## Generalised Additive Mixed Modelling

In the additive model in Eq. 8.8, we also assume independence and homogeneity. In the previous section, we aimed to improve the lack of fit due to the time effect (Series), but the model still assumes that the variation is the same everywhere; for different treatment effects, at the beginning of the experiment and at the end of the experiment, and also for different Larval stages. However, in Fig. 8.6a we saw heterogeneity in the residuals obtained from the GAM, and this is something we have to improve. One option is to transform length, but we prefer to avoid transformations. Another possible solution is generalised additive mixed modelling. In the same way as linear mixed effects modelling is an extension of linear regression, so is generalised additive mixed modelling an extension of generalised additive model. Pinheiro and Bates (2000) discuss a wide range of options to model heterogeneity. If, for example, the length variation increases over time, we can use:

The unknown variance parameters are s and d. For positive d, the variance in the residuals increases for larger Series values, and vice versa. If 8 = 0, we obtain our familiar variance from Eq 8.4 again. It is also possible to assume that one particular drug treatment, say Propofol, causes more variation. This can be modelled with:

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