Example of fitting a linear response surface model
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The following examples use data from [3]. The experiment uses three factors - nitrogen, phosphoric acid, and potash - all ingredients in fertilizer. The effect of the fertilizer on snap bean yield was studied in a central composite design.

The actual units for the -1 and +1 levels are 2.03 and 5.21 for nitrogen, 1.07 and 2.49 for phosphoric acid, 1.35 and 3.49 for potash. To reduce the impact of non-orthogonal terms, Minitab fits the model in coded units.

Step 1: Generating the central composite design

1    Choose Stat > DOE > Response Surface > Create Response Surface Design.

2    Under Type of Design, choose Central composite.

3    From Number of factors, choose 3.

4    Click Designs. To create the design, click OK.

5    Click Factors. In the Name column, enter Nitrogen Phosphoric Acid Potash in rows one through three, respectively. Click OK in each dialog box.

Step 2: Fitting a linear model

1    Open the worksheet CCD_EX1.MTW. (The design from the previous step and the response data have been saved for you.)

2    Choose Stat > DOE > Response Surface > Analyze Response Surface Design.

3    In Responses, enter BeanYield.

4    Click Terms.

5    From Include the following terms, choose Linear. Click OK in each dialog box.

Session window output

Response Surface Regression: BeanYield versus Nitrogen, PhosAcid, Potash

 

 

Analysis of Variance

 

Source         DF   Adj SS  Adj MS  F-Value  P-Value

Model           3   7.7886  2.5962     1.08    0.387

  Linear        3   7.7886  2.5962     1.08    0.387

    Nitrogen    1   4.4960  4.4960     1.86    0.191

    PhosAcid    1   0.4593  0.4593     0.19    0.668

    Potash      1   2.8332  2.8332     1.17    0.295

Error          16  38.5965  2.4123

  Lack-of-Fit  11  36.0569  3.2779     6.45    0.026

  Pure Error    5   2.5396  0.5079

Total          19  46.3851

 

 

Model Summary

 

      S    R-sq  R-sq(adj)  R-sq(pred)

1.55315  16.79%      1.19%       0.00%

 

 

Coded Coefficients

 

Term      Effect    Coef  SE Coef  T-Value  P-Value   VIF

Constant          10.198    0.347    29.36    0.000

Nitrogen  -1.148  -0.574    0.420    -1.37    0.191  1.00

PhosAcid   0.367   0.183    0.420     0.44    0.668  1.00

Potash     0.911   0.455    0.420     1.08    0.295  1.00

 

 

Regression Equation in Uncoded Units

 

BeanYield = 10.01 - 0.361 Nitrogen + 0.258 PhosAcid + 0.426 Potash

 

 

Fits and Diagnostics for Unusual Observations

 

Obs  BeanYield     Fit   Resid  Std Resid

  7      8.260  11.163  -2.903      -2.17  R

 16     13.190  10.500   2.690       2.03  R

 

R  Large residual

Interpreting the results

It is important to check the adequacy of the fitted model, because an incorrect or under-specified model can lead to misleading conclusions. By checking the fit of the linear (first-order) model you can tell if the model is under specified. The small p-value (p = 0.026) for the lack of fit test indicates the linear model does not adequately fit the response surface. The F-statistic for this test is (Adj MS for Lack of Fit) / (Adj MS for Pure Error).

The VIFs are all close to 1, which indicates that the predictors are not correlated. VIF values greater than 5-10 suggest that the regression coefficients are poorly estimated due to severe multicollinearity.

Because the linear model does not adequately fit the response surface, you need to fit a quadratic (second-order) model.