Environmental variability often has substantial impacts on natural populations and communities through its effects on the performance of individuals. Because organisms’ responses to environmental conditions are often nonlinear (e.g., decreasing performance on both sides of an optimal temperature), the mean response is often different from the response in the mean environment. Ye et al. (Ann Appl Stat 14(3):1326–12341, 2020) proposed testing for the presence of such variance effects on individual or population growth rates by estimating the “Jensen Effect”, the difference in average growth rates under varying versus fixed environments, in functional single index models for environmental effects on growth. In this paper, we extend this analysis to effects of environmental variance on reproduction and survival, which have count and binary outcomes. In the standard generalized linear models used to analyze such data the direction of the Jensen Effect is tacitly assumed a priori by the model’s link function. Here we extend the methods of Ye et al. (2020) using a generalized single index model to test whether this assumed direction is contradicted by the data. We show that our test has reasonable power under mild alternatives, but requires sample sizes that are larger than are often available. We demonstrate our methods on a long-term time series of plant ground cover in Idaho sagebrush steppe.

环境多变性通常通过影响个人绩效对自然人口和社区产生重大影响。由于生物对环境条件的响应通常是非线性的（例如，最佳温度两侧的性能下降），因此平均响应通常不同于平均环境中的响应。Ye等。（Ann Appl Stat 14（3）：1326–12341，2020）建议通过估算“ Jensen效应”（在可变环境与固定环境下平均增长率的差异）来测试这种差异对个人或人口增长率的影响，在功能性单一指数模型中研究环境对增长的影响。在本文中，我们将此分析扩展到环境差异对繁殖和生存的影响，这些影响具有计数和二进制结果。在用于分析此类数据的标准广义线性模型中，模型的链接函数默认先验地假设了詹森效应的方向。在这里，我们扩展了Ye等人的方法。（2020年）使用广义的单一指数模型来测试该假定的方向是否与数据相矛盾。我们表明，在温和的替代条件下，我们的测试具有合理的功效，但所需要的样本量大于通常可用的样本量。我们在爱达荷州鼠尾草草原的植物地被植物的长期时间序列上论证了我们的方法。（2020年）使用广义的单一指数模型来测试该假定的方向是否与数据相矛盾。我们表明，在温和的替代条件下，我们的测试具有合理的功效，但所需要的样本量大于通常可用的样本量。我们在爱达荷州鼠尾草草原的植物地被植物的长期时间序列上论证了我们的方法。（2020年）使用广义的单一指数模型来测试该假定的方向是否与数据相矛盾。我们表明，在温和的替代条件下，我们的测试具有合理的功效，但所需要的样本量大于通常可用的样本量。我们在爱达荷州鼠尾草草原的植物地被植物的长期时间序列上论证了我们的方法。