Random errors are neither: On the interpretation of correlated data

Abstract

Many statistical models currently used in ecology and evolution account for covariances among random errors. Here, I address five points: (i) correlated random errors unite many types of statistical models, including spatial, phylogenetic and time-series models; (ii) random errors are neither unpredictable nor mistakes; (iii) diagnostics for correlated random errors are not useful, but simulations are; (iv) model predictions can be made with random errors; and (v) can random errors be causal? These five points are illustrated by applying statistical models to analyse simulated spatial, phylogenetic and time-series data. These three simulation studies are paired with three types of predictions that can be made using information from covariances among random errors: predictions for goodness-of-fit, interpolation, and forecasting. In the simulation studies, models incorporating covariances among random errors improve inference about the relationship between dependent and independent variables. They also imply the existence of unmeasured variables that generate the covariances among random errors. Understanding the covariances among random errors gives information about possible processes underlying the data. Random errors are caused by something. Therefore, to extract full information from data, covariances among random errors should not just be included in statistical models; they should also be studied in their own right. Data are hard won, and appropriate statistical analyses can make the most of them.