# Design of Experiments (DOE) and Analysis of Variation (ANOVA)

Document Sample

```					ANOVA & Design of
Experiments

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ANALYSIS OF VARIANCE

(ANOVA)

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ANALYSIS OF VARIANCE (ANOVA)
 Analysis of variance is a technique for examining whether
there exist significant differences between two or more
population averages.
 Analysis of Variance is a powerful tool applicable for two
different purposes.
a. To identify the dominant 'factors' from a list of suspects;
b. To estimate the contribution of different factors in the variation
of the characteristic under study.

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EXAMPLE-ANOVA
• An experiment was conducted to determine the effect of
temperature on the UTS of Steel. The material was tested at four
different temperatures and the data were recorded as follows (Data
are coded):The object of this experiment is to test whether the
effects of the temperature are the same or not.
Temperatures
1           2           3            4
66          74         55            52
65          71         56            49
72          60         55            55
69          65         49            53
70          66         53            51
342         336         268         260
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EXAMPLE-ANOVA

• The null hypothesis is that there is no real difference between
temperatures. This hypothesis is tested by the F statistic for the
ratio between temperature mean square to residual mean
square. The calculated value of F = 29.8. For 3 and 16 degrees
of freedom exceeds the tabular value 5.29 of F, even at 1% level
of significance.

• Therefore the null hypothesis is rejected and it is concluded
that real differences between temperatures - exist.

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ANOVA FROM MINITAB
 Open MINITAB Worksheet.
 Enter the data in one column and its corresponding level
of the factor in another column
 Choose STAT > ANOVA > ONE-WAY
 In response, enter the data column.
 In factor, enter the levels of the factor column.
 Enter OK.
 If, P-value < 0.05; Reject the Null Hypothesis and
conclude that there is a difference in means for all the
levels of the factor.                                  6
One-way ANOVA: t1, t2, t3, t4

Source     DF       SS          MS       F       P
Factor      3   1135.0    378.3      29.79   0.000
Error     16     203.2     12.7
Total     19    1338.2
S = 3.564     R-Sq = 84.82%         R-Sq(adj) = 81.97%
Individual 95% CIs For Mean Based on
Pooled StDev
Level   N     Mean     StDev   ---------+---------+---------+---------+
t1      5   68.400     2.881                                        (-----*-----)
t2      5   67.200   5.450                                    (-----*-----)
t3      5   53.600   2.793        (----*-----)
t4      5   52.000   2.236     (-----*----)
---------+---------+---------+---------+
54.0         60.0       66.0        72.0
Pooled StDev = 3.564                                     7
Main Effects Plot (fitted means) for UTS
70

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Mean of C6

60

55

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t1                t2                 t3         t4
Temperature
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General Linear Model: UTS versus Temp.
Factor                Type      Levels    Values
Temperature           fixed      4        t1, t2, t3, t4
Analysis of Variance for UTS, using Adjusted SS for Tests
Temp       3      1135.00    1135.00    378.33   29.79       0.000
Error     16       203.20     203.20     12.70
Total     19      1338.20

S = 3.56371         R-Sq = 84.82%       R-Sq(adj) = 81.97%
Unusual Observations for UTS
Obs     Temp         Fit       SE Fit    Residual       St Resid
6     74.0000     67.2000    1.5937      6.8000            2.13 R
8     60.0000     67.2000    1.5937     -7.2000            -2.26 R
R denotes an observation with a large standardized residual.
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DESIGN OF
EXPERIMENTS
(DOE)

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STATISTICALLY DESIGNED EXPERIMENTS
   A statistically designed experiment permits
simultaneous consideration of all the possible
factors that are suspected to have bearing on
the quality problem under investigation and as
such even if interactions effect exist, a valid
evaluation of the main effect can be made.
Scanning a large number of variables is one of
the ready and simpler objectives that a
statistically designed experiment would fulfill in
many problem situations.

   Even a limited number of experiments would
enable the experimenter to uncover the vital
factors as which further trials would yield useful
results. The approach has number of merits, it
is quick, reliable and efficient.

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Objectives Of Experimentation
The following are some of the objectives of
experimentation in an industry :
   Improving efficiency or yield
   Finding optimum process settings
   Locating sources of variability
   Correlating process variables with product
characteristics
   Comparing      different   processes, machines,
materials etc
   Designing new processes and products.

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NEED FOR DESIGNED EXPERIMENT

• A design engineer is interested in studying the effect Bronze and
Synthetic material on life of Bush.

•Here, the objective is to determine the material that produces the
maximum life (minimum wear) for this particular Bush.

•The procedure thought of by the engineer is to make a number of Bush
specimens with each material and to measure the life of the specimens
after running the pump.

•The results thus obtained will be averaged and the average life of the
specimens made with each material will be used to determine which
material is best.

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NEED FOR DESIGNED EXPERIMENT
• As we think about the experiment, a number of important
questions come to mind:
a) Are these two materials the only materials of potential interest?
b) Are there any other factors that might affect life that should be
investigated or controlled in this experiment?
c) How many specimens of Bush should be tested with each other
material?
d) How should the specimens be made with different materials and in
what order should the data be collected?
e) What method of data analysis should be used?
f) What difference in average observed life between the two materials
will be considered important?
• All of these questions and perhaps many others will have to be
satisfactorily answered before the experiment is performed.
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PROOF OF THE NEED FOR EXPERIMENTATION
 After having selected the area for experimentation we have
to ensure that the problem is of ‘Break through’ or
‘Improvement’ nature and not a problem of ‘control’ nature.
For this purpose past data should be suitably analyzed and
plotted on some process control chart to check whether the
process is within statistical control or not. If the analysis
shows lack of control or statistical instability, then it is a
problem of ‘control’ nature and experimentation may not be
needed.

 However if the problem is of chronic nature and there is
stability in the process, then it establishes the need for
experimentation.    Before    deciding    to   carry   out
experimentation the need for experimentation must be
established.                                             15
IF CONTROL ONLY IS YOUR PROBLEM

WE SEEK ANSWERS TO THE FOLLOWING
QUESTIONS

Which variables (Input & Process) are important
?
Are they controllable ?
Do they change      by    chance   or   by   intent
(assignable) ?

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IF CONTROLLABLE AND CHANGE BY CHANCE

E.g: You find that finish (Y) deteriorates
with time.   You    find  that   this  is
because of Tool Wear.
   Does the tool wear by chance
   Do you have some control over the above
- yes, by changing the tool
   In all such cases where „X‟ changes by
chance but some control is exercisable

USE CONTROL CHARTS ON YOUR Y
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IF CONTROLLABLE & CHANGE BY INTENT
E.g.: Supplier of sheets        for     vacuum
forming process is      an           important
factor for shift of mean.

   The change in material on line is done by
intent and knowingly.

   In such cases where you know when a
critical X has changed levels. Find the
optimum setting of the process for each
level of the X.

D.O.E. IS THE TOOL FOR IT
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IF VARIATION IS OUR PROBLEM

   Find at what level each of the X has
to be set on i.e. the optimum
process settings.

   D.O.E. is the tool for it

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Design of experiments
Design of experiments (DOE) is a valuable tool to
optimize product and process designs, to accelerate
the development cycle, to reduce development costs,
to improve the transition of products from research
and development to manufacturing and to effectively
trouble shoot manufacturing problems. Today, Design
of Experiments is viewed as a quality technology to
achieve product excellence at lowest possible overall
cost.

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One-factor-At-A-Time
This is a traditional method of experimentation which tests, then changes, one factor
at a time to allow for observation and comparison. Note on the example below, all 8
factors are varied one-at-a-time . It is efficient because it takes only 16 runs.

•A1 and A2 are evaluated by comparing Result - 1 and Result - 2
B1, B2 and B3 are evaluated by comparing Result-2, Result-3 and Result-4.
C1, C2, and C3 are evaluated by comparing Result-4, Result-5 and Result-6
Etc.
Run No.     A    B    C    D    E    F    G    H    Re sult
1        1    1    1    1    1    1    1    1   Result 1
2        2    1    1    1    1    1    1    1   Result 2
3        2    2    1    1    1    1    1    1   Result 3
4        2    3    1    1    1    1    1    1   Result 4
5        2    3    2    1    1    1    1    1   Result 5
6        2    3    3    1    1    1    1    1   Result 6
7        2    3    3    2    1    1    1    1   Result 7
8        2    3    3    3    1    1    1    1   Result 8
9        2    3    3    3    2    1    1    1   Result 9
10        2    3    3    3    3    1    1    1   Result 10
11        2    3    3    3    3    2    1    1   Result 11
12        2    3    3    3    3    3    1    1   Result 12
13        2    3    3    3    3    3    2    1   Result 13
14        2    3    3    3    3    3    3    1   Result 14
15        2    3    3    3    3    3    3    2   Result 15
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16        2    3    3    3    3    3    3    3   Result 16
 Problem: Current Car gas mileage is 20
mpg. Would like to get 30 mpg.
 We might try:
 Change brand of gas
 Change octane rating
 Drive Slower
 Tune-up Car
 Wash and wax car
 Change Tire Pressure
   What if it works?
   What if it doesn’t?

“Survey Says” These variables greatly effect MPG   22
One-Factor-At-A-Time
Problem: Fuel economy we want is 30 MPG
Try changing each input variable at two settings believed to be
associated with dramatically changing fuel economy. See what
happens.
Speed          Octane     Tire Pressure         MPG
55             85             30               23
60             85             30               29
60             90             30               23
60             85             35               24
How many more Combinations would you need to figure out the best
combination of variables? (3 Variables at two settings; 2x2x2 = 8 total)
How can you explain the above results? (Combination 2 is the answer)
If there were more variables, how long would it take to get a good solution?
(Multiply by another 2 for each one)
What if there’s a specific combination of two or more variables that leads to
the best mileage? (Too hard for me to figure out; What do you think?)

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OFAT Studies and Interactions
• Suppose we looked at one factor at a time. We conduct a total of 8
runs
Cube Plot (data means) for Miles per Gallon

30                           36

37                  23
90

Octane         37                           24
35

Tire Pressur
23                  29
85                               30
55                60
Speed

While we would have made significant improvement, we would
have missed the optimum point!                                  24
Full Factorial Experiment
Problem: Want 30 MPG

Speed        Octane   Tire Pressure    MPG
55           85           30          23
60           85           30          29
55           90           30          37
60           90           30          23
55           85           35          37
60           85           35          24
55           90           35          30
60           90           35          36
OFAT Runs

What conclusion do you make now?
(Murphy is alive and well!)         25
TERMINOLOGY USED IN D.O.E.
EXPERIMENT: A planned set of operations which leads to a
corresponding set of observations. The purpose of
experimentation is to ensure that the experimenter obtains
the data relevant to the task of decision making in an
economical way.

OUTCOME (RESPONSE): The numerical result of a trial based
on a given treatment combination is called Outcome or
Response.

The response may be :
◦ Continuous or measurement type and follows a normal
distribution
◦ Continuous or measurement type but does not follow normal
distribution
◦ Discrete or count type and does not follow normal distribution

E.g.: diameter of a shaft, No. of rejected cylinders etc.

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TERMINOLOGY USED IN D.O.E.
FACTOR   (X) - The parameters of the process which are
deliberately varied from trial to trial. This could be
qualitative or quantitative. e.g. Speed, feed, coolant rate,
operator skill.
LEVELS OF A FACTOR - The alternative values of a factor
considered in the experiment are called its levels.
e.g.: Speed 400 rpm, circular wheel etc.
TREATMENT COMBINATION - The set of levels of all factors
employed in a given trial is called treatment or treatment
combination.
EXPERIMENTAL UNIT : It is a generic term used to denote the
group of material to which a treatment is applied in a single
trial of the experiment.

BALANCED TEST - Where number of samples in each
treatment combination is same.

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TERMINOLOGY USED IN D.O.E.
EFFECT OF FACTOR :
MAIN EFFECT: The change in the average response
produced by a change in the level of the factor is
called “Main Effect” of that factor.

INTERACTION EFFECT : If the effect of one factor is
different at different levels of another factor, the two
factors are said to interact (or) to have interaction.

The interaction between factors A and B, is termed as
“First Order Interaction” or “Two Factor Interaction”
and is denoted by AxB.

If the interaction between two factors A and B, is
different at different levels of a third factor C, then
there is said to be interaction among three factors.
This is referred to as “Second Order Interaction” or
“Three Factor Interaction” and is denoted by AxBxC.8 2
TERMINOLOGY USED IN D.O.E.
Interactions
 Y = f (X1, X2). But if X2 = f (X1)
Then changing X1 will give other than predicted Y
since X2 also automatically changes.
 The same holds true for change of x2

e.g: leakage of dome welded components is a
function of current and electrode thickness but
current also depends on electrode thickness.

Hence there is interaction between electrode and
current
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TERMINOLOGY USED IN D.O.E.
An example to understand interaction

F
I
N
I
S
H
Speed X              Speed Y

Changing feed from level A to level B betters finish.
But this effect is more predominant speed level Y than speed level X.
Hence there is an interaction between speed & feed
REPLICATION: Replication is a repetition of the whole experiment in order
to estimate experimental error, increase precision (detect smaller changes).

EXPERIMENTAL ERROR: The failure of two identically treated              30

experimented units to give the same value.
STEPS IN DESIGNING AND ANALYZING
1. Statement of the problem.
2. Formulation of hypothesis.
3. Planning of the experiment.
a) Choosing an appropriate experimental technique.
b) Examination of possible outcomes to make sure that the experiment
provides the required information.
c) Consideration of possible results from the point of view of statistical
analysis.
4. Collection of data, after performing the experiment according
to the plan.
5. Statistical Analysis of the data.
6. Drawing conclusions with appropriate level of significance.
7. Verification or evaluation of results (conclusions).
8. Drawing final conclusions and recommendations.
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PLANNING FOR EXPERIMENTATION

The various steps to be followed in this
direction are listed below :
 Selection   of area of study : Pareto analysis
 Proof   of the need for experimentation
 Brain storming and Cause & Effect diagram :
To list all the possible factors
 Classification   of factors
 Interactions   to be studied
 Response    and type of model for analysis

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Classification of factors
Tools like brainstorming and cause & effect diagrams
helps in identification of factors and preparing a
complete list of the factors involved in any
experiment. Factors listed can be classified into three
categories :
1. Experimental Factors
Experimental factors are those which we really
experiment with by varying them at various levels.
2. Control Factors
Control Factors are those which are kept at a
constant (controlled) level throughout
experimentation.
3. Error or Noise Factors
Error or Noise factors are those which can neither be
changed at our will nor can be fixed at one particular
level. Effect of these factors causes the error
component in the experiment and as such these
factors are termed as error or noise factors.
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Note : At the planning stage itself all the factors viz. Experimental, Control and error should
be recognized. This will help to tackle them appropriately during experimentation.
PLANNING FOR EXPERIMENTATION
•   State what do you want
• What is my response(s)
• What are my factors
• Choose the level of the factors
• Decide on the design
• Run the design and collect the data
• Analysis the data and obtain results
•   Run confirming test on settings

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Requisites of DOE

UNBIASEDNESS
PRECISION
INDUCTIVE SCOPE
CLEARLY DEFINED OBJECTIVES

Fulfillment of the requirements
1. RANDOMISATION
2. REPLICATION AND
3. LOCAL CONTROL OR ERROR CONTROL

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Replication
Definition
means repeating all the
Replication
experimental conditions (or running a
combination) two or more times.
◦ This does not mean measuring an experimental unit
twice
◦ It does mean repeating a certain set of conditions
and measuring the new output
◦ Two replicates means that for an 8-run design you
will do 16 runs in one experiment
 Minitab will randomize all the runs (including replicates) at
the same time
 If for some reason you cannot, or choose not to, do all the
runs at the same time, you need to be concerned about
blocking (a topic we’ll discuss later in this module)
◦ One replicate really means no replication

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Why Do Replicates?
◦ To measure pure error: the amount of variability
among runs performed at the same experimental
conditions (this represents common cause
variation)
◦ To see more clearly whether or not a factor is
important—
is the difference between responses due to a
change in factor conditions (an induced special
cause) or is it due to common cause variability?
◦ To see the effect of changing factor conditions not
only on the average response, but also on
response variability, if desired (two responses can
be analyzed: the mean and the st. dev.)

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Why Randomize: Example
   Background
   Suppose that the plating thickness on printed circuit
boards is the response of interest. Notice that this
value tends to decline over the month.

Thickness vs. Day of Month

200

Plating thickness in microns
190

180

170

160

150

140

5   10   15   20   25   30
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What might explain this decline?                                              Day of Month
Why Randomize: Example, cont.
   Suppose in an experiment to evaluate the
effect of soak temperature, the company
tested 50°C first, then 70°C .
Thickness vs. Day of Month
50°C                   70 °C

200
Plating Thickness in Microns

190

180

170

160

150

140

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5      10       15    20       25   30

Day of Month
Why Randomize: Example, cont.
   Alternatively, what if both temperatures
were tested randomly throughout the
month?           Thickness vs . Day of Month

50°C
70 °C
200
Plating Thickness in Microns

190

180

170

160

150

140

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5   10   15    20   25       30

Day of Month
Randomization: the Experimenter‟s
Insurance
Definition
To  assign the order in which the experimental
trials will be run using a random mechanism
◦ It is not the standard order
◦ It is not running trials in an order that is convenient
◦ To create a random order, you can “pull numbers from
a hat” or have Minitab randomize the sequence of
trials for you
Why?
◦ Averages the effect of any lurking variables over all of
the factors in the experiment
 Prevents the effect of a lurking variable from being mistakenly
attributed to another factor
◦ Helps validate statistical conclusions made from the
experiment
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Local control

   By local control is meant blocking, grouping or
balancing the experimental units. Balancing is done by
replicating all the treatment combination, the same
number of times under different conditions. Local
control makes the test more sensitive and powerful, by
reducing the experimental error.

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Formulation of experimental problem
1.   Process Flow chart prepared
2.   Measures Established
3.   Stability of the process verified
4.   Cause & Effect Analysis made involving process experts and Operators
5.   Technical Justification of factors and it’s level completed.

Factor Unit of Present Investigation Nature of       Levels       Operations
Measure Status range          Factor                       Implications   Remarks
(E/C/N)     1     2      3

* E : Experimental factors, C : Control factors, N : Noise factors                   43

• Suspected interactions between Factors:
•   It is observed that only 2 to 6 variables
end up being vital few.
•   Try to keep the design simple by
utilizing your experience to decide which
are the most likely factors unless you
know nothing of the process.
•   The above calls for judgement which
sometimes can be wrong.

REMEMBER:
The Experiment is Run to Understand
Reality, Not the Data
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WHY EXPERIMENTS CAN FAIL / ABORTED
   You are not clear what you want.
   Measurement systems differ at different points in
time.
   Lack of understanding of DOE tools / Strategies .
   No management support
   Need instant results
   No resources
   Lack of time
   Cost of experiment may be high

45
Classification of
Design Block
•Completely Randomized
•Randomised
•Balanced incomplete block
•Partially balanced incomplete   block
•Latin Square
•Factorial
•Blocked factorial
•Fractional factorial
•Youden square
•Nested
•Response surface
•Mixture designs
•Taguchi’s Design
•EVOP

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Classification of
Design : Completely Randomized
Design
Type of Application : Appropriate when one experimental
factor is being investigated.
Structures :
Basic : One factor is investigated by allocating
experimental units at random to treatments (Levels of a
factor)
Blocking : None
Information Sought :
1. Estimate and compare treatment effects
2. Estimate Variance.

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Completely Randomized Design : Example
Example : Suppression of Bacterial Growth on Stored Meats
The shelf life of stored meats is the time a prepackaged cut
salable, safe and nutritious. There are four type of packaging
and the management is interested about the best packaging
alternatives.
The packaging types are
- Commercial plastic Wraps
- Vacuum packaged
- 1% CO, 40% O2, 59% N
-100% CO2
How to plan the experiment? We also decided to carry out the
trials in 3 samples each.

48
Completely Randomized Design : Example
The data collected and presented below;
Psychotrophic Bacteria
Pckaging Condition      Log(Count/cm2) Total      Average
Commercial plastic wrap 7.66, 6.98, 7.80 22.44        7.48
Vacuum packaged         5.26, 5.44, 5.80 16.50        5.50
1% CO, 40% O2, 59% N 7.41, 7.33, 7.04 21.78           7.26
100% CO2                3.51, 2.91, 3.66   10.08     3.36

Do the following;
-Write down the test of hypothesis
-Carry out ANOVA
-Compare treatment mean

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Completely Randomized Design : Example
Ho: Null Hypothesis: Psychotropic Bacteria Growth at four
different packing material is same.
H1: Alternate Hypothesis: Psychotropic Bacteria Growth at least
one packing material is different than others.
Analysis of Variance Table
Degrees of Sum of   Mean
Source    Freedom    Square   Square      F-ratio F-tab     P value
Treatment           3 32.8728     10.9576     94.58     4.07 0.0000
Error             8     0.9268     0.1158
Total            11    33.7996

From ANOVA table we conclude that Ho is rejected, i.e., there
exist difference among the packaging conditions.

Now the next question is which mean(average) is different than
others. And how to do it.
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Completely Randomized Design : Example
The average and standard deviation is presented in the following
table along with data.
CO, O2, N 100% CO2 Commercial Vacuum
7.41     3.51      7.66     5.26
7.33     2.91      6.98     5.44
7.04     3.66      7.80     5.80
Average             7.26     3.36      7.48     5.50
Stdev             0.1947   0.3969    0.4386   0.2750

Effect ( i)     1.36     -2.54     1.58   -0.40

The overall average is 5.9. Hence the effect of each condition is
calculated as (Treatment mean – Overall mean)
The model followed here is
yij =  + i + ij
where yij is the ith treatment and jth observation
 is the overall mean
i is the effect of ith treatment
ij is the error follows ND.
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Completely Randomized Design : Example
The two treatment mean is different when the difference between
the two mean is larger than Least Significant Difference (LSD).
The LSD can be calculated using the following formula

LSD( ) = t  /2,  s2 [ 1/ri + 1/rj]
However this is applicable when treatment are significant in
ANOVA.                              Commercial Vacuum CO, O , N 100% CO
2              2

7.48   5.50     7.26         3.36
Pckaging Condition      Average
Commercial plastic wrap   7.48     -       1.98         0.22        4.12
Vacuum packaged           5.50            -            -1.76        2.14
1% CO, 40% O2, 59% N     7.26                      -                3.90
100% CO2                 3.36                                   -
S = Sqrt(MSE) = SQRT(0.1158)
= 0.3403
LSD = 2.306 X SQRT(0.1158X2/3)
= 0.6407

The model is accurate when
- The variances among treatment are same
- The error follows Normal Distribution          52

In case it is not, then we need to transform data and analysis again
Completely Randomized Design : Example
CO, O2, N 100% CO2 Commercial Vacuum
7.41          3.51             7.66      5.26
7.33          2.91             6.98      5.44
7.04          3.66             7.80      5.80
Average              7.26          3.36             7.48      5.50
Stdev              0.1947     0.3969             0.4386     0.2750
Effect (   )         1.36          -2.54            1.58     -0.40

The test we carry out is comparing S2 max to S2 min and and F-test
The test statistics is
F0 Max = Max (si2)/Min(si2)
The test details given below.
Stdev Max =      0.4386 F cal = 5.08
Stdev Min =      0.1947 F tab = 7.18

As the Fcal is lower than Ftab we conclude that the variation within
treatment are same.
The NORMALIRTY of error and transformation will be discussed
53
in the class.
Classification of
Design
Design :Randomised Block
Type of Application : Appropriate when one factor is being
investigated and experimental material or environment can be
divided into blocks or homogeneous groups.
Structures :
Basic : Each treatment or level of factor is run in each block.
Blocking : Usually with respect to one variable.
Information Sought :
1. Estimate and compare effects of treatments free of block
effects.
2. Estimate block effects.
3. Estimate variance.

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Classification of
Design
Design :Balanced incomplete block
Type of Application : Appropriate when all the treatment
cannot be accommodated in a block.
Structures :
Basic : Prescribed assignment of treatments to blocks are
made. Every treatments will appear at least once in the
experimental design, but each block will contain only a subset
of pairs.
Information Sought :
1. Estimate and compare effects of several factors.
2. Estimate certain interaction effect (some may not be
possible).
3. Certain small fractional factorial designs may not provide
sufficient information for estimating the variance.

55
Classification of
Design
Design : Partially balanced incomplete block
Type of Application : Appropriate if a balanced incomplete
block requires a larger number of blocks than is practical.
Structures :
Basic : Prescribed assignment of treatments to blocks are
Information Sought :
1. Estimate and compare effects of several factors.
2. Estimate certain interaction effect (some may not be
possible).
3. Certain small fractional factorial designs may not provide
sufficient information for estimating the variance.
(All treatments are not estimated with equal precision)

56
Classification of
Design
Design :Latin Square
Type of Application : Appropriate when one primary factor is under
investigation and results may be affected by two other experimental
variables or by two sources of nonhomogenity. It is assumed that no
interaction exists.
Structures :
Basic : Two cross grouping of the experimental units are made
corresponding to the square and row of the square. Each treatment
occurs once in every row and every column. Number of treatments
must equal number of rows and number of columns.
Information Sought :
1. Estimate and compare effects of several factors.
2. Estimate certain interaction effect (some may not be possible).
3. Certain small fractional factorial designs may not provide
sufficient information for estimating the variance.

57
Classification of
Design
Design : Factorial
Type of Application : Appropriate several experimental
factor are to be investigated at two or more levels and
interaction of factors may be important.
Structures :
Basic : Several factors are investigated at several levels
by running all combinations of factors and levels.
Blocking : None
Information Sought :
1. Estimate and compare effect of several factors.
2. Estimate possible interaction effects.
3. Estimate Variance.

58
Classification of
Design
Design : Blocked factorial
Type of Application : Appropriate when number of runs
required for factorial is too large to be carried out under
homogeneous conditions.
Structures :
Basic : Full set of combinations of factors and levels is
divided into subsets so that high order interactions are
equated into blocks. Each subsets constitutes a block. All
blocks are run.
Blocking : Block are usually units in space of time. Estimate
of certain interaction are sacrificed to provide blocking.
Information Sought :
1. Same as factorial except certain high order interactions
cannot be estimated.

59
Classification of
Design
Design :Fractional factorial
Type of Application : Appropriate when there are many factors
and many levels and it is impractical to run all combinations.
Structures :
Basic : Several factors are investigated at several levels but
only a subset of the full factorial is run.
Blocking : Sometimes possible.
Information Sought :
1. Estimate and compare effects of several factors.
2. Estimate certain interaction effect (some may not be
possible).
3. Certain small fractional factorial designs may not provide
sufficient information for estimating the variance.

60
Classification of
Design
Design : Youden square
Type of Application : Same as Latin square but number
of rows, columns and treatments need not be same
Structures :
Basic: Each treatment occurs once in every row. Number
of treatments must equal number of columns.
Blocking: With respect to other variables is a two-way
layout
Information Sought :
1.       Same as Latin square

61
Classification of
Design
Design : Nested
Type of Application : Appropriate when objective is to
study relative variability instead of mean effect of
sources of tests on the same sample and variance of
different samples
Structures :
Basic: Factors are strata in some hierarchical structure,
units are tested from each stratum
Information Sought :
1. Relative variation in various strata, components of
variance.

62
Classification of Design
Design : Response surface
Type of Application : Appropriate several experimental
factor are to be investigated at two or more levels and
interaction of factors may be important.
Structures :
Factor settings are viewed as defining points in the
factor space (may be multidimensional) at which the
response will be recorded
Information Sought :
1. Maps illustrating the nature of the response surface .

63
Classification of Design
Design : Mixture designs
Type of Application : Objective is to provide empirical
maps contour diagrams) illustrative of how factors
under the experimenter’s control influence the response
Structures :
May unique arrays, Factor settings are constrained.
Factor levels are often percentages that must sum to
100%. Other factor level constraints are possible
Information Sought :
1. Estimate and compare effect of several factors.
2. Estimate possible interaction effects.
3. Estimate Variance.

64
Classification of Design
Taguchi’s Design

Let us say   y = f(x,z)
where y = Product Response
x’s = controllable factors
z’s = noise factors
The objective is to choose settings of x that will make
the product’s response insensitive to variability
associated with both x and z and still meet target
specifications with least variability.

65
Orthogonal Arrays
The symbology for an orthogonal array is La(bc)
where;
L = Latin Square
a = The number of test trials
b = The number of levels for each column, and
c = The number of columns in the array.
2n Series Orthogonal Arrays
L4(23) Orthogonal Array
No.   1     2     3
1    1      1    1
2    1      2    2
3    2      1    2
4    2      2    1                  66
FACTORIAL

EXPERIMENTS

67
Full Factorial Experiments

 Wear of pin is an important criteria in
affecting field life of a component.
is believed that hardness of pin is
 It
an important parameter affecting it.
 Henceexperiments are carried out to
check wear on :
◦ Pin of hardness in range of 60 - 62 RC
◦ Pin of hardness in range of 66 - 68 RC

68
Seek the answers to the following questions
   What is your response ?
   How many Factors [f] ?
   How many Levels [L] ?
   The experiment is Lf
   How many combinations/runs are
possible ?
   How many runs do you plan to carry
out ?

69
SEEK THE ANSWERS TO THE FOLLOWING QUESTIONS
   What is the response ?
Wear
   How many Factors [f] ?                           1
   How many Levels [L] ?
2
   The experiment is Lf
21
   How many combinations/runs are possible ?        2
   How many runs do we plan to carry out ?          2

HENCE IT IS A 21 FULL FACTORIAL.
70
22 Full factorial experiments

   It is believed that pin wear depends on
◦ Hardness
◦ Oil flow

   The levels of hardness are
◦ 60 - 62 Rc
◦ 66 - 68 Rc

   The levels of oil flow is
◦ 20 cc / min
◦ 120 cc / min

71
22 Full factorial experiments

   Number of Factors :
2
   Number of Levels :
2
   Possible Runs :
22
   Nos. we plan to carry out:
4

Hence it is a 22 full factorial experiment.

Similarly you have 23 and 24 full factorial
experiments for 3 and 4 factors
respectively.                             72
EXAMPLE- 22 FACTORIAL EXPERIMENT

Consider a chemical process of Silicate Mfg. It is
felt that Temperature and Concentration are
the contributors to increase residue.

The factors and levels are as below
Factor      -1          +1
Temp.             40ºC       80ºC
Conc.       Low         High

-1 signifies one level (normally lower) and +1
signifies the other level (normally higher)

73
22 Full factorial experiments
   It is now believed that residue depends on concentration of Acid and
Temperature of bath.

RUN           CONC.           TEMP.         RESIDUE
1            Low              40            20.4
2            Low              40            19.3
3            Low              40            17.6
4            Low              40            16.3
5            Low              80             9.7
6            Low              80            16.4
7            Low              80            14.8
8            Low              80            12.3
9            High             40            17.4
10            High             40            17.7
11            High             40            23.2
12            High             40            20.4
13            High             80             15
14            High             80             24
15            High             80            15.6
16            High             80            15.2
74
WHAT DO WE WANT TO FIND ?

   We want to find that
◦ Does concentration and temperature
have any effect on residue.
◦ Of concentration and temperature which
is more important .
◦ What is the ideal and feasible level of the
process settings.
◦ Does any interaction exist between
temperature and concentration.
◦ Is there any problem with data or model

How Do We Find This. ?
75
EXAMPLE- 22 FACTORIAL EXPERIMENT

Club all values of all 4 possible combinations and represent as below

9.7             15
16.4            24
80
Temperature.

14.8            15.6
12.3            15.2

20.4            17.4
40    19.3            17.7
17.6            23.2
16.3            20.4

Low           High
Concentration
76
EXAMPLE- 22 FACTORIAL EXPERIMENT

Calculate the average and the std.dev of each block as below
Low              High
80         13.3               17.4               15.4
Temperature

(2.93)             (4.37)

40
18.4               19.7                19
(1.81)             (2.71)

AVG.       15.8               18.6               17.2

The value in bkt. is the std.dev. while the other value is the average
S.D. (pooled), SP =  {( 2.932 + 4.372 + 1.812 +2.712 ) / 4 } = 3.09
77
WHAT DO YOU INTERPRET. ?
Low          High

80
13.3   4.1
17.4     15.4

Temp.   5.1                   2.3   3.6
40
18.4             19.7     19
1.3

AVG.     15.8       2.8   18.6     17.2

78
The above can be graphically represented as
1. MAIN EFFECTS PLOT :
19

17

15
Low         High    40           80
Concentration        Temperature      79
2. INTERACTION PLOT :

19

18

17

15

13

Low   Concentration   High
80
EXAMPLE- 23 FACTORIAL EXPERIMENT

Consider another setup of surface cleaning. It is
felt that Time, Temp. and Conc. are the
contributors.
The factors and levels are as below
Factor      -1          +1
Temp.             R.T.       90ºC
Time        3 mins      10 mins
Conc.       Low         High
-1 signifies one level (normally lower) and +1
signifies the other level (normally higher)

81
EXAMPLE- 23 FACTORIAL EXPERIMENT

HOW MANY FACTORS?                         3

HOW MANY LEVELS?                          2

HOW MANY RUNS WOULD BE THERE IDEALLY?     8

HOW MANY YOU PLAN TO RUN?                 8

WHICH EXPERIMENT?
23 FULL FACTORIAL EXPERIMENT

82
EXAMPLE: THE PROBABLE COMBINATIONS ARE

NO. TEMP.            TIME        CONC.
1  RT              3 mins       Low
2  90              3 mins       Low
3  RT             10 mins       Low
4  90             10 mins       Low
5  RT              3 mins       High
6  90              3 mins       High
7  RT             10 mins       High
8  90             10 mins       High

•   This is called an array
•   Since it contains all possible combinations. It is a full factorial
array
•   It is also called orthogonal array
•   If columns are orthogonal we can estimate the effect of a
variable independent of the other variables

83
Designing the Experiment
Minitab Steps for Designing Full Factorial Experiments:
1. GO TO STAT > DOE> Factorial > Create Factorial Design > Type of Design > No of
Factors> Click Designs > Select Full Factorial > Enter no of Replications > OK>Click
Factors > Enter Factor names and Levels>OK > Click Options >Select Randomization
as required > OK>OK
Select type of Design

Input number of Factors

Click Design

Select Full Factorial

Select No. of Replicates
84
Click OK
Designing the Experiment
Minitab Steps for Designing Full Factorial Experiments:
1. GO TO STAT > DOE> Factorial > Create Factorial Design > Type of Design > No of
Factors> Click Designs > Select Full Factorial > Enter no of Replications > OK>Click
Factors > Enter Factor names and Levels>OK > Click Options >Select Randomization as
required > OK>OK

Enter Factor name & Levels

Deactivated if you don’t want to randomize it.

85
Designing the Experiment
The Design out put along with the data obtained after conducting experiment.

StdOrder RunOrder CenterPt Blocks Tempareture Time Concentration Response
1        1        1        1       RT       3mins   Low          65
11       2        1        1       RT      10 mins  Low          43
13       3        1        1       RT       3mins   High         61
12       4        1        1       90      10 mins  Low          45
5        5        1        1       RT       3mins   High         58
15       6        1        1       RT      10 mins  High         50
3        7        1        1       RT      10 mins  Low          50
7        8        1        1       RT      10 mins  High         52
10       9        1        1       90       3mins   Low          42
8        10       1        1       90      10 mins  High         41
14       11       1        1       90       3mins   High         43
9       12        1      1        RT      3mins      Low         65
16      13        1      1        90     10 mins     High        45
6       14        1      1        90      3mins      High        45
4       15        1      1        90     10 mins     Low         41
2       16        1      1        90      3mins      Low         44

Note here the second column gives the run order on which the experiment
has to be conducted.
86
Detail of Data Analyze

   Three phases of data analysis

A: Look for              B: Identify large        C: View effects
problems with            effects                  on response
the data or the
model
• Pareto chart of        • Main effects plot
• Time plot of             effects
• Interaction plots
response(s)
• Normal probability
• Residuals plots          plot of effects        • Cube plots
• p-values of effects

87
Look for Problems
   Background: Before you analyze your own data, we will walk
through an earlier example together, using the data in the file listed
above (so we will all get the same results). You will analyze your
own MSD data (from exercise) when the analysis demonstration is
complete.

1.Make a time plot of the response:
Graph > Time Series Plot > (Select ‘Response’ for Y)

88
Look for Problems cont.
2.      Interpret the plot by looking for:
a. “Defects” in the data
◦ Missing values; outliers caused by typos or mistakes
◦ If there is no correction or explanation for an outlier, proceed with
analysis (do not “throw it out”)
b. Trends or cycles that indicate lurking variables
associated with time
◦ This is not a plot of common cause variation—it (hopefully)
contains special causes induced by the factor settings of the
experiment, so it probably won’t “look” random
◦ However, this plot helps us see if the results are influenced by
variables not being tested in the experiment (that we did not
anticipate or control)
◦ If a trend caused by another variable is identified, see a statistician
for help with the analysis (use time order as a covariate in the
model)
Conclusion
No problems;       proceed with the analysis.

89
Residuals
 Definition
   Residual = (Observed Y) – (Average of Ys at that experimental
condition)

   A residual is the difference between a
response and what we “expect” it to be (the
expected value is the average of all
replicates for a particular combination of
factor settings).
◦ We hope most variation in the Ys is accounted for
by deliberate changes we’re making in the
factor settings
◦ Whatever variation is left over is residual
 The assumption is that this residual variation reflects the
common cause variation in the experiment

90
Residuals: An Example

   Residuals for 22 Design with 3 Replicates

Response or
Experimental Conditions   Observed (Y)   Average            Residuals
Std. order   A         B        (3reps)

1        –        –       9, 11, 7        9.0            0.0, 2.0, –2.0

2        +        –       10, 6, 8        8.0            2.0, –2.0, 0.0

3        –        +        15, 19, 20    18.0            –3.0, 1.0, 2.0

4        +        +        15, 18, 12    15.0            0.0, 3.0,

What is the residual
for this observation?
91
Assumptions of DoE Analysis

   The Residuals
Residual = (Observed Y) – (Average at each experimental
condition)

   We assume the residuals are:
◦ Normal: bell-shaped with a mean of 0
◦ Constant: do not increase as averages
of each experimental condition increase
◦ Stable: do not change over time
◦ Not related to the Xs (factors)             0

◦ Random: represent common causes of variation
◦ Independent

92
   Residuals plots must be checked to ensure
the assumptions hold. Otherwise conclusions
Residuals Plot                                Good                                                Bad                    Meaning / Actions
Residual
Residual
3
3
1.   Time Plot of               2
2                                         Any pattern visible over
Residuals*                 1                                                1                                         time means another
Used to check for          0                                                0                                         factor, related to time,
stability over time    -1                                                  -1                                         influences Y. Try to
-2                                                  -2                                         discover it.**
-3                                                  -3
0        10        20       30                      0        10       20     30

Time Order                                          Time Order

2. Residuals vs.                Residual                                        Residual
3                                               3
Fitted Value                 2
This fan shape means
2
(average of each             1                                               1
the variation increases
condition)                   0
as the average of each
0
condition increases (it’s
Used to check for            -1                                              -1
not constant). Try a
constancy;                   -2                                              -2
square root, log, or
variation does not           -3
60        65    70           75
-3
inverse transformation
50      70   90   110 130 150 170
increase as                        Fitted Value (Average)                              Fitted Value (Average)           on Y.
average increases                 8 conditions, 2 reps                             8 conditions, 5 reps
(2 reps will appear
as a mirror image)

Nscore                                          Nscore
3. Normal                  3                                               3
The residuals are not
Probability Plot        2                                               2
normal. Try a
of Residuals            1                                               1
transformation on Y.
0
Used to check that      0

-1                                              -1
residuals are
-2                                              -2
Normal                                                                  -3
-3
-3       -2    -1   0        1        2             -5        0        5    10   15
93
Residual                                        Residual

The three plot of residual does
not show any pattern. In the
Normality test of residual, we
get p-value is 0.717. Hence we
conclude that the residual
follows Normal distribution and

94
Plot of the effects

Normal probability Plot of the effects shows
that the main effect of A, B and Interaction
AB is Significant

Pareto Chart of the effects shows that the
main effect of A, B and Interaction AB is
Significant

95
P-value of effects
Fractional Factorial Fit: Response versus Tempareture, Time, ...
Estimated Effects and Coefficients for Response (coded units)

Term                    Effect      Coef       SE Coef        T        P
Constant                          49.375        0.7512    65.73    0.000
Temparet              -12.250     -6.125        0.7512    -8.15    0.000
Time                   -7.000     -3.500        0.7512    -4.66    0.001 Significant
Concentr                0.000      0.000        0.7512     0.00    1.000 Effects
Temparet*Time           6.500      3.250        0.7512     4.33    0.002
Temparet*Concentr       0.500      0.250        0.7512     0.33    0.747
Time*Concentr           2.250      1.125        0.7512     1.50    0.168

Analysis of Variance for Response (coded    units)
Main Effects           3      796.25        796.25       265.417   29.40   0.000
2-Way Interactions     3      190.25        190.25        63.417    7.02   0.010
Residual Error         9       81.25         81.25         9.028
Lack of Fit          1       30.25         30.25        30.250    4.75   0.061
Pure Error           8       51.00         51.00         6.375
Total                 15     1067.75

96
Plot of Significant effects

97
FRACTIONAL FACTORIALS
   If you have 5 factors, each at two levels,
number of runs will be 25 i.e. 32 runs for
a full factorial.

   If you want to carry out the same in less
runs (say 16) you call it fractional
factorial.

   You are carrying out ½ the number runs,
hence you call it 25 Fractional Factorial
run of order ½.

98
EXAMPLE-FRACTIONAL FACTORIALS

If the residue is dependent on 5 factors viz..

NO.       FACTOR               -1       +1

1        Conveyor Speed(m/min)         500
1500
2        Height of Acid (m)    1        3
3        Agitation rate (rpm) 100      120
4        Temperature (0C)     40       90
5        Conc. %               3        6

HOW MANY EXPERIMENTS REQUIRED? 25 = 32
99
THE RESULTS OF THE EXPTS ARE AS BELOW

Variables         -     +
1       Height of Acid (mtr)   1     3
2       Conveyor speed         500   1500
3       Agitation rate(rpm)    100   120
4       Temperature            40    90
5       Concentration (%)      3     6

RUN       1                     2    3       4    5        Y

1       -                     -    -       -    -       6.1
*2       +                     -    -       -    -       5.3
*3       -                     +    -       -    -       6.3
4       +                     +    -       -    -       6.1
*5       -                     -    +       -    -       5.3
6       +                     -    +       -    -       5.6
7       -                     +    +       -    -       5.4
*8       +                     +    +       -    -       6.1
*9       -                     -    -       +    -       6..9
10       +                     -    -       +    -       6.1
11       -                     +    -       +    -       9.4
*12       +                     +    -       +    -       9.3
13       -                     -    +       +    -       6.6
*14       +                     -    +       +    -       6.0
*15       -                     +    +       +    -       9.5
16       +                     +    +       +    -       9.8
*17       -                     -    -       -    +       5.6
18       +                     -    -       -    +       6.3
19       -                     +    -       -    +       7.0
*20       +                     +    -       -    +       6.5
*21       -                     -    +       -    +       5.9
*22       +                     -    +       -    +       5.5
*23       -                     +    +       -    +       6.7
24       +                     +    +       -    +       6.5
25       -                     -    -       +    +       4.4
*26       +                     -    -       +    +       4.5
*27       -                     +    -       +    +       7.8
28       +                     +    -       +    +       7.7
*29       -                     -    +       +    +       4.9
30       +                     -    +       +    +       4.2
31       -                     +    +       +    +   100 8.1
*32       +                     +    +       +    +       8..2
MAIN EFFECTS PLOT

75

R 70
e
s
65
i
d
u 60
e
50
Ht. Acid Conv.speed   agitate      temp     conc
101
INTERACTION PLOTS

Residue                                              Residue
90                         87
90                                     78
Temperature=90ºC
80                                                     Concentration = 3%
80                                     73
70                          64                  70
60   56                                                    59                      Concentration=6%
60
Temperature=40ºC                                    51
50    54
50
500                   1500                             500                          1500
Conv. speed                                             Conv. speed

90                                79
Concentration = 3%
80
Residue
70                                     64

61                Concentration=6%
60
58
50     40                         90              Temperature (ºC)
102
THIS WAS A FULL 25 FACTORIAL EXPT.

•   It requires 32 Runs.

•   We would like to reduce the Runs.

•   This is done by running a fractional
factorial expt.
i.e. instead of 32 Runs, 16 (1/2
fraction) or 8 Runs (1/4 fraction)
are performed.

Do you lose any information capabilities by
fractionalising ?
103
YES
We use fractionalizing when there are more
(say 5) factors.
Effects that are estimated
Overall average              1
Main Effect             5
2 way Interactions           10
3 way Interactions           10
4 way Interactions           5
5 way Interactions           1
TOTAL      32 Runs

THE HIGH ORDER INTERACTIONS NORMALLY
ARE NOT SIGNIFICANT
104
WHEN YOU USE A FRACTIONAL FACTORIAL

   The effect of High Order Interactions get
mixed with those of main effects and Low
Order Interactions.
e.g. Main Effect 1  (Main Effect of 1)actual +
(5 way interaction)
   Since High Order Interactions are lower, the
difference in interpretations may not be much
different.
   This combining which occurs when you use
fractional factorials is called CONFOUNDING

HIGHER THE FRACTIONAL - MORE THE
CONFOUNDING
105
EXAMPLE- THE FACTORS AND THEIR LEVELS ARE AS BELOW

FACTOR                -1                +1
Voltage               50 kV                   100 k
Coating Hanger        0 mm                    <1.5
mm
Side                  Front (FR)        Back (BK)
Fluoridation pressure 0.5 Kg/cm2             2
kg/cm2
Supplier            Ployplast (PP)      Ploycoat
(PC)

106
EXAMPLE-FRACTIONAL FACTORIALS

•   How many Factors                    =       5
•   How many Levels                     =       2
•   How many Runs Ideally               =       32
•   No. of Runs We want to Run =        8
•   What Experiment do we Use?

25 Fractional factorial expt. of 1/4 fraction

107
THE EXPERIMENT AND ITS RESULTS
RUN VOLTAGE COAT HNG   SIDE   FLPR   SUP   DFT
1    50       0        FR     0.5    PP    46
2    50       0        BK      2     PC    65
3    50      1.5       FR     0.5    PC    20
4    50      1.5       BK      2     PP    25
5   100      1.5       FR      2     PP    100
6   100      1.5       BK     0.5    PC    72
7   100       0        FR      2     PC    100
8   100       0        BK     0.5    PP    100

108
ANALYSIS OF RESULTS
AT +1 LEVELS :
100    1.5     BK     2      PC
100    20      65     65     65
72     25      25     25     20
100    100     72     100    72
100    100     100    100    100
AVG.        93     54.25   65.5   72.5   64.25

AT -1   LEVELS :
50     0       FR     0.5    PP
46     46      46     46     46
65     65      20     20     25
20     100     100    72     100
25     100     100    100    100
AVG.        39     77.75 66.5     59.5   67.75
EFFECT DIFFERENCE
54     -23.5   -1     13     -3

109
MAIN EFFECT PLOT
100
90
80
70
60
50
40
30
20
10
50   100   0    1.5   FR     BK   0.5   2     PP   PC

VOLTAGE    COAT HNG        SIDE    FL PRESS   SUPPLIERS
110
In 2 **5 experiment with 2 replications :

Total Degrees of Freedom = 63
Out of that Error DF = 32
Main Effects D F = 5

2 Factor Interactions : ( AB,AC, AD, AE, BC, BD, BE, CD, CE, DE ) : DF = 10

3 Factor Interaction : DF = 10

4 Factor Interaction : DF = 5

5 Factor Interaction : DF =1

For estimating the main effects only, we require 5 degrees of freedom and we
can get 5 degrees of freedom from 6 observations. Thus minimum no of
experiments = RDF(required degree of freedom) + 1.
In an experiment, if we have 7 factors each at 2 levels. If we conduct full
factorial experiments it will be 27 = 128 experimental combinations called as trials
or runs. Suppose we intend to carry out experiments only to estimate Main Effects,
we need 7 DF. Then MNE(min no of expr.)= 7 + 1 = 8. We sacrifice here the
knowledge of interactions.
This is done through Taguchi Designs.
111
Classification of Design

Taguchi’s Design

Let us say   y = f(x,z)
where y = Product Response
x’s = controllable factors
z’s = noise factors
The objective is to choose settings of x that will make
the product’s response insensitive to variability
associated with both x and z and still meet target
specifications with least variability.

112
Orthogonal Arrays

The symbology for an orthogonal array is La(bc)
where;
L = Latin Square
a = The number of test trials
b = The number of levels for each column, and
c = The number of columns in the array.
2n Series Orthogonal Arrays
L4 (23) Orthogonal Array
No.    1     2     3
1     1     1     1
2     1     2     2
3     2     1     2
4     2     2     1               113
L12 (211) Orthogonal Array

1   2   3   4   5   6   7   8   9 10 11
1   1   1   1   1   1   1   1   1 1 1
1   1   1   1   1   2   2   2   2 2 2
1   1   2   2   2   1   1   1   2 2 2
1   2   1   2   2   1   2   2   1 1 2
1   2   2   1   2   2   1   2   1 2 1
1   2   2   2   1   2   2   1   2 1 1
2   1   2   2   1   1   2   2   1 2 1
2   1   2   1   2   2   2   1   1 1 2
2   1   1   2   2   2   1   2   2 1 1
2   2   2   1   1   1   1   2   2 1 2
2   2   1   2   1   2   1   1   1 2 2
2   2   1   1   2   1   2   1   2 2 1

This is a special array and no interaction is estimated.
114
L8 (27) Orthogonal Array
No.    1         2       3       4       5       6       7
1     1         1       1       1       1       1       1
2     1         1       1       2       2       2       2
3     1         2       2       1       1       2       2
4     1         2       2       2       2       1       1
5     2         1       2       1       2       1       2
6     2         1       2       2       1       2       1
7     2         2       1       1       2       2       1
8     2         2       1       2       1       1       2

L8 (41 x 24) Orthogonal Array
No.       1-2-3   4       5       6       7
1          1     1       1       1       1
2          1     2       2       2       2
3          2     1       1       2       2
4          2     2       2       1       1
5          3     1       2       1       2
6          3     2       1       2       1
7          4     1       2       2       1
8          4     2       1       1       2       115
L16 (215) Orthogonal Array
No.   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15
1    1   1   1   1   1   1   1   1   1    1    1    1    1    1    1
2    1   1   1   1   1   1   1   2   2    2    2    2    2    2    2
3    1   1   1   2   2   2   2   1   1    1    1    2    2    2    2
4    1   1   1   2   2   2   2   2   2    2    2    1    1    1    1
5    1   2   2   1   1   2   2   1   1    2    2    1    1    2    2
6    1   2   2   1   1   2   2   2   2    1    1    2    2    1    1
7    1   2   2   2   2   1   1   1   1    2    2    2    2    1    1
8    1   2   2   2   2   1   1   2   2    1    1    1    1    2    2
9    2   1   2   1   2   1   2   1   2    1    2    1    2    1    2
10    2   1   2   1   2   1   2   2   1    2    1    2    1    2    1
11    2   1   2   2   1   2   1   1   2    1    2    2    1    2    1
12    2   1   2   2   1   2   1   2   1    2    1    1    2    1    2
13    2   2   1   1   2   2   1   1   2    2    1    1    2    2    1
14    2   2   1   1   2   2   1   2   1    1    2    2    1    1    2
15    2   2   1   2   1   1   2   1   2    2    1    2    1    1    2
16    2   2   1   2   1   1   2   2   1    1    2    1    2    2    1

116
3n Series Orthogonal Arrays

L9 (34) Orthogonal Array
No.     1   2   3   4
1      1   1   1   1
2      1   2   2   2
3      1   3   3   3
4      2   1   2   3
5      2   2   3   1
6      2   3   1   2
7      3   1   3   2
8      3   2   1   3
9      3   3   2   1

117
Mixed Series Orthogonal Arrays
L18 (21 x 37) Orthogonal Array
No.   1   2   3   4   5   6   7   8
1    1   1   1   1   1   1   1   1
2    1   1   2   2   2   2   2   2
3    1   1   3   3   3   3   3   3
4    1   2   1   1   2   2   3   3
5    1   2   2   2   3   3   1   1
6    1   2   3   3   1   1   2   2
7    1   3   1   2   1   3   2   3
8    1   3   2   3   2   1   3   1
9    1   3   3   1   3   2   1   2
10    2   1   1   3   3   2   2   1
11    2   1   2   1   1   3   3   2
12    2   1   3   2   2   1   1   3
13    2   2   1   2   3   1   3   2
14    2   2   2   3   1   2   1   3
15    2   2   3   1   2   3   2   1
16    2   3   1   3   2   3   1   2
17    2   3   2   1   3   1   2   3
18    2   3   3   2   1   2   3   1   118
Selection Of Array - Guidelines

Array      Series    Effects can be estimated                       Remarks
L4(23)      Pure 2K Main effect & Interaction effect     Only 2 level Factors
7
L8(2 )      Pure 2K Main effect & Interaction effect     Only 2 level Factors
15            K
L16(2 )     Pure 2 Main effect & Interaction effect      Only 2 level Factors
L32(231)    Pure 2K Main effect & Interaction effect     Only 2 level Factors
L12(211)    Pure 2K Only Main effect & no Interaction
effect                              Only 2 level Factors
19            K Only Main effect & no Interaction
L20(2 )     Pure 2
effect                              Only 2 level Factors
L24(223)    Pure 2K Only Main effect & no Interaction
effect                              Only 2 level Factors
4              K Main effect & Interaction effect
L9(3 )      Pure 3                                       Only 3 level Factors
13            K
L27(3 )      Pure 3 Main effect & Interaction effect     Only 3 level Factors
L18(21 X 37) Mixed Main effect & Interaction effect      Only 1 interaction between 2 and 3
factor can be estimated.

For other arrays and requirements, discuss with MBB
119
SELECTION OF DESIGN LAYOUT

 Compute Total Degrees of Freedom Required to estimate all
Factorial Effects of Interest (TDF).
   Minimum number of Experiments (MNE) = (TDF)+1
 Choose an OA nearest to the size of Run from MINITAB.
 Let us design and analyze a experiment using Taguchi’s method

120
Reaction Example
 Objective: To design and analyze a
fractional factorial experiment using Minitab
[Taguchi Design]
 Output Variable: % Reacted
 Inputs:
• Feed Rate (liters/minute)[A]           10, 15
• Catalyst (%)        [B]          1,2
• Agitation Rate (rpm)      [C]        100, 120
• Temperature (C)     [D]        140,180
• Concentration (%) [E]          3, 6
 Use Minitab to setup the Design Matrix
 All the main effects and the following interaction to
be estimated.
•   Feed Rate & Temperature
•   Feed Rate & Agitation rate
•   Temperature & Agitation rate
•   Catalyst & Concentration
121
Reaction Example
•NO. OF TRIALS FOR FULL FACTORIAL : 2 5 = 32 NOS.
•STUDY THE EFFECT OF :
A, B, C, D, E  : MAIN EFFECTS
AXD, AXC, DXC & BXE : INTERACTIONS

•DEGREES OF FREEDOM REQUIRED:

FACTORS          DOF
A          1
B          1
C          1
D          1
E          1
AXC        1X1 = 1
AXD        1X1 = 1
DXC        1X1 = 1
BXE        1X1 = 1

TOTAL          9

Minimum number of experiment = 9 + 1 = 10                                        122
The nearest OA table to select is L16(2 15), as we need to assess the interaction.
Experiment
Go to Stat>DOE>Taguchi>Create Taguchi Design

123
Reaction Example- Designing the experiment

124
Reaction Example- Designing the experiment

125
Reaction Example- Designing the experiment

Carry out the trials randomly and with at least 1 replicates. Enter the data in
next 2 columns and then analyse it.

126
Reaction Example- Analyse the experiment
Select Stat>DOE>Taguchi>Analyse Taguchi Design and click.

Enter the response

127
Reaction Example- Analyse
the experiment

128
Reaction Example- Interpret the experiment
MINITAB Output

Agitation Rate and Temperature is most critical factor as the delta is
high in both cases. In case of conflict tradeoff.
129
Reaction Example- Analyse the experiment

Agitation Rate and Temperature is most critical factor as the slope is high.
130
Reaction Example- Analyse the experiment

Interaction exists between Agitation Rate and Temperature, because the lines
131

are not parallel.
Reaction Example- Analyse the experiment
How to get ANOVA
Step –1 Copy the 2nd response and stack it below
the 1st response.
Step-2 : Copy the design Matrix and paste it below.
Step – 3: Run MINITAB Command
Stat>ANOVA>General Linear Model
In Response enter the response column
In Model Enter all the main effects and selected
interaction as decided earlier
Click Okay and the ANOVA will be as below.

Check the p value and if p < 0.05 then
the effect is significant.
132
Reaction Example- Optimisation and Prediction
•Select Optimum , trade off if necessary .
• The significant effects are main effect of Agitation, Temperature and
interaction effect of them.
•The selected level is Agitation rate 120 and Temperature 140
•Predict the optimum response.
•Predict the optimum response at Agitation rate of 120 and Temperature
of 140.
MINITAB Command
Goto STAT>DoE>Taguhi>Predict Taguchis result and click. And get the
window

133
Selct the terms, Optimum levels of selected terms and click ok.

•Confirm , Confirm , Confirm the results by actual Trial.       134
Class Exercise
Time Limit : 45 mins
   Objective: To design and analyze a fractional
factorial experiment using Minitab
   Output Variable: Taste Level (Higher Is Better)
   Inputs:
◦   Oak Type (A)                     Allier, Troncais
◦   Stems     (B)                    None, All
◦   Barrel Toast ( C)                Light Medium
◦   Temperature (D)                  75 , 92
◦   Age of Barrel (E)                Old, New
   The interaction AB, CD is important.
   Use Minitab to setup the design matrix

Modeled After Problem 9-26 In Montgomery

135
   The experiment has been performed and the

Run 1      56    54
Run 2      53    52
Run 3      63    65
Run 4      65    63
Run 5      53    55
Run 6      55    57
Run 7      67    69
Run 8      61    59
Run 9      69    71
Run 10     45    48
Run 11     78    79
Run 12     93    94
Run 13     49    51
Run 14     60    61
Run 15     95    97
Run 16     82    85
Analyse the data and present. Time limit 45 mins   136

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