1. Statistical Design of Experiments
SECTION VI
RESPONSE SURFACE METHODOLOGY
Monday, Aug 13, 2007

2. TYPE OF 3D RESPONSE SURFACES
Sample Maximum or Minimum
Stationary Ridge
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

3. TYPE OF 3D RESPONSE SURFACES
Rising Ridge
Saddle or Minimax
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

4. TYPE OF CONTOUR RESPONSE SURFACES
Sample Maximum or Minimum:
Stationary Ridge
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

5. TYPE OF CONTOUR RESPONSE SURFACE
Rising Ridge:
Saddle or Minimax:
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

6. 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

7. 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 = β constant
+ β1X1 + β2X main effects
+ β3X12 + β4X curvature
+ β5X1X interaction
+ ε error
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

8. RESPONSE SURFACE MODEL FOR THREE FACTORS TWO LEVELS
Y = β constant
+ β1X1 + β2X2 + β3X main effects
+ β11X12 + β22X22 + β33X curvature
+ β12X1X2 + β13X1X3 + β23X2X3 interactions
+ ε error
(Note that higher order interactions are not included.)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

9. LACK OF FIT
Before deciding whether to build a response surface model, it is important to assess the adequacy of a linear model:
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

10. COMPONENTS OF ERROR
The error term ε in the model is comprised of two parts:
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

11. GRAPHICAL EXAMPLE OF LACK OF FIT IN ONE FACTOR
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

12. 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

13. ESTIMATING THE PURE ERROR
Suppose we have n repeat points at some Xj, then
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

14. ESTIMATING THE ERROR DUE TO LOF
If there are m points available (m>>n), with grand mean ,
SSLOF = SSE – SSPE
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

15. 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

16. 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

17. PLOT OF DATA
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

18. TYPES OF RSM DESIGN Three Level Factorial Experiments
Central Composite Designs (CCD)
Box Behnken Designs
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

19. THREE LEVEL FACTORIAL EXPERIMENTS FOR TWO FACTORS
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

20. RESPONSE SURFACE MODEL FOR TWO FACTOR EXPERIMENT
Y = β constant
+ β1X1 + β2X main effects
+ β3X12 + β4X curvature
+ β5X1X interaction
+ ε error
All the other terms are dropped into the error term.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

21. THREE LEVEL FACTORIAL EXPERIMENTS FOR THREE FACTORS
Geometric Presentation X3 X2 X1
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

22. 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

23. RESPONSE SURFACE MODEL FOR THREE FACTOR EXPERIMENT
Y = β constant
+ β1X1 + β2X2 + β3X main effects
+ β11X12 + β22X22 + β33X 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

24. RSM EXAMPLE FOR TWO FACTER EXPERIMENT
Problem:
Predict the rate of solid API dissolution in a liquid solvent.
Factors Levels
API/Solvent Ratio
Agitation (100 RPM)
Response Variable
Rate of API dissolution (mg/min)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

25. DATA FROM THE THREE LEVEL FACTORIAL EXPERIMENT
Dissolution rate (mg/min)
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

26. JMP ANALYSIS OF DATA
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

27. 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

28. CENTRAL COMPOSITE DESIGNS
2 Factor Central Composite Design =
Factorial + Star points = CCD
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

29. 3 FACTOR CENTRAL COMPOSITE DESIGNS+
Factorial Star points =
CCD
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

30. CENTRAL COMPOSITE DESIGN
In a central composite design, each factor has 5 levels
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
Although the star point levels cannot always be controlled as well as one would like, there exist formulas to obtain the “best” (in a statistical sense) star point levels. In order to calculate the levels of the star points for a general factor x1 defined in the range x1L≤ x1≤ x1u
One calculation for is shown below:
where m = number of points in factorial or fractional portion of design
c = number of center points
s = number of star points
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

31. 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

32. 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

33. 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

34. VALUES OF α
Beware! Many software packages place heavy emphasis on star point placement and calculation! This is one reason that automated designs request high numbers of center points. Use your process knowledge as your guide.
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

35. DISSOLUTION STUDY IN THREE FACTOR EXPERIMENT
A 3 factor experiment with 5 center points is conducted for an orthogonal design, α = Below are the factors and the response.
Pressure (P) Ton to 1 Ton
Punch Distance (D) mm to 2 mm
API/Binder Ratio (R) 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

36. CALCULATING THE STAR POINTS
Upper star level
= *.25 =
Lower star level
= *.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

37. TABLET DISSOLUTION DATA
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

38. 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

39. 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

40. 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

41. 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

42. COMPARISON TABLE FOR NUMBER OF RUNS
* One third replicate is used for a 3k factorial design and one-half 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

43. 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

44. 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

45. 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

46. 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 =  Pressure
 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 : or : 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

47. PROCEDURE FOR BOX WILSON METHOD
Use a first order model (factorial experiment or fractional factorial) in the neighborhood of the current conditions
Test for lack of fit
If no significant lack of fit, then locate path of steepest ascent
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

48. 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

49. 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

50. 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

51. DETERMINE THE MAXIMUM % DISSOLUTION BY RSM
Dr. Gary Blau, Sean Han
Monday, Aug 13, 2007

52. 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

53. 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