We aimed to address two common challenges for scientists working with observational data: "how to quantify the relation between two observed variables", and, "how to account for censored observations" (i.e., observations whose value is only known to fall within a range). Quantifying the relationship between observed variables, and predicting one measured quantity from another (and vice versa), violates the assumption of standard regression regarding the existence of an independent, explanatory variable that is measured with no (or limited) measurement error. To overcome this challenge, we developed and tested a Bayesian error-in-variables, EIV, regression model which accounts for measurement uncertainty 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 observations in one or both variables. The Bayesian implementation allowed for such application as independent likelihoods for both censored and uncensored data could be combined. 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 measured variables, and make predictions from one variable to the next while accounting for measurement uncertainties and censored data.