When variability varies: Heteroscedasticity and variance functions

Abstract

Variability is inherent to the world around us. Its quantification is essential to understand processes of interest in environmental and social sciences, such as adaptation of species to climate change or social inequality. Variance, one of the parameters of the normal distribution, is commonly used to quantify variability. Classical linear models assume that variance is constant (homoscedasticity assumption), while focusing only on changes in average trends. It is possible to extend classical models and relax the assumption of homoscedasticity through variance functions. However, these functions are scarcely used and we often lack examples in the Spanish-wri�en scientific literature. In this paper, we introduce variance functions in linear models from a theoretical-applied approach. We begin by introducing a real problem where heteroscedasticity is expected, which is accompanied by one simulated example. Subsequently, we formulate the classical linear model and discuss how it can be extended to model heteroscedasticity. Then, we explain some of the variance functions and apply them to the real case and the simulated data. We use the gls() function of the nlme package in R, and provide scripts that make data analyses reproducible. Additionally, we describe other options available in R for dealing with heteroscedastic data. We expect this paper will provide a guide for using variance functions and will expand the toolbox of scientists with basic statistical knowledge.