Abstract

The expectation part of a linear model is often presented as a single equation with unknown parameters, and the reader is supposed to know that this is shorthand for a whole family of expectation models (for example, is there interaction or not?). It is helpful to list the whole family of models separately and then represent them on a Hasse diagram. This shows which models are sub-models of others, which helps the user to respect marginality when choosing the most parsimonious model to explain the data. Each row in an analysis-of-variance table corresponds to an edge in the Hasse diagram. In the scaled version of the Hasse diagram, the length of each edge is proportional to the appropriate mean square. This gives a visual display of the analysis of variance (ANOVA). For some people, this is easier to interpret than the standard analysis-of-variance table. Moreover, the scaled Hasse diagram makes clear the difficulties in model choice that can occur under non-orthogonality. The ideas are illustrated using some familiar families of models defined by crossed and nested factors, possibly including polynomial terms for quantitative factors, as well as some more recently introduced families of models for experiments in biodiversity.

摘要

线性模型的期望部分通常表示为具有未知参数的单个方程，读者应该知道这是整个期望模型系列的简写（例如，是否存在交互作用？）。单独列出整个模型系列，然后在 Hasse 图上表示它们是有帮助的。这显示了哪些模型是其他模型的子模型，这有助于用户在选择最简约的模型来解释数据时尊重边缘性。方差分析表中的每一行对应于哈斯图中的一条边。在 Hasse 图的缩放版本中，每条边的长度与适当的均方成正比。这给出了方差分析 (ANOVA) 的直观显示。对于有些人，这比标准的方差分析表更容易解释。此外，缩放的 Hasse 图清楚地表明了在非正交性下可能出现的模型选择困难。使用由交叉和嵌套因子定义的一些熟悉的模型系列来说明这些想法，可能包括定量因子的多项式项，以及一些最近引入的用于生物多样性实验的模型系列。