendix with 10 runs (maximum should be easily identified by
then). After this run is completed, analysis of the response surface would need to be executed to
identify the correct second-order equation for the model.
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Appendix
Raw Data in Standard Order
Pattern
Treatment
Combination
Run
Order
Flour
(A)
Baking
Powder
(B) Milk (C)
Temperature
(D)
Cake
Height (in
cm)
−−−− (1) 20 -1 -1 -1 -1 2.75
+−−− a 3 1 -1 -1 -1 4
−+−− b 10 -1 1 -1 -1 2.5
++−− ab 18 1 1 -1 -1 4.75
−−+− c 2 -1 -1 1 -1 2.5
+−+− ac 7 1 -1 1 -1 3.75
−++− bc 19 -1 1 1 -1 3.25
+++− abc 4 1 1 1 -1 4.5
−−−+ d 15 -1 -1 -1 1 2
+−−+ ad 12 1 -1 -1 1 4
−+−+ bd 5 -1 1 -1 1 2
++−+ abd 11 1 1 -1 1 4.75
−−++ cd 9 -1 -1 1 1 2
+−++ acd 1 1 -1 1 1 3.5
−+++ bcd 14 -1 1 1 1 2.5
++++ abcd 16 1 1 1 1 5
0000 0000 6 0 0 0 0 3.5
0000 0000 8 0 0 0 0 3.25
0000 0000 13 0 0 0 0 3.75
0000 0000 17 0 0 0 0 3.5
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Raw and Transformed Data in Experimental Run Order
Pattern Flour
Baking
Powder Milk Temperature
Cake
Height (in
cm)
Residual
Cake Height
(in cm)
Cake
Height (in
cm) X
Residual Cake
Height (in cm)
X
+−++ 1 -1 1 1 3.5 -0.278125 1.97713983 -0.2823434
−−+− -1 -1 1 -1 2.5 -0.153125 1.02590294 -0.120072
+−−− 1 -1 -1 -1 4 0.096875 2.52140747 0.1285797
+++− 1 1 1 -1 4.5 -0.153125 3.1081455 -0.1645449
−+−+ -1 1 -1 1 2 -0.278125 0.62543701 -0.2104426
0000 0 0 0 0 3.5 0.1125 1.97713983 0.03803975
+−+− 1 -1 1 -1 3.75 -0.153125 2.24382887 -0.1489989
0000 0 0 0 0 3.25 -0.1375 1.72164538 -0.2174547
−−++ -1 -1 1 1 2 -0.028125 0.62543701 -0.0102096
−+−− -1 1 -1 -1 2.5 -0.403125 1.02590294 -0.3580808
++−+ 1 1 -1 1 4.75 -0.153125 3.41681619 -0.1694981
+−−+ 1 -1 -1 1 4 0.221875 2.52140747 0.26192422
0000 0 0 0 0 3.75 0.3625 2.24382887 0.30472879
−+++ -1 1 1 1 2.5 0.221875 1.02590294 0.19002337
−−−+ -1 -1 -1 1 2 -0.028125 0.62543701 -0.0102096
++++ 1 1 1 1 5 0.096875 3.73539316 0.14907889
0000 0 0 0 0 3.5 0.1125 1.97713983 0.03803975
++−− 1 1 -1 -1 4.75 0.096875 3.41681619 0.14412574
−++− -1 1 1 -1 3.25 0.346875 1.72164538 0.33766162
−−−− -1 -1 -1 -1 2.75 0.096875 1.2456278 0.09965283
Residual Plots for the Final Model with Only Significant Terms
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Proposed Experimental Run Plan for Steepest Ascension Method
Coded Variables Natural Variables Response
Steps
Flour
(cup)
Baking
Powder
(tsp)
Temperature
(F) Flour (cup)
Baking
Powder
(tsp)
Temperature
(F)
Cake Height
(cm)
Origin 0 0 0 1.0 1.00 375
1.00 0.32 0.15 0.5 0.25 5
Origin + ∆ 1.00 0.32 0.15 1.5 1.25 380
Origin + 2∆ 2.00 0.64 0.30 2.0 1.50 385
Origin + 3∆ 3.00 0.96 0.45 2.5 1.75 390
Origin + 4∆ 4.00 1.28 0.60 3.0 2.00 395
Origin + 5∆ 5.00 1.60 0.75 3.5 2.25 400
Origin + 6∆ 6.00 1.92 0.90 4.0 2.50 405
Origin + 7∆ 7.00 2.24 1.05 4.5 2.75 410
Origin + 8∆ 8.00 2.56 1.20 5.0 3.00 415
Origin + 9∆ 9.00 2.88 1.35 5.5 3.25 420
Origin + 10∆ 10.00 3.20 1.50 6.0 3.50 425
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