Return to the data presented in Exercise 2 in Chapter 1.

(a) Fit the following model to these data: Exercises 239 Response variate: available chlorine Fixed-effect model: linear, quadratic and cubic effects of time Random-effect model: deviation of mean available chlorine at each time from the value predicted by the fixed-effect model.

(b) Investigate whether any terms can reasonably be dropped from this model.

(c) Obtain diagnostic plots of the residuals, and consider whether the assumptions on which the analysis is based are reasonable.

(d) Plot the estimate of each random effect against the corresponding time, and consider whether there is any evidence of a trend over time that is not accounted for by the fixed-effect model. Note that, as the level of available chlorine declines, the rate of decline becomes lower, and this exponential decay process is more naturally modelled by a nonlinear function than by the polynomial function used for convenience here. The model fitted by Draper and Smith (1998, Chapter 24, Section 24.3, pp 518–519) is Y = α + (0.49 − α)e−β(X−8) + ε where Y = available chlorine X = time, α and β are parameters to be estimated.