Linear regression in astronomy. II

A wide variety of least-squares linear regression procedures used in observational astronomy, particularly investigations of the cosmic distance scale, are presented and discussed. We emphasize that different regression procedures represent intrinsically different functionalities of the data set under consideration and should be used only under specific conditions. Discussion is restricted to least-squares approaches, and for most methods computer codes are located or provided. The classes of linear models considered are (1) unweighted regression lines, some discussed earlier in Paper 1 of this series, with bootstrap and jackknife resampling; (2) regression solutions when measurement error, in one or both variables, dominates the scatter; (3) methods to apply a calibration line to new data; (4) truncated regression models, which apply to flux-limited data sets; and (4) censored regression models, which apply when nondetections are present. For the calibration problem we develop two new procedures: a formula for the intercept offset between two parallel data sets, which propagates slope errors from one regression to the other; and a generalization of the Working-Hotelling confidence bands to nonstandard least-squares lines. They can provide improved error analysis for Faber-Jackson, Tully-Fisher, and similar cosmic distance scale relations. We apply them to a recently published data set, showing that the distance ratio between the Coma and Virgo clusters can be determined to ∼ 1% accuracy. The paper concludes with suggested strategies for the astronomer in dealing with linear regression problems. Precise formulation of the scientific question and scrutiny of the sources of scatter are crucial for optimal statistical treatment.