  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. 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 effect 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 on the Idaho steppe.
