RM_05_DOE.pdf

DESIGN OF EXPERIMENTS
(DOE)
Dr. Sonu Rajak
Assistant Professor
National Institute of Technology Patna
Patna- 800005

The objectives of the experiment
1. Determining which variables are most influential on the
response y
2. Determining where to set the influential x’s so that y is almost
always near the desired nominal value
3. Determining where to set the influential x’s so that variability
in y is small
4. Determining where to set the influential x’s so that the effects
of the uncontrollable variables z1, z2, . . . , zq are minimized.

EXPERIMENT:
Is defined as a study in which certain
independent variables are manipulated, their effect
on one or more dependent variables is determined .
PURPOSE:
To discover something about a particular
process (or) to compare the effect of several factors
on some phenomena

TERMINOLOGY
DEPENDENT VARIABLE:
 It is an outcome or response of an experiment.
 It is also called as response variable.
 Criterion used is also a dependent variable.
INDEPENDENT VARIABLE (OR) FACTORS:
 Variables, which are varied in the experiment.
 Can be controlled at fixed levels.
 Can be varied or set at levels of our interest.
 Randomized.
 Can be qualitative (or) quantitative.

LEVELS OF A FACTOR:
The variation of independent variable under each factor
(or)number of different possible values of a factor.
The relative intensities at which a treatment will be set during
the experiment
EFFECT OF A FACTOR:
Defined as the change of response produced by a change in the
level of that factor.
TREATMENT:
Different combinations of conditions that one wish to test.

• Responses – outcomes that will be elicited from
experimental units after treatments have been applied
• Analysis of Variance (ANOVA) – principal statistical
means for evaluating potential sources of variation in the
responses.
• Replication – observing individual responses of multiple
experimental units under identical experimental conditions

• Design is defined as the selection of parameters and
specification of features that would help the creation of a
product or process with a predefined expected performance.
• Robust design aims at finding parameter settings, which
would ensure that performance is on target, minimizing
simultaneously the influence of any adverse factors (the
noise) that the user may be unable to control economically or
eliminate.

SUMMARY OF DESIGN OF
EXPT. PROCEDURE
EXPERIMENT
• Statement of the problem
• Choice of dependent or response variable
• Selection of factors to be varied
• Choice of levels of these factors

DESIGN
• No of observations
• Order of experimentation
• Method of randomization
• Mathematical model to describe the experiment
• Hypothesis to be tested.
ANALYSIS
• Data collection and processing
• Computation of test statistics
• Interpretation of results

STATISTICAL DESIGN OF AN EXPERIMENT
The process of planning the experiment so that
appropriate data collected which shall be analyzed by
statistical methods resulting in valid and objective
conclusions.
Two aspects of experimental problem:
The design of the experiment
The statistical analysis of the data

Three basic principles of design:
Replication:
Repetition of the basic experiment i.e. obtaining the
response from the same experimental set-up once again
– Used to obtain experimental error
– Permits the experimenter to obtain a precise estimate of the
factor.

Randomization:
The allocation of the experimental units
(material) and the order of experimentation
(trails) are randomly determined.
Blocking:
A block is a portion of the experimental material
that should be more homogeneous than the entire
set of material.

CONVENTIONAL TEST STRATERGIES
1. One factor experiments
Determining the effect of one factor keeping all other
factors constant
Trial Factor
Level
Test Result Average
1 A1 * * Y1
2 A2 * * Y2

2. Several Factors one at a time
Trial Factors Result Average
A B C
1 1 1 1 * * * Y1
2 2 1 1 * * * Y2
3 1 2 1 * * * Y3
4 1 1 2 * * * Y4

3. Several factors all at the same time
Trial Factor and Level Result Average
A B C D
1 1 1 1 1 * * * Y1
2 2 2 2 2 * * * Y2
4. Factorial Experiments
5. Fractional Factorial Experiments

FACTORIAL EXPERIMENTATIONS
A factorial design is one in which all possible combinations
of the levels of factors are investigated
E.g. Factor A at 2 levels and Factor B at 3 levels
Total possible combinations
B1 B1
A1 B2 A2 B2
B3 B3

Factor A at a levels and Factor B at b levels
Total possible combinations ab
• Factorial designs are more efficient designs
• Factorial designs are necessary when interactions are present
• Factorial designs allow effects of a factor to be estimated at
several levels of the other factors
TYPES OF FACTORIAL DESIGNS
1. Full Factorial Designs
2. Fractional Factorial Designs

FULL FACTORIAL EXPERIMENTATION
• To conduct a full factorial experiment with two
factors each at 2 levels, it is required to do 4 trials
(22)
• In general, Total number of trials to be conducted for
full factorial experiments with ‘a’ factors at ‘b’ levels
each is ba

• Imagine a case with 7 factors at 2 levels each. It
would require
27 trials = 128 trails
• Usual time and financial limitations preclude the use
of Full Factorial experimentation most of the time
• How then can engineer efficiently and economically
investigate these factors ?

FRACTIONAL FACTORIAL EXPERIMENTS
• Fractional Factorial Experiments (FFE) are more efficient
test plans
• FFE’s use only a portion of the total possible combinations
to estimate the Main factor effects and some, not all of the
interactions
• Simplest FFE designs are those with factors are at 2 levels
example One-half FFE
One-quarter FFE
One-eighth FFE etc.

FACTORIAL DESIGNS
MAIN EFFECT: Change in response produced by a change in the
level of the factor
B B
A A
L
L
L
L
H
H
H
H
. .
. .
. .
. .
20 20
40
40
30
50
12
52
FIG. 1 FIG. 2
FIG 1
Avg. effect of A = [(40-20)+(52 –30)]/2 = 21
Avg. effect of B = [(52-40)+(30-20)]/2 = 11

INTERACTION EFFECT
The difference in response between the levels of one factor is not the same
at all levels of the other factors.
At low level of ‘B’, the ‘A’ effect is : 50-20=30
At high level of ‘B’, the ‘A’ effect is : 12-40=-28
The avg. interaction effect ‘AB’ = (-28-30)/2 = -29
NO SIGNIFIANT INTERACTION SIGNIFIANT INTERACTION

The main effect of A:
The main effect of B:
Interaction effect AB:

• In experiments involving 2k designs, it is always important to
examine the magnitude and direction of the factor effects to
determine which variables are likely to be important.
• The analysis of variance can generally be used to confirm this
interpretation.
• Consider determining the sums of squares for A, B, and AB.
Note from Equation that a contrast is used in estimating A,
namely,

In general, SST has 4n - 1 degrees of freedom. The error
sum of squares, with 4(n - 1) degrees of freedom, is usually
computed by subtraction as

F0.05,1,8= 5.32
We conclude that the main effects are statistically significant
and that there is no interaction between these factors.

The Regression Model
For the above experiment (chemical process):- The Regression
model is:
where x1 is a coded variable that represents the reactant
concentration, x2 is a coded variable that represents the amount of
catalyst, and the β’s are regression coefficients. ε is random error.

The relationship between the natural variables, the reactant
concentration and the amount of catalyst, and the coded
variables is
Thus, if the concentration is at the high level (Conc = 25%),
then x1 = +1; if the concentration is at the low level (Conc =
15%), then x1 = -1. Furthermore,
When the natural variables have only two levels, this coding
will produce the familiar ±1 notation for the levels of the coded
variables. To illustrate this for our example, note that

Thus, if the catalyst is at the high level (Catalyst = 2 pounds),
then x2 = +1; if the catalyst is at the low level (Catalyst = 1
pound), then x2 = -1.
The fitted regression model is:
where the intercept is the grand average of all 12 observations,
and the regression coefficients β1 and β2 are one-half the
corresponding factor effect estimates.

Residuals and Model Adequacy
The regression model can be used to obtain the predicted or fitted
value of y at the four points in the design. The residuals are the
differences between the observed and fitted values of y.
• For example, when the reactant concentration is at the low
level (x1 = -1) and the catalyst is at the low level (x2 =-1), the
predicted yield is
There are three observations at this treatment combination, and
the residuals are

The remaining predicted values and residuals are calculated
similarly. For the high level of the reactant concentration and the
low level of the catalyst,
For the low level of the reactant concentration and the high level
of the catalyst,

Finally, for the high level of both factors,

23 Factorial Design
• Suppose there are three factors A, B and C
each at two levels, the design is called a 23
factorial and the eight treatment
combinations can now be displayed
graphically as a cube

Three different notations are widely used for the runs in the 2k design. The
first is the + and - notation, often called the geometric coding (or the
orthogonal coding or the effects coding). The second is the use of
lowercase letter labels to identify the treatment combinations. The final
notation uses 1 and 0 to denote high and low factor levels, respectively,
instead of + and -. These different notations are illustrated below for the 23
design:

Example
An engineer is interested in the effects of
• Cutting speed (A)
• Tool geometry (B)
• Cutting angle (C)
in the life (in hours) of a Cutting tool. Two levels of each
factor are chosen, and three replicates of a 23 factorial
design are run. The results follow:

A
Cutting
speed
B
Tool Geometry
C
Cutting
angle
Treatment
Combination
Response
Tool life (in hours)
Replicate
I
Replicate
II
Replicate
III
- - - (1) 22 31 25
+ - - a 32 43 29
- + - b 35 34 50
+ + - c 55 47 46
- - + ab 44 45 38
+ - + bc 40 37 36
- + + ac 60 50 54
+ + + abc 39 41 47

• Subtracting 40 from the output values
Treatment
Combination
Response Replicate Response
Sum
I II III
(1) -18 -9 -15 -42
a -8 3 -11 -16
b -5 -6 10 -1
c 4 5 -2 7
ab 15 7 6 28
bc 20 10 14 44
ac 0 -3 -4 -7
abc -1 1 7 7

• Effect of factor A =
• Effect of factor B =
 
 
  33
0
4
12
1
7
7
44
28
7
1
16
42
12
1
1
4
1
.
abc
ac
bc
ab
c
b
a
)
(
n


















 
 
  33
.
11
136
12
1
7
7
44
28
7
1
16
42
12
1
)
1
(
4
1

















 abc
ac
bc
ab
c
b
a
n

• Effect of factor C=
• Effect of Interaction
AB =
 
 
  83
.
6
82
12
1
7
7
44
28
7
1
16
42
12
1
)
1
(
4
1

















 abc
ac
bc
ab
c
b
a
n
 
 
  67
.
1
20
12
1
7
7
44
28
7
1
16
42
12
1
)
1
(
4
1




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
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

 abc
ac
bc
ab
c
b
a
n

 
 
  83
.
2
34
12
1
7
7
44
28
7
1
16
42
12
1
)
1
(
4
1




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
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




 abc
ac
bc
ab
c
b
a
n
• Effect of BC =
• Effect of AC =  
 
  83
.
8
106
12
1
7
7
44
28
7
1
16
42
12
1
)
1
(
4
1


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





 abc
ac
bc
ab
c
b
a
n

Sum of Squares
 SSA =
 SSB =
 SS C =
 SSAB =
    667
.
0
24
4
8
2
2


n
contrast
  667
.
770
24
136
2

  167
.
280
24
82
2

  667
.
16
24
20
2



• SSBC =
• SSAC =
• SSABC =
• SSTotal = = 2095.33
• SSError = SST – SSA – SSB – SSC – SSAB – SSAC – SSBC
– SSABC
=483.164
  167
.
48
24
34
2


  167
.
468
24
106
2


  167
.
28
24
26
2


abcn
y
y
a
i
b
j
c
k
n
l
ijkl
2
...
1 1 1 1
2


   

Number of degrees of freedom
• Degrees of freedom for main effect A =( a-1)
• Degrees of freedom for interaction effect AB
= (a-1)(b-1)
• Total degrees of freedom = abcn – 1
• Degrees of freedom for error = abc(n-1)
Where
 a – Number of levels of factor A
 b – Number of levels of factor B
 c – Number of levels of factor C
 n – Number of Replications

Mean Square
• Sum of Squares divided by its degrees of
freedom is the mean square.
• MSA =
• MSAB=
• MSError=
)
1
( 
a
SSA
)
1
)(
1
( 
 b
a
SSAB
)
1
( 
n
abc
SSError

F Test
• To test the significance of both the main effects
and their interactions divide the mean square
by the error mean square to get the F value.
• For the main effect A, F =
• For the interaction effect AB, F =
Error
A
MS
MS
Error
AB
MS
MS

ANOVA Table
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Square
F F0.05
Cutting Speed (A) 0.667 1 0.667 0.022 4.49
Tool Geometry (B) 770.667 1 770.667 25.520 4.49
Cutting Angle (C) 280.167 1 280.167 9.278 4.49
AB 16.167 1 16.167 0.535 4.49
BC 48.167 1 48.167 1.595 4.49
AC 468.167 1 468.167 15.503 4.49
ABC 28.167 1 28.167 0.933 4.49
Error 483.164 16 30.198
Total 2095.333 23

Regression Model
• The main effects B and C and the
interaction effect AC are significant
• Regression model is

THE TWO FACTOR AND THREE LEVEL
FACTORIAL DESIGN
The effect model:
Yijk = µ + Ai + Bj + ABij + C(ij)k
where A and B are the two Factors, µ is overall mean effect, C(ij)k
is a random error component.
i = 1,2………a levels of A
j = 1,2………b levels of B
k = 1,2…….. n observations per cell
SSTotal = SSA+ SSB+ SSAB+ SSE
df: (abn-1) = (a-1) + (b-1) + (a-1)(b-1) + ab(n-1)

FORMULA FOR SS:
Correction factor (CF) = T2/N ; (N=abn) and T= Grand total
SSTotal = Σ Σ Σ(Yijk)2 - CF ;
i,j,k ranges from 1 to a,b,n respectively
SSA = Σ (Ti )2/bn – CF ; i ranges from 1 to a
SSB = Σ (Tj )2/an – CF ; j ranges from 1 to b
SSAB = Σ (Tij )2/n – CF – SSA - SSB
SSE = SSTotal – (SSA+ SSB+ SSAB)

Source of
variation
Sum of
squares
Degrees of
freedom
Mean
square
F0
A SSA a-1 MSA=SSA/a-1 MSA/MSE
B SSB b-1 MSB=SSB/b-1 MSB/MSE
AB SSAB (a-1)(b-1) MSAB=SSAB/
(a-1)(b-1)
MSAB/MSE
Error SSE ab(n-1) MSE=SSE/
ab(n-1)
TOTAL abn-1
H0 : No significant difference
If F0 > Fx,v1,v2 , reject H0

TWO FACTOR EXPERIMENT : ILLUSTRATION
Material
Type (B)
Temperature (A) T.j.
15 70 125
1
130 155
539
34 40
229
20 70
230 998
74 180 80 75 82 58
2
150 188
623
136 122
479
25 70
198 1300
159 126 106 115 58 45
3
138 110
576
174 120
583
96 104
342 1501
168 160 150 139 82 60
Ti 1738 1291 770 T=3799
Life Data (Hrs) for a battery Design
What effect do material type and temperature have on the life of
the battery? (Given that F0.05, 2, 27 = 3.35 and F0.05, 4, 27 = 2.73).

SSTotal = sum of Y2ijk – CF i,j,k = 1…………a,b,x respectively
= 1302+1552+ 742+………+602- 37992/36 = 77646.97
SSA = ((1738)2+(1291)2+(770)2)/12) – ((3799)2/36) = 39118.72
SSB = ((998)2+(1300)2+(1501)2)/12) – ((3799)2/36) = 10683.72
SSAB =¼*(5392+2292+………3422)–((3799)2/36)-SSA -SSB =
9613.78
SSE = SSTOTAL- SSA- SSB- SSAB = 18230.75

ANOVA : Battery Life
Source of variation Sum of Squares Degrees
of
freedom
Mean Square FO
Temperature (A) 39118.72 2 19559.36 28.97 S
Material Type (B) 10683.72 2 5341.86 7.91 S
Interaction (AB) 9613.78 4 2403.44 3.56 S
Error 18230.75 27 675.21
Total 77646.97 35
F0.05, 2,27 = 3.35 F0.05, 4,27 = 2.73

Regression Analysis
In a simple regression analysis the relationship
between the dependent variable y and some
independent variable x can be represented by a
straight line
y= a+bx
Where, b is the slope of the line
a is the y-intercept
∑y= Na +b∑x , (i)
∑xy= a∑x + b∑x2 , (ii)
59

Example: The following data gives the sales of
the company for various years. Fit the straight
line. Forecast the sales for the year 2022.
60
year 2013 2014 2015 2016 2017 2018 2019 2020 2021
Sales
(000)
13 20 20 28 30 32 33 38 43

Year Sale (y) Deviation (x)
1 13 -4
2 20 -3
3 20 -2
4 28 -1
5 30 0
6 32 1
7 33 2
8 38 3
9 43 4
N=9 ∑y= 257 ∑x=0
61

Year Sale (y) Deviation (x) x2 xy
1 13 -4 16 -52
2 20 -3 9 -60
3 20 -2 4 -40
4 28 -1 1 -28
5 30 0 0 0
6 32 1 1 32
7 33 2 4 66
8 38 3 3 114
9 43 4 16 172
N=9 ∑y= 257 ∑x=0 ∑x2 =60 ∑xy = 204
62
a = 28.56, b= 3.4
The equation of the straight line of best fit is
y= 28.56 + 3.4 x
So, sale for the year 2022 = 28.56 + 3.4 X 5 = 45.56=
45560

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