Download Document

Statistical Design of
Experiments
SECTION VI
RESPONSE SURFACE
METHODOLOGY
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
TYPE OF 3D RESPONSE
SURFACES
• Sample Maximum or Minimum
• Stationary Ridge
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
TYPE OF 3D RESPONSE
SURFACES
• Rising Ridge
• Saddle or Minimax
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
TYPE OF CONTOUR RESPONSE
SURFACES
• Sample Maximum or Minimum:
• Stationary Ridge
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
TYPE OF CONTOUR RESPONSE
SURFACE
• Rising Ridge:
• Saddle or Minimax:
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
RESPONSE SURFACE MODEL
• Models are simple polynomials
• Include terms for interaction and
curvature
• Coefficients are usually established by
regression analysis with a computer
program
• Insignificant terms are discarded
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
RESPONSE SURFACE MODEL
FOR TWO FACTORS
Response Surface Model for two
factors X1 and X2 and measured
response Y (Regardless of number of
levels):
Y = β0
+ β1X1 + β2X2
+ β3X12 + β4X22
+ β5X1X2
+ ε
Dr. Gary Blau, Sean Han
constant
main effects
curvature
interaction
error
Monday, Aug 13, 2007
RESPONSE SURFACE MODEL FOR
THREE FACTORS TWO LEVELS
Y = β0
constant
+ β1X1 + β2X2 + β3X3
main effects
+ β11X12 + β22X22 + β33X32
curvature
+ β12X1X2 + β13X1X3 + β23X2X3 interactions
+ ε
error
(Note that higher order interactions are not
included.)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
LACK OF FIT
Before deciding whether to build a
response surface model, it is important to
assess the adequacy of a linear model:
Y= 0 +
N
N
N
  X   
i
i=1
i
i 1 j 1
ij
Xi X j
The lack of fit method presented below is
general and can be considered for any
model:
Y = f(β,Xi) + ε,
where f(β,Xi) is an arbitrary function of the
factors and the statistical parameters.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
COMPONENTS OF ERROR
•
The error term ε in the model is comprised of
two parts:
1.
2.
modeling error, (lack of fit, LOF)
experimental error, (pure error, PE), which can
be calculated from replicate points
•
The lack of fit test helps us determine if the
modeling error is significant different than
the pure error.
•
In the method compare LOF and PE by using
F ratios calculated from sum of squares.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
GRAPHICAL EXAMPLE OF LACK
OF FIT IN ONE FACTOR
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
CALCULATING THE F RATIO FOR
LACT OF FIT
The F ratio for the test is the ratio between the
estimate of error due to lack of fit (LOF) and the
estimate of error due to pure error (PE). The
estimates are obtained from the two components
which make up the total sum of squares for error
(SSE):
SSE = SSPE + SSLOF
where SSE
= Total sum of squares for error
or Residual sum of squares
SSPE = Sum of squares due to pure error
SSLOF = Sum of squares due to lack to fit
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
ESTIMATING THE PURE ERROR
Suppose we have n repeat points at some
Xj, then
N
SSPE   ( yi  y ) 2
i 1
where yi ‘s are the n different measured
value at Xj
Then the estimate of pure error is
MSPE = SSPE / ( n -2)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
ESTIMATING THE ERROR DUE TO
LOF
If there are m points available (m>>n),
with grand mean Y ,
SSLOF = SSE – SSPE
M
N
k 1
i 1
SSLOF   ( yk  y ) 2   ( yi  y ) 2
MSLOF = SSLOF / (m-n)
Fobs= MSLOF / MSPE with m-n and n-2
degree of freedom respectively
If Fobs >Fcal(DFLOF,DFPE,α)
(from tables),
then there is a lack of fit.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
LACK OF FIT TEST FOR SEVERAL
POINTS REPEATED
When several points are repeated, the
general approach is to determine the
SSPE for each set of replicates and then
“pool” these sum of squares by forming
an overall SSPE weighted by the degree
of freedom for each set. Then the
estimate of PE is obtained by dividing
the SSPE by the appropriate number of
degree of freedom and continuing as
above.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
LACK OF FIT EXAMPLE
Suppose there are 5 data points. Fit different
lines to show the effects of lack of fit.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
PLOT OF DATA
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
TYPES OF RSM DESIGN
• Three Level Factorial Experiments
• Central Composite Designs (CCD)
• Box Behnken Designs
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
THREE LEVEL FACTORIAL
EXPERIMENTS FOR TWO FACTORS
• 32 Factorial Experiments
– Geometric Presentation
X2
X1
– Mathematical Model
Y = β0 + β1X1+ β2X2 + β3X12 + β4X22 + β5X1X2
+ β6X12X2 + β7X1X22 + β8X12X22 + ε
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
RESPONSE SURFACE MODEL
FOR TWO FACTOR EXPERIMENT
Y = β0
+ β1X1 + β2X2
+ β3X12 + β4X22
+ β5X1X2
+ ε
constant
main effects
curvature
interaction
error
All the other terms are dropped into the
error term.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
THREE LEVEL FACTORIAL
EXPERIMENTS FOR THREE FACTORS
• 33 FACTORIAL EXPERIMENT
– Geometric Presentation
X3
X2
X1
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
THREE LEVEL FACTORIAL FOR
THREE FACTOR EXPERIMENT
• Mathematical Model
Y = β0 + β1X1+ β2X2 + β3X3 + β4 X1X2 + β5X1X3 + β6X2X3
+ β7X12 + β8X22 + β9X32 + β10X12X2 + β11X12X3
+ β12X1X22 + β13X22X3 + β14X1X32 + β15X2X32
+ β16X12X22 + β17X12X32 + β18X22X32 + β19X1X2X3
+ β20X12X2X3 + β21X1X22X3 + β22X1X2X32 + β23X12X22X3
+ β24X12X2X32 + β25X1X22X32 + β26X12X22X32 + ε
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
RESPONSE SURFACE MODEL FOR
THREE FACTOR EXPERIMENT
Y = β0
constant
+ β1X1 + β2X2 + β3X3
main effects
+ β11X12 + β22X22 + β33X32
curvature
+ β12X1X2 + β13X1X3 + β23X2X3 interaction
+ ε
error
All the other terms are dropped into the error term.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
RSM EXAMPLE FOR TWO FACTER
EXPERIMENT
• Problem:
Predict the rate of solid API dissolution in a
liquid solvent.
•
Factors
API/Solvent Ratio
Agitation (100 RPM)
Levels
1.0 1.5 2.0
2.0 2.25 2.5
• Response Variable
Rate of API dissolution (mg/min)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
DATA FROM THE THREE LEVEL
FACTORIAL EXPERIMENT
Dissolution rate (mg/min)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
JMP ANALYSIS OF DATA
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
NUMBER OF RUNS FOR A 3k
FACTORIAL EXPERIMENT
• The number inside [brackets] is the number
of runs needed for a third replicate of the
full 3k factorial experiment
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
CENTRAL COMPOSITE DESIGNS
2 Factor Central Composite Design
=
Factorial + Star points =
Dr. Gary Blau, Sean Han
CCD
Monday, Aug 13, 2007
3 FACTOR CENTRAL COMPOSITE
DESIGNS
+
Factorial
+
Star points
=
CCD
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
CENTRAL COMPOSITE DESIGN
•
In a central composite design, each factor
has 5 levels
1.
2.
3.
4.
5.
extreme high (star point)
high
center
low
extreme low (star point)
•
The “hidden” factorial or fractional factorial
experiment should be run first and analyzed
•
Depending on the results of a LOF test, the
star points should be run next
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
COMPARISON OF CCD WITH 3k
FACTORIAL EXPERIMENTS
• Are as efficient as 3k factorial experiments
- minimum number of trials for estimating
main effects and quadratic terms
• Require less runs than 3k factorial
experiments
• Allow sequential experimentation, which
provides flexibility in running the
experiment
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
ORTHOGONAL CCD’S
Orthogonal CCD’s can be constructed by taking
α1 = α2 = …=αn and suitably choosing α . (Using
JMP) Here α is the distance from the star
points to the center point. (All star points lie a
specific equal distance from the center of the
circumscribing sphere.) Orthogonal CCD’s
assure no correlation among the effects being
estimated. The value of α depends on whether
or not the design is orthogonally blocked. That
is, the question is whether or not the design is
divided into blocks such that the block effects
do not affect the estimates of the coefficients in
the 2nd order model.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
ROTATABLE CCD’S
• Rotatable CCD’s are such that all points lie an
equal distance from the center. (Hicks, 1964)
(Star points lie on the sphere which
circumscribes the factorial design.) This type
of CCD provides equal error prediction R units
from the center, independent of direction.
• In most cases, rotatable designs have a small
correlation between the curvature terms. This
correlation can be lessened by adding more
center points. With enough center points, the
design can be both orthogonal and rotatable.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
VALUES OF α
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
DISSOLUTION STUDY IN THREE
FACTOR EXPERIMENT
A 3 factor experiment with 5 center points is
conducted for an orthogonal design, α = 1.47. Below
are the factors and the response.
Pressure (P)
Punch Distance (D)
API/Binder Ratio (R)
.5 Ton to 1 Ton
1 mm to 2 mm
.05 to .15
Response measurement: % Dissolution after 80
minutes
Calculate the star points for pressure trials:
(Plow, o, o) and (Phigh, o, o):
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
CALCULATING THE STAR POINTS
• Upper star level
Phigh  P   ( RangeT / 2) = .75+1.47*.25 = 1.1175
• Lower star level
Phigh  P   ( RangeT / 2) = .75-1.47*.25 = .3825
• The corresponding star points for temperature
are the following:
(1.1175, 1.5, .1) and (.3825, 1.5, .1)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
TABLET DISSOLUTION DATA
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
JMP ANALYSIS OF TABLET
DISSOLUTION DATA
A first order model with first order interactions is run
in JMP:
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
JMP ANALYSIS OF TABLET
DISSOLUTION DATA
Since the p-value is 0094, there is a significant
lack of fit and the star points should be run.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
BOX – BEHNKEN EXPERIMENT
• 3 Factor Experiment
This Box-Behnken experiment for 3 factors consists
of twelve “edge” points all lying on a single sphere
about the center of the experimental region, plus
replicates of the center point.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
BOX-BEHNKEN EXPERIMENT
•
•
•
•
Is actually a portion of a 3k factorial
Three levels of each factor are used
Center points should always be included
It is possible to estimate main effects and
second order terms
• The experiments cannot be run sequentially as
with CCD’s
• Box-Behnken experiments are particularly
useful if some boundary areas of the design
region are infeasible, such as the extremes of
the experiment region.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
COMPARISON TABLE FOR
NUMBER OF RUNS
* One third replicate is used for a 3k factorial design and onehalf replicate is used for a 2k factorial design with the CCD for 5,
6 and 7 factors.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
PROCESS OPTIMIZATION
• Response Surface Methodology (RSM) allows
the researcher to approximate the behavior of a
process in the vicinity of the optimum.
• The challenge is to find the region within the
range of the factors for which this RSM model
is a good approximation and then locate the
optimum.
• A sequential approach of experimentation
followed by analysis can be used to find the
region of interest.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
BOX–WILSON OPTIMUM SEEKING
METHOD
Box–Wilson optimum seeking method is an
interactive procedure for finding the optimum
of a response surface by
1) using factorial or fractional factorial
experiments to find the best way to change
the levels of the factors to search out the
region which is close to the optimum
2) using RSM to incorporate curvature into
the surface and help you decide whether you
have reached the optimum.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
STEEPEST DESCENT DIRECTION
Let us assume the optimum we are
looking for is a maximum. The steepest
ascent direction is the path which gives
the maximum increase in response as
estimated from the coefficients of the
mathematical model associated with a
factorial or fractional factorial
experiment. (i.e. the first order term of a
Taylor’s series.)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
CALCULATING THE STEEPEST
DESCENT DIRECTION
• The increments of the factors on the path are directly
proportional to their coefficients.
• Example: after analysis of a factorial experiment in two
factors pressure and distance from a tableting machine,
the model was found to be:
%Dissolution = 65.2 + 1.96  Pressure
+ 2.82  Distance
where both the pressure and distance were found to be
significant. (The interaction term is always dropped out.)
• Then, the path of steepest descent for pressure and
distance maximizing %dissolution would be in the
proportion of:
1.96 : 2.82
or
1 : 1.44
(i.e. for each unit increase in Pressure. there could be a
1.44 unit increase in Distance)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
PROCEDURE FOR BOX WILSON
METHOD
1.
Use a first order model (factorial experiment
or fractional factorial) in the neighborhood of
the current conditions
2.
Test for lack of fit
3.
If no significant lack of fit, then locate path of
steepest ascent
4.
Run a series of experiments along path until
no additional increase in response is evident
(This a one dimensional search procedure)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
PROCEDURE FOR BOX WILSON
METHOD
5.
Repeat steps 1 – 4
6.
If lack of fit is present, then use response
surface design to investigate curvature
7.
If curvature is present, use RSM to locate the
optimum (either graphically or by setting
derivatives = 0). Beware of saddle points!
8. Once a maximum has been found, make sure
that all excursions from the point result in
decreased function values (sensitivity analysis)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
DETERMINE THE MAXIMUM %
DISSOLUTION BY RSM
How to get to the maximum region from a
starting point: (P0, D0) ?
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
DETERMINE THE MAXIMUM %
DISSOLUTION BY RSM
• A full factorial experiment is run to yield a
model. The response Y is the %Dissolution.
Suppose we are starting far from the
maximum area, we use a first-order model
as the approximating function:
Y = β0 + β1P + β2D + ε
• We test the validity of the model near the
region by doing a lack-of-fit test.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
DETERMINE THE MAXIMUM %
DISSOLUTION BY RSM
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
DETERMINE THE MAXIMUM %
DISSOLUTION BY RSM
• Keep using the deepest ascent method
until we reach the area where there is a
significant lack of fit and curvature must
be added into the model using star
points (See CCD Design).
• Then fit a response surface model:
Y = β0 + β1X1+ β2X2 + β11X12 + β22X22
+ β12X1X2 + β112X12X2 + β122X1X22 + ε
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007
DETERMINE THE MAXIMUM %
DISSOLUTION BY RSM
Once get the response surface model, predict the location of
the maximum by taking derivatives of the model and setting
them to zero.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007