Errors in variables models: Diagnosing parameter estimability and MCMC convergence using empirical characteristic functions
Subhash Lele (February 22, 2012)
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Most ecological models are constructed to understand the relationship between environmental variables and an ecological response, be it site occupancy or population abundance or changes to them. The usual regression models take into account the environmental variation in the response but in many cases, the measurement of the environmental variables themselves are made with error. This is called an errors-in-variables model. Measurement error in the covariates leads to substantial issues with parameter estimability and likelihood-based inference is computationally challenging. Bayesian inference using Markov Chain Monte Carlo methods also runs into trouble because of the convergence issues with the MCMC algorithm. These convergence issues are severe especially with non-informative priors.
Errors in variables models, linear and non-linear, can be formulated as hierarchical models. Data cloning is a recently developed computational technique to conduct likelihood- based analysis for general hierarchical models. In this paper, we show that data cloning coupled with informative priors can circumvent the convergence issues with MCMC. We develop a new testing procedure to compare multivariate distributions using empirical characteristic functions and show its usefulness in diagnosing convergence of MCMC algorithm in these tricky situations. More importantly, we show that data cloning not only facilitates parameter estimation but also diagnosing which parameters are estimable and which ones are not. This is essential for drawing scientifically meaningful inferences. We illustrate the method using various linear and non-linear regression models useful in ecology. We report a somewhat surprising result that a widely used population dynamics model, the Hasell model, is non-identifiable but a closely related Generalized Beverton-Holt model is identifiable.
Work done in collaboration with Khurram Nadeem.