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Title: DESIGN OF EXPERIMENTS by R. C. Baker

1
DESIGN OF EXPERIMENTSbyR. C. Baker
• How to gain 20 years of experience in one short
week!

2
Role of DOE in Process Improvement
• DOE is a formal mathematical method for
systematically planning and conducting scientific
studies that change experimental variables
together in order to determine their effect of a
given response.
• DOE makes controlled changes to input variables
in order to gain maximum amounts of information
on cause and effect relationships with a minimum
sample size.

3
Role of DOE in Process Improvement
• DOE is more efficient that a standard approach of
changing one variable at a time in order to
observe the variables impact on a given
response.
• DOE generates information on the effect various
factors have on a response variable and in some
cases may be able to determine optimal settings
for those factors.

4
Role of DOE in Process Improvement
• DOE encourages brainstorming activities
associated with discussing key factors that may
affect a given response and allows the
experimenter to identify the key factors for
future studies.
• DOE is readily supported by numerous statistical
software packages available on the market.

5
BASIC STEPS IN DOE
• Four elements associated with DOE
• 1. The design of the experiment,
• 2. The collection of the data,
• 3. The statistical analysis of the data, and
• 4. The conclusions reached and recommendations
made as a result of the experiment.

6
TERMINOLOGY
• Replication repetition of a basic experiment
without changing any factor settings, allows the
experimenter to estimate the experimental error
(noise) in the system used to determine whether
observed differences in the data are real or
just noise, allows the experimenter to obtain
more statistical power (ability to identify small
effects)

7
TERMINOLOGY
• .Randomization a statistical tool used to
minimize potential uncontrollable biases in the
experiment by randomly assigning material,
people, order that experimental trials are
conducted, or any other factor not under the
control of the experimenter. Results in
averaging out the effects of the extraneous
factors that may be present in order to minimize
the risk of these factors affecting the
experimental results.

8
TERMINOLOGY
• Blocking technique used to increase the
precision of an experiment by breaking the
experiment into homogeneous segments (blocks) in
order to control any potential block to block
variability (multiple lots of raw material,
several shifts, several machines, several
inspectors). Any effects on the experimental
results as a result of the blocking factor will
be identified and minimized.

9
TERMINOLOGY
• Confounding - A concept that basically means that
multiple effects are tied together into one
parent effect and cannot be separated. For
example,
• 1. Two people flipping two different coins would
result in the effect of the person and the effect
of the coin to be confounded
• 2. As experiments get large, higher order
interactions (discussed later) are confounded
with lower order interactions or main effect.

10
TERMINOLOGY
• Factors experimental factors or independent
variables (continuous or discrete) an
investigator manipulates to capture any changes
in the output of the process. Other factors of
concern are those that are uncontrollable and
those which are controllable but held constant
during the experimental runs.

11
TERMINOLOGY
• Responses dependent variable measured to
describe the output of the process.
• Treatment Combinations (run) experimental trial
where all factors are set at a specified level.

12
TERMINOLOGY
• Fixed Effects Model - If the treatment levels
are specifically chosen by the experimenter, then
conclusions reached will only apply to those
levels.
• Random Effects Model If the treatment levels
are randomly chosen from a population of many
possible treatment levels, then conclusions
reached can be extended to all treatment levels
in the population.

13
PLANNING A DOE
• Everyone involved in the experiment should have a
clear idea in advance of exactly what is to be
studied, the objectives of the experiment, the
questions one hopes to answer and the results
anticipated

14
PLANNING A DOE
• Select a response/dependent variable (variables)
that will provide information about the problem
under study and the proposed measurement method
for this response variable, including an
understanding of the measurement system
variability

15
PLANNING A DOE
• Select the independent variables/factors
(quantitative or qualitative) to be investigated
in the experiment, the number of levels for each
factor, and the levels of each factor chosen
either specifically (fixed effects model) or
randomly (random effects model).

16
PLANNING A DOE
• Choose an appropriate experimental design
(relatively simple design and analysis methods
are almost always best) that will allow your
experimental questions to be answered once the
data is collected and analyzed, keeping in mind
tradeoffs between statistical power and economic
efficiency. At this point in time it is
generally useful to simulate the study by
generating and analyzing artificial data to
insure that experimental questions can be
answered as a result of conducting your
experiment

17
PLANNING A DOE
• Perform the experiment (collect data) paying
particular attention such things as randomization
and measurement system accuracy, while
maintaining as uniform an experimental
environment as possible. How the data are to be
collected is a critical stage in DOE

18
PLANNING A DOE
• Analyze the data using the appropriate
statistical model insuring that attention is paid
to checking the model accuracy by validating
underlying assumptions associated with the model.
Be liberal in the utilization of all tools,
including graphical techniques, available in the
statistical software package to insure that a
maximum amount of information is generated

19
PLANNING A DOE
• Based on the results of the analysis, draw
interpret the physical meaning of these results,
determine the practical significance of the
findings, and make recommendations for a course
of action including further experiments

20
SIMPLE COMPARATIVE EXPERIMENTS
• Single Mean Hypothesis Test
• Difference in Means Hypothesis Test with Equal
Variances
• Difference in Means Hypothesis Test with Unequal
Variances
• Difference in Variances Hypothesis Test
• Paired Difference in Mean Hypothesis Test
• One Way Analysis of Variance

21
CRITICAL ISSUES ASSOCIATED WITH SIMPLE
COMPARATIVE EXPERIMENTS
• How Large a Sample Should We Take?
• Why Does the Sample Size Matter Anyway?
• What Kind of Protection Do We Have Associated
with Rejecting Good Stuff?
• What Kind of Protection Do We Have Associated

22
Single Mean Hypothesis Test
• After a production run of 12 oz. bottles, concern
is expressed about the possibility that the
average fill is too low.
• Ho m 12
• Ha m ltgt 12
• level of significance a .05
• sample size 9
• SPEC FOR THE MEAN 12 .1

23
Single Mean Hypothesis Test
• Sample mean 11.9
• Sample standard deviation 0.15
• Sample size 9
• Computed t statistic -2.0
• P-Value 0.0805162
• CONCLUSION Since P-Value gt .05, you fail to
reject hypothesis and ship product.

24
Single Mean Hypothesis Test Power Curve
25
Single Mean Hypothesis Test Power Curve
26
Single Mean Hypothesis Test Power Curve -
Different Sample Sizes
27
DIFFERENCE IN MEANS - EQUAL VARIANCES
• Ho m1 m2
• Ha m1 ltgt m2
• level of significance a .05
• sample sizes both 15
• Assumption s1 s2
• Sample means 11.8 and 12.1
• Sample standard deviations 0.1 and 0.2
• Sample sizes 15 and 15

28
DIFFERENCE IN MEANS - EQUAL VARIANCES Can you
detect this difference?
29
DIFFERENCE IN MEANS - EQUAL VARIANCES
30
DIFFERENCE IN MEANS - unEQUAL VARIANCES
• Same as the Equal Variance case except the
variances are not assumed equal.
• How do you know if it is reasonable to assume
that variances are equal OR unequal?

31
DIFFERENCE IN VARIANCE HYPOTHESIS TEST
• Same example as Difference in Mean
• Sample standard deviations 0.1 and 0.2
• Sample sizes 15 and 15
• Null Hypothesis ratio of variances 1.0
• Alternative not equal
• Computed F statistic 0.25
• P-Value 0.0140071
• Reject the null hypothesis for alpha 0.05.

32
DIFFERENCE IN VARIANCE HYPOTHESIS TEST Can you
detect this difference?
33
DIFFERENCE IN VARIANCE HYPOTHESIS TEST -POWER
CURVE
34
PAIRED DIFFERENCE IN MEANS HYPOTHESIS TEST
• Two different inspectors each measure 10 parts on
the same piece of test equipment.
• Null hypothesis DIFFERENCE IN MEANS 0.0
• Alternative not equal
• Computed t statistic -1.22702
• P-Value 0.250944
• Do not reject the null hypothesis for alpha
0.05.

35
PAIRED DIFFERENCE IN MEANS HYPOTHESIS TEST -
POWER CURVE
36
ONE WAY ANALYSIS OF VARIANCE
• Used to test hypothesis that the means of several
populations are equal.
• Example Production line has 7 fill needles and
you wish to assess whether or not the average
fill is the same for all 7 needles.
• Experiment sample 20 fills from each of the 9
needles and test at 5 level of sign.
• Ho m1 m2 m3 m4 m5 m6 m7

37
RESULTS ANALYSIS OF VARIANCE TABLE
38
SINCE NEEDLE MEANS ARE NOT ALL EQUAL, WHICH ONES
ARE DIFFERENT?
• Multiple Range Tests for 7 Needles

39
VISUAL COMPARISON OF 7 NEEDLES
40
FACTORIAL (2k) DESIGNS
• Experiments involving several factors ( k of
factors) where it is necessary to study the joint
effect of these factors on a specific response.
• Each of the factors are set at two levels (a
low level and a high level) which may be
qualitative (machine A/machine B, fan on/fan off)
or quantitative (temperature 800/temperature 900,
line speed 4000 per hour/line speed 5000 per
hour).

41
FACTORIAL (2k) DESIGNS
• Factors are assumed to be fixed (fixed effects
model)
• Designs are completely randomized (experimental
trials are run in a random order, etc.)
• The usual normality assumptions are satisfied.

42
FACTORIAL (2k) DESIGNS
• Particularly useful in the early stages of
experimental work when you are likely to have
many factors being investigated and you want to
minimize the number of treatment combinations
(sample size) but, at the same time, study all k
factors in a complete factorial arrangement (the
experiment collects data at all possible
combinations of factor levels).

43
FACTORIAL (2k) DESIGNS
• As k gets large, the sample size will increase
exponentially. If experiment is replicated, the
runs again increases.

44
FACTORIAL (2k) DESIGNS (k 2)
• Two factors set at two levels (normally referred
to as low and high) would result in the following
design where each level of factor A is paired
with each level of factor B.

45
FACTORIAL (2k) DESIGNS (k 2)
• Estimating main effects associated with changing
the level of each factor from low to high. This
is the estimated effect on the response variable
associated with changing factor A or B from their
low to high values.

46
FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT
• Neither factor A nor Factor B have an effect on
the response variable.

47
FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT
• Factor A has an effect on the response variable,
but Factor B does not.

48
FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT
• Factor A and Factor B have an effect on the
response variable.

49
FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT
• Factor B has an effect on the response variable,
but only if factor A is set at the High level.
This is called interaction and it basically means
that the effect one factor has on a response is
dependent on the level you set other factors at.
Interactions can be major problems in a DOE if
you fail to account for the interaction when

50
EXAMPLEFACTORIAL (2k) DESIGNS (k 2)
• A microbiologist is interested in the effect of
two different culture mediums medium 1 (low) and
medium 2 (high) and two different times 10
hours (low) and 20 hours (high) on the growth
rate of a particular CFU.

51
EXAMPLEFACTORIAL (2k) DESIGNS (k 2)
• Since two factors are of interest, k 2, and we
would need the following four runs resulting in

52
EXAMPLEFACTORIAL (2k) DESIGNS (k 2)
• Estimates for the medium and time effects are
• Medium effect (1539)/2 (17 38)/2
-0.5
• Time effect (3839)/2 (17 15)/2 22.5

53
EXAMPLEFACTORIAL (2k) DESIGNS (k 2)
54
EXAMPLEFACTORIAL (2k) DESIGNS (k 2)
• A statistical analysis using the appropriate
statistical model would result in the following
information. Factor A (medium) and Factor B
(time)

55
EXAMPLECONCLUSIONS
• In statistical language, one would conclude that
factor A (medium) is not statistically
significant at a 5 level of significance since
the p-value is greater than 5 (0.05), but factor
B (time) is statistically significant at a 5
level of significance since this p-value is less
than 5.

56
EXAMPLECONCLUSIONS
• In layman terms, this means that we have no
evidence that would allow us to conclude that the
medium used has an effect on the growth rate,
although it may well have an effect (our
conclusion was incorrect).

57
EXAMPLECONCLUSIONS
• Additionally, we have evidence that would allow
us to conclude that time does have an effect on
the growth rate, although it may well not have an
effect (our conclusion was incorrect).

58
EXAMPLECONCLUSIONS
• In general we control the likelihood of reaching
these incorrect conclusions by the selection of
the level of significance for the test and the
amount of data collected (sample size).

59
2k DESIGNS (k gt 2)
• As the number of factors increase, the number of
runs needed to complete a complete factorial
experiment will increase dramatically. The
following 2k design layout depict the number of
runs needed for values of k from 2 to 5. For
example, when k 5, it will take 32 experimental
runs for the complete factorial experiment.

60
2k DESIGNS (k gt 2)
61
Interactions for 2k Designs (k 3)
• Interactions between various factors can be
estimated for different designs above by
multiplying the appropriate columns together and
then subtracting the average response for the
lows from the average response for the highs.

62
Interactions for 2k Designs (k 3)
63
2k DESIGNS (k gt 2)
• Once the effect for all factors and interactions
are determined, you are able to develop a
prediction model to estimate the response for
specific values of the factors. In general, we
will do this with statistical software, but for
these designs, you can do it by hand calculations
if you wish.

64
2k DESIGNS (k gt 2)
• For example, if there are no significant
interactions present, you can estimate a response
by the following formula. (for quantitative
factors only)

65
ONE FACTOR EXAMPLE
• Simple one factor example where the factor is
the number of hours a student studies for an exam
(LOW 10 HRS, HIGH 20 HRS) and the response
variable is their grade. Estimate the model for
prediction a students grade as a function of the
number of hours they study.

66
ONE FACTOR EXAMPLE
67
ONE FACTOR EXAMPLE
• The output shows the results of fitting a
general linear model to describe the relationship
between GRADE and HRS STUDY. The equation of
the fitted general model is
• GRADE 29.3 3.1 (HRS STUDY)
• The fitted orthogonal model is
• GRADE 75 15 (SCALED HRS)

68
Two Level Screening Designs
• Suppose that your brainstorming session resulted
in 7 factors that various people think might
have an effect on a response. A full factorial
design would require 27 128 experimental runs
without replication. The purpose of screening
designs is to reduce (identify) the number of
factors down to the major role players with a
minimal number of experimental runs. One way to
do this is to use the 23 full factorial design
and use interaction columns for factors.

69
Note that Any factor d effect is now
confounded with the ab interaction Any factor
e effect is now confounded with the ac
interaction etc. What is the de interaction
confounded with????????
70
Problems that Interactions Cause!
• Interactions If interactions exist and you fail
to account for this, you may reach erroneous
conclusions. Suppose that you plan an experiment
with four runs and three factors resulting in the
following data

71
Problems that Interactions Cause!
• Factor A Effect 0
• Factor B Effect 0
• Factor C Effect 5
• In this example, if you were assuming that
larger is better then you would set Factor C at
the high level and it appears to make no
difference where you set factors A and B. In
this case there is a factor A interaction with
factor B and this interaction is confounded with
the factor C effect.

72
Problems that Interactions Cause!
73
Resolution of a Design
• The above design would be called a resolution III
design because main effects are aliased
(confounded) with two factor interactions.

74
Resolution of a Design
• Resolution III Designs No main effects are
aliased with any other main effect BUT some (or
all) main effects are aliased with two way
interactions
• Resolution IV Designs No main effects are
aliased with any other main effect OR two factor
interaction, BUT two factor interactions may be
aliased with other two factor interactions
• Resolution V Designs No main effect OR two
factor interaction is aliased with any other main
effect or two factor interaction, BUT two factor
interactions are aliased with three factor
interactions.

75
Common Screening Designs
• Fractional Factorial Designs the total number
of experimental runs must be a power of 2 (4, 8,
16, 32, 64, ). If you believe first order
interactions are small compared to main effects,
then you could choose a resolution III design.
Just remember that if you have major
interactions, it can mess up your screening
experiment.

76
Common Screening Designs
• Plackett-Burman Designs Two level, resolution
III designs used to study up to n-1 factors in n
experimental runs, where n is a multiple of 4 (
of runs will be 4, 8, 12, 16, ). Since n may
be quite large, you can study a large number of
factors with moderately small sample sizes. (n
100 means you can study 99 factors with 100 runs)

77
Other Design Issues
• May want to collect data at center points to
estimate non-linear responses
• More than two levels of a factor no problem
(multi-level factorial)
• What do you do if you want to build a non-linear
model to optimize the response. (hit a target,
maximize, or minimize) called response surface
modeling

78
Other Design Issues
• What do you do if the factors levels are
categorical and not quantitative, or some are
categorical and some are quantitative?
• What do you do if the structure of you experiment
is nested? These are called heirarchical
designs and will allow you to partition the total
variability among the different levels of the
design (called variance components)

79
Response Surface Designs Box-BehnkenAfter
screening designs identify major factors Next
step.
• Design class Response Surface
• Design name Box-Behnken design
• Base Design
• -----------
• Number of experimental factors 3 Number of
blocks 1
• Number of responses 1
• Number of runs 15 Error degrees
of freedom 5
• Randomized No
• Factors Low High
Units Continuous
• --------------------------------------------------
----------------------
• Factor_A -1.0 1.0
Yes
• Factor_B -1.0 1.0
Yes
• Factor_C -1.0 1.0
Yes

80
Response Surface Designs Box-Behnken
FACTOR A FACTOR B FACTOR C
0 0 0
-1 -1 0
1 -1 0
-1 1 0
1 1 0
-1 0 -1
1 0 -1
0 0 0
-1 0 1
1 0 1
0 -1 -1
0 1 -1
0 -1 1
0 1 1
0 0 0
81
Response Surface Designs Central Composite
• Design class Response Surface
• Design name Central composite blocked cube-star
• Number of experimental factors 3 Number of
blocks 2
• Number of responses 1
• Number of runs 16 Error degrees
of freedom 5
• Randomized No
• Factors Low High
Units Continuous
• --------------------------------------------------
----------------------
• Factor_A -1.0 1.0
Yes
• Factor_B -1.0 1.0
Yes
• Factor_C -1.0 1.0
Yes

82
Response Surface Designs Central Composite
FACTOR A FACTOR B FACTOR C
-1 -1 -1
1 -1 -1
-1 1 -1
1 1 -1
0 0 0
-1 -1 1
1 -1 1
-1 1 1
1 1 1
-1.76383 0 0
1.76383 0 0
0 -1.76383 0
0 0 0
0 1.76383 0
0 0 -1.76383
0 0 1.76383
83
Multilevel Factorial Designs
• Design class Multilevel Factorial
• Number of experimental factors 3 Number of
blocks 1
• Number of responses 1
• Number of runs 27 Error degrees
of freedom 17
• Randomized No
• Factors Low High
Levels Units
• --------------------------------------------------
-----------------------
• Factor_A -1.0 1.0
3
• Factor_B -1.0 1.0
3
• Factor_C -1.0 1.0
3

84
Multilevel Factorial Designs
85
Nested Design
• Design class Variance Components
• Number of experimental factors 3
• Number of responses 1
• Number of runs 27
• Randomized No
• Factors Levels Units
• -----------------------------------------------
• Factor_A 3
• Factor_B 3
• Factor_C 3
• You have created a variance components design
which will estimate the contribution of 3 factors
to overall process variability. The design is
hierarchical, with each factor nested in the
factor above it. A total of 27 measurements are
required.

86
Nested Design
87
Response Surface Designs Box-BehnkenEXAMPLE -
RECAP
• Design class Response Surface
• Design name Box-Behnken design
• Base Design
• -----------
• Number of experimental factors 3 Number of
blocks 1
• Number of responses 1
• Number of runs 15 Error degrees
of freedom 5
• Randomized No
• Factors Low High
Units Continuous
• --------------------------------------------------
----------------------
• Factor_A 10 30
Yes
• Factor_B 30 60
Yes
• Factor_C 40 60
Yes

88
(No Transcript)
89
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1 (NOISE)
RUN F1 F2 F3
1 10 45 60
2 30 45 40
3 20 30 40
4 10 30 50
5 20 45 50
6 30 60 50
7 20 45 50
8 30 45 60
9 20 45 50
10 20 60 40
11 10 45 40
12 30 30 50
13 20 60 60
14 10 60 50
15 20 30 60
90
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
RUN F1 F2 F3 Y0
1 10 45 60 11800
2 30 45 40 8800
3 20 30 40 8400
4 10 30 50 9300
5 20 45 50 9400
6 30 60 50 8300
7 20 45 50 9400
8 30 45 60 10800
9 20 45 50 9400
10 20 60 40 8400
11 10 45 40 9800
12 30 30 50 11300
13 20 60 60 10400
14 10 60 50 12300
15 20 30 60 10400
91
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
92
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
93
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
94
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
RUN F1 F2 F3 Y100
1 10 45 60 11825
2 30 45 40 8781
3 20 30 40 8413
4 10 30 50 9216
5 20 45 50 9288
6 30 60 50 8261
7 20 45 50 9329
8 30 45 60 10855
9 20 45 50 9205
10 20 60 40 8538
11 10 45 40 9718
12 30 30 50 11308
13 20 60 60 10316
14 10 60 50 12056
15 20 30 60 10378
95
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
96
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
97
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
98
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
• Optimize Response
• -----------------
• Goal maximize Y
• Optimum value 13139.4
• Factor Low High
Optimum
• --------------------------------------------------
---------------------
• Factor_A 10.0 30.0
10.1036
• Factor_B 30.0 60.0
60.0
• Factor_C 40.0 60.0
60.0

99
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
RUN F1 F2 F3 Y100
1 10 30 40 8270
2 20 30 40 8272
3 30 30 40 10324
4 10 45 40 9928
5 20 45 40 8520
6 30 45 40 8973
7 10 60 40 11082
8 20 60 40 8377
9 30 60 40 7410
10 10 30 50 9191
11 20 30 50 9331
12 30 30 50 11131
13 10 45 50 10615
100
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
RUN F1 F2 F3 Y100
14 20 45 50 9302
15 30 45 50 9723
16 10 60 50 12088
17 20 60 50 9343
18 30 60 50 8260
19 10 30 60 10313
20 20 30 60 10363
21 30 30 60 12267
22 10 45 60 11763
23 20 45 60 10534
24 30 45 60 10791
25 10 60 60 13281
26 20 60 60 10349
27 30 60 60 9497
101
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
102
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
• Optimize Response
• -----------------
• Goal maximize Y
• Optimum value 13230.6
• Factor Low High
Optimum
• --------------------------------------------------
---------------------
• Factor_A 10.0 30.0
10.0
• Factor_B 30.0 60.0
60.0
• Factor_C 40.0 60.0
60.0

103
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
104
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
105
CLASSROOM EXERCISE
• STUDENT IN-CLASS EXPERIMENT Collect data for
experiment to determine factor settings (two
factors) to hit a target response (spot on wall).
• Factor A height of shaker (low and high)
• Factor B location of shaker (close to hand and
close to wall)
• Design experiment would suggest several
replications

106
CLASSROOM EXERCISE
• Conduct Experiment student holds 3 foot pin
the tail on the donkey stick and attempts to hit
the target. An observer will assist to mark the
hit on the target.
• Collect data students take data home for week
and come back with what you would recommend AND
why.
• YOU TELL THE CLASS HOW TO PLAY THE GAME TO WIN.

107
CLASSROOM EXERCISE
108
CLASSROOM EXERCISE
109
CLASSROOM EXERCISE
• HOMEWORK
• .Determine the effects marker stick and
vertical pole have on the mean location of the
hit.
• .Determine the effects marker stick and
vertical pole have on the standard deviation
of the hit.
• .Which factor would you say affects the mean
location of the hit?
• .Which factor would you say affects the standard
deviation of the hit?
• OPTIMAL SETTINGS Where would you recommend we
locate the vertical pole and the marker stick
IF we wish to (a) MINIMIZE THE VARIABILITY OF THE
HIT and (b) HIT THE TARGET LOCATED AT 0?

110
PIN THE TAIL DATA INPUT
111
ESTIMATE OF EFFECTS (MEAN HIT)
• Estimated effects for MEAN
• --------------------------------------------------
--------------------
• average 0.875
• AMARKER STICK 1.906
• BVERTICAL POLE 12.969
• AB 4.625
• --------------------------------------------------
--------------------
• No degrees of freedom left to estimate standard
errors.

112
ESTIMATE OF EFFECTS (MEAN HIT)
113
ESTIMATE OF EFFECTS (MEAN HIT)
114
INTERACTION PLOT (MEAN HIT)
115
3-D PLOT OF RESPONSE (MEAN HIT)
116
CONTOUR PLOT OF RESPONSE (MEAN HIT)
117
ANALYSIS OF VARIANCE TABLE (MEAN HIT)
• Analysis of Variance for MEAN
• --------------------------------------------------
------------------------------
• Source Sum of Squares Df
Mean Square F-Ratio P-Value
• --------------------------------------------------
------------------------------
• AMARKER STICK 3.63284 1
3.63284 0.17 0.7511
• BVERTICAL POLE 168.195 1
168.195 7.86 0.2181
• Total error 21.3906 1
21.3906
• --------------------------------------------------
------------------------------

118
ESTIMATED LINEAR RESPONSE MODEL (MEAN HIT)
• Regression coeffs. for MEAN
• --------------------------------------------------
--------------------
• constant 0.875
• AMARKER STICK 0.953
• BVERTICAL POLE 6.4845
• --------------------------------------------------
--------------------
• ---------------
• This pane displays the regression equation
which has been fitted to
• the data. The equation of the fitted model is
• MEAN 0.875 0.953MARKER STICK
6.4845VERTICAL POLE

119
OPTIMAL FACTOR SETTINGS (MEAN HIT)
• Optimize Response
• -----------------
• Goal maintain MEAN at 0.0
• Optimum value 0.0
• Factor Low High
Optimum
• --------------------------------------------------
---------------------
• MARKER STICK -1.0 1.0
0.03311
• VERTICAL POLE -1.0 1.0
-0.139803

120
ESTIMATE OF EFFECTS (STD DEV HIT)
• Estimated effects for STD DEV
• --------------------------------------------------
--------------------
• average 2.63275
• AMARKER STICK 2.7605
• BVERTICAL POLE 0.3735
• AB -0.0895

121
ESTIMATE OF EFFECTS (STD DEV HIT)
• Analysis of Variance for STD DEV
• --------------------------------------------------
------------------------------
• Source Sum of Squares Df
Mean Square F-Ratio P-Value
• --------------------------------------------------
------------------------------
• AMARKER STICK 7.62036 1 7.62036
951.33 0.0206
• BVERTICAL POLE 0.139502 1 0.139502
17.42 0.1497
• Total error 0.00801025 1
0.00801025
• --------------------------------------------------
------------------------------
• Total (corr.) 7.76787 3

122
OPTIMAL FACTOR SETTINGS (STD DEV HIT)
• Optimize Response
• -----------------
• Goal minimize STD DEV
• Optimum value 1.06575
• Factor Low
High Optimum
• --------------------------------------------------
---------------------
• MARKER STICK -1.0 1.0
-1.0
• VERTICAL POLE -1.0 1.0
-1.0

123
INTERACTION (STD DEV HIT)
124
CONTOUR PLOT OF RESPONSE (STD DEV HIT)
125
• I WOULD
• 1. SET THE MARKER STICK AT LOW (CLOSE TO THE
WALL)
• 2. SET THE VERTICAL POLE AT A VALUE THAT WILL
HIT THE TARGET.

126