Modelling Heteroskedasticity, Overdispersion and the Impacts of Climate Change for Long-Term Hay Yield Data

Non-constant residuals, also known as heteroskedastic errors, are a common challenge in statistical modelling. One common method for coping with heteroskedastic errors for non-negative data is through the use of a Gamma likelihood function, often with the inclusion of an additional overdispersion parameter. However, modelling heteroskedasticity in this way restricts the mean-variance relationship assumed for the data to follow that of the Gamma distribution. This study was concerned with the modelling of the impact of climate change on the Park Grass hay yields from 1901 to 2016, a response that showed such heteroskedastic errors. Our approach replaced the fixed mean-variance relationship of a Gamma likelihood function (including the additional overdispersion parameter), instead modelling the relationship using a non-linear log-logistic function. Model fitting was performed in a Bayesian environment, where we compared the posterior predictive accuracy of the proposed method, with values for previously well-established methods for dealing with heteroskedastic errors of continuous variables. The model was then extended to incorporate the effects of monthly weather summaries on these yields, thus identifying the potential impacts of climate change. This extended model included a varying intercept multiple regression component, which modelled hay yield in terms of meteorological covariates, and an autoregressive lag one (AR1) process. The results are important in the context of identifying the impacts of climate change on agricultural production, with annual mean air temperatures now 1.43◦C above the 20th century average and changes in rainfall patterns. They confirm previous findings that variation in rainfall partly explains year-to-year variation in hay yield, but also show that warmer temperatures are associated with lower hay yields. Our work demonstrates the benefits of directly modelling the mean-variance relationship of a Gamma likelihood function to cope with heteroskedastic error, rather than applying an overdispersion parameter to the fixed relationship or considering transformations of the response.