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DESIGN OF EXPERIMENTS by R. C. Baker

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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
    conclusions/inferences about the results,
    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
    with Accepting Bad Stuff?

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
    designing your experiment.

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
  • --------------------------------------------------
    --------------------
  • The StatAdvisor
  • ---------------
  • 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
SO, WHATS THE ANSWER?
  • 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
SO, WHATS THE ANSWER?
  • FROM REGRESSION FOR MEAN HIT, SET MARKER STICK
    AT -1, HIT AT 0, AND SOLVE FOR VP
  • HIT .0875 .953MS 6.4845VP
  • 0 .875 .953(-1) 6.4845VP
  • Resulting in
  • VP .012 and MS -1

127
Contour Plots for Mean and Std. Dev.
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