## Model Validation of the GAM

Instead of presenting the numerical and graphical output of the optimal GAM model, we go immediately to the model validation and show that there are still some serious problems even though we are increasing the complexity of the model. As in linear regression, with additive models we need to verify the underlying assumptions.

L1 L2 L3 L3P 5 10 15

### Series

Fig. 8.6 (a) Residuals versus Larvae stage. Note the differences in residual spread. Larvae stage (L3) has the most variation. (b) Residuals versus Series. Independence is a valid assumption as the points are equally distributed above and bellow the zero line, indicating the same number of positive and negative residuals but some discrepancies can be expected

L1 L2 L3 L3P 5 10 15

### Series

Fig. 8.6 (a) Residuals versus Larvae stage. Note the differences in residual spread. Larvae stage (L3) has the most variation. (b) Residuals versus Series. Independence is a valid assumption as the points are equally distributed above and bellow the zero line, indicating the same number of positive and negative residuals but some discrepancies can be expected

There is no doubt that much progress has been made in terms of the independence assumption as the smoother of Series seems to capture the patterns over time quite well (Fig. 8.6b). The smoother had 8.3 degrees of freedom, and was highly significant (F = 698.4, p < 0.001).

Although the GAM showed an improvement in terms of independence, the heterogeneity problems are still there, see Fig. 8.6a; it shows a boxplot of the residuals versus larvae stage. Hence, we are still violating the homogeneity assumption, and therefore further model improvement is required.

Because the GAM has less residual patterns than the linear regression model, it turned out to be an improvement. The good news is that there is life beyond GAMs and we will extend it to generalised additive mixed modelling (GAMM) to allow for the heterogeneity.

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