Quantifying the relationship between observed variables with censored values using Bayesian error-in-variables regression

Abstract

We aimed to address two common challenges for scientists working with observational data: “how to quantify the relationship between two observed (or measured) variables”, and, “how to account for censored values” (i.e., observations or measures whose value is only known to fall within a range). Quantifying the relationship between observed variables, and predicting one variable from the other (and vice versa), violates the assumption of standard regression regarding the existence of an independent, explanatory variable that is observed with no (or limited) uncertainty. To overcome this challenge, we developed and tested a Bayesian error-in-variables, EIV, regression model which accounts for uncertainty in variables orthogonally. Moreover, parameter estimation using Bayesian inference allowed the full parameter uncertainty to be propagated into probabilistic model predictions suitable for decision making. Alternative model formulations were applied to a dataset containing measured concentrations of organic pollutants in mothers and their eggs from the freshwater turtle Malaclemys terrapin and validated against an independent dataset of the turtle Chelydra serpentina. The best performing EIV model was then applied to the dataset again after censoring measurements in one or both variables. Here, independent likelihoods for both censored and uncensored data were formulated and then easily combined following the Bayesian implementation of the model. The EIV model performed well, as revealed by posterior predictive checks around 85%, and obtained comparable parameter estimates in both censored and uncensored cases. The resulting model allows scientists and decision-makers to quantitatively link variables, and make predictions from one variable to the next while accounting for uncertainties and censored data.