Goodness of fit for censored regression in internal dosimetry

Radiation doses from incorporated radionuclides are typically estimated from monitoring data using the maximum-likelihood method followed by a check of the goodness of fit. The likelihood function depends not only on the assumed biokinetic model but also on the information provided by the measurement process. Measurements recorded as below a known censoring threshold (type-I left censoring) are still encountered in internal dosimetry, particularly in historical databases used for epidemiological studies. This paper considers a goodness of fit method for datasets that include censored observations. We focus on censored regression, commonly used in internal dosimetry to estimate intakes from measurements. The method relies on the Tobit likelihood function assuming that both observed and censored data follow a specified probability distribution. We propose the use of the deviance, a generalisation of the familiar χ 2 quantity, as a measure of discrepancy between measurements and the model predictions. Goodness of fit is then evaluated by comparing the observed deviance to its reference probability distribution. Although this reference distribution does not have generally a closed analytical form it is simple to compute via Monte Carlo simulations. As a computationally more efficient alternative, we derive analytical approximations for the distribution of the deviance of normally and lognormally distributed data. These approximations are expressed in terms of the gamma distribution, which includes the χ 2 distribution as a special case. The derived gamma approximations have been validated against Monte Carlo simulations for several exposure scenarios. While censoring should be avoided when possible, providing a simple method for assessing the goodness of fit should further encourage internal dosimetrists to rely on the censored regression technique rather than using substitution methods.