Unreplicated Two-Level Full Factorial Experiment
Process Development Example

Data Entry:  Do  Stat-->DOE-->Factorial-->Create Factorial Design(images/process1.gif here)
Then  enter  responses and edit run order as the data set has the run order available:(images/process2.gif here)

Initial Analysis:
obtain and store all effects and produce normal plot of effects
 (images/process7.gif here)

Session Output (edited with abbreviated effect names added)

Factorial Fit: y versus catalyst, temperature, pressure, concentration 


Estimated Effects and Coefficients for y (coded units)

Term                                Effect    Coef  EffectName
Constant                                    72.250
catalyst                            -8.000  -4.000   K
temperature                         24.000  12.000   T
pressure                            -2.250  -1.125   P
concentration                       -5.500  -2.750   C
catalyst*temperature                 1.000   0.500   KT
catalyst*pressure                    0.750   0.375   KP
catalyst*concentration              -0.000  -0.000   KC
temperature*pressure                -1.250  -0.625   TP
temperature*concentration            4.500   2.250   TC
pressure*concentration              -0.250  -0.125   PC
catalyst*temperature*pressure       -0.750  -0.375   KTP
catalyst*temperature*concentration   0.500   0.250   KTC
catalyst*pressure*concentration     -0.250  -0.125   KPC
temperature*pressure*concentration  -0.750  -0.375   TPC
catalyst*temperature*pressure*      -0.250  -0.125   KTPC
  concentration

S = *  <-- No error degree of freedom

Analysis of Variance for y (coded units)

Source              DF   Seq SS   Adj SS   Adj MS  F  P
Main Effects         4  2701.25  2701.25  675.313  *  *
2-Way Interactions   6    93.75    93.75   15.625  *  *
3-Way Interactions   4     5.75     5.75    1.438  *  *
4-Way Interactions   1     0.25     0.25    0.250  *  *
Residual Error       0        *        *        *
Total               15  2801.00

Estimated Coefficients for y using data in uncoded units

Term                                        Coef
Constant                                 44.3333
catalyst                                 55.9333
temperature                             0.500000
pressure                                 1.73333
concentration                           -3.00000
catalyst*temperature                   -0.263333
catalyst*pressure                      -0.566667
catalyst*concentration                  -7.06667
temperature*pressure                  -0.0100000
temperature*concentration            -0.00833333
pressure*concentration                 -0.350000
catalyst*temperature*pressure         0.00266667
catalyst*temperature*concentration     0.0316667
catalyst*pressure*concentration        0.0733333
temperature*pressure*concentration    0.00166667
catalyst*temperature*pressure*      -3.33333E-04
  concentration


Effects Plot for y
(images/process6.png here)  


Alias Structure
I
catalyst
temperature
pressure
concentration
catalyst*temperature
catalyst*pressure
catalyst*concentration
temperature*pressure
temperature*concentration
pressure*concentration
catalyst*temperature*pressure
catalyst*temperature*concentration
catalyst*pressure*concentration
temperature*pressure*concentration
catalyst*temperature*pressure*concentration


At this point, it is also helpful to enter the respective effect names in the data window:(images/process8.gif here)
We will use these two columns and apply Lenth's method for the identification of significant effects later.

Factorial Plots are also created by doing  Stat-->DOE-->Factorial-->Factorial Plots

Main Effect Plot
(images/process3.png here)

Interaction Plot
(images/process4.png here)

Cube Plot
(images/process5.png here)
 

Assuming 3-fi's and 4-fi are negligible:  the Terms dialog box is shown below, you need only to change the `include terms in the model up through order'  from 4 to 2
(images/process9.gif here)

Session Output

Factorial Fit: y versus catalyst, temperature, pressure, concentration 

Estimated Effects and Coefficients for y (coded units)

Term                       Effect    Coef  SE Coef       T      P
Constant                           72.250   0.2739  263.82  0.000
catalyst                   -8.000  -4.000   0.2739  -14.61  0.000
temperature                24.000  12.000   0.2739   43.82  0.000
pressure                   -2.250  -1.125   0.2739   -4.11  0.009
concentration              -5.500  -2.750   0.2739  -10.04  0.000
catalyst*temperature        1.000   0.500   0.2739    1.83  0.127
catalyst*pressure           0.750   0.375   0.2739    1.37  0.229
catalyst*concentration     -0.000  -0.000   0.2739   -0.00  1.000
temperature*pressure       -1.250  -0.625   0.2739   -2.28  0.071
temperature*concentration   4.500   2.250   0.2739    8.22  0.000
pressure*concentration     -0.250  -0.125   0.2739   -0.46  0.667

The significant effects are (in the descending order of respective absolute T ratios)  T, K, C, TC, and perhaps P.

S = 1.09545   R-Sq = 99.79%   R-Sq(adj) = 99.36%

Analysis of Variance for y (coded units)

Source              DF   Seq SS   Adj SS   Adj MS       F      P
Main Effects         4  2701.25  2701.25  675.313  562.76  0.000
2-Way Interactions   6    93.75    93.75   15.625   13.02  0.006
Residual Error       5     6.00     6.00    1.200
Total               15  2801.00


Unusual Observations for y

Obs  StdOrder        y      Fit  SE Fit  Residual  St Resid
 12        12  83.0000  81.7500  0.9083    1.2500      2.04R

R denotes an observation with a large standardized residual.

Estimated Coefficients for y using data in uncoded units

Term                              Coef
Constant                       418.000
catalyst                      -6.85000
temperature                   -1.25417
pressure                      0.850000
concentration                 -53.9583
catalyst*temperature         0.0200000
catalyst*pressure            0.0100000
catalyst*concentration        0.000000
temperature*pressure       -0.00416667
temperature*concentration     0.225000
pressure*concentration      -0.0083333


Alias Structure
I
catalyst
temperature
pressure
concentration
catalyst*temperature
catalyst*pressure
catalyst*concentration
temperature*pressure
temperature*concentration
pressure*concentration

Lenth's Method
Click session window to make it current then do  Editor-->Enable Commands. Then the MINITAB prompt   MTB >   appears and MINITAB is ready to accept type-in commands.  Assume you have downloaded  lenth.mac  (make sure the file is in plain text format and with trailing extension  mac  (not txt)).

Session Output (commands you need to type in are in red color)
 
MTB > %'a:\Lenth' c10;
SUBC>  Names c11.
Executing from file: a:\Lenth.MAC
32 rows read.
14 rows read.

 
Lenth Statistics 
 
Data Display 

Row  EffectName  EFFE1     tpse  pIER     pEER
  1  K           -8.00  -7.1111  <=0.001  0.00653
  2  T           24.00  21.3333  <=0.001  <=0.001
  3  P           -2.25  -2.0000  0.06333  > 0.400
  4  C           -5.50  -4.8889  0.00327  0.02909
  5  KT           1.00   0.8889  0.35057  > 0.400
  6  KP           0.75   0.6667  > 0.400  > 0.400
  7  KC          -0.00  -0.0000  > 0.400  > 0.400
  8  TP          -1.25  -1.1111  0.25389  > 0.400
  9  TC           4.50   4.0000  0.00699  0.06187
 10  PC          -0.25  -0.2222  > 0.400  > 0.400
 11  KTP         -0.75  -0.6667  > 0.400  > 0.400
 12  KTC          0.50   0.4444  > 0.400  > 0.400
 13  KPC         -0.25  -0.2222  > 0.400  > 0.400
 14  TPC         -0.75  -0.6667  > 0.400  > 0.400
 15  KTPC        -0.25  -0.2222  > 0.400  > 0.400

MTB > 


Halfnormal Plot
Assume you have downloaded hnorm.mac  (make sure the file is in plain text format and with trailing extension  mac  (not txt)).

Session Output
 
MTB > %'a:\hnorm' c10;
SUBC>  names c11.
Executing from file: a:\hnorm.MAC
(images/process13.png here)

It appears that 4 to 5 effects are significant.  Let's choose 5 and you can replot halfnormal plot as follows:
MTB > %'a:\hnorm' c10;
SUBC>  identify 5;
SUBC>  names c11.
Executing from file: a:\hnorm.MAC
(images/process14.png here)



Final model diagnostics (Assuming  K,  T,  C,  and TC  only)
(images/process10.gif here)

(images/process11.gif here)

Session Output

Factorial Fit: y versus catalyst, temperature, concentration 


Estimated Effects and Coefficients for y (coded units)

Term                       Effect    Coef  SE Coef       T      P
Constant                           72.250   0.4707  153.48  0.000
catalyst                   -8.000  -4.000   0.4707   -8.50  0.000
temperature                24.000  12.000   0.4707   25.49  0.000
concentration              -5.500  -2.750   0.4707   -5.84  0.000
temperature*concentration   4.500   2.250   0.4707    4.78  0.001

S = 1.88294   R-Sq = 98.61%   R-Sq(adj) = 98.10%

Analysis of Variance for y (coded units)

Source              DF   Seq SS   Adj SS   Adj MS       F      P
Main Effects         3  2681.00  2681.00  893.667  252.06  0.000
2-Way Interactions   1    81.00    81.00   81.000   22.85  0.001
Residual Error      11    39.00    39.00    3.545
  Lack of Fit        3     5.00     5.00    1.667    0.39  0.762
  Pure Error         8    34.00    34.00    4.250
Total               15  2801.00

Unusual Observations for y

Obs  StdOrder        y      Fit  SE Fit  Residual  St Resid
 12        12  83.0000  79.7500  1.0526    3.2500      2.08R

R denotes an observation with a large standardized residual.

Estimated Coefficients for y using data in uncoded units

Term                           Coef
Constant                    415.750
catalyst                   -1.60000
temperature                -1.27500
concentration              -54.5000
temperature*concentration  0.225000

Alias Structure
I
catalyst
temperature
concentration
temperature*concentration

 
Residual Plots for y
 
(images/process12.png here)