Return to the data set concerning the effect of oil type on the amount of wear suffered by piston rings, introduced in Exercise 2 in Chapter 2.

(a) For each oil type, plot the value of wear against the ring number (1 to 4). Omit the values from the ‘oil ring’ from this plot. Is there evidence of a trend in the amount of wear from Ring 1 to Ring 4?

(b) Repeat the mixed-model analysis performed on these data previously, but exclude the values from the ‘oil ring’.

(c) Now specify ‘ring’ as a variate instead of a factor. Fit this new model to the data by mixed modelling. Do the results confirm that there is a linear trend from Ring 1 to Ring 4? Does this trend vary significantly depending on the oil type?

(d) Note that when this change is made to the model, DFring is reduced from 3 to 1, and DFoil.ring from 6 to 2. What source of variation is included in these terms in the previous model, but excluded in the present model? Now suppose that the three types of oil tested are a representative sample of the types used in practice.

(e) When this change is made, which parts of the expression ‘oil ∗ ring’ should be regarded as fixed-effect terms, and which as random-effect terms?

(f) Fit this new model to the data by mixed modelling. Does the new model indicate that the linear trend from Ring 1 to Ring 4 varies significantly depending on the oil type?