4.33 Removing oil from a water/oil mix. Refer to the Journal of Colloid and Interface Science study of water/oil mixtures, Exercise 4.14 (p. 189). Recall that three of the seven variables used to predict voltage (y) were volume (x1), salinity (x2), and surfactant concentration (x5). The model the researchers ﬁt is
E(y) = β0 +β1×1 +β2×2 +β3×5 +β4x1x2 +β5x1x5.
(a) Note that the model includes interaction between disperse phase volume (x1) and salinity (x2) as well as interaction between disperse phase volume (x1) and surfac- tant concentration (x5). Discuss how these interaction terms affect the hypothetical rela- tionship between y and x1. Draw a sketch to support your answer.
(b) Fit the interaction model to the WATEROIL data. Does this model appear to ﬁt the data better than the ﬁrst-order model in Exercise 4.14? Explain.
(c) Interpret the β estimates of the interaction model.
Figure 4.16 Potential y
misuse of quadratic model
40 60 80 100 120
Maximal oxygen uptake (ml/kg)