Abstract
We introduce a new type of threshold regression models called upper hinge models. Under this type of threshold models, there only exists an association between the predictor of interest and the outcome when the predictor is less than some threshold value. Just like hinge models, upper hinge models can be seen as a special case of the more general segmented or two-phase regression models. The importance of studying upper hinge models is that even though they only have one fewer degree of freedom than segmented models, they can be estimated with much greater efficiency. We develop a new fast grid search algorithm to estimate upper hinge linear regression models. The new algorithm reduces the computational complexity of the search algorithm dramatically and renders the existing fast grid search algorithm inadmissible. The fast grid search algorithm makes it feasible to construct bootstrap confidence intervals for upper hinge linear regression models; for upper hinge generalized linear models of non-Gaussian family, we derive asymptotic normality to facilitate construction of model-robust confidence intervals. We perform numerical experiments and illustrate the proposed methods with two real data examples from the ecology literature.
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Acknowledgements
The authors are grateful to the Editor, the AE and two referees for their constructive comments. We are also indebted to Dr. Helen Sofaer of the US Geological Survey for suggesting the paramecium population growth dataset, to Dr. Takehiro Sasaki of Yokohama National University, Japan for providing the Mongolian rangelands vegetation dataset, and to Lindsay N. Carpp for help with editing. This work was supported by the National Institutes of Health (R01-AI122991; UM1-AI068635).
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Handling Editor: Pierre Dutilleul.
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Elder, A., Fong, Y. Estimation and inference for upper hinge regression models. Environ Ecol Stat 26, 287–302 (2019). https://doi.org/10.1007/s10651-019-00428-1
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DOI: https://doi.org/10.1007/s10651-019-00428-1