Analysis of variance

In a quadratic response surface model for three factors, ten parameters are used to model the mean response,

A single run of a central composite design has 8 factorial runs, 6 star points and r replicates of the centre point. The analysis of variance table for the model therefore has the following form.

  Source of variation      SSq   df   MSq      F      p-value  
  Linear X    ? 1 ? ? ?
  Linear Z ? 1 ? ? ?
  Linear W ? 1 ? ? ?
  Quadratic X ? 1 ? ? ?
  Quadratic Z ? 1 ? ? ?
  Quadratic W ? 1 ? ? ?
  X-Z interaction ? 1 ? ? ?
  X-W interaction ? 1 ? ? ?
  Z-W interaction ? 1 ? ? ?
  Lack of fit ? 5 ? ? ?
  Residual (pure error)   r - 1      
  Total ? r + 13      

The residual sum of squares comes only from the replicates of the centre point of the design and is the best estimate of the error variance for the experiment.

The p-values in the anova table indicate the significance of the various terms in the model.

If the lack-of-fit sum of squares is significant, it could indicate either that the relationships are nonlinear in a way that is not well modelled by a quadratic, or that there is a 3-factor interaction.

Wastewater treatment in a refinery

The analysis of variance table for the wastewater treatment experiment is shown below.

  Source of variation      SSq   df   MSq      F      p-value  
  Linear Alum    408.4   1   408.4   19.08 0.022
  Linear Poly 3,761.9   1     3,761.9     175.79   0.001
  Linear pH 100.6   1   100.6   4.70 0.119
  Quadratic Alum 1,424.5   1   1,424.5   66.57  0.004
  Quadratic Poly 5.8   1   5.8   0.27 0.639
  Quadratic pH 731.5   1   731.5   34.18 0.010
  Alum-Poly interaction 23.5   1   23.5   1.10 0.372
  Alum-pH interaction 205.0   1   205.0   9.58 0.054
 Poly-pH interaction 754.7   1   754.7   35.27 0.010
  Lack of fit 1,343.2   5   268.64     12.55 0.032
  Residual (pure error) 64.2    3   21.4      
  Total   8,823.4     17        

The p-value for lack of fit is 0.032, giving evidence that there are some problems with the model. We will not investigate the lack of fit here, but a transformation of the response may fit a quadratic model better than the raw data.

The diagram below shows the full fitted response surface.

The quadratic term in Polyelectrolyte and the Aluminium-Polyelectrolyte interaction have high p-values, so use the checkboxes to remove these terms from the model. Use the slider and observe that this model behaves in a similar way to the full quadratic model.

The best quality (blue) arises when

However the Aluminium-pH interaction is also not significant at the 5% level. Try also deleting this terms from the model and investigating the shape of the response surface.

Without the interaction between Aluminium sulphate and pH, the greatest turbidity reduction occurs when pH and Polyelectrolyte are both high.

The difference between these sets of conclusions gives a warning about interpreting response surfaces when they are based on limited experimental data.

Be careful not to over-interpret the shape of response surfaces if some parameters are not significantly different from zero.