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DESIGN OF EXPERIMENTS by R. C. Baker

- How to gain 20 years of experience in one short

week!

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.

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.

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.

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.

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)

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.

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.

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.

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.

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.

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.

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

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

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).

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

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

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

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

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

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?

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

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.

Single Mean Hypothesis Test Power Curve

Single Mean Hypothesis Test Power Curve

Single Mean Hypothesis Test Power Curve -

Different Sample Sizes

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

DIFFERENCE IN MEANS - EQUAL VARIANCES Can you

detect this difference?

DIFFERENCE IN MEANS - EQUAL VARIANCES

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?

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.

DIFFERENCE IN VARIANCE HYPOTHESIS TEST Can you

detect this difference?

DIFFERENCE IN VARIANCE HYPOTHESIS TEST -POWER

CURVE

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.

PAIRED DIFFERENCE IN MEANS HYPOTHESIS TEST -

POWER CURVE

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

RESULTS ANALYSIS OF VARIANCE TABLE

SINCE NEEDLE MEANS ARE NOT ALL EQUAL, WHICH ONES

ARE DIFFERENT?

- Multiple Range Tests for 7 Needles

VISUAL COMPARISON OF 7 NEEDLES

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).

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.

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).

FACTORIAL (2k) DESIGNS

- As k gets large, the sample size will increase

exponentially. If experiment is replicated, the

runs again increases.

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.

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.

FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT

- Neither factor A nor Factor B have an effect on

the response variable.

FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT

- Factor A has an effect on the response variable,

but Factor B does not.

FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT

- Factor A and Factor B have an effect on the

response variable.

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.

EXAMPLE FACTORIAL (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.

EXAMPLE FACTORIAL (2k) DESIGNS (k 2)

- Since two factors are of interest, k 2, and we

would need the following four runs resulting in

EXAMPLE FACTORIAL (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

EXAMPLE FACTORIAL (2k) DESIGNS (k 2)

EXAMPLE FACTORIAL (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)

EXAMPLE CONCLUSIONS

- 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.

EXAMPLE CONCLUSIONS

- 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).

EXAMPLE CONCLUSIONS

- 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).

EXAMPLE CONCLUSIONS

- 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).

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.

2k DESIGNS (k gt 2)

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.

Interactions for 2k Designs (k 3)

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.

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)

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.

ONE FACTOR EXAMPLE

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)

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.

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????????

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

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.

Problems that Interactions Cause!

Resolution of a Design

- The above design would be called a resolution III

design because main effects are aliased

(confounded) with two factor interactions.

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.

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.

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)

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

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)

Response Surface Designs Box-Behnken After

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

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

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

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

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

Multilevel Factorial Designs

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.

Nested Design

Response Surface Designs Box-Behnken EXAMPLE -

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

(No Transcript)

Response Surface Designs Box-Behnken REAL

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

Response Surface Designs Box-Behnken REAL

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

Response Surface Designs Box-Behnken REAL

MODEL Y 40F1200F2100F3-10F1F29F1F1

(NOISE) Example std. dev. of noise 0

Response Surface Designs Box-Behnken REAL

MODEL Y 40F1200F2100F3-10F1F29F1F1

(NOISE) Example std. dev. of noise 0

Response Surface Designs Box-Behnken REAL

MODEL Y 40F1200F2100F3-10F1F29F1F1

(NOISE) Example std. dev. of noise 0

Response Surface Designs Box-Behnken REAL

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

Response Surface Designs Box-Behnken REAL

MODEL Y 40F1200F2100F3-10F1F29F1F1

(NOISE) Example std. dev. of noise 100

Response Surface Designs Box-Behnken REAL

MODEL Y 40F1200F2100F3-10F1F29F1F1

(NOISE) Example std. dev. of noise 100

Response Surface Designs Box-Behnken REAL

MODEL Y 40F1200F2100F3-10F1F29F1F1

(NOISE) Example std. dev. of noise 100

Response Surface Designs Box-Behnken REAL

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

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

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

Response Surface Designs Three Level Factorial

Design (33) REAL MODEL Y 40F1200F2100F3-10F1F

29F1F1 (NOISE) Example std. dev. of noise

100

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

Response Surface Designs Three Level Factorial

Design (33) REAL MODEL Y 40F1200F2100F3-10F1F

29F1F1 (NOISE) Example std. dev. of noise

100

Response Surface Designs Three Level Factorial

Design (33) REAL MODEL Y 40F1200F2100F3-10F1F

29F1F1 (NOISE) Example std. dev. of noise

100

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

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.

CLASSROOM EXERCISE

CLASSROOM EXERCISE

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?

PIN THE TAIL DATA INPUT

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.

ESTIMATE OF EFFECTS (MEAN HIT)

ESTIMATE OF EFFECTS (MEAN HIT)

INTERACTION PLOT (MEAN HIT)

3-D PLOT OF RESPONSE (MEAN HIT)

CONTOUR PLOT OF RESPONSE (MEAN HIT)

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 - --------------------------------------------------

------------------------------

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

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

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

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

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

INTERACTION (STD DEV HIT)

CONTOUR PLOT OF RESPONSE (STD DEV HIT)

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.

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

Contour Plots for Mean and Std. Dev.