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.

中文翻译：

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分析长期农业实验的贝叶斯方法


有效且灵活的统计分析是充分利用长期实验 (LTE) 的关键。在这里，我们的目标是将贝叶斯分析引入更广泛的 LTE 社区，并展示建模过程与传统统计分析的不同之处。由于模型开发更加灵活，可以更好地访问统计软件和采样算法，贝叶斯方法变得越来越流行。使用贝叶斯定理，通过结合我们可能拥有的有关模型项的任何先验知识来估计模型系数。以这种方式包含先验知识需要对拟合模型采用不同的估计过程。贝叶斯模型系数通常是从马尔可夫链的一次或多次运行的数千个样本中采样的。我们通过三个例子展示贝叶斯分析的使用。示例 1 说明了使用 Broadbalk 长期实验进行的有因素和无因素的单一回归，显示了估计模型如何随着我们对模型系数的先验知识的不确定性而变化。示例 2 演示了多元回归的使用，根据因子变量和季节性天气变量预测谷物产量。实施例3显示了使用瑞典土壤肥力实验的分层时间序列模型对作物轮作和施肥处理下的土壤碳变化的估计。