solution

Qian et al. [2003a] proposed a “nonparametric deviance reduction” method for detecting ecological threshold. The method is based on the CART model, but uses only one predictor representing the environmental gradient. The first split point is used as the threshold. In the paper, the authors suggested that a χ ² test can be used to test whether the resulting split point is “statistically significant.” Because the split point is the point that results in the largest difference in deviance, it is highly likely that such a test will have a highly inflated type I error probability. Design a simulation to estimate the type I error probability of such a test. In the simulation, we can assume that the response variable is a normal random variable, such that the χ ² test is reduced to a two-sample t-test. As the method is used to detect a threshold, the null hypothesis should be that a threshold does not exist, or the response variable distribution does not change along the gradient.