Response Surface Methodology #1

Chemical Example (Box, Hunter, and Hunter)

Create First Order Design

Do Stat → DOE → Factorial → Create Factorial and fill in the dialog box to create a 22 design with 3 center points:
(rsm01.png here)

(rsm02.png here)

Then we got the design schedule:
(rsm03.png here)

De-select the 'Randomize runs' in the Options sub-dialog box:
(rsm04.png here)

Then fill in the corresponding responses to get:
(rsm05.png here)

Analyze the First-Order Design Experimental Data

Do Stat → DOE → Factorial → Analyze Factorial Design:
(rsm06.png here)

(rsm07.png here)

Session Output

 
Factorial Fit: yield versus R, T 

Estimated Effects and Coefficients for yield (coded units)

Term       Effect     Coef  SE Coef      T      P
Constant           61.8000    1.000  61.80  0.000
R          4.7000   2.3500    1.000   2.35  0.143
T          9.0000   4.5000    1.000   4.50  0.046
R*T       -1.3000  -0.6500    1.000  -0.65  0.582
Ct Pt               0.5000    1.528   0.33  0.775

Note: Curvature effect in MINITAB is calculated by MEANc−MEANf = 0.5000.
S = 2           PRESS = *
R-Sq = 92.93%   R-Sq(pred) = *%   R-Sq(adj) = 78.80%

Analysis of Variance for yield (coded units)

Source              DF   Seq SS   Adj SS   Adj MS      F      P
Main Effects         2  103.090  103.090  51.5450  12.89  0.072
  R                  1   22.090   22.090  22.0900   5.52  0.143
  T                  1   81.000   81.000  81.0000  20.25  0.046
2-Way Interactions   1    1.690    1.690   1.6900   0.42  0.582
  R*T                1    1.690    1.690   1.6900   0.42  0.582
  Curvature          1    0.429    0.429   0.4286   0.11  0.775
Residual Error       2    8.000    8.000   4.0000
  Pure Error         2    8.000    8.000   4.0000
Total                6  113.209

Unusual Observations for yield

                                                      St
Obs  StdOrder    yield      Fit  SE Fit  Residual  Resid
  1         1  54.3000  54.3000  2.0000    0.0000      * X
  2         2  60.3000  60.3000  2.0000   -0.0000      * X
  3         3  64.6000  64.6000  2.0000    0.0000      * X
  4         4  68.0000  68.0000  2.0000    0.0000      * X

X denotes an observation whose X value gives it large leverage.

Estimated Coefficients for yield using data in uncoded units

Term            Coef
Constant    -714.450
R             7.2300
T            5.70000
R*T       -0.0520000
Ct Pt        0.50000

Alias Structure
I
R
T
R*T

Stored Columns in Worksheet

(rsm08.png here)

Note that column 'EFFE1' stores main effects R, T, and interaction RT; column 'COEF1' stores the grand mean, the coefficients for the main effects, interaction, and curvature effect.

First-Order Model Fit

(rsm09.png here)

(rsm10.png here)

Now, contour plot and surface plot can be created for the first-order model.

Contour/Surface Plot

Do Stat → DOE → Factorial → Contour/Surface Plot (rsm11.png here)

Contour Plot

(rsm12.png here)

Surface Plot

(rsm13.png here)

Second-Order Model Near Maximum

Create Response Surface Design

Do Stat → DOE → Response Surface → Create Response Surface Design
(rsm21.png here)

(rsm22.png here)

Then enter the responses:
(rsm23.png here)

Analyze Response Design

Do Stat → DOE → Response Surface → Analyze Response Surface Design
(rsm24.png here)

(rsm25.png here)

Session Output
Response Surface Regression: yield versus Block, R, T 

The analysis was done using coded units.

Estimated Regression Coefficients for yield

Term         Coef  SE Coef       T      P
Constant  87.3750   0.8780  99.515  0.000
Block      0.8500   0.5069   1.677  0.154
R         -1.3837   0.6208  -2.229  0.076
T          0.3620   0.6208   0.583  0.585
R*R       -2.1437   0.6941  -3.088  0.027
T*T       -3.0937   0.6941  -4.457  0.007
R*T       -4.8750   0.8780  -5.552  0.003

S = 1.75601    PRESS = 136.493
R-Sq = 92.74%  R-Sq(pred) = 35.70%  R-Sq(adj) = 84.02%

Analysis of Variance for yield

Source          DF   Seq SS   Adj SS  Adj MS      F      P
Blocks           1    8.670    8.670   8.670   2.81  0.154
Regression       5  188.189  188.189  37.638  12.21  0.008
  Linear         2   16.366   16.366   8.183   2.65  0.164
    R            1   15.318   15.318  15.318   4.97  0.076
    T            1    1.048    1.048   1.048   0.34  0.585
  Square         2   76.760   76.760  38.380  12.45  0.011
    R*R          1   15.504   29.412  29.412   9.54  0.027
    T*T          1   61.256   61.256  61.256  19.87  0.007
  Interaction    1   95.062   95.062  95.062  30.83  0.003
    R*T          1   95.062   95.062  95.062  30.83  0.003
Residual Error   5   15.418   15.418   3.084
  Lack-of-Fit    3   10.713   10.713   3.571   1.52  0.421
  Pure Error     2    4.705    4.705   2.353
Total           11  212.277

Estimated Regression Coefficients for yield using data in uncoded units

Term            Coef
Constant    -3958.53
Block       0.850000
R            17.8579
T            44.7349
R*R       -0.0214375
T*T        -0.123750
R*T       -0.0975000

Coefficient Column in Worksheet
(rsm26.png here)

Note that the coefficient column contains, in this order, the grand mean, block coefficient, linear coefficients, quadratic coefficients, and interaction coefficient.

Contour/Surface Plot

Do Stat → DOE → Response Surface → Contour/Surface Plot
(rsm27.png here)

(rsm28.png here)

(rsm29.png here)

Graphics Output
(rsm30.png here)

(rsm31.png here)

Canonical Analysis

First enable minitab commands by doing Editor → Enable Commands:
(rsm32.png here)

Now enter MINITAB commands to create a constant for the grand mean, a column for the linear coefficients, and a 2 by 2 matrix of quadratic coefficients. Then execute the canonical analysis macro (downloadable from canonical.MAC):

MTB > let k1 = 87.3750
MTB > set c9
DATA> -1.3837 0.3620
DATA> end
MTB > read 2 2 m1
DATA> -2.1437 -2.4375
DATA> -2.4375 -3.0937
2 rows read.
MTB > 
MTB > %canonical k1 c9 m1
Executing from file: C:\Program Files (x86)\Minitab\Minitab 16\English\Macros\canonical.MAC

==============================
=     CANONICAL ANALYSIS     =
==============================

<EIGENVALUES and STATIONARY POINT (in original coordinates)>:

                  Stationary
Row  Eigenvalues       Point
  1     -5.10205    -3.73837
  2     -0.13535     3.00394

  <EIGENVECTORS>:

 Matrix Eigenvectors

0.635896   0.771775
0.771775  -0.635896

  <CONSTANT IN TRANSFORMED COORDINATES>:

YSHAT    90.5051

========================================
The sorted absolute eigen values are:

|Eigenvalues|
   0.13535   5.10205

Do you want a rising (falling) ridge analysis? (yes/no)
yes

=====================================
=     ANALYSIS OF A RISING RIGE     =
=====================================

<POINT ON THE RIDGE NEAREST TO THE DESIGN CENTER POINT>:

SPRIME
   -0.0374221   -0.0454186

   <ESTIMATED RESPONSE AT S'>:

YHATSP    87.3927

   <LINEAR COEFFICIENT ALONG RIDGE>:

LINRIDGE
   -1.29810

========================================
MTB > 


Note that you are prompted for a rising (falling) ridge analysis. Inspect the eigenvalues and the coordinates of the stationary point, then you know you need a rising ridge analysis, so you answer with an yes and then you get the above rising ridge analysis.
Canonical Form of the Rising Ridge
The canonical form A is
y.hat−87.3927=−5.10205X12−1.29810X2
and
X1=0.635896[x1−(−3.73837)]+0.771775[x2−3.00394]=0.635896x1+0.771775x2+0.058849

X2=0.771775[x1−(−3.73837)]−0.635896[x2−3.00394]=0.771775x1−0.635896x2+4.795374
Moreover, the equation for the ridge is X1=0. That is,
0.635896x1+0.771775x2+0.058849=0