A Bayesian approach to analyzing long-term agricultural experiments

Effective and flexible statistical analyses are key to getting the most out of long-term experiments (LTEs). Here, we aim to introduce Bayesian analysis to the wider LTE community and show how the modelling process differs from traditional statistical analyses. Bayesian methods have become increasingly popular due to more flexibility in model development with better access to statistical software and sampling algorithms. Using Bayes' Theorem, model coefficients are estimated by incorporating any prior knowledge we may have on model terms. Including prior knowledge in this way requires a different estimating procedure for a fitted model. Bayesian model coefficients are usually sampled from thousands of samples from one or more runs of a Markov Chain. We present the use of Bayesian analyses through three examples. Example 1 illustrates a single regression with and without factors using the Broadbalk Long-Term Experiment, showing how the estimated model changes with more uncertainty in our prior knowledge of model coefficients. Example 2 demonstrates the use of multiple regression, predicting grain yield from factor variables and seasonal weather variables. Example 3 shows an estimation of soil carbon changes under crop rotation and fertilization treatments with a hierarchical time series model using a Swedish soil fertility experiment.