This chapter considers alternatives to standard regression schemes such as normal linear regression and generalised linear models when underlying assumptions are not met. For example, the assumptions of the normal linear model are often violated by the data, and inferences regarding regression impacts may be affected. These include the linearity assumption, namely that the conditional means of the response are a linear function of the predictors, and the assumptions that the error terms are normally distributed and have constant variance. More general formulations such as skew normal and heteroscedastic regression (Section 4.2), discrete mixture regression (Section 4.5), and non-linear regression (Section 4.6) are then needed.

Assumptions made under the generalised linear modelling approach are summarised by Breslow (1996) namely ‘the statistical independence of the observations, the correct specification of the link and variance functions, the correct scale for measurement of the explanatory variables, and the lack of undue influence of individual observations on the fitted model’. Of these issues, those considered in detail in this chapter are specification of the variance function in terms of representing overdispersion (Section 4.3), and flexible link specification in Section 4.4.

Bayesian computing options for such generalised regression techniques include BUGS, R-INLA (e.g. Wang, 2013), and R packages such ...

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