Introduction to advanced models

What to do when linear regression is not suitable for our problem? When the phenomena under study do not conform to the premises of simple linear regression, for instance when the response departs clearly from normality, when the variance changes systematically, or when the data are temporally or spatially correlated, we need to extend the modelling framework to methods that explicitly accommodate such complexity. In these cases, the idea of a model as a simplified mathematical representation of reality still applies, but the form of that representation changes: instead of assuming independent, normally distributed errors, we may specify alternative error structures (e.g. Poisson or binomial for counts and proportions), use transformations or variance functions to stabilise heteroscedastic data, or introduce correlation structures to account for temporal or spatial autocorrelation. The model is then assessed in a similar way, by inspecting residuals, information criteria and diagnostic plots, but now the diagnostics must match the chosen structure, so that we verify not only the adequacy of the predictors but also that the assumed distribution and correlation pattern describe the data reasonably well. From that basis, we can expand to more elaborate mixed models, generalised linear models or time-series and spatial models, and still keep the same objective: to capture the main signal in the data while keeping violations of the underlying assumptions under control.