In ecological community studies it is often of interest to study the effect of species related trait variables on abundances or presence-absences. Specifically, the interest may lay in the interactions between environmental and trait variables. An increasingly popular approach for studying such interactions is to use the so-called fourth-corner model, which explicitly posits a regression model where the mean response of each species is a function of interactions between covariate and trait predictors (among other terms). On the other hand, many of the fourth-corner models currently applied in the literature are too simplistic to properly account for variation in environmental and trait response and any residual covariation between species. To overcome this problem, we propose a fourth-corner latent variable model which combines the following three features: latent variables to capture the correlation between species, fourth-corner terms to account for environment-trait interactions, and species-specific random slopes for modeling excess heterogeneity between species in their environmental response. We perform an extensive numerical study comparing a variety of fourth-corner models available in the literature which account for the aforementioned sources of variation to varying degrees. Simulation results demonstrate that the proposed fourth-corner latent variable models performed well when testing for the fourth-corner (interaction) coefficients, across both Type I error and power. By comparison, some models that do not full account for all relevant sources of variation suffer from inflated Type I error leading to potentially misleading inference. The proposed method is illustrated by an example on ground beetle data.

中文翻译：

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用第四角潜变量模型分析生态群落中的环境-特征相互作用

在生态群落研究中，研究物种相关性状变量对丰度或存在缺失的影响通常很有趣。具体来说，兴趣可能在于环境和特征变量之间的相互作用。研究此类相互作用的一种越来越流行的方法是使用所谓的第四角模型，该模型明确提出了一个回归模型，其中每个物种的平均响应是协变量和性状预测变量（以及其他术语）之间相互作用的函数。另一方面，目前文献中应用的许多第四角模型过于简单，无法正确解释环境和性状响应的变化以及物种之间的任何残余协变。为了克服这个问题，我们提出了一个第四角潜在变量模型，它结合了以下三个特征：捕获物种之间相关性的潜在变量，解释环境-性状相互作用的第四角项，以及用于模拟物种之间过度异质性的物种特异性随机斜率。他们的环境反应。我们进行了广泛的数值研究，比较了文献中可用的各种第四角模型，这些模型在不同程度上解释了上述变化来源。仿真结果表明，在测试第四角（交互）系数时，所提出的第四角潜在变量模型在 I 类误差和功效方面表现良好。通过比较，一些未充分考虑所有相关变异来源的模型会遭受夸大的 I 类错误，从而导致潜在的误导性推理。通过地面甲虫数据的示例说明了所提出的方法。