Linear Regression for Astronomical Data with Measurement Errors and Intrinsic Scatter

Abstract

Two new methods are proposed for linear regression analysis for data with measurement errors. Both methods are designed to accommodate intrinsic scatter in addition to measurement errors. The first (BCES) is a direct extension of the ordinary least squares (OLS) estimator to allow for measurement errors. It is quite general, allowing a) for measurement errors on both variables, b) the measurement errors for the two variables to be dependent, c) the magnitudes of the measurement errors to depend on the measurements, and d) other `symmetric' lines such as the bisector and the orthogonal regression can be constructed. The second method is a weighted least squares (WLS) estimator, which applies only in the case where the `independent' variable is measured without error and the magnitudes of the measurement errors on the 'dependent' variable are independent from the measurements. Several applications are made to extragalactic astronomy: The BCES method, when applied to data describing the color-luminosity relations for field galaxies, yields significantly different slopes than OLS and other estimators used in the literature. Simulations with artificial data sets are used to evaluate the small sample performance of the estimators. Unsurprisingly, the least-biased results are obtained when color is treated as the dependent variable. The Tully-Fisher relation is another example where the BCES method should be used because errors in luminosity and velocity are correlated due to inclination corrections. We also find, via simulations, that the WLS method is by far the best method for the Tolman surface-brightness test, producing the smallest variance in slope by an order of magnitude. Moreover, with WLS it is not necessary to ``reduce'' galaxies to a fiducial surface-brightness, since this model incorporates intrinsic scatter.