Main considerations in experimental design
Design of experiments relies heavily on the statistical principles discussed in
previous section. At the outset it is necessary to emphasise the statistical principles
involved in a consolidated manner. Two basic principles involved are randomisation
and replication. Also blocking may be considered as an additional principle involved.
Randomization is a key issue in a designed experiment which makes sure that
responses and errors are independently distributed random variables. This is one of
the important assumptions for statistical methods. Randomization should be done at
each stage of experiment with regard to test specimens, lubricant formulations, and
testing order. It evenly distributes the experimental error in design space leading to
unbiased results. Random order can be obtained either by using random number
tables or software.
Replication means performing the experiment many times independently at selected
experimental design points. Replication may be considered as a repetition that is
made in random order. For example consider 5 experiments repeated twice at five
selected conditions involving a total of 10 experiments. If we randomise all the ten
tests the repetitions are done in random order and not one after the other. It differs
from repetition in which the tests are done one after the other at the given condition.
For example hardness measurement at a given condition three times is a repetition.
Here the same test is done three times in sequence. Replication serves mainly two
purposes:
(i) Estimation of experimental error on which analysis of designed experiment
is based.
(ii) Increases the ability to differentiate between the influence of various factors
due to increase in sample size as discussed in earlier section.
Blocking may be considered as a technique that reduces the influence of nuisance
factors. For example if wear comparisons are done between two lubricants in a
reciprocating tester roughness variations in test pieces may influence the
performance. On the other hand if we restrict the roughness to a specific value the
two lubricants can be clearly distinguished. In effect we have blocked a factor and
this is referred to as blocking. Roughness is a nuisance factor because in our study
we are not interested in its influence on wear. In a broader sense blocking involves
special methodology to minimise the influence of nuisance factor. This enables
comparison between interested factors more clear. Let us consider the same example
in which one wants to know whether two lubricants are different in terms of wear by
conducting experiments in a reciprocating rig. If the roughness of different lower test
specimens vary too much it will create a noise in comparing lubricants as roughness
affects wear. If we treat each specimen as a block and conduct two tests in a
2
reciprocating rig with both lubricants on the same specimen at different locations in a
random way the variability due to roughness will be reduced. For analysis, t 01 test
statistic mentioned in Table 6.1 may be used. It can be computed by recording data
as d i  y1i  y 2i where y1i and y 2i are the wear coefficients due to lubricants 1 and
2 of ith specimen. The null and alternative hypotheses can be taken as H0:  d  0
and H1:  d  0 respectively. If one wants to compare more than two lubricants
randomized complete block design should be used which have not been discussed in
the book. The details of use of blocking in experimental design may be obtained
from literature.
In blocking randomisation should be done within the block. Also blocking is
effective only if the variability within the block is less than between blocks. If this
condition is not satisfied then blocking will be wrong choice and test should be
conducted in a completely random way in the entire design space.
Factorial design
This section first compares one-factor-at-a-time approach with factorial design. The
coding formulae for levels of a factor are also given which is normally used in
modelling. Analysis of variance (ANOVA) is then discussed.
Comparison between one-factor-at-a-time and factorial
design experiments
In the usual one-factor-at-a-time-approach, one factor is varied while other factors
are kept constant. In DOE factorial design is used so that all possible combinations
of the levels of the factors are investigated in the experiment. Fig. 6.2 shows the
comparison between two approaches. In this figure coded value of five levels for
concentrations of two additives ZDDP and MoDTC are shown in horizontal and
vertical axes. The five levels are -1.4, -1.0, 0, 1.0 and 1.4. The coding values in this
example can be decoded in % (w/v) from the formula given below.
xi 
zi  z0
Δz
(6.14)
where
x i = Coded value of additive concentration at ith experiment
z i = Additive concentration in % (w/v) at ith experiment
z 0 = The additive concentration in % (w/v) at mid level i.e. at centre point
z
 z min
z
 z min z max  z min
z = The step value = max
= max
=
x max  x min 1.4  (1.4)
2.8
3
z max , z min = Maximum and minimum values of additive concentrations in
% (w/v)
x max , x min = Maximum and minimum values of additive concentrations in
coded form
In the above coding variation in the range of levels is linear. The nonlinear variation
can also be taken into account. One form is given below.
log e z max  log e z min
log e z i  log e z 0
where z 
x max  x min
Δz
MoDTC
MoDTC
xi 
1.4
1.0
1.4
1.0
ZDDP
-1.4 -1.0
0
(a)
1.0 1.4
(6.15)
ZDDP
-1.4 -1.0
0
-1.0
-1.0
-1.4
-1.4
1.0 1.4
(b)
Fig. 6.2. Comparison between one-factor-at-a-time and factorial design approaches (a)
Experimental points for two factors with five levels in one-factor-at-a-time experiment (b)
Experimental points for two factors with five levels in factorial design experiment
Fig. 6.2 (a) depicts one-factor-at-a-time method. Using this method the output
response of wear is observed by varying concentration of one additive keeping
concentration of other additive constant at central level. Total 9 experiments have to
be performed. The solid circular points in the figure are the experimental points. For
both additives if the experimenter wants to observe the variation in wear five times
in the entire range total number of experiments will be 45. Now suppose 52 full
factorial design is used in which the number of experiments to be performed by
factorial deign = levelfactor . Therefore, number of experiments to be performed is
52 = 25. Fig. 6.2 (b) shows the above design. From the figure we can easily see that
one can obtain variation in wear five times on varying concentration of one additive
from -1.4 to 1.4 while keeping concentration of other additive constant at different
level of -1.4, -1, 0, 1 and 1.4. In the second approach we have to perform only 25
experiments compared to 45 in the first approach for the same objective. The second
4
approach has also the advantage of studying interaction of factors. In the present
example one can see whether there is synergy between two additives in terms of
antiwear property.
Analysis of variance (ANOVA) for factorial design
The ANOVA is used to quantify the significance of factors and their interactions. In
order to understand the concept behind it, consider the example of 52 factorial design
given earlier with two times repetition of experiments at all 25 combinations of
levels of two factors. The total 50 experiments have to be performed in a randomized
way so that experimental error is uniformly distributed in the design space. Table 6.2
shows the wear rate wijk at different combinations of factors’ level and replication.
The nomenclature of symbols used in table are given after the Table 6.2.
Table 6.2. Two factor 52 factorial design having two replications with wear rate as response
Factor 1:
ZDDP
Concentration
(coded value)
-1.4
-1.0
0
1.0
1.4
Sum of
wear rates
in column
Average wear
rate in column
w111
w112
Factor 2: MoDTC Concentration (coded value)
Sum of
-1.0
0
1.0
1.4
wear rates
in row
w121
w131
w141
w151
w122
w132
w142
w152
Wf 1
w11
w12
w13
w14
w15
w211
w212
w221
w222
w231
w232
w241
w242
w251
w252
w21
w22
w23
w24
w25
w311
w312
w321
w322
w331
w332
w341
w342
w351
w352
w31
w32
w33
w34
w35
w411
w412
w421
w422
w431
w432
w441
w442
w451
w452
w41
w42
w43
w44
w45
w511
w512
w521
w522
w531
w532
w541
w542
w551
w552
w51
w52
w53
w54
w55
W f 21
W f2 2
Wf2 3
W f2 4
Wf2 5
-1.4
w f 21
w f2 2
w f2 3
wf2 4
w f2 5
1
Average
wear rate
in row
w f11
W f1 2
w f1 2
W f1 3
w f1 3
W f1 4
w f1 4
W f1 5
w f1 5
Overall sum
of wear rate
=W
Overall
average
wear rate
=w
5
where,
wijk
= wear rate at ith level of factor 1, jth level of factor 2 and kth replication.
l1
l2
n
wij
1  i  l1 ; 1  j  l2 ; 1  k  n
= number of levels in factor 1. In the present example it is 5
= number of levels in factor 2. In the present example it is 5
= number of replications. In the present example it is 2
= average wear rate at ijth cell. It represents the expected wear rate due
to combined effect of factor 1, factor 2 and interaction at ijth
combination of level of factors in replicated space
l2
W f 1i
= sum of wear rates in ith row =
n
 w
ijk
j
k
l1
n
 w
W f 2 j = sum of wear rates in jh column =
ijk
i
k
w f1i
= average wear rate in ith row = W f1i /(l2n) representing the expected
w f2 j
wear rate due to factor 1 at ith level in design space of factor2
= average wear rate in jth column = W f 2 j /(l1n) representing the expected
wear rate due to factor 2 at jth level in design space of factor1
l1
W
= overall sum of wear rates =
l2
n
 w
ijk
i 1 j 1 k 1
w
= overall average wear rate = W/(l1 l2 n). It represents expected wear
rate due to factor 1, factor 2, interaction and experimental error in
entire design space
The variability in wear rates in the entire 50 experiments quantified by total sum of
squares SST expressed in Eqn. (6.16) is due to various influences. These are the
concentration effects of ZDDP and MoDTC, their interactions, and experimental
errors.
l1
Total sum of squares SST =
l2
n
 (w
ijk
 w )2
(6.16)
i 1 j 1 k 1
SST may be expressed as summation of sum of squares (SS) due to various factors
mentioned above. It is the decomposition of total variability into various components
which is expressed as
SST = SS1 + SS2 + SSinteraction + SSE
(6.17)
6
The above decomposition can also be expressed as summation of SS due to
combined effect of factor 1, factor 2, their interaction, and experimental error.
SST = SS1+2+interaction + SSE
(6.18)
where,
l1
= Sum of squares due to factor1: ZDDP = l2 n
SS1
 (w
f 1i
 w )2
i 1
l2
= Sum of squares due to factor2: MoDTC = l1n
SS2
 (w
f2 j
 w )2
j 1
SS1+2+interaction = Sum of squares due to factor1, factor2 and interaction
l1
= n
l2
 (w
ij
 w )2
i 1 j 1
SSinteraction
SSE
= Sum of squares due to interaction between factors =
SS1+2+interaction – SS1 – SS2
= Sum of squares due to experimental error
l1
=
l2
n
 (w
ijk
 wij ) 2
i 1 j 1 k 1
Now significance test can be done by generating ANOVA Table 6.3. In this table the
mean squares are obtained by dividing SS with degree of freedom. It estimates the
variance of source. In ANOVA the variance of factor is compared with variance of
experimental error. If F-ratio is greater than Fα, ν1 , ν 2 obtained from F-distribution
table available in statistical handbooks the factor is called significant. The  value is
the level of significance, 1 and 2 are degree of freedom of factor and experimental
error respectively. The level of significance  normally taken as 0.05 which signifies
that the chances of error in conclusion of significance is 5%.
Table 6.3 ANOVA table for wear rate data of two factor 52 factorial design
Source
Factor 1:
ZDDP
Factor 2:
MoDTC
Sum of
squares
Degree of freedom
Mean
squares
F-ratio
SS1
l1 – 1
MS1
MS1/MSE
SS2
l2 – 1
MS2
MS2/MSE
Interaction
SSinteraction
Error
Total
SSE
SST
(l1 l2 – 1) – ( l1 – 1) – ( l2 – 1)
= (l1 – 1) (l2 – 1)
l1 l2(n – 1)
l1 l2n – 1
MSinteraction
MSE
MSinteraction/MSE
7
Fractional factorial design
The full factorial design is not economically viable if number of factors increases. In
that case fractional factorial design is used which is a fraction of full factorial design.
The scheme to obtain the design and inherent aliasing of factorial effect are
discussed below. The last subsection discusses the dealiasing techniques and an
appropriate methodology for selection of fractional design.
Fractional factorial design – 2 levels
2-level factorial design is expressed as +1 and –1 levels. Total number of
experiments is 2k where k is number of factors. As number of factors increases
number of experiments increases rapidly. It should be noted here that out of 2k – 1
k!
k!
degree of freedom (df) only k C1 
and k C 2 
df are used for
1!(k  1)!
2!(k  1)!
main and two factor interactions. Other df obtained from doing large number of
experiments is for higher order interaction terms that are normally insignificant.
Hence fractional factorial design can be constructed so that expected levels of
interaction only are included.
The two level fractional design is constructed by choosing proper generator/s I =
ab…c which has all levels of +1 or –1. ab…c represents multiplication of
corresponding level of three or more factors a, b, …, and c . The examples shown in
this section have generator/s of +1 levels. In 1 2 p
th
fraction of 2 k design which is
expressed as 2 k - p , p independent generators are required to construct 2 k - p design.
Out of these p generators the generator Ismallest having smallest number of factors
defines the resolution of the design. The number of factors in Ismallest, n(Ismallest) is
called as resolution of the design expressed in roman number.
Let us consider 2 5- 2 design shown in Table 6.4. This design has 8 combinations of
levels. First three factors have levels according to 23 full factorial design, 4th factor’s
levels are found by multiplying 1st and 2nd column, 5th factor’s levels are found by
multiplying 1st and 3rd column. 4th and 5th column generation in short may be
expressed in equation form as 4 = 12 and 5 = 13. It gives two generators I = 124 and
I = 135 respectively on multiplying 4 on both sides of first relation 4 = 12 and 5 on
both sides of second relation 5 = 13 as 42 and 52 will give level +1 level. This design
is called resolution III design as both generators in this design have same number of
factors 3 so n(Ismallest) = 3 . From the relations 4 = 12 and 5 = 13 we see that it will be
difficult to distinguish between main effect of 4 and interaction effect between 1& 2.
Similarly it will be difficult to find whether the behaviour in response is due to factor
5 or due to interaction between 1&3. Factor 4 is said to be aliased with 12 interaction
8
and factor 5 is aliased with 13 interaction. The word alias refers to the situation
where interaction between a given number of factors gets confounded with other
interactions or main factors. All aliases for this design can be obtained from the
above two generators and are given below.
1 = 24 = 35 = 12345, 2 = 14 = 345 = 1235, 3 = 15 = 245 = 1234, 4 = 12 = 235 =
1345, 5 = 13 = 234 = 1245, 23 = 45 = 125 = 134, 25 = 34 = 123 = 145
From the above aliases we can say that the above resolution III design is suitable
only if the interest is to find the main factor effects. This is so provided it is known
that two-factor and higher order interactions are insignificant. Other resolution III
design can be generated by choosing any 2 independent generators from 6
generators: I = 124 = 134 = 234 = 125 = 135 = 235.
th
Table 6.4. Construction of 1 4 fraction of 25 design i.e. 2 5- 2 design
S.N.
Factor 1
Factor 2
Factor 3
1
2
3
4
5
6
7
8
–1
+1
–1
+1
–1
+1
–1
+1
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
–1
–1
+1
+1
+1
+1
Factor 4
I = 124
4 = 12
+1
–1
–1
+1
+1
–1
–1
+1
Factor 5
I = 135
5 = 13
+1
–1
+1
–1
–1
+1
–1
+1
The ANOVA for above design can be obtained from the procedure given in previous
section in which 7 effects: 1 + 24 + 35 + 12345, 2 + 14 + 345 + 1235, 3 + 15 + 245 +
1234, 4 + 12 + 235 + 1345, 5 + 13 + 234 + 1245, 23 + 45 + 125 + 134, 25 + 34 +
123 + 145 each having one df is estimated. Of course as stated earlier to do ANOVA
the above design must have  2 replications to estimate experimental error.
If the experimenters’ interest is to find main factor effect with confidence then
resolution IV design is useful. In this design main factors are not aliased with two
factor interaction. The example is 2 4 -1 design in which I = 1234 generator is used to
get 4th factor’s levels. ANOVA of this design will estimate following seven effects in
which each has one df: 1 + 234, 2 + 134, 3 + 124, 4 + 123, 12 + 34, 13 + 24, and 14
+ 23. From these aliases we can say that if we have knowledge that three factor
interactions are insignificant as normally happens, main factor effects can be
investigated clearly.
9
To find the main factor and two factor interaction effect without any ambiguity the
resolution V design is useful. In this design main factor and two factor interactions
are not aliased with any main factor and two factor interaction. The example is 2 5 -1
design in which I = 12345 generator is used to get levels of 5th factor. ANOVA of
this design will estimate the following 15 effects each having one df: 1 + 2345, 2 +
1345, 3 + 1245, 4 + 1235, 5 + 1234, 12 + 345, 13 + 245, 14 + 235, 15 + 234, 23 +
145, 24 + 135, 25 + 134, 34 + 123, 35 + 124, 45 + 123. These aliases clearly indicate
that if higher order  3 interactions are insignificant the main and two factor
interaction effect can be estimated with confidence.
Fractional factorial design – 3 levels
3-level factorial design allows taking into account the response curvature if it exists
in modelling. Three levels in this design are usually represented numerically by
designating low, medium and high level as 0, 1 and 2. Fractional design in this case
is developed by taking generator I as summation of level of factors. Sum value, s m3
in generator equation is represented as remainder of Sum/3. This is called modulus 3
(mod 3). The value of s m3 may be 0, 1 or 2. Let us consider 1/3 fraction of 33
factorial design represented as 33-1 design. Let the factors be represented by A, B
and C. The generator I can be chosen from any four orthogonal components of three
factor interactions: ABC, AB2 C , ABC 2 , and AB2 C 2 . The respective generators are
x1  x 2  x 3  s m3 , x1  2 x 2  x 3  s m3 , x1  x 2  2x 3  s m3 , and x1  2 x 2  2 x 3  s m3 .
Exponent 2 in B and C represent two times addition in generator equation hence
expressed as coefficients in respective factors’ level x in the generator equation. As
s m3 can be 0, 1 or 2 total 12 different 33-1 design can be generated. Let us select last
generator with s m3 = 1. In 33-1 design first two factors A and B have levels
according to 32 full factorial design. The levels of 3rd factor C are found from
equation x1  2x 2  2x 3  1 (mod 3). As s m3 represents modulus 3, a multiple of 3
is added in right hand side of the equation whenever required in order to find the
level of third factor. Following the above procedure the 9 different combinations of
levels found are: 002, 011, 020, 100, 112, 121, 201, 210, and 222. All aliases of this
design can be obtained from I and I2. ANOVA of this design estimates the following
four effects each having two df: A+BC+ABC, B+ AC 2 + ABC 2 , C+ AB2 + AB2 C ,
and AB+AC + BC 2 . Seeing these aliases care should be taken in analysing the main
effect. If the factors are of independent nature and does not have interaction with
other factors this design is suitable to investigate the effect of factors with possible
nonlinear nature of response. The modelling of response curvature can also be done
by using RSM. The selection of 3-level design has been discussed in detail in chapter
7.
10
Dealiasing and selection of optimum fractional design
The previous subsections on 2 and 3 level factorial designs did consider the issue of
selection between alternate designs. The present subsection provides a more formal
methodology.
The best fractional design selection and ambiguity removal of aliased terms becomes
easier if the following empirical principles are considered. These are hierarchy,
sparsity, and heredity. Hierarchy principle states that the effect of factors having
lower order has more probability to be significant than of higher order. It suggests
that if resources are limited then priority should be given to lower order factor.
Sparsity principle indicates that out of several variables the process is likely to be
affected by few factors only. Heredity principle says that interaction terms can be
significant only if at least one of its parent factors is significant.
Dealiasing Technique
The ambiguity between aliased factors can be removed by following considerations
given below:
(i) If factors of different orders are aliased then using hierarchy principle
higher order factors may be ignored in comparison to lower order.
(ii) If the aliased factors are of same order then prior knowledge of non
significant factor may dealiase the effect.
(iii) In study of whole process the factors can be grouped into different
subprocesses. Taking wear process as an example lubricant formulation,
surface formation, and operating conditions are the subprocesses. The factor
interaction within subprocesses have more probability to be significant than
factor interaction between subprocesses.
(iv) More experiments other than the original can be planned that provide
additional information to dispel the confusion among aliased factors. The
details of conducting such experiments are available in literature.
Optimum selection of fractional design
Selection of optimum fraction is normally obtained by using two criteria namely (i)
maximum resolution, and (ii) minimum aberration. In order to understand these
criteria let us define a pattern vector W  (A 3 , A 4 ,..., A k ) for 2 k - p design. In this
11
vector A i is the number of generators having ‘i’ factors. The smallest r, such that Ar
 1, is known as resolution of a 2 k - p design. The maximum resolution criterion
selects the fractional design which has maximum resolution. Minimum aberration
criterion states that for any two 2 k - p designs d1 and d2, let r be the smallest integer
such that Ar(d1)  Ar(d2). Then d1 is said to have less aberration than d2 if Ar(d1) <
Ar(d2). The reason is that large number of generators having ‘r’ factors will give
more aliases hence more aberration. If there is no design with less aberration than d1
then d1 has minimum aberration.
The best fractional design depends mainly on the objective of experiments. The
following tabulation of some important general objectives, recommendations, and its
limitations may help in selection and design of fractional experiments.
Table 6.5. Recommendation and limitation to select a fractional design for different objectives
Objectives
Recommendation to select
fractional design
Linear modelling of response
Nonlinear modelling of
response
Designs having 2-levels
Screening of main factors
Design having III resolution
Clear effect of main factors
Design having IV resolution**
Clear effect of main factors
and two factor interactions
Design having V resolution
Limitation
Designs having  3-levels
Factors having  2 order
are insignificant*
Factors having  3 order
are insignificant***
Factors having  3 order
are insignificant***
Some 2 k - p designs may be generated by selecting generators in such a way that
resolution III design give some clear main effects along with some clear two factor
interaction while resolution IV design give all clear main effects without any clear
two factor interaction. Now if objective is to investigate some main effects along
with some two factor interaction rather all main effects then resolution III design will
be best choice compared to resolution IV design. Maximum resolution criteria in this
case will give wrong recommendation as stated in.
*
th
Among many resolution IV designs with given factor k and 1 2 p fraction,
design having largest number of clear two-factor interactions are best. For this case
minimum aberration criteria does not give better design.
**
***
Three factor interaction usually remains insignificant and the special cases where
they are significant are considered in the last section.
12
Regression modelling
The results of a designed experiment can be used to develop an empirical model that
relates the input variables with the response. Such a model is useful for prediction
and control. This modelling is usually done by regression analysis. This model also
helps in selecting conditions that lead to optimum response. The polynomial model
normally used is given below.
k
y  b0 
b x
i i
ε
(6.19)
i 1
where,
y = response of the experiment
b i = coefficients found by regression, 0  i  k
k = number of regressor variables
x i = regressor representing a linear e.g. main factors or non linear
variables e.g. interaction and quadratic terms of main factors.
ε = error or residual
Although in the above model variables may be linear or nonlinear, the model is
linear in coefficients b i and is called linear regression model. The coefficients are
found by linear least square method which minimises the error ε . In order to
develop the method to find the coefficients let us express the model in terms of all
experimental observations in matrix form as
Y  Xb  ε
(6.20)
where,
Y  Column matrix of ‘N’ observed responses having order Nx1
 y1 y 2 . . . y N T
X  Regressor variable matrix of order Nxp where p = k+1 represents total
number of coefficients in the model including constant
th
1 x11 x12 . . . x1k  where x ij represents level of j
th
1 x
x 22 . . . x 2 k  regressor variable at i experiment.
21

.


 It should be noted that if the levels
.
 are expressed in coded form in the
.
 model relative importance of the

 factor can be easily deduced.
1 x N1 x N 2 . . . x Nk 
b  Coefficient matrix of order px1  b 0 b1 . . . b k T
ε  Column matrix of errors or residuals at ‘N’ experiments of order Nx1
13
 1  2 . . .  N T
The criteria used in least square method to find the coefficients is to minimize the
deviation of predicted value from experimental value of response. This is achieved
by minimizing the following function
N
f=

2
i
 Y  Xb T Y  Xb 
(6.21)
i 1
Using matrix operations the above equation can be expressed as
f = Y T Y  2b T X T Y  b T X T Xb
(6.22)
Using minimization principle i.e. the partial differentiation of function f with respect
to unknown coefficients and equating to zero will give the relationship
 2X T Y  2X T Xb  0 . Rearranging it gives the coefficient matrix as

b  XT X

1
XT Y
(6.23)
The above equation can also be used in the model expressed as
y  b 0 x1b1 x 2 b2 x 3 b3
(6.24)
where x1 , x 2 , and x 3 are main factors. Taking log of both sides the model becomes
log e y  log e b 0  b1log e x1  b 2 log e x 2  b 3log e x 3
(6.25)
If we take log e y as response then this equation is linear in coefficients log e b 0 , b1 ,
b 2 , and b 3 hence Eqn.(6.24) and (6.25) represent linear regression model. Therefore
linear least square equation Eqn. (6.23) can be used to estimate coefficients of the
above model.
In some engineering applications model may be nonlinear in the coefficients. One
form is given as
y  b 0 (1  x1b1 )  b 2 x 2
(6.26)
The coefficients in the above equations are determined by using non-linear least
square method. One such method developed by authors [12] has been used in
characterization of wear which is discussed in the next chapter.
14
Significance of the model
Significance of the model is tested by doing variance analysis. The ANOVA table is
given in Table 6.6. The nomenclature of symbols used are given after the table.
Table 6.6. ANOVA table for regression model
Sum of
squares
Source
Degree of freedom
(df)
Mean Squares
Regression
SSreg
k
MSreg  SSreg k
Residual
SSres
N–k–1
MSres  SS res ( N  k  1)
Lack of fit
SSlof
Pure error
SS pe
df lof  df res  df pe
MSlof  SSlof df lof
F-ratio
MSreg
MSres
MSlof
MSpe
np
 (n  1)  N  n
i
MSpe  SSpe ( N  n p )
p
i 1
SST
Total
N–1
where,
SSreg = Sum of squares (SS) due to regression model = b T X T Y  g 2 N .
where g is grand total of all experimental observations y, and N is
total number of observations. b, X, and Y have already been
defined in Eqn. (6.20)
SSres = Residual SS representing sum of squared deviation between
experimental value and predicted value f as defined in Eqn. (6.21)
n p , n i , N = Number of experimental points, number of replication/s at ith
experimental point, and total number of experiments respectively
np
SSpe = SS due to pure error =
ni
 (y
ij
 y i ) 2 . y ij is the observation at
i 1 j1
th
SSlof
i experimental point and jth replicate, y i is the average of
observations at ith experimental point
= SS for lack of fit representing weighted sum of squared deviation
between expected response y i at ith location and predicted value
np
ŷ i at this location =
 n (y
i
i 1
i
 ŷ i ) 2
15
SST = Y T Y  g 2 N
It should be noted that in the above ANOVA table SST is partitioned into SS reg and
SSres while SSres is further partitioned into SSlof and SSpe . This information can
be expressed by the following two equations.
SST = SS reg + SSres
(6.27)
SSres = SSlof + SSpe
(6.28)
Model is tested in two ways:
(i) Significance of regression is tested with reference to residual i.e. MSres . If
F-ratio is found to be greater than F, k, N  k 1 the model is said to be
significant which means out of k regressor variables at least one regressor is
significant.
(ii) Non significance of lack of fit is tested with reference to pure error i.e.
MSpe . If F-ratio is found to be less than F, df lof , df pe then model is linear. If
lack of fit is found to be significant then model should be discarded and
another model should be tried out.
Individual coefficients are tested by computing F-ratio using the formula
Fi 
b i 2 Cii
where Cii is the diagonal element of (X T X) 1 corresponding to ith
MSres
regressor variable x i . If Fi the F-ratio of ith coefficient of b i corresponding to
regressor variable x i is found to be greater than F,1, N  k 1 then that coefficient is
said to be significant. In other words it can be said that variable x i is significant. It
should be noted that this is not a confirmatory test as value of coefficient depends on
the other regressor variables in the model.
Testing the adequacy of the model
Adequacy of the model is checked by plotting residuals, computing externally
studentised residual t i of all experiments, quantifying and comparing R-squared
2
R 2 , adjusted R-squared R adj
, and measuring the predictive capability of the model
R 2prediction .
16
Residual plots are helpful in checking whether the residuals are normally distributed
as the testing hypothesis is based on this assumption.
Another information required is to find outliers. The outliers are the experimental
points at which residual y i  ŷ i goes beyond the specified limit e.g.  3 MSE . The
best way to find the outlier is to compute externally studentised residual t i at all
experimental points and check outlier through hypothesis testing described in section
6.2 as the statistic t i is t-distributed random variable with N-p-1 degree of freedom.
The formula to calculate t i is given below
ti 
y i  ŷ i
2
(1  h ii )S(i)
(6.29)
where,
h ii = Diagonal element of hat matrix H of order NxN defined as
H  X(X T X) 1 X T . It can be easily shown that predicted response
column matrix Ŷ is easily obtained from Ŷ  HY .
2
= Estimate of variance of residual  2 by removing the ith observation
S(i)
=
N  pMSres  yi  ŷi 2 1  h ii 
N  p 1
The outliers found should be critically examined. This may be due to serious fault in
experiments, measurements or wrong assumptions in modelling.
How close the experimental observations are fitted with model is quantified by
coefficient of determination R 2 and can be obtained from the following equation.
R2  1
SSRes
SST
(6.30)
By adding higher order terms whether it is significant or not R 2 value will increase
but it does not mean that model is good. For good model having large number of
2
significant terms adjusted R-squared R adj
should be estimated. In this statistic an
adjustment is done for the corresponding df of SSres and SST . The equation is given
below.
17
2
R adj
1
SSres ( N  p)
(N  1)
 1
(1  R 2 )
SST ( N  1)
(N  p)
(6.31)
where N and p represent total no. of experiments and total number of coefficients in
the model including constant respectively.
The large difference in R 2 and R adj2 imply that nonsignificant terms have been
added in the model.
The capability of prediction of new observation by the model is quantified by
R 2prediction which is obtained by the formula given below.
R 2prediction  1 
PRESS
SST
(6.32)
where,
N
PRESS = Prediction sum of squares =
 yi  ŷ i 


1  h ii 
i 1 

2
(6.33)
2
There should not be much difference between R adj
and R 2prediction . The above terms
described to check the adequacy of the model are given to understand the concept
and are only indicative. One has to refer to the literature for more detailed analysis.
The empirical model obtained must be verified by the experiments at other locations
within design space.