Some Aspects of Measurement Error in Linear Regression of Astronomical Data

I describe a Bayesian method to account for measurement errors in linear regression of astronomical data. The method allows for heteroscedastic and possibly correlated measurement errors and intrinsic scatter in the regression relationship. The method is based on deriving a likelihood function for the measured data, and I focus on the case when the intrinsic distribution of the independent variables can be approximated using a mixture of Gaussian functions. I generalize the method to incorporate multiple independent variables, nondetections, and selection effects (e.g., Malmquist bias). A Gibbs sampler is described for simulating random draws from the probability distribution of the parameters, given the observed data. I use simulation to compare the method with other common estimators. The simulations illustrate that the Gaussian mixture model outperforms other common estimators and can effectively give constraints on the regression parameters, even when the measurement errors dominate the observed scatter, source detection fraction is low, or the intrinsic distribution of the independent variables is not a mixture of Gaussian functions. I conclude by using this method to fit the X-ray spectral slope as a function of Eddington ratio using a sample of 39 z ≲ 0.8 radio-quiet quasars. I confirm the correlation seen by other authors between the radio-quiet quasar X-ray spectral slope and the Eddington ratio, where the X-ray spectral slope softens as the Eddington ratio increases. IDL routines are made available for performing the regression.