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Title: Diapositive 1

1
Good Morning
2
RESPONSE SURFACE METHODOLOGY (R S M)

Par
Mariam MAHFOUZ

3
Planning

Part I
A - Introduction to the RSM method
B - Techniques of the RSM method
C - Terminology
D - A review of the method of least squares
Part II
A - Procedure to determine optimum
conditions Steps of the RSM method
B Illustration of the method against an example

4

Part I
5
A Introduction to the RSM method

The experimenter frequently faces the task of
exploring the relationship between some response
y and a number of predictor variables x (x1,
x2, , xk). Various degrees of knowledge or
ignorance may exist about the nature of such
relationships.
Most exploratory-type investigations are set up
with a twofold purpose
To determine and quantify the relationship
between the values of one or more measurable
response variable(s) and the sittings of a group
of experimental factors presumed to affect the
response(s) and
To find the sittings of experimental factors that
produce the best value or best set of values of
the response (s).

6
Example

The following is an example of seeking an
optimal value of the response in drug
manufacturing.
Combinations of two drugs, each known to reduce
blood pressure in humans, are to be studied.
A series of clinical trials involving 100 high
blood pressure patients is set up, and each
patient is given some predetermined combination
of the two drugs.
The purpose of administering the different
combinations of the drugs to the individuals is
to find the specific combination that results the
greatest reduction in the patients blood
pressure reading within some specified interval
of time.

7
B - Techniques of the Response surface
methodology (RSM)

Setting up a series of experiments (designing a
set of experiments) that will yield adequate and
reliable measurements of the response of
interest,
Determining a mathematical model that best fits
the data collected from the design chosen in (1),
by conducting appropriate tests of hypotheses
concerning the models parameters, and
Determining the optimal settings of the
experimental factors that produce the optimum
(maximum, minimum or close to a specific value)
value of the response.

8
C Terminology

Factors
Response
The response function
The polynomial representation of a response
surface
The predicted response function
The response surface
Contour representation of a response surface
The operability region and the experimental
region

9
Factors

Factors are processing conditions or input
variables whose values or settings can be
controlled by experimenter. Factors in a
regression analysis can be qualitative or
quantitative.
The specific factors whose levels are to be
studied in detail in this course are those that
are quantitative in nature, and their levels are
assumed to be fixed or controlled by the
experimenter.
Factors and their levels will be denoted by X1,
X2,,Xk respectively.

10
Response

The response variable is the measured quantity
whose value is assumed to be affected by changing
the levels of the factors. The true value of the
response is denoted by ?.
However, because experimental error is present
in all experiments involving measurements, the
response value that is actually observed measured
for any particular combination of the factor
levels differs from ?.
This difference from the true value is written
as Y ? ?, where Y represents the observed
value
of the response and ? denotes experimental error.

11
Factors or input variables
Response or output variable
Experiences
12
Response function

When we say that the value of the true response
? depends upon the levels X1, X2,,Xk of k
quantitative factors, we are saying that there
exists some function of theses levels, ? ?(X1,
X2,,Xk).
The function ? is called the true response
function (unknown), and is assumed to be a
continuous , smooth function of the Xi.

13
The polynomial representation of a response
surface

Let us consider the response function ? ?(X1)
for a 1313single factor. If ? is a continuous,
smooth function, then it is possible to represent
it locally to any required degree of
approximation with a Taylor series expansion
about some arbitrary point X1,0
where are
respectively, the first, second, derivatives of
?(X1) with respect to X1 .

The expansion (1) reduces to a polynomial of the
form
where the coefficients ?0, ?1, ?11 are
parameters which depend on X1,0 and the
derivatives of ?(X1) at X1,0.
First order model with one factor
Second order model with one factor
The second order model with two factors is in
the form (equation 1)
And so on

15
Predicted response function

The structural form of ? is usually unknown and
therefore an approximating form is sought using a
polynomial or some other type of empirical model
equation.
The steps, taken in obtaining the approximating
model, are as follows
First, an assumed form of model equation in the
k input variables is proposed. Then, associated
with the proposed model, some number of
combinations of the levels X1, X2, , Xk of the k
factors are selected for use as the design. At
each factor level combination chosen, one or more
observations are collected.

The observations are used to obtain estimates of
the parameters in the proposed model as well as
to obtain an estimate of the experimental error
variance.
Tests are then performed on the magnitudes of the
coefficient estimates as well as on the model
form itself, and, if the model is considered to
be satisfactory, it can be used as a prediction
equation.
Let us assume the true response function is
represented by equation 1. Estimates of the
parameters ?0, ?1, are obtained using the
method of least squares.
If these estimates, denoted by b0, b1,
respectively, are used instead of the unknown
parameters ?0, ?1, , we obtain the prediction
equation
where , called Y hat, denotes the
predicted response value for given values of X1
and X2.

.
17
The response surface

With k factors, the response surface is a subset
of (k1)-dimensional Euclidean space, and have as
equation
where xi, i1, , k, are called coded variables.

18
Contour representation of a response surface

A technique used to help visualize the shape of
a three-dimensional response surface (case of two
factors), is to plot the contours of the response
surface.
In a contour plot, lines or curves of constant
response values are drawn on a graph or plane
whose coordinate axes represent the coded levels
x1 and x2 of the factors.
The lines (or curves) are known as contours of
the surface. Each contour represents a specific
value for the height of the surface (i.e., a
specific value of ) above the plane defined
for combinations of the levels of the factors.

Geometrically, each contour is a projection onto
the x1x2 plane of a cross-section of the response
surface made by a plane, parallel to the x1x2
plane, cutting through the surface.

20
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22

The plotting of different surface height values
enables one to focus attention on the levels of
the factors at which the changes occur in the
surface shape.
Contour plotting is not limited to three
dimensional surfaces.
The geometrical representation for two and
three factors enables the general situation for k
gt 3 factors to be more readily understood,
although they cannot be visualized geometrically.

23
Operability and experimental regions

Let us call the region in the factor space in
which the experiments can actually be performed
the operability region O. For some
applications the experimenter may wish to explore
the whole region O, but this is usually rare.
Instead, a particular group of experiments is set
up to explore only a limited region of interest,
R, which is entirely contained within the
operability region O. The region is called
experimental region.
24

In most experimental programs, the design points
are positioned inside or on the boundary of the
region R.
Typically R is defined as, a cubical region, or
as a spherical region.

25
D A review of the method of least squares

Let us assume provisionally that N observations
of the response are expressible by means of the
first-order model in k variables
Yu denotes the observed response for the uth
trial, Xui represents the level of factor i at
the uth trial, ?0 and ?i are unknown parameters,
?u represents the random error in Yu and N is
the number of observations (experiences).

Assumptions made about the errors are
Random errors ?u have zero mean and common
variance ?2.
Random errors ?u have zero mean and common
variance ?2.
For tests of significance (T- and F_statistics),
and confidence interval estimation procedures, an
additional assumption must be satisfied
Random errors ?u are normally distributed.

27
Parameter estimates and properties

The method of least squares selects as
estimates for the unknown parameters in Eq. (1),
those values, b0, b1,, bk respectively, which
minimize the quantity
Over N observations, the first-order model in
Eq. (1) can be expressed, in matrix notation, as
YX? ?, where

The parameter estimates b0, b1,, bk which
minimize R(?0, ?1, , ?k) are the solutions to
the (k1) normal equations, which can be
expressed, in matrix notation, as
X X b X Y,
where X is the transpose of the matrix X,
and b(b0, b1,, bk).
The matrix X is assumed to be of full column
rank. Then
b(XX)-1 X Y, where (XX)-1 is the inverse of
X X.
If the used model is correct, b is an
unbiased estimator of ?.
The variance-covariance matrix of the vector
of estimates, b, is Var(b) ?2(X X)-1.

It is easy to show that the least squares
estimator, b, produces minimum variance estimates
of the elements of ? in the class of all linear
unbiased estimators of ?.
As stated by the Gauss-Markov theorem, b is the
best linear unbiased estimator (BLUE) of ?.

30
Predicted response values

Let denote a 1x p vector whose elements
correspond to the elements of a row of the matrix
X (pgtk). The expression for the predicted value
of the response, at point in the
experimental region is
Hereafter we shall use the notation
to denote the predicted value of Y at the point
.
A measure of the precision of the prediction,
defined as the variance of , is
expressed as

31
Estimation of ?2

Let , u1, ,Nnumber
of experiments.
is called the uth residual.
For the general case where the fitted model
contains p parameters, the total number of
observations is N (Ngtp) and the matrix X is
supposed of full column rank, the estimate, s2 of
?2, is computed from
SSE is the sum of squared residuals. The
divisor N-p is the degrees of freedom of the
estimator s2.
When the true model is given by YX??, then s2
is an unbiased estimator of ?2.

32
The Analysis of variance table

The entries in the ANOVA table represent
measures of information concerning the separate
sources of variation in the data.
The total variation in a set of data is called
the total sum of square (SST)
where is
the mean of Y.
The total sum of squares can be partitioned
into two parts The sum of squares dues to
regression, SSR (or sum of squares explained by
the fitted model) and the sum of squares
unaccounted for by the fitted model, SSE (or the
sum of squares of the residuals).
and

If the fitted model contains p parameters, then
the number of degrees of freedom associated with
SSR is p-1, and this associated with SSE, is N-p.
Short-cut formulas for SST, SSR, and SSE are
possible using matrix notation. Letting 1 be a
1xN vector of ones, we have
Note that SST SSR SSE

The usual test of the significance of the
fitted regression equation is a test of the null
hypothesis
H0 all values of ?i (excluding ?0) are zero.
The alternative hypothesis is
Ha at least one value of ?i (excluding ?0)
is not zero.
Assuming normality of the errors, the test of H0
involves first calculating the value of the
F-statistic where
is called the mean square regression, and
is called the
mean square residual.

If the null hypothesis is true, the F-statistic
follows an F-distribution with (p-1) and (N-p)
degrees of freedom in the numerator and in the
denominator, respectively.
The second step of the test of H0 is to compare
the value of F to the table value, F?,p-1,N-p,
which is the upper 100? percent point of the
F-distribution with (p-1) and (N-p) degrees of
freedom, respectively.
If the observed value of F exceeds F?,p-1,N-p,
then the null hypothesis is rejected at the ?
level of significance

An accompanying statistic to the F-statistic is
the coefficient of determination
The value of R2 is a measure of the proportion
of total variation of the values of Yu about the
mean explained by the fitted model.
A related statistic, called the adjusted R2
statistic, is
or

37
ANOVA table
Source of variation Degrees of freedom (df) Sum of square (SS) Mean square (MS) F-statistic
Due to regression (fitted model) p - 1 SSR MSR SSR / (p-1) F MSR / MSE
Residual (error) N p SSE MSE SSE / (N-p)
Total (variations) N 1 SST
38
Tests of hypotheses concerning the individual
parameters in the model

In general, tests of hypotheses concerning
parameters in the proposed model are performed by
comparing the parameter estimates in the fitted
model to their respective estimated standard
errors.
Let us denote the least squares estimate of ?i
by bi and the estimated standard error of bi by
est.s.e.(bi).
Then a test of the null hypothesis H0 ?i0, is
performed by calculating the value of the test
statistic
and
comparing the value of t against a table value,
t?, from the student-table.

The choice of the table value, t?, depends on
the alternative hypothesis, Ha, the level of
significance, ?, and the degrees of freedom for
t.
If the alternative hypothesis is Ha ?i ? 0,
the test is called a two-sided test, and the
value of t? is taken from the column
corresponding to t?/2 in the table.
If, on the other hand, the alternative
hypothesis is Ha ?igt0 or Ha ?ilt0, the test is a
one sided test, and the value of t? is taken from
the column t? in the table.
The degrees of freedom for t are the degrees of
freedom of s2 used in est.s.e.(bi).

or
40
Testing lack of fit of the fitted model using
replicated observations

In general, to say the fitted model is
inadequate or is lacking in fit is to imply the
proposed model does not contain a sufficient
number of terms.
This inadequacy of the model is due to either
or both of the following causes
Factors (other than those in the proposed model)
that are omitted from the proposed model but
which affect the response, and or,
The omission of higher-order terms involving the
factors in the proposed model which are needed to
adequately explain the behavior of the response.

Since in most modeling situations it is far
easier from a design and analysis standpoint to
upgrade (add terms to) the model in the factors
already considered than to introduce new factors
to the program, we shall assume also, upon
detecting inadequacy of the fitted model, that
the inadequacy is due to the omission of
higher-order terms in the fitted model, case (2).

The test of adequacy (or zero lack of fit) of
the fitted model requires two conditions be met
regarding the collection (design) of the data
values
The number of distinct design points, n, must
exceed the number of terms in the fitted model.
If the fitted model contains p terms, then n gt p.
An estimate of the experimental error variance
that does not depend on the form of the fitted
model is required. This can be achieved by
collecting at least two replicate observations at
one or more of the design points and calculating
the variation among the replicates at each point.

In addition, we shall assume the random errors
are normal and independently distributed with a
common variance ?2.
When conditions (1) and (2) are met, the
residual sum of squares, SSE, can be partitioned
into two sources of variation
the variation among the replicates at those
design points where replicates are collected,
and the variation arising from the lack of fit of
the fitted model.

The sum of squares due to the replicate
observations is called the sum of squares for
pure error (abbreviated, SSPE) and once
calculated, it is then subtracted from the
residual sum of squares to produce the sum of
squares due to lack of fit (SSLOF).

To illustrate the partitioning of the residual
sum of squares, let us first give a formula for
calculating the pure error sum of squares.
Denote the uth observation at the lth design
point by Yul, where u1,2,,rl ?1, l1,2,,n.
Define to be the average of the rl
observations at the lth design point. Then the
sum of squares for pure error is calculated
using
The degrees of freedom associated with SSPE is
where N is the total number of observations.

The sum of squares due to lack of fit is found
by subtraction SSLOF SSE - SSPE .
The degrees of freedom associated with
obtained by subtraction is (N-p)-(N-n) n-p.
An expanded analysis-of-variance table that
displays the partitioning of the residual sum of
squares is the following one

47
Source Df SS MS F-statistic
Due to regression p - 1 SSR SSR / (p-1)
Residual N-p SSE SSE / (N-p)
Lack of fit n-p SSLOF MSLOF SSLOF / (n-p) MSLOF / MSPE
Pure error N-n SSPE MSPE SSPE / (N-n)
Total (variations) N-1 SST
48

The test of the null hypothesis of adequacy of
fit (or lack of fit is zero) involves calculating
the value of the F-ratio
and comparing the value of F with a table value
of the Fisher distribution. Lack of fit can be
detected, at the ? level of significance, if the
value of F exceeds the table value,
where the latter quantity is the upper 100?
percentage point of the central F-distribution.

49
The use of the coded variables in the fitted model

The use of coded variables in place of the input
variables facilitates the construction of
experimental design.
Coding removes the units of measurement of the
input variables and as such distances measured
along the axes of the coded variables in
k-dimensional space are standardized (or defined
in the same metric).
Another advantages to using coded variables
rather than the original input variables, when
fitting polynomial models, are computational
ease and increased accuracy in estimating the
model coefficients, and enhanced interpretability
of the coefficient estimates in the model.

50

Part II
51
A - Procedure to determine optimum conditions
steps of the method

This method permits find the settings of the
input variables which produce the most desirable
response values.
These response values may be the maximum yield
or the highest level of quality coming off the
production line.
Similarly, we may seek the variables settings
that minimize the cost of marking the product.
In any case, the set of values of the input
variables which result in the most desirable
response values is called the set of optimum
conditions.

52
Steps of the method

The strategy in developing an empirical model
through a sequential program of experimentation
is as follows
The simplest polynomial model is fitted (a
first-order model) to a set of data collected at
the points of a first-order design. If extra
points are included from which data are collected
and an estimate of the error variance is
available, the model is tested for adequacy of
fit.
If the fitted first-order model is adequate, the
information provided by the fitted model is used
to locate areas in the experimental region, or
outside the experimental region, but within the
boundaries of the operability region, where more
desirable values of the response are suspected to
be.

3. In the new region, the cycle is repeated in
that the first-order model is fitted and testing
for adequacy of fit. If nonlinearity in the
surface shape is detected through the test for
lack of fit of the first-order model, the model
is upgraded by adding cross-product terms and /
or pure quadratic terms to it. The first-order
design is likewise augmented with points to
support the fitting of the upgraded model.

4. If curvature of the surface is detected and a
fitted second-order model is found to be
appropriate, the second-order model is used to
map or describe the shape of the surface, through
a contour plot, in the experimental region. If
the optimal or most desirable response values are
found to be within the boundaries of the
experimental region, then locating the best
values as well as the settings of the input
variables that produce the best response values
in the next order of business.

5. Finally, in the region where the most
desirable response values are suspected to be
found, additional experiments are performed to
verify that this is so. Once the location of the
most desirable response values is determined, the
shape of the response surface in the immediate
neighborhood of the optimum is described.

56
B- Illustration of the method against an example

For simplicity of presentation we shall assume
there is only one response variable to be studied
although in practice there can be several
response variables that are under investigation
simultaneously.

57
Experience

In a particular chemical reaction setting, the
temperature, X1 , and the length of time, X2 , of
the reaction are known to affect the reaction
rate and thus the percent yield.
An experimenter, interested in determining if
an increase in the percent yield is possible,
decides to perform a set of experiments by
varying the reaction temperature and reaction
time while holding all other factors fixed.

The initial set of experiments consists of
looking at two levels of temperature (70 and
90) and two levels of time (30 sec and 90 sec).
The response of interest is the percent yield,
which is recorded in terms of the amount of
residual material burned off during the reaction
resulting in a measure of the purity of the end
product.
The process currently operates in a range of
percent purity between 55 and 75 , but il is
felt that a higher percent yield is possible.

59
Design 1 Fitting first order model

For the initial set of experiments, the
two-variable model to be fitted is
Each of the four temperature-time settings,
70-30 sec, 70-90 sec, 90-30 sec, 90-90 sec,
is replicated twice and the percent yield
recorded for each of the eight trials.
The measured yield values associated with each
temperature-time combination are listed in the
following table

60

Original variables Original variables Coded variables Coded variables Percent yield
Temperature X1 (C) Time X2 (sec.) x1 x2 Y
70 30 -1 -1 49.8 48.1
90 30 1 -1 57.3 52.3
70 90 -1 1 65.7 69.4
90 90 1 1 73.1 77.8
x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as

61
Representation of the first design
62
First-order model

Expressed in terms of the coded variables, the
observed percent yield values are modeled as
The remaining term, ?, represents random error
in the yield values.
The eight observed percent yield values, when
expressed as function of the levels of the coded
variables, in matrix notation, are
Y X ? ?

63
Matrix form

64
Estimations

The estimates of the coefficients in the
first-order model are found by solving the normal
equations
The estimates are
The fitted first-order model in the coded
variables is

65
ANOVA table design 1

Source Degrees of freedom d.f. Sum of squares SS Mean square F
Model 2 864.8125 432.4063 63.71
Residual 5 33.9363 6.7873
Lack of fit 1 2.1013 2.1013 0.264
Pure error 4 31.8350 7.9588
66
Test of adequacy

To perform a test on the adequacy of the fitted
model, the errors in the observed percent yield
values are assumed to be distributed normally
with mean zero and variance and variance ?2.
The value of the lack of fit test statistic is
F 0.264. Since this value not exceed the table
value F140.05 7.71, we do not have sufficient
evidence to doubt the adequacy of the fitted
model.
In the next steep the fitted model is tested
to see if it explains a significant amount of the
variation in the observed percent yield values.

67
Global test of parameters

This test is equivalent to testing the null
hypothesis,
or that both temperature and time have zero or
no effect on percent yield.
The test is highly significant since the
corresponding value of the test statistic is F
63.71 gt F250.01 13.27.
Hence, one or both of the parameters, ?1 and ?2
, are non zero.

68
Individual tests of parameters

At this point in the model development, tests
are performed on the magnitudes of the separate
effects of temperature and time on percent yield
to see if both terms b1x1 and b2x2, are needed in
the fitted model.
To do that the Student-test is used.
For the test of we have
And for we have
Each of the null hypotheses is rejected at the ?
0.05 level of significance owing to the
calculated values, 3.73 and 10.65, being greater
in absolute value than the tabled value,
T50.025 2.571.

We infer, therefore, that both temperature and
time have an effect on percent yield.
Furthermore, since both b1 and b2 are
positive, the effects are positive.
Thus, by raising either the temperature or time
of reaction, this produced a significant increase
in percent yield.

70
Second stage of the sequential program

At this point in the analysis and in view of
the objective of the experiment, which is to find
the temperature and time settings that maximize
the percent yield, the experimenter quite
naturally might ask, If additional experiments
can be performed, at what settings of temperature
and time should the additional experiments be
run?
To answer this question, we enter the second
stage of our sequential program of
experimentation.

71
Contour plots

The fitted model
can now be used to map values of the estimated
response surface over the experimental region.
This response surface is a hyper-plane their
contour plots are lines in the experimental
region.
The contour lines are drawn by connecting two
points (coordinate settings of x1 and x2) in the
experimental region that produce the same value
of

In the figure above are shown the contour
lines of the estimated planar surface for percent
yield corresponding to values of 55, 60,
65 and 70 .

The direction of tilt of the estimated percent
yield planar surface is indicated by the
direction of the arrow which is drawn
perpendicular to the surface contour lines.
The arrow points upward and to the right
indicating that higher values of the response are
expected by increasing the values of x1 and x2
each above 1.
This action corresponds to increasing the
temperature of the reaction above 90C and
increasing the time of reaction above 90 sec.
These recommendations comprise the beginning
steps in a series of single experiments to be
performed along the path of steepest ascent up
the surface.

74
Performing experiments along the path of steepest
ascent

The steepest ascent procedure consists of
performing a sequence of experiments along the
path of maximum increase in response. (Reminder
the direction is dependent on the scale of the
coded variables).
The procedure begins by approximating the
response surface using an equation of a
hyper-plane. The information provided by the
estimated hyper-plane is used to determine a
direction toward which one may expect to observe
increasing values of the response.
As one moves up the surface of increasing
response values and approaches a region where
curvature in the surface is present, the increase
in the response values will eventually level off
at the highest point of the surface in the
particular direction.
If one continues in this direction and the
surface height decreases, a new set of
experiments is performed and again the
first-order model is fitted. A new direction
toward increasing values of the response is
determined from which another sequence of
experiments along the path toward increasing
values is performed. This sequence of trials
continues until it becomes evident that little or
no additional increase in response can be
achieved from the method.

75
Description of the method of steepest ascent

To describe the method of steepest ascent
mathematically, we begin by assuming the true
response surface can be approximated locally with
an equation of a hyper-plane
Data are collected from the points of a
first-order design and the data are used to
calculate the coefficient estimates to obtain the
fitted first-order model

The next step is to move away from the center
of the design, a distance of r units, say, in the
direction of the maximum increase in the
response.
By choosing the center of the design in the
coded variable to be denoted by O(0, 0, , 0),
then movement from the center r units away is
equivalent to find the values of
which maximize
subject to the constraint
Maximization of the response function is
performed by using Lagrange multipliers. Let
where ? is the Lagrange multiplier.

To maximize subject to the
above-mentioned constraint, first we set equal to
zero the partial derivatives
i1,,k and
Setting the partial derivatives equal to zero
produces
i
1,,k, and
The solutions are the values of xi satisfying
or i 1,,k, where the
value of ? is yet to be determined. Thus the
proposed next value of xi is directly
proportional to the value of bi.

Let us the change in Xi be noted by ?i , and
the change in xi be noted by ?i. The coded
variables is obtained by these formulas
where
(respectively si) is the mean (respectively the
standard deviation) of the two levels of Xi .
Thus ,
then
or

Let us illustrate the procedure with the
first-order model
that was fitted early to the percent yield
values in our example.
To the change in X2, ?245 sec. corresponds the
change in x2, ?245/301.5 units.
In the relation , we can
substitute ?i to xi
, thus
and ?1 0.526, so
?10.526105.3C .

The first point on the path of steepest ascent,
therefore, is located at the coordinates (x1,
x2)(0.53, 1.5), which corresponds to the
settings in the original variables of (X1, X2)
(85.3, 105).
Additional experiments are now performed along
the path of steepest ascent at points
corresponding to the increments of distances 1.5
?i, 2 ?i, 3 ?i, and 4 ?i (i1,2).
The table below lists the coordinates of these
points and the corresponding observed percent
yield values.

81
Points along the path of steepest ascent and
observed percent yield values at the points

Temperature X1 (C) Time X2 (sec.) Observed percent yield
Base 80.0 60
?i 5.3 45
Base ?i 85.3 105 74.3
Base 1.5 ?i 87.95 127.5 78.6
Base 2 ?I 90.6 150 83.2
Base 3 ?i 95.9 195 84.7
Base 4 ?i 101.2 240 80.1
82

The observed percent yield values increase to a
value of 84.7 at the setting in X1 and X2 of
95.9 C and 195 sec, respectively, and then the
value drop to 80.1 at X1 101.2 C and X2
240 sec.
Our thinking at this moment is that either the
temperature of 101.2 C is too high or the length
of time of 240 sec is too long and therefore
additional experimentation along the path at
higher values of X1 and X2 would not be useful.
The decision is made to conduct a second group
of experiments and again fit a first-order model.
The table below list the points of the design
two with two replicate yield values were
collected at each of the four factorial
combinations along with a second replicated
observation at the center point.

83
Sequence of experimental trials performed in
moving to a region of high percent yield values

Design two For this design the coded
variables are defined as

x1 x2 X1 X2 yield
-1 -1 85.9 165 82.9 81.4
1 -1 105.9 165 87.4 89.5
-1 1 85.9 225 74.6 77.0
1 1 105.9 225 84.5 83.1
0 0 95.9 195 84.7 81.9
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85

The fitted model corresponding to the group of
experiments of design two is
The corresponding analysis of variance is

Source d.f. SS MS F
Model 2 162.745 81.372 42.34
Residual 7 13.455 1.922
Lack of fit 2 2.345 1.173 0.53
Pure error 5 11.110 2.222
Total (variations) 9 176.2
86

The test for lack of fit of this model produced
an F value of F 0.53, which is not significant.
The test of significance of the fitted model
produced a highly significant F42.34 value.
Thus, the information obtained from this fitted
model is used to obtain a new direction in which
to perform additional experiments in seeking
higher percent yield values.
The table below lists the sequence of
experimental trials that were performed in the
direction two

87
sequence of experimental trials that were
performed in the direction two

Steps x1 x2 X1 X2 yield
1 Base ?I 1 - 0.77 105.9 171.9 89.0
2 Base 2 ?I 2 - 1.54 115.9 148.8 90.2
3 Base 3 ?I 3 - 2.31 125.9 125.7 87.4
4 Base 4 ?I 4 - 3.08 135.9 102.6 82.6
88
Retreat to center 2 ?i and proceed in
direction three
Steps x1 x2 X1 X2 yield
5 Replicated 2 2 - 1.54 115.9 148.8 91.0
6 3 - 0.77 125.9 171.9 93.6
7 4 0 135.9 195 96.2
8 5 0.77 145.9 218.1 92.9

89
Set up design three using points of steps 6, 7,
and 8 along with the following two points

Steps x1 x2 X1 X2 yield
9 3 0.77 125.9 218.1 91.7
10 5 - 0.77 145.9 171.9 92.5
11 Replicated 7 4 0 135.9 195 97.0
Center of design is (X1 X2)(135.9 195) Fitted model using percent yield values in steps 6 11 This model is considered not adequate. Center of design is (X1 X2)(135.9 195) Fitted model using percent yield values in steps 6 11 This model is considered not adequate. Center of design is (X1 X2)(135.9 195) Fitted model using percent yield values in steps 6 11 This model is considered not adequate. Center of design is (X1 X2)(135.9 195) Fitted model using percent yield values in steps 6 11 This model is considered not adequate. Center of design is (X1 X2)(135.9 195) Fitted model using percent yield values in steps 6 11 This model is considered not adequate. Center of design is (X1 X2)(135.9 195) Fitted model using percent yield values in steps 6 11 This model is considered not adequate.
90
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91
Moore explanations

The figure above shows the sequence of
experiments performed by numbering the points
which are listed as steps 1 11 in the least
table, where steps 1 4 represent the
experimental trials taken along the second
direction of steepest ascent.
Step 5 denotes a return to the point in step 2
and replicating the experiment to validate the
previously high percent yield value.
In fact, at step 6 the coded values of the
temperature and time combinations are 1, 1,
respectively, in a ¾ replicate of a factorial
design consisting of the points at steps 1, 3,
and 6, with the point at step 5 as center.
Upon observing a higher percent yield at step 6
than at step 5, steps 7 and 8 represent
additional experiments performed along a third
direction defined by the line joining the points
of steps 5 and 6.
The choice of this third direction represents a
deviation from the conventional steepest ascent
(or descent) approach and was undertaken in an
attempt to reduce the amount of work required in
setting up a complete factorial experiment with
center at point 5 and the subsequent fitting of
another first-order model.

Design three was set up using the point at step
7 as its center. It includes steps 6 11. If we
redefine the coded variables
and
then the fitted first-order model is
The corresponding analysis of variance table
is

93
ANOVA table
source d.f. SS MS F
Model 2 0.5650 0.2825 0.04
Residual 3 22.1833 7.3944
total 5 22.7483
It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface.

94
Fitting a second-order model

A second-order model in k variables is of the
form
The number of terms in the model above is
p(k1)(k2)/2 for example, when k2 then p6.
Let us return to the chemical reaction example
of the previous section. To fit a second-order
model (k2), we must perform some additional
experiments.

95
Central composite rotatable design

Suppose that four additional experiments are
performed, one at each of the axial settings
(x1,x2)
These four design settings along with the four
factorial settings (-1,-1) (-1,1) (1,-1)
(1,1) and center point comprise a central
composite rotatable design.
The percent yield values and the corresponding
nine design settings are listed in the table
below

96
Central composite rotatable design
97
Percent yield values at the nine points of a
central composite rotatable design

The fitted second-order model, in the coded
variables, is
The analysis is detailed in this table, using
the RSREG procedure in the SAS software

99
SAS output 1

100
Moore explanations

The test for adequacy of fit of the fitted
model produced an F value ( lack of fit mean
square/pure error mean square) less than 1, which
is clearly not significant.
The pure quadratic coefficient estimates are
each highly significant (plt0.001), which
indicates that surface curvature is present in
the observed percent yield values.
With the fitted second-order model, we can
predict percent yield values for values of x1 and
x2 inside the region of experimentation.
The table below show some predicted percent
yield values and their variances

101
SAS output 2
102
Response surface and the contour plot
103
Moore explanations

The contours of the response surface, showing
above, represent predicted yield values of 95.0
to 96.5 percent in steps of 0.5 percent.
The contours are elliptical and centered at the
point
(x1 x2)(- 0.0048 - 0.0857)
or (X1 X2)(135.85C 193.02 sec).
The coordinates of the centroid point are called
the coordinates of the stationary point.
From the contour plot we see that as one moves
away from the stationary point, by increasing or
decreasing the values of either temperature or
time, the predicted percent yield (response)
value decreases.

104
Determining the coordinates of the stationary
point

A near stationary region is defined as a region
where the surface slopes (or gradients along the
variables axes) are small compared to the
estimate of experimental error.
The stationary point of a near stationary region
is the point at which the slope of the response
surface is zero when taken in all direction.
The coordinates of the stationary point
are calculated by differentiating the estimated
response equation with respect to each xi,
equating these derivatives to zero, and solving
the resulting k equations simultaneously.

105

Remember that the fitted second-order model in
k variables is
To obtain the coordinates of the stationary
point, let us write the above model using matrix
notation, as

106

where
and

107
Some details

The partial derivatives of with respect
to x1, x2, , xk are

108
Moore details

Setting each of the k derivatives equal to zero
and solving for the values of the xi, we find
that the coordinate of the stationary point are
the values of the elements of the kx1 vector x0
given by
At the stationary point, the predicted response
value, denoted by , is obtained by
substituting x0 for x

109
Return to our example

The fitted second-order model was
so the stationary point is
In the original variables, temperature and time
of the chemical reaction example, the setting at
the stationary point are temperature135.85C
and time193.02 sec.
And the predicted percent yield at the
stationary point is

110
Moore details

Note that the elements of the vector x0 do not
tell us anything about the nature of the surface
at the stationary point.
This nature can be a minimum, a maximum or a
mini_max point.
For each of these cases, we are assuming that
the stationary point is located inside the
experimental region.
When, on the other hand, the coordinates of the
stationary point are outside the experimental
region, then we might have encountered a rising
ridge system or a falling ridge system, or
possibly a stationary ridge.

111
Nest Step

The next step is to turn our attention to
expressing the response system in canonical form
so as to be able to describe in greater detail
the nature of the response system in the
neighborhood of the stationary point.

112
The canonical Equation of a Second-Order Response
System

The first step in developing the canonical
equation for a k-variable system is to translate
the origine of the system from the center of the
design to the stationary point, that is, to move
from (x1,x2,,xk)(0,0,,0) to x0.
This is done by defining the intermediate
variables (z1,z2,,zk)(x1-x10,x2-x20,,xk-xk0)
or zx-x0.
Then the second-order response equation is
expressed in terms of the values of zi as

113

Now, to obtain the canonical form of the
predicted response equation, let us define a set
of variables w1,w2,,wk such that W(w1,w2,,wk)
is given by
where M is a kxk orthogonal matrix whose
columns are eigenvectors of the matrix B.
The matrix M has the effect of diagonalyzing B,
that is, where ?1,?2,,?k are the corresponding
eigenvalues of B.
The axes associated with the variables
w1,w2,,wk are called the principal axes of the
response system.
This transformation is a rotation of the zi
axes to form the wi axes.

114

So we obtain the canonical equation
The eigenvalues ?i are real-valued (since the
matrix B is a real-valued, symmetric matrix) and
represent the coefficients of the terms in the
canonical equation.
It is easy to see that if ?1,?2,,?k are
1) All negative, then at x0 the surface is a
maximum.
2) All positive, then at x0 the surface is a
minimum.
3) Of mixed signs, that is, some are positive
and the
others are negative, then x0 is a saddle
point of
the fitted surface.
The canonical equation for the percent yield
surface is

115
Moore details

The magnitude of the individual values of the
?i tell how quickly the surface height changes
along the Wi axes as one moves away from x0.
Today there are computer software packages
available that perform the steps of locating the
coordinates of the stationary point, predict the
response at the stationary point, and compute the
eigenvalues and the eigenvectors.

116

For example, the solution for optimum response
generated from PROC RSREG of the Statistical
Analysis System (SAS) for the chemical reaction
data, is in following table

117
Recapitulate
Process to optimize
Contours and optimal direction

Input and output variables
Experiments in the Optimal direction
Experimental and Operational regions
Locate a new Experimental region
Series of experiments
New series of experiments
Yes
Fitting First-order model
Fitting First-order model
Model Adequate ?
Model Adequate ?
Yes
Fitting a Second-order model
No
No
118
Bibliography

André KHURI and John CORNELL Response Surfaces
Designs and Analyses , Dekker, Inc., ASQC
Quality Press, New York.
Irwin GUTTMAN Linear Models An Introduction,
John Wiley Sons, New York.
George BOX, William HUNTER J. Stuart HUNTER
Statistics for experimenters An Introduction to
Design, Data Analysis, and Model Building , John
Wiley Sons, New York.
George BOX Norman DRAPPER Empirical
Model-Building and Response Surfaces , John
Wiley Sons, New York.