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I describe a Bayesian method to account for measurement errors in linearregression of astronomical data. The method allows for heteroscedastic andpossibly correlated measurement errors, and intrinsic scatter in the regressionrelationship. The method is based on deriving a likelihood function for themeasured data, and I focus on the case when the intrinsic distribution of theindependent variables can be approximated using a mixture of Gaussians. Igeneralize the method to incorporate multiple independent variables,non-detections, and selection effects (e.g., Malmquist bias). A Gibbs sampleris described for simulating random draws from the probability distribution ofthe parameters, given the observed data. I use simulation to compare the methodwith other common estimators. The simulations illustrate that the Gaussianmixture model outperforms other common estimators and can effectively giveconstraints on the regression parameters, even when the measurement errorsdominate the observed scatter, source detection fraction is low, or theintrinsic distribution of the independent variables is not a mixture ofGaussians. I conclude by using this method to fit the X-ray spectral slope as afunction of Eddington ratio using a sample of 39 z < 0.8 radio-quiet quasars. Iconfirm the correlation seen by other authors between the radio-quiet quasarX-ray spectral slope and the Eddington ratio, where the X-ray spectral slopesoftens as the Eddington ratio increases.

Some Aspects of Measurement Error in Linear Regression of Astronomical Data